Shant Tondon brings a diverse background blending financial markets analysis, consulting, and entrepreneurship. He holds a Bachelor's in Commerce with a focus on Financial Markets from Narsee Monjee College of Commerce and Economics, and completed his High School Diploma in Business/Commerce at Mayo College, Ajmer.

Professionally, Shant gained early analytical experience as an intern at Teach For India and KPMG in Mumbai, followed by a role as an Analyst at PwC.

Project Abstract

This project builds and evaluates a market-neutral pairs trading strategy focusing on 25 NSE large-cap stocks spanning the Banking, IT, Pharma, Cement, and Auto sectors. The pairs are selected using a residual stationarity test, specifically the ADF(0) with MacKinnon p-value, on a training sample. To ensure statistical robustness and control for false discoveries, the Benjamini–Hochberg False Discovery Rate (FDR) at 5% is applied.

The strategy trades mean-reversion via z-scores of the spread using a walk-forward train/test split. It represents a clean, defendable academic implementation with no look-ahead bias, explicit transaction costs (5 bps per leg per side), equal capital per active pair (₹5,00,000), and comprehensive portfolio-level risk metrics.

Introduction & Project Motivation

Pairs trading is a classic statistical arbitrage strategy that seeks to exploit temporary price divergences between two related assets while maintaining a market-neutral stance. This project applies this concept to the Indian equity market between January 1, 2015, and June 30, 2025.

The primary motivation was to build a rigorous prototype that addresses common algorithmic trading pitfalls, such as look-ahead bias, incomplete Profit and Loss (PnL) calculations, and inadequate multiple testing controls.

Strategy & Implementation Methodology (Technical Breakdown)

The strategy relies on a rolling walk-forward methodology utilizing a 252 trading-day training window and a 21-day test step.

Python Implementation Code

Below is a conceptual Python snippet demonstrating the core mathematical logic utilized in Shant's strategy:

import pandas as pd
import statsmodels.api as sm

def calculate_signals(train_data, test_data, stock_a, stock_b):
    # 1. Estimate Hedge Ratio (Beta) using OLS on Training Window
    model = sm.OLS(train_data[stock_a], train_data[stock_b]).fit()
    beta = model.params

    # 2. Calculate Out-of-Sample Spread
    # Spread formula: S_t = A_t - beta * B_t
    spread = test_data[stock_a] - (beta * test_data[stock_b])

    # 3. Calculate Z-Score strictly avoiding look-ahead bias
    # z_t = (S_t - mu_{t-1}) / sigma_{t-1}
    rolling_mean = spread.rolling(window=30).mean().shift(1)
    rolling_std = spread.rolling(window=30).std().shift(1)
    z_score = (spread - rolling_mean) / rolling_std

    # 4. Generate Trading Signals based on Z-Score Thresholds
    # Enter when absolute z-score > 1.5, Exit when it crosses 0
    long_entry = z_score < -1.5
    short_entry = z_score > 1.5
    exit_signal = (z_score.shift(1) * z_score <= 0)

    return z_score, long_entry, short_entry, exit_signal

3. Portfolio & Risk Management:

• Sizing: Equal-weight capital allocation, assigning ₹5,00,000 per active pair.
• Costs: Transaction costs are explicitly modeled at 5 bps per leg per side for entry and exit.
• PnL Calculation: PnL is mapped from both legs. Any open position is force-closed on the final backtest day to ensure complete reporting.

Key Findings & Portfolio Performance

The out-of-sample backtest generated the following portfolio-level performance metrics over the test period:

Challenges & Limitations

1. Sizing Constraints: The allocation is educational (equal capital per pair); it does not dynamically model capacity limits or real margin constraints.
2. Transaction Costs: Modeled cleanly at 5 bps per leg per side, but real-world execution slippage and bid-ask spreads can differ.
3. ADF(0) Approximation: The model uses a lag-0 ADF for computational speed. A full ADF test with optimized lags is recommended for future iterations.
4. Multiple Testing: While the FDR method reduces false discoveries, it does not completely eliminate them.
5. Survivorship Bias: The 25-stock universe is fixed and does not dynamically account for historical index reconstitution.

Next steps

Improving Strategy Performance

While the current strategy provides a clean academic baseline, several targeted enhancements can meaningfully improve its risk-adjusted returns and real-world applicability:

1. Optimise the ADF Lag Selection

Replace the current ADF(0) shortcut with an information-criterion-based lag selector (AIC or BIC). This reduces the risk of spurious cointegration signals and improves pair selection quality, leading to more stable and reliable trade entries.

2. Expand the Universe and Diversify Pairs

The current three-pair portfolio is highly concentrated. Extending the stock universe beyond 25 large-caps to include mid-cap NSE stocks across additional sectors (Energy, FMCG, Metals) would yield a broader set of cointegrated candidates, improve diversification, and reduce the impact of any single pair breaking down.

3. Introduce Dynamic Position Sizing

The strategy currently uses a fixed ₹5,00,000 per pair. Replacing this with volatility-scaled sizing (e.g., inverse-volatility or Kelly-criterion weighting) would allocate more capital to pairs showing stronger mean-reversion signals and tighter spreads, improving overall Sharpe ratio and reducing drawdowns.

4. Refine Entry/Exit Thresholds Adaptively

The fixed z-score thresholds of ±1.5 for entry and 0 for exit are static across all market regimes. An adaptive threshold model; where entry and exit levels are calibrated to each pair’s rolling volatility or regime classification (trending vs. mean-reverting), can filter out low-quality signals and improve the win ratio beyond the current 63.47%.

5. Incorporate Stop-Loss Rules to Control Drawdown

The current maximum drawdown of -34.31% is high relative to the annualised return of 0.30%. Adding a pair-level stop-loss (e.g., exit when the z-score breaches ±3.0 or when unrealised loss exceeds a fixed percentage of allocated capital) would cap downside on regime-breaking events and substantially improve the Sharpe ratio.

6. Address Survivorship Bias with a Rolling Universe

The fixed 25-stock universe inflates historical performance by only including companies that survived the full 2015–2025 period. Using a point-in-time NSE Nifty 50 or Nifty 100 constituent list that reflects actual index composition at each training window would eliminate this bias and produce more realistic forward-looking performance estimates.

Steps for continuous learning:
To build on the concepts covered in this blog, such as statistical arbitrage, cointegration testing, and mean-reversion strategy development, you can explore advanced resources and structured learning paths that focus on algorithmic trading.

Start with foundational application guides like Python for Trading Basics and Mean Reversion Trading Strategy by Dr Ernest P Chan, which walk through how statistical models are built and evaluated in live financial contexts.

For those looking to go beyond supervised models, Learning Tarck on Advanced Algorithmic Trading  is ideal for complex quantitative techniques, while Factor Based Investing offers insight into strategies that adapt over time and across market regimes.

To further strengthen your modelling and evaluation skills, refer to Portfolio & Risk Management and Backtesting Trading Strategies. These resources offer focused guidance on the types of statistical models Shant Tondon used in his EPAT project.

If you’re ready for hands-on learning with industry guidance, explore the Quantitative Trading and Artificial Intelligence in Trading learning tracks. These curated paths offer end-to-end training from data handling and feature engineering to model deployment.

Finally, if you’re inspired by Shant Tondon’s structured approach and want to replicate a similar end-to-end project, consider the Executive Programme in Algorithmic Trading (EPAT). It provides a comprehensive curriculum covering Python, statistics, machine learning, backtesting, and real-world trading applications, all essential components behind this EPAT final project.

Disclaimer: The information in this project is true and complete to the best of our Student’s knowledge. All recommendations are made without guarantee on the part of the student or QuantInsti ^®. The student and QuantInsti ^® disclaim any liability in connection with the use of this information. All content provided in this project is for informational purposes only and we do not guarantee that by using the guidance you will derive a certain profit.

Artificial intelligence is no longer a peripheral tool in quantitative finance. From machine learning models that uncover subtle market regimes to large language models that interpret unstructured news in real time, AI is increasingly embedded in how modern trading strategies are researched, tested, and deployed.

Yet for many practitioners, the real challenge is not whether to use AI, but how to apply it rigorously, realistically, and at scale.

This is precisely the gap addressed by Hands-On AI Trading with Python, QuantConnect, and AWS, a new book published by Wiley. Rather than focusing on abstract theory, the book emphasizes end-to-end, deployable AI trading strategies, built in a professional research environment. In this article, we outline what makes the book distinctive, who it is for, and how it fits into the broader QuantConnect–QuantInsti learning ecosystem.

This blog covers:

• Why This Book Exists
• Author Context: Why the Perspective Matters
• What Makes This Book Different
• Built on QuantConnect: From Research to Deployment
• Free Download: Book Summary (written by Jiri Pik)
• Who Should Read This Book?
• What Readers Are Saying (Early Feedback)
• What You Can Do Next
• Contribute and Collaborate

📌 Key Takeaways

• A hands-on, strategy-first guide to applying AI in real trading workflows
• Over 20 fully coded strategies, spanning ML, deep learning, NLP, and reinforcement learning
• Built entirely on QuantConnect’s institutional-grade research and execution platform
• Emphasis on intuition, interpretation, and decision-making, not just model accuracy
• Designed for practitioners who want working reference implementations, not toy examples

Why This Book Exists

The AI-in-trading landscape has expanded rapidly. Academic papers, blog posts, GitHub repositories, and notebooks are plentiful but fragmented. What is often missing is a coherent, practitioner-oriented path that connects AI ideas to tradable strategies under realistic constraints: data quality, execution frictions, and risk management.

This book was written to bridge that gap.

Rather than treating AI models as black boxes, the authors focus on:

• Why a specific model is appropriate for a given trading problem
• How model outputs should be interpreted by a trader or portfolio manager
• What failure modes look like in live trading, and how to diagnose them

The result is a guide that mirrors how professional quants actually work iterating between hypotheses, models, backtests, and risk evaluation.

Author Context: Why the Perspective Matters

Authority matters in quantitative finance, and this book benefits from a rare combination of perspectives across trading, platforms, and AI infrastructure.

The authors include:

• Jiri Pik – Founder of RocketEdge, with over 20 years of experience building trading and risk systems across banks and hedge funds
• Ernest P. Chan – Quantitative trading expert and founder of PredictNow.ai, widely known for his work on ML-driven trading and risk management
• Jared Broad – Founder and CEO of QuantConnect, whose LEAN engine underpins all strategies in the book
• Philip Sun – Former portfolio manager at WorldQuant and Renaissance Technologies, now CEO of Adaptive Investment Solutions
• Vivek Singh – Senior AI leader at AWS, specializing in large-scale ML and generative AI systems

This blend ensures the material is technically rigorous, operationally realistic, and aligned with modern institutional workflows.

What Makes This Book Different

1. Strategy-First, Not Model-First

Each chapter begins with a trading objective, not an algorithm. Models are introduced only when they add economic or operational value.

Readers learn how to reason about questions such as:

• When does supervised learning outperform rule-based logic?
• How should regime classifiers influence allocation decisions?
• What does overfitting look like after transaction costs?

This philosophy closely mirrors how AI is used in professional quant research.

Read about trading strategies here.

2. 20+ Fully Implemented AI Trading Strategies

At the core of the book are over twenty complete, end-to-end strategies, each including:

• Feature engineering and data preparation
• Model training and validation
• Portfolio construction and risk controls
• Backtest results and performance diagnostics

Representative examples include:

• Crypto trend detection using ML-based trend scanning
• Volatility regime modeling with Hidden Markov Models
• Dynamic asset allocation via neural-network regime classifiers
• Event-driven strategies around stock splits
• Fundamental ML models for dividend yield forecasting
• CNN-based pattern recognition in price time series
• Reinforcement learning for adaptive hedging
• LLM-based news sentiment signals using GPT-style models

Each strategy is written as a deployable QuantConnect algorithm, not a standalone notebook.

Download a detailed book summary.

Visit blog to download

Built on QuantConnect: From Research to Deployment

All strategies in the book are implemented on QuantConnect’s cloud platform, allowing readers to focus on research rather than infrastructure.

Key benefits include:

• Immediate access to multi-asset historical data
• Institutional-grade backtesting and execution logic
• Seamless transition from research to paper or live trading

This setup reflects real-world constraints such as contract rolls, slippage, margin, and execution costs; making the learning experience directly transferable to professional environments.

For readers new to Quant trading, the free Quantra learning track “Quantitative Trading for Beginners” provides a solid foundation before diving into the book.

Strategy Themes Covered

Volatility & Risk-Aware Strategies

• Volatility forecasting for position sizing
• Regime-aware stop-loss and drawdown control
• ML-driven futures allocation

Regime Detection & Market States

• Momentum vs. mean-reversion classifiers
• PCA-based macro regime modeling
• HMM-based market state inference

Alpha Across Data Types

• Technical signals via deep learning
• Fundamental and event-driven ML models
• Statistical arbitrage enhanced with clustering

NLP, LLMs, and Alternative Data

• Financial news sentiment using FinBERT and GPT models
• Practical considerations for using LLM APIs in trading systems

Readers interested in NLP applications can begin with the free Quantra course “Introduction to Machine Learning in Trading.”

Free Download: Book Summary (written by Jiri Pik)

To help readers quickly evaluate whether this book fits their needs, we’re offering a free downloadable summary based on the full draft version of the book.

📥 Download the free Hands-On AI Trading summary (≈ 5000 words)
(Includes strategy overview, learning outcomes, and practical takeaways)

Visit blog to download

Who Should Read This Book?

This book is ideal for:

• Quantitative traders and researchers
• Algorithmic trading developers
• ML practitioners entering finance
• Portfolio managers exploring AI-driven signals
• Graduate students preparing for quant or fintech roles

If your goal is to apply AI to real trading decisions, this book is designed for you.

What Readers Are Saying (Early Feedback)

“A rare combination of depth and practicality,  these are strategies you can actually build on.”

“Bridges the gap between machine learning theory and real trading systems.”

“Particularly strong on intuition and decision-making, not just code.”

What You Can Do Next

• 📘 Read the book to explore all strategies in depth
• 🎓 Explore free Quantra courses to strengthen specific AI or trading foundations
• 🚀 Apply the strategies on QuantConnect, adapting them to your own research

Contribute and Collaborate

At QuantInsti, we believe the future of algorithmic trading depends on shared learning and open collaboration. Our mission is to make advanced tools and research in quantitative finance accessible to all, helping both individuals and institutions navigate complex markets with confidence.

If the ideas explored in this blog speak to you, we invite you to contribute to the global community of quants. Whether you are building strategies, developing tools, conducting research, or applying AI in new ways, your work can add real value. To get started, read our Blog Contribution Guidelines. Every contribution helps grow the shared knowledge base and supports the evolution of quantitative trading. Let’s build the future together.

Disclaimer: This blog post is for informational and educational purposes only. It does not constitute financial advice or a recommendation to trade any specific assets or employ any specific strategy. All trading and investment activities involve significant risk. Always conduct your own thorough research, evaluate your personal risk tolerance, and consider seeking advice from a qualified financial professional before making any investment decisions.

At QuantInsti, our commitment to delivering quality education and empowering technology is reflected in our collaborations with renowned institutions and industry leaders. Together, we drive the future of algorithmic trading by fostering innovation and knowledge-sharing through strategic alliances and impactful events.

Here is a list of Announcements, Webinars and Workshops by QuantInsti, Industry Events, and Academic Collaborations, in 2026 so far!

Announcements

Algorithmic Trading & Options Risk Management | April 9, 2026

Free Live Workshop on Python for Trading Using AI

Move from discretionary ideas to systematic, rule-based decision-making. Explore how Python, AI, and professional risk management come together in real-world trading. Learn from iwith Dr Gaurav Raizada co-founder, iRage, Kelvin Foo, founder of Elemen79 and Vivek Krishnamoorthy, Head of Research, QuantInsti and gain a practical introduction to algorithmic trading and options risk management, core to QuantInsti’s flagship EPAT programme.
Register Now

How AI is Reshaping the EPAT Learning Experience?

AI is helping make the EPAT learning journey more practical, interactive, and aligned with how modern traders and quants actually work.

• Accelerated Coding and Analysis: AI-powered tools significantly speed up the process of writing, debugging, and analysing code for participants.
• Instant query resolution inside the LMS: Learners can get immediate answers, with references drawn directly from EPAT course content, through AI support available on the EPAT LMS desktop version.
• Advanced Project Work: Participants are moving beyond basic models, focusing on sophisticated quantitative techniques and leveraging diverse data sources, including market news and price action, to achieve a market edge.
• AI-enabled live trading workflows: Strategies can be deployed across multiple platforms using languages such as MQL5 and Pine Script, and integrated with broker APIs, including IB API, Kite API, and Alpaca API.
• Exploration of Emerging Trading Tech: The curriculum includes experimenting with LLM use cases in trading and multi-agent AI orchestration,

pushing the boundaries of technology-driven trading strategy development.

Explore EPAT’s AI-enhanced learning experience in the upcoming batch. Explore EPAT!

AI WhatsApp agent for personalised assistance

Try WhatsApp agent

New on Quantra: Agentic AI for Trading

If you have ever wished your trading workflow could run like a disciplined research desk, that is what agents help you build. In trading, that usually means faster iteration without losing rigor.

“Fantastic course. Really enjoyed building my own agentic quant team and already have plans to expand.”
Jackie Pineda, EDI Specialist, United States

With an agentic setup, you can:

• Split complex work into roles: researcher, data checker, strategist, risk reviewer, execution planner
• Reduce blind spots by having agents cross check logic, data quality, and bias in your assumptions
• Turn “ideas” into a workflow: prompts become reusable steps, not one-off chats
• Improve consistency with guardrails, checkpoints, and structured outputs you can audit later
• Scale experimentation: test more variations while keeping your reasoning organised

Ready to explore? Start for free Agentic AI for Trading

AI AlgoTrader Bootcamp

From Zero to Quantitative Trading Strategist

QuantInsti’s AI Algo Trader Bootcamp is now underway as a live, hands-on programme built around AI-driven strategy development, backtesting, automation, and broker API workflows. With speakers including Stefan Jansen, Rajib Borah, Ishan Shah, Satoshi Ido of Alpaca, and Nivas K of Zerodha, the bootcamp also signals strong relevance across both the learning and brokerage ecosystem.

Packt x QuantInsti - GenAI and LLMs for Futures Trading

QuantInsti collaborated with Packt on GenAI and LLMs for Futures Trading, a learning bundle designed to make AI more practical and relevant for real trading workflows. From a branding perspective, this collaboration strengthens QuantInsti’s positioning at the intersection of trading education, applied AI, and industry-led learning. With a live virtual workshop by Dr Paul Bilokon and Quantra’s Agentic AI for Trading course as part of the bundle, the initiative helped reinforce the brand’s role in bringing structured, real-world AI use cases closer to traders and market learners.

What’s new in EPAT?

EPAT Placements: Recent graduates secured roles such as Quant Analyst, Quant Trader, Quant Strategist, Quant Developer, Junior Trader, and Algo Trading Specialist at firms including COFCO International, ProAlpha Capital, NeoTrader Research LLP, and Alpha Alternatives.

Curriculum Updates: The curriculum now expanded with two new additions: a dedicated module for options backtesting and a foundational track covering Quant & HFT Systems.

EPAT Learning Portal Enhancements: Easier, uninterrupted learning with cross-module search, AskAI, auto-captioned videos, and organised, timely session recordings.

Alumni Meet-Up in Singapore

21 January 2026, The EPAT Alumni Meet-Up in Singapore was built around a simple idea: the programme ends, but the connections should not.

“The meet-up in Singapore reflected the shared dreams, challenges and strength of the quant/trader community. We stayed back to continue chatting late in the evening. EPAT is designed to go beyond the classroom and we are happy to be on the right track. ”
- Rohan Mathews , Global Head of Business, QuantInsti

More EPAT alumni meet-ups and community sessions are planned soon. Write to us at collaborate@quantisnti.com and help us plan the next EPAT Meetup in your city.

Academic and Industry Collaborations

Face2Face Multi-Asset Trading Conclave 2026

12th March 2026 | Nitesh Khandelwal spoke on “Algo Trading: Why Not!”, while Ishan Shah covered demystifying algo trading, from discretionary chaos to AI-powered automation.

“As AI becomes a bigger part of trading, these discussions matter because they help traders move beyond hype and understand where technology can genuinely add structure, speed, and better decision-making.”- Nitesh Khandelwal

Zerodha Varsity

28th Feb 2026 | Our collaboration with Zerodha Varsity brought algo trading learning to a wider audience through a practical 2-day programme focused on fundamentals, backtesting, and AI-assisted workflows.

IIT Mandi

At Xpecto 2026, IIT Mandi, QuantInsti came on board as a Knowledge Partner for the Quant Hackathon, while also contributing an educational session on algorithmic trading led by Mahavir B. The collaboration helped position QuantInsti within a high-potential student and innovation-led environment, while further reinforcing the brand’s focus on practical learning, market relevance, and academic engagement.

Knowledge Session at SGX

23 January 2026, SGX hosted a learning session for its employees, conducted by QuantInsti. The online session focused on Agentic AI, showing how AI agents can support the trading workflow from idea to backtest.

SPARQ at Integration 2026 | ISI Kolkata (Knowledge Partner)

We partnered as a knowledge partner for SPARQ, a quantitative reasoning event under Integration 2026, organised by students at ISI Kolkata.

This collaboration brings together strong quantitative foundations and industry-relevant exposure, supporting the next generation of analysts, quants, and problem-solvers.

Finance Club IIT Madras: Machine Learning in Trading

A session with Finance Club IIT Madras focused on introducing systematic trading, quantitative strategies, and how algorithms are shaping modern markets, with “Machine Learning in Trading” as the theme.

News & Press Update

Moneycontrol | Risk, Speed and Fairness in Derivatives

At the Moneycontrol Global Wealth Summit 2026, Nitesh Khandelwal shared his perspective on risk, market access, and fairness in derivatives trading. The mention places QuantInsti within a wider market conversation around informed participation and responsible trading.
Read here.

Moneycontrol | Retail Traders and Options Losses

In Moneycontrol’s coverage of retail participation in options markets, Nitesh Khandelwal highlighted how unrealistic return expectations can push smaller traders toward high-risk behaviour. The article adds to QuantInsti’s voice in discussions around trading behaviour, market structure, and risk awareness.
Read her.

India Market Update:

The NSE’s shift to nanosecond-level response times from April 11 marks a major infrastructure upgrade that puts even greater focus on execution quality and market plumbing in intraday trading. Meanwhile, the RBI’s tighter credit rules for capital market intermediaries from April 1 are expected to push the ecosystem toward more disciplined leverage and collateral practices.

Algorithmic trading market to reach $1.55 billion by 2033  The Hindu BusinessLine

16 Jan 2026, A market-size outlook story shared by BusinessLine, citing IMARC Group. Their LinkedIn post mentions the market rising from ~$562M (2024) to ~$1.27B (2033) (the headline uses $1.55B by 2033).
Read More

Conclusion

2026 has already been a strong start for QuantInsti, shaped by one clear focus: making algorithmic trading education more practical, more current, and more connected to real industry workflows. From launching Agentic AI for Trading and bringing live learning to the forefront through the AI AlgoTrader Bootcamp, to strengthening EPAT with curriculum additions, learning portal enhancements, and visible career outcomes, our priority remains building capability, not just curiosity.

Alongside this, our academic collaborations and knowledge sessions with institutions such as SGX and leading campuses reflect our commitment to supporting the next generation of quants and problem-solvers. And as the broader market conversation around algorithmic trading continues to grow, we aim to contribute with grounded insights, credible education, and a community that keeps learning together.

More sessions, partnerships, and community meet-ups are on the way. If you would like to collaborate, host a session, or help us bring an EPAT alumni meet-up to your city, write to us at collaborate@quantinsti.com.

Next Steps

• Explore Announcements, Webinars and Workshops by QuantInsti, Industry Events, and Academic Collaborations from 2025: QuantInsti in 2025
• Explore 2024 webinars: Review key collaborations and takeaways from the year’s sessions: 2024 QuantInsti Webinars.
• Learn with EPAT: A career-oriented programme in quantitative & algorithmic trading: EPAT.
• Explore Quantra: Self-paced, hands-on Quantitative Finance Courses. Try Blueshift: Research, backtest, and deploy strategies: Blueshift.
  

Markets do not move in a straight line. There are phases where trends are strong, phases where volatility rises, and periods where markets remain range-bound. Identifying these phases early can help traders adjust risk and position sizing. This is where machine learning for market regime detection becomes relevant.

Watch the Full Video

Add the full walkthrough below to see how the model, features, and strategy were built step by step.

Download the Code

Access the implementation and test the model using the code below:

Visit blog to download

This project, developed by an EPAT learner from QuantInsti, focuses on building a regime detection framework using market breadth data and a Random Forest model. The objective is to classify market regimes and adjust capital allocation based on those regimes.

About the Author:
Aparna Singhal is a quantitative research and trading professional with 3+ years of experience across equities, commodities, and cryptocurrency markets, as well as equity research and market analysis. She also has a strong foundation in credit analysis from her previous role in Wholesale Banking at IDFC FIRST Bank. Aparna has successfully completed the Executive Programme in Algorithmic Trading (EPAT) with QuantInsti, with a focus on quantitative trading systems, portfolio optimization, and machine learning.

Why Market Regime Detection Matters

A trading strategy that performs well in a bull market may struggle during high volatility or bear phases. Detecting the current regime allows traders to:

• Adjust exposure
• Manage drawdowns
• Improve risk-adjusted returns
• Maintain consistency across market conditions

Instead of reacting after losses, regime detection helps in preparing for changing market environments.

Data and Feature Creation

The project uses historical data from the Nifty 500 index to represent broad market behaviour across large-cap, mid-cap, and small-cap stocks.

Market breadth indicators were created to capture:

• Momentum across stocks
• Trend strength
• Volatility participation
• Percentage of stocks moving above key moving averages

These features help measure whether the broader market supports index movement or shows divergence.

Defining Market Regimes

Four regimes were defined:

• Bull market
• Bear market
• High volatility
• Low volatility

Adaptive thresholds were used instead of fixed values to account for changing market environments. A persistence filter was also applied to avoid frequent regime shifts caused by short-term noise.

Model Training with Random Forest

A Random Forest classifier was used to detect regimes. The model was trained on historical market breadth features and tested on unseen data using time-series validation.

Random Forest works as a collection of decision trees that collectively classify the current market condition. This approach helps capture relationships between multiple features without relying on a single indicator.

Strategy and Capital Allocation

Once regimes are identified, position sizing is adjusted based on market conditions.
For example:

• Higher allocation during low-volatility bull phases
• Reduced exposure during high-volatility or bear phases

The focus is on reducing drawdowns and improving the Sharpe ratio rather than only increasing returns. Transaction costs and signal smoothing were also considered to keep the strategy realistic.

Conclusion

Market regime detection using machine learning provides a structured way to adapt trading decisions to changing market conditions. Combining market breadth indicators with models such as Random Forest allows traders to adjust exposure, manage risk, and build more stable strategies.

This project shows how Python and machine learning can be applied to regime detection and capital allocation using a clear, step-by-step workflow.

Is it possible to predict where the Gold price is headed?

Yes, let’s use machine learning regression techniques to predict the price of one of the most important precious metal, the Gold.

Gold is a key financial asset and is widely regarded as a safe haven during periods of economic uncertainty, making it a preferred choice for investors seeking stability and portfolio diversification.

We will create a machine learning linear regression model that takes information from the past Gold ETF (GLD) prices and returns a Gold price prediction the next day.

GLD is the largest ETF to invest directly in physical gold. (Source)

This project prioritizes establishing a solid foundation with widely used machine learning techniques instead of immediately turning to advanced models. The objective is to build a robust and scalable pipeline for predicting gold prices, designed to be easily adaptable for incorporating more sophisticated algorithms in the future.

We will cover the following topics in our journey to predict gold prices using machine learning in python.

• Import the libraries and read the Gold ETF data
• Define explanatory variables
• Define dependent variable
• Non-stationary variables in linear regression
• Split the data into train and test dataset
• Create a linear regression model
• Predict the Gold ETF prices
• Plotting cumulative returns
• How to use this model to predict daily moves?

Import the libraries and read the Gold ETF data

First things first: import all the necessary libraries which are required to implement this strategy. Importing libraries and data files is a crucial first step in any data science project, as it ensures you have all dependencies and external data sources ready for analysis.

Then, we read the past 14 years of daily Gold ETF price data from a file and store it in Df. This data set includes a date column, which is essential for time series analysis and plotting trends over time. We remove the columns which are not relevant and drop NaN values using dropna() function. Then, we plot the Gold ETF close price.

Output:

Define explanatory variables

An explanatory variable, also known as a feature or independent variable, is used to explain or predict changes in another variable. In this case, it helps predict the next-day price of the Gold ETF.

These are the inputs or predictors we use in a model to forecast the target outcome.

In this strategy, we start with two simple features: the 3-day moving average and the 9-day moving average of the Gold ETF. These moving average serve as smoothed representations of short-term and slightly longer-term trends, helping capture momentum or mean-reversion behavior in prices. Before using these features in modeling, we eliminate any missing values using the .dropna() function to ensure the dataset is clean and ready for analysis. The final feature matrix is stored in X.

However, this is just the beginning of the feature engineering process. You can extend X by incorporating additional variables that might improve the model's predictive power. These may include:

• Technical indicators such as RSI (Relative Strength Index), MACD (Moving Average Convergence Divergence), Bollinger Bands, or ATR (Average True Range).
• Cross-asset features, such as the price or returns of related ETFs like the Gold Miners ETF (GDX) or the Oil ETF (USO), which may influence gold prices through macroeconomic or sector-specific linkages.
• Macroeconomic indicators such as inflation data (CPI), interest rates, and USD index movements can influence gold prices because gold is perceived as a safe-haven asset during times of economic uncertainty.

The process of identifying and constructing such variables is called feature engineering. Separately, selecting the most relevant variables for a model is known as feature selection.

The better your features reflect meaningful patterns in the data, the more accurate your forecasts are likely to be.

Define dependent variable

The dependent variable, also known as the target variable in machine learning, is the outcome we aim to predict. Its value is assumed to be influenced by the explanatory (or independent) variables. In the context of our strategy, the dependent variable is the price of the Gold ETF (GLD) on the following day.

In our dataset, the Close column contains the historical prices of the Gold ETF. This column serves as the target variable because we are building a model to learn patterns from historical features (such as moving averages) and use them to predict future GLD prices. We assign this target series to the variable y, which will be used during model training and evaluation.

To create the target variable, we apply the shift(-1) function to the Close column. This shifts the price data one step backward, making each row's target the next day's closing price. This approach enables the model to use today's features to forecast tomorrow's price.

Clearly defining the target variable is essential for any supervised learning problem, as it shapes the entire modelling objective. In this case, the goal is to forecast future movements in gold prices using relevant financial and economic signals.

Alternatively, instead of predicting the absolute price of gold, we can use gold returns as the target variable. Returns represent the percentage change in gold prices over a specified time period, such as daily, weekly, or monthly intervals.

Non-stationary variables in linear regression

In time series analysis, it's common to work with raw financial data such as stock or commodity prices. However, these price series are typically non-stationary, meaning their statistical properties like mean and variance change over time. This poses a significant challenge because many analytical techniques rely on the assumption that the data behaves consistently. When the data is non-stationary, its underlying structure shifts. Trends evolve, volatility varies, and historical patterns may not hold in the future.

Working with non-stationary data can lead to several problems:

• Spurious Relationships: Variables may appear to be related simply because they share similar trends, not because there's a genuine connection.
• Unstable Insights: Any patterns or relationships identified may not hold over time, as the data’s behaviour continues to evolve.
• Misleading Forecasts: Predictive models built on non-stationary data often struggle to perform reliably in the future.

The core issue is that non-stationary processes do not follow fixed rules. Their dynamic nature makes it difficult to draw conclusions or make predictions that remain valid as conditions change. Before performing any serious analysis, it's crucial to test for stationarity and, if needed, transform the data to stabilize its behaviour.

Two Ways to Work with Non-Stationary Data

Rather than discarding non-stationary variables, there are two reliable strategies to handle them in linear regression models:

1. Make Variables Stationary (Differencing Approach)

One common method is to transform the data to make it stationary. This is often done by focusing on changes in values. For example, price series can be converted into returns or differences. This transformation helps stabilize the mean and reduces trends or seasonality. Once the data is transformed, it becomes more suitable for linear modeling because its statistical properties remain consistent over time.

2. Use Original Non-Stationary Series (Cointegration Approach)

The second strategy allows us to use the original non-stationary series without transformation, provided certain conditions are met. Specifically, it involves checking whether the variables, when combined in a particular way, share a long-term equilibrium relationship. This concept is known as cointegration.

Even if the individual variables are non-stationary, their linear combination might be stationary. If this is the case, the residuals from the regression (the differences between actual and predicted values) remain stable over time. This stability makes the regression valid and meaningful, as it reflects a genuine relationship rather than a statistical coincidence.

In our analysis, we will use this second method by testing for residual stationarity to confirm that the regression setup is appropriate.

Output:

Cointegration p-value between S_3 and next_day_price: 3.1342217460742354e-16

Cointegration p-value between S_9 and next_day_price: 1.268049574487298e-15

S_3 and next_day_price are cointegrated.

S_9 and next_day_price are cointegrated.

The time series S_3 (3-day moving average) and next_day_price, as well as S_9 (9-day moving average) and next_day_price, are cointegrated. Thus, we can proceed with running a linear regression directly without transforming the series to achieve stationarity.

Why You Can Run the Regression Directly?

Cointegration implies that there is a stable, long-term relationship between the two non-stationary series. This means that while the individual series may each contain unit roots (i.e., be non-stationary), their linear combination is stationary and running an Ordinary Least Squares (OLS) regression will not lead to a spurious regression. This is because the residuals of the regression (i.e., the difference between the predicted and actual values) will be stationary.

Key Points to Remember

As cointegration already ensures a valid statistical relationship, making OLS appropriate for estimating the parameters, there is no need to difference the series to make them stationary before running the regression

The regression run between S_3 (or S_9) and next_day_price will capture a valid long-term equilibrium relationship, which cointegration confirms.

Split the data into train and test dataset

In this step, we split the predictors and output data into train and test data. The training data is used to create the linear regression model, by pairing the input with expected output.

Model training is performed on the training dataset, where the model learns from the features and labels.

The test data is used to estimate how well the model has been trained. Comparing different models and evaluating their training time and accuracy is an important part of the model selection process. Model evaluation, including the use of validation sets and cross-validation, ensures the model generalizes well to unseen data.

1. First 80% of the data is used for training and remaining data for testing
2. X_train & y_train are training dataset
3. X_test & y_test are test dataset

Create a linear regression model

We will now create a linear regression model. But, what is linear regression?

Linear regression is one of the simplest and most widely used algorithms in machine learning for supervised learning tasks, where the goal is to predict a continuous target variable based on input features. At its core, linear regression captures a mathematical relationship between the independent variables (x) and the dependent variable (y) by fitting a straight line that best describes how changes in x affect the values of y.

When the data is plotted as a scatter plot, linear regression identifies the line that minimizes the difference between the actual values and the predicted values. This fitted line represents the regression equation and is used to make future predictions.

To break it down further, regression explains the variation in a dependent variable in terms of independent variables. The dependent variable - ‘y’ is the variable that you want to predict. The independent variables - ‘x’ are the explanatory variables that you use to predict the dependent variable. The following regression equation describes that relation:

Y = m1 * X1 + m2 * X2 + C
Gold ETF price = m1 * 3 days moving average + m2 * 9 days moving average + c

Then we use the fit method to fit the independent and dependent variables (x’s and y’s) to generate coefficient and constant for regression.

Output:

Linear Regression model

Gold ETF Price (y) = 1.19 * 3 Days Moving Average (x1) + -0.19 * 9 Days Moving Average (x2) + 0.28 (constant)

Predict the Gold ETF prices

Now, it’s time to check if the model works in the test dataset. We predict the Gold ETF prices using the linear model created using the train dataset. The predict method finds the Gold ETF price (y) for the given explanatory variable X.

Output:

The graph shows the predicted prices and actual prices of the Gold ETF. Comparing predicted prices to actual prices helps evaluate the performance of the trained model and shows how closely the predictions match real-world values. Functions like evaluate_model() can be used to generate diagnostic plots and further evaluate the model's quality.

Now, let’s compute the goodness of the fit using the score() function.

Output:

99.70

As it can be seen, the R-squared of the model is 99.70%. R-squared is always between 0 and 100%. A score close to 100% indicates that the model explains the Gold ETF prices well.

On the surface, this seems impressive. It shows a near-perfect fit between the model's outputs and real market values.

However, translating this predictive accuracy into a profitable trading strategy is not straightforward. In practice, you need to make critical decisions such as:

• When to enter a trade (signal generation)
• How long to hold the position
• When to exit (e.g., based on a predicted reversal or fixed threshold)
• And how to manage risk (e.g., using stop-loss or position sizing)

To illustrate this challenge, we attempted to use predicted prices to generate a simple long-only trading signal.

A position is taken only if the next day's predicted price is higher than today’s closing price. This creates a unidirectional signal with no shorting or hedging. The position is exited (and potentially re-entered) whenever the signal condition is no longer met.

Plotting cumulative returns

Let’s calculate the cumulative returns of this strategy to analyse its performance.

1. The steps to calculate the cumulative returns are as follows:
2. Generate daily percentage change of gold price
3. Shift the daily percentage change ahead by one day to align with our position when there is a signal.
4. Create a buy trading signal represented by “1” when the next day's predicted price is more than the current day price. No position is taken otherwise
5. Calculate the strategy returns by multiplying the daily percentage change with the trading signal.
6. Finally, we will plot the cumulative returns graph

The output is given below:

We will also calculate the Sharpe ratio.

The output is given below:

'Sharpe Ratio 1.82'

Given the model’s high predictive accuracy, the Sharpe Ratio of the resulting trading strategy is only 1.82, which is not very good for a scalable and practical trading system.

This disparity highlights a crucial point: good price prediction doesn’t always lead to extremely profitable or risk-adjusted trading performance. Several factors may explain the lower Sharpe Ratio:

The strategy may suffer from unidirectional bias, ignoring shorting or range-bound periods.

• It might not adapt well to market volatility, leading to sharp drawdowns.
• The trading rules are too simplistic, failing to capture timing nuances or noise in the predictions.

In summary, while the model performs well in predicting price levels, converting this into a robust trading strategy requires thoughtful design. Signal logic, timing, position management, and risk controls all play a significant role in enhancing actual strategy performance.

Suggested Reads:

• Top 10 machine learning algorithms which can be used in your trading strategy
• Learn all about Exchange-Traded Funds (ETF)

How to use this model to predict daily moves?

You can use the following code to predict the gold prices and give a trading signal whether we should buy GLD or take no position.

The output is as shown below:

Congrats! You've just implemented a simple yet effective  machine learning technique using linear regression to forecast gold prices and derive trading signals. You now understand how to:

• Engineer features from raw price data (using moving averages),
• Build and fit a predictive model,
• Use the model for making forward-looking forecasts, and
• Translate those forecasts into actionable signals.

What’s Next?

Linear regression is a great starting point due to its simplicity and interpretability. But in real-world financial modeling, more complex patterns and nonlinear relationships often exist that linear models might not fully capture.

To improve accuracy, you can explore more powerful machine learning regression models, such as:

• Random Forest Regression
• Gradient Boosted Trees (like XGBoost or LightGBM)
• Support Vector Regression (SVR)
• Neural Networks (MLPs for tabular data)

The core structure of your pipeline remains the same: data preprocessing, feature engineering, forecasting, and signal generation. The only change is the model itself. You simply replace the .fit() and .predict() methods with those from your chosen algorithm, possibly adjusting a few additional hyperparameters.

Keep Exploring

Want to dive deeper into using machine learning for trading? Learn step by step how to build your first ML-based trading strategy with our guided course. If you're ready to take it to the next level, explore our Learning Track. Experts like Dr. Ernest Chan will guide you through the entire lifecycle, from idea generation and backtesting to live deployment, using advanced machine learning techniques.

File in the download:

Gold Price Prediction Strategy - Python Notebook

Visit blog to download

Disclaimer: All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.

At QuantInsti, our commitment to delivering quality education and empowering technology is reflected in our collaborations with renowned institutions and industry leaders. Together, we drive the future of algorithmic trading by fostering innovation and knowledge-sharing through strategic alliances and impactful events.

Here is a list of Announcements, Webinars and Workshops by QuantInsti, Industry Events, and Academic Collaborations, in 2025 so far!

Following SEBI guidelines for Algo Trading in India

SEBI’s new algo guidelines (e.g., NSE/INVG/69255) require client algos to be hosted within broker-controlled infrastructure, with execution routed through the broker. That shifts both compliance and operational responsibility towards brokers, and demands:

• cleaner, reviewable research-to-production pipelines
• infrastructure ready for audits, risk and monitoring
• learning paths that reflect how live trading systems actually run

QuantInsti is committed to equipping market participants including both brokers and retail traders with comprehensive knowledge and fully compliant technological solutions. As Nitesh Khandelwal, Co-founder & Director at QuantInsti, put it in his recent Outlook India interview, SEBI’s framework “brings structure, clarity and accountability without stifling innovation,” balancing investor protection with room for tech-savvy traders to operate responsibly.

INDUSTRY UPDATES

QuantInsti's industry events in 2025 so far showcased innovation and expertise in algorithmic trading, bringing together professionals, institutions, and market leaders to discuss advancements and real-world applications of trading strategies.

Blueshift News: FnO Algos in Code-Free Research

The FnO wizard lets you go from idea to working prototype without writing code, while staying close to an institutional-style infrastructure.

Explore Blueshift: https://www.quantinsti.com/blueshift

Rajib Borah Decoding Algos 101 at Global Fintech Fest 2025

iRage Co-Founder & CEO, Mr. Rajib Ranjan Borah, presented "Decoding Algos 101," at The Global Fintech Fest 2025, exploring algorithmic trading's principles, complexities, and pitfalls. Attendees then engaged with him for further discussions and Q&A. GFF2025 featured addresses from PM Narendra Modi and UK PM Sir Keir Starmer, alongside leaders from banking and fintech.

Prodipta Ghosh on Managing Portfolios of Algo Strategies at 5paisa Algo Convention 2025

Our VP, Prodipta Ghosh, was a featured speaker at the 5paisa Algo Convention 2025, held on September 27, 2025, at the Bombay Stock Exchange (BSE) in Mumbai. He led a session on "Managing Portfolios of Algo Strategies" for over 700 traders, investors, fintech innovators, quants, and finance professionals. The convention provided a platform to explore the rapidly evolving world of algorithmic trading.

Rajib at Mumbai Financial Markets Forum

Rajib Ranjan Borah, Co-Founder and CEO of iRage, participated in the Mumbai Financial Markets Forum at The Indus Club on April 1st, speaking on the Alpha Panel: Leading Traders on Market Trends and the Future of Trading.
The unique, moderator-free session saw lively cross-examinations and direct exchanges between panelists and an engaged international audience. Rajib shared perspectives on how evolving regulations, market structures, and business models are reshaping trading strategies.
Read the detailed insights here.

iRage participated in Asia Pacific’s premier trading infrastructure event, the FIX Trading Conference

April 2, Mumbai | Organized by the FIX Trading Community Asia Pacific, the FIX Conference 2025 in Mumbai brought together more than 500 professionals from 200+ organizations across major global financial hubs. Since its inception in 2008, the FIX Conference has served as one of the most influential platforms for stakeholders in exchange-traded products.

Rajib Ranjan Borah, CEO & Co-Founder of iRage and QuantInsti, contributed to the high-impact panel discussion on "Infrastructure & High-Frequency Trading – The Next Frontier." The session explored the impact of market structure shifts and regulatory developments on the future of high-frequency trading and trading infrastructure in India and beyond. Read More

Participation in FIA Mumbai: Sustainability of Market Liquidity in the Indian Market

20 February 2025 | At the FIA, Inc. Forum in Mumbai, Rajib Ranjan Borah, CEO of iRage, Director at QuantInsti, participated in a panel discussion on the sustainability of market liquidity in India and refining margin rules—issues that continue to shape the future of trading. Dive deeper into his take on this discussion here → FIA Mumbai Discussion

QuantInsti at QUANT EDGE - VII by PhillipCapital India

18th Jan 2025 | We had the privilege of participating in QUANT EDGE - VII, a specialized seminar for options traders. Our very own Rekhit P , Senior Quant Associate, QuantInsti, delivered a session on Backtesting an Options Trading Strategy. This collaboration with PhillipCapital aligns perfectly with our mission to make algorithmic trading knowledge and technology accessible to all.

GUEST SPEAKERS & WEBINARS

QuantInsti hosts Free Learning Sessions with leading industry experts to make algorithmic trading accessible to all. Attracting thousands worldwide, most sessions are open to everyone, ensuring broad access to cutting-edge knowledge.

Our Guest Speakers

Our First Advanced AI for Traders & Asset Managers Bootcamp

Our November bootcamp brought quants and traders together to apply AI in real markets with:

• Dr Ernest P. Chan – deep learning & transformers in trading
• Dr Thomas Starke – LSTM-based allocation & robust testing
• Ishan Shah & Rekhit Pachanekar – agentic AI & research workflows
• Vivek Krishnamoorthy – sentiment and text-driven strategies using LLMs

Learners showcased projects like an automated backtesting platform in VS Code, an LSTM-based portfolio model, and a sentiment strategy on event announcements. As one participant put it:

“Very useful class. You should have more classes like this on an ongoing basis to keep us updated on the latest developments.”

Core takeaway: learning in quant and AI is a long-term journey, not a one-time event.
Full recap: https://www.linkedin.com/feed/update/urn:li:activity:7396192927128989696

Algorithmic Trading Conference 2025

On our 15th anniversary, more than 13,000 aspiring quants, finance professionals, coders, students, and career switchers registered to learn how AI, LLMs, and GenAI are moving from promise to production. By publicly reported figures for peer events, ATC 2025 is likely among the largest quant finance conferences.

What did the speakers teach us?

Tucker Balch: "Don't use AI to trade Apple. Traders should tune the AI to pay attention to areas they are already knowledgeable about (e.g., stocks in Hong Kong, Singapore, or India) . This focuses the search on corners of the market where inefficiency is left ".

Peter Cotton argues that the playing field is levelled because everyone is "starting from scratch" with this new technology; large firms may have existed for decades, but they haven't had these specific tools before.

Dimitri Bianco: Quality remains the difficult aspect; processing large amounts of data requires a reasonable process to check for data quality.
Watch the full workshop and panel
YouTube replay: https://www.quantinsti.com/#webinar-recording

Build Your Quant Portfolio: First Steps into Algorithmic Trading

24 June 2025 | QuantInsti hosted a live session that demystified how professionals craft data-driven quant portfolios and chart rewarding careers in algorithmic trading. Designed for finance practitioners, developers, students, and career-switchers, the webinar served as a practical gateway into the quantitative world—pairing real-world examples with clear, step-by-step guidance from seasoned mentors.

The session blended hands-on market-microstructure insight, machine-learning-driven asset allocation, macro-quant research, and career-transition guidance into one cohesive roadmap for aspiring quants.

Speakers: Jay Parmar, Mrinall Mahajan, Tomás V. García-Purriños, Rohan Mathews

Gen AI and Automated Trading Summit: Algo Trading on MT5 and Trading View

10 April 2025 | A power-packed webinar on AI-driven trading and automation brought together 6,000+ participants from 110+ countries. Dr. Ernest P. Chan explained how Generative AI goes beyond LLMs in trading applications, highlighting its diverse use cases in finance.

Speakers: Dr. Ernest P. Chan, Varun Pothula, Akshay Chaudhary

Key highlights:

• Session 1: GenAI in Finance & Trading – Dr. Ernest P. Chan shared how Generative AI goes beyond LLMs, with practical applications in finance. View Presentation
  
• Session 2: Algorithmic Trading on MT5 & TradingView – Live demos on building trading bots using Python and Pine Script. View Presentations:
  1. Code a Strategy on Pine Script With the Help of ChatGPT by Akshay Chaudhary
  2. Why Use Python to Automate Your Trading? by Varun Pothula

You can download the Python Notebook shared by Varun by clicking the button below:

Visit blog to download

If you have any questions or feedback related to the webinar, You can drop a comment on our post on our community forum.

MasterClass: High-Frequency Trading, Algo Desk Setup and Quant Career Growth

January 2025 | This webinar offered an in-depth look at high-frequency trading, practical guidance on setting up an algorithmic trading desk, and career-focused insights for aspiring quants. Participants gained clarity on the skills HFT firms value, actionable steps for building a trading desk, and global opportunities in quantitative trading. The session was especially valuable for finance professionals, traders, and entrepreneurs aiming to accelerate their growth.

Speakers: Rajib Ranjan Borah, Richard Harrison, Rohan Mathews

Check out a snippet of the webinar where Rajib Ranjan Borah explains different roles in a High Frequency Trading Firm:

Watch the Full Webinar Recording here:

Check out the esteemed guest speakers who've shared their insights at QuantInsti's webinars and workshops. Explore insightful sessions from leading industry experts, offering invaluable knowledge in algorithmic trading.

ACADEMIC COLLABS & CONTRIBUTIONS

QuantInsti's collaborations with educational institutions in 2025 so far focused on bridging the gap between theoretical concepts and practical trading applications, empowering the next generation of quantitative analysts and enthusiasts with actionable skills.

State of Algorithmic Trading Education 2025

How are practitioners learning, building, and automating trading strategies in 2025?
Vivek Krishnamoorthy recently wrote a report on how practitioners are learning, building, and automating trading strategies in the algo trading field in 2025.
It draws on data from over 13,000 registrants and attendees of the Algo Trading Conference held in September 2025.
Vivek captured how quants are using AI across the trading workflow - what works, where they struggle, and what skills people want to acquire.
There was a consistent theme in my findings: AI speeds up research, code writing, documentation, and early testing. But there are hard parts which we grapple with - judgment that comes with experience, market intuition, data readiness, and validating AI output.
The main gap we see is not so much AI usage, but in supervising it effectively.

Read Publication

Get PDF:

Visit blog to download

@report{krishnamoorthy2025state,
  title        = {State of Algorithmic Trading Education 2025: What Practitioners Are Learning and Automating},
  author       = {Krishnamoorthy, V.},
  collaborators = {Khandelwal, N. and Bhattacharya, M. and Gonz{\'a}les Tanaka, J. C.},
  institution  = {QuantInsti},
  year         = {2025},
  month        = nov,
  type         = {Research Report},
  doi          = {10.5281/zenodo.17732323},
  url          = {https://doi.org/10.5281/zenodo.17732323}
}

QuantInsti x Finance Club IIT Madras

We partnered with the Finance Club, IIT Madras for the first QuantInsti Online Hackathon, inviting students to design and backtest trading strategies for a practical introduction to systematic trading.
Top winners received Quantra courses, and shortlisted teams earned participation rewards. Their projects showed strong curiosity and data usage. As a follow-up, Mohak, Senior Quant Researcher, will lead a webinar for the Finance Club, IIT Madras, early next year.

IIT Patna | ICTC & Quant-a-thon

March 29, 2025 | QuantInsti proudly partnered as a knowledge partner for the Intra College Tech Championship (ICTC) at IIT Patna, powering the second edition of the Quant-a-thon competition. The event brought together over 400 students for a highly competitive and engaging experience.

Akshay Choudhary, Quantitative Analyst at QuantInsti, conducted a live session on Momentum-Based Strategies, giving students practical exposure to quantitative analysis and its relevance in real market scenarios. The competition encouraged participants to think critically and apply technical concepts to solve trading challenges.

IIT Mandi | Xpecto 2025 Hackathon

March 24, 2025 | As the Knowledge Partner for Xpecto 2025—IIT Mandi’s flagship tech festival—QuantInsti engaged over 5,000 students, researchers, and industry professionals during the last week of March. The hackathon segment featured a QuantInsti-curated problem statement designed to test applied quant thinking and coding skills.

Mahavir Bhattacharya, EPAT faculty at QuantInsti, conducted a foundational session on Introduction to Algorithmic Trading. The competition encouraged application-based problem-solving, and winners were awarded internship opportunities at QuantInsti, providing a pathway into the world of algorithmic trading.

IICPC CODEFEST: India’s Biggest Coding Fest

12 February 2025 | QuantInsti is committed to bridging academic learning with industry-ready skills. As a knowledge partner for IICPC (Inter IIT Competitive Programming Conclave) CODEFEST, we are awarding a 100% EPAT Scholarship to the winner. The event so far has drawn over 45,000 participants from 32 countries and 250+ universities. The top 400 will advance to regionals at eight IITs, while the top 2,000 will receive Certificates of Merit.

Trading Challenge, Finance Club, Indian Institute of Technology, Madras

30 January 2025 | Mahavir Bhattacharya, EPAT faculty led a session on Momentum Trading with Machine Learning, as a beginning of a trading challenge at Finance Club - IIT, Madras. Winners of the challenge received special Quantra pricing and the opportunity to intern at QuantInsti, further bridging academic learning with practical experience in quantitative trading.

Collaborative Workshop with IIT Guwahati - Finance and Economics Club

21st January 2025 | We recently hosted an engaging session on Introductory Options: Trading Strategies Using Combinations in partnership with the Finance & Economics Club – IIT Guwahati. A big thank you to Varun Kumar Pothula (Quantitative Analyst, QuantInsti) for leading an interactive discussion around non-directional options trading, volatility-based strategies (like straddles and strangles), risk management, and market conditions.

“Events like these are critical for sharing knowledge and fostering a better understanding of financial concepts.”
– Varun Kumar Pothula

Our collaborations with academic institutions continue to be a cornerstone of our commitment to advance financial education globally.

Want to know more? Check out our past Academic Collaborations.

Research Spotlight: Trade Informativeness of Foreign Investors in India by Dr Gaurav Raizada

We are proud to highlight the research of Dr. Gaurav Raizada, whose paper "Trade Informativeness of Foreign Investors in India," co-authored with Samarpan Nawn, CFA, was recently published in the Journal of Asset Management.

The study presents he performance of foreign investors’ (FI) portfolio alphas in the Indian stock market. It adopt calendar-time portfolio approaches on a dataset spanning the period 2003-2019. Holdings-based portfolios do not generate abnormal performance, while transactions-based portfolios show buys of foreign investors underperform sells. It investigate whether Direct Market Access (DMA) can help decrease the underperformance of FI. Though DMA reduces the underperformance of FI portfolios on shorter holding periods, trade informativeness remains negative across all trading holding periods, from one day to one year. The trading behavior continues to be positive feedback, indicating informational disadvantage to FI.
Read the Full Paper: https://link.springer.com/article/10.1057/s41260-024-00387-8

PARTNERSHIPS

QuantInsti x Dhan

On 22 November 2025, we kicked off our first collaboration with Dhan. This started with a webinar titled “Build & Understand an Algorithmic Options Trading Workflow”.
In this session, we walked through what a realistic options workflow looks like, from idea generation and data to research, testing and execution.
If you could not attend, you can watch the recording here: https://www.youtube.com/live/3GFaS4adbUs?feature=share

QuantInsti x Zerodha Varsity: Introduction to Algo Trading with Ishan Shah

The Zerodha Varsity x QuantInsti 'Introduction to Algo Trading' sessions on September 13th and 14th (English & Hindi) went housefull. Ishan Shah, Head of Quantra Content, demonstrated AI's role in financial market data analysis and strategy backtesting using tools like Kite MCP, Google Finance, Sheets, Claude, and VS Code. We appreciate the collaboration and are developing further initiatives to democratise algorithmic trading education and technology.

Not your broker? No worries, we’ve got you covered. Head over to our live trading page, which covers other brokers/platforms.
Live Trading: https://quantra.quantinsti.com/live-trading

Lambda Quant x QuantInsti: Indian Institutional Quant Conference 2025

17 January 2025 | We were proud to join forces with Lambda Quant at the Indian Institutional Quant Conference 2025 hosted at St. Regis, Mumbai.
Our Vice President, Prodipta Ghosh, spoke on a panel discussing “Backtesting Methods and Drawbacks.” He was joined by a stellar lineup of industry experts, Kusal Kansara (Nuvama PCG), and Kannan Singaravelu (CQF, Fitch Learning). The session was moderated by Sonam Srivastava (Wright Research).

Cornell Financial Engineering Manhattan presents the Future of Finance AI Conference 2025

September 19, New York | Organized by Cornell Financial Engineering Manhattan, the Future of Finance AI Conference 2025 brought together leading professionals and innovators in quantitative research, AI, and financial technology. The full-day event featured discussions on large language models, predictive analytics, AI infrastructure, and practical strategies reshaping global capital flows.

Since its inception, the conference has served as a platform for pioneering dialogue on the future of finance, connecting industry leaders, academics, and practitioners to explore emerging trends and transformative technologies.

Collaboration with Quantpedia

2025 Quantpedia Research Award | 1st Place

The paper “The Unintended Consequences of Rebalancing” by Campbell R. Harvey, Michele G. Mazzoleni, and Alessandro Melone examined how predictable month-end rebalancing by 60/40 funds can reduce equity prices by approximately 17 bps the following day, resulting in an estimated annual cost of $16 billion for investors. The research also highlighted how informed traders may gain an advantage by positioning ahead of these institutional flows.

Along with other awards, one of the authors received a fully sponsored EPAT® scholarship (worth $9,499) from QuantInsti, supporting excellence in quantitative finance and algorithmic trading.

IN THE NEWS

Trading, quant professionals relying on AI for research

A PTI-syndicated report referencing QuantInsti’s State of Algorithmic Trading Education 2025 notes that trading and quant professionals are increasingly using AI not just for speed, but to turn ideas into testable strategy code.
The report highlights that 76% of respondents use AI for early-stage research and code scaffolding, and that the findings are based on survey/poll inputs from a large global audience (over 13,000 registrants across 160 countries).
It also flags persistent capability gaps 54% cite skills/know-how as the biggest barrier—alongside data readiness and compute constraints.
Read More: https://www.ptinews.com/story/business/trading-quant-professionals-relying-on-ai-for-research-says-report/3176234

Algo Trading In India Is Moving From Learning To Real-World Execution

In an interview with Outlook India, Nitesh Khandelwal explains how algorithmic trading in India is moving beyond learning the basics toward deploying strategies in live markets, and what this shift means for the next generation of quantitative traders.
Source: https://www.outlookindia.com/hub4business/algo-trading-in-india-is-moving-from-learning-to-real-world-execution-nitesh-khandelwal-quantinsti-co-founder

GenAI is democratising algo trading, but experts warn on quality and regulation risks

Generative AI is lowering entry barriers for algorithmic trading, letting smaller firms and retail traders process complex data and test strategies faster. Experts caution that weak data hygiene, opaque models, and overfitting can trigger blow-ups and attract tougher regulatory scrutiny. The message: adopt AI, but pair it with strict validation, governance, and explainability.
Source: GenAI is democratising algo trading, but experts warn on quality and regulation risks - Moneycontrol

From algorithms to intelligence: How AI is reshaping quantitative finance education

AI is pushing quant training beyond rule-based “algorithms” to intelligence-driven workflows, feature engineering, robust modelling, and portfolio optimisation, delivered through hands-on, project-based learning. The piece highlights how programmes like QuantInsti’s make these skills accessible to both beginners and working professionals, bridging classroom concepts with live markets.

Source: From algorithms to intelligence: How AI is reshaping quantitative finance education - Times of India (ANI).

Gold Is Unpredictable. Your Trading Doesn’t Have to Be

In his piece, Ishan Shah (Lead, Research & Content, QuantInsti) explains that gold is hard to predict because of shocks like geopolitics and inflation, so treat it as a probabilistic problem. Use ML to estimate regime/trend probabilities and act with risk-first rules (position sizing, validation, periodic retraining) rather than point predictions.
Source: Ishan Shah on “Gold Is Unpredictable. Your Trading Doesn’t Have to Be” - Republic

Interns are offered $14000 a month pay by quant firms in India

Markets are getting sharper, and edges fade quicker. As iRage CEO Rajib Borah points out, strategies that once ran profitably for six months now last about two, with “alpha decay” accelerating as more firms crowd similar opportunities. The takeaway is clear: shorten research-to-deployment cycles, monitor live performance closely, and be ready to rotate or retrain fast.
Source: Rajib Ranjan Borah on “Interns are offered $14000 a month pay by quant firms in India” - Bloomberg

India’s Systematic Awakening: Nitesh Khandelwal on the Rise of Retail Algo Trading

In this featured interview with BusinessWorld, QuantInsti CEO and Co-founder Nitesh Khandelwal reflects on the changing face of India’s trading landscape. The piece highlights how retail investors are increasingly adopting algorithmic trading, fueled by improved, read more on BW Businessworld.

India Today NE – “Why Algo Trading Is Vital for an Efficient Capital Market”

A macro look at how algorithmic execution boosts transparency and lowers costs, echoing our mission to strengthen markets through technology and education. Read the full article here.

Conclusion

QuantInsti’s 2025 collaborations encapsulated the latest trends, strategies, and technologies in algorithmic and quantitative trading. Stay Tuned as we get you more excited webinars

Next Steps

• Explore 2024 webinars: Review key collaborations and takeaways from the year’s sessions: 2024 QuantInsti Webinars.
• Learn with EPAT: A career-oriented programme in quantitative & algorithmic trading: EPAT.
• Explore Quantra: Self-paced, hands-on Quantitative Finance Courses. Try Blueshift: Research, backtest, and deploy strategies: Blueshift.
  

What we cover in the Video

The indicator in plain English. Donchian Channels track the highest high and lowest low over a lookback window. To see how channel-based breakout tools compare to oscillators and trend indicators, our tutorial on how to build technical indicators in Python covers a full range of indicator types including moving averages, RSI, Bollinger Bands, and ATR with complete Python code. That gives you upper and lower channels, a middle channel is then calculated as an average of the two. I also show a small but important step: shift the bands by one bar so your signals do not peek into the future.

Three strategy shapes

1. Long-short. Go long when the price closes above the upper channel(entry window), exit long when it closes below the lower lower channel(exit window)And vice-versa for the short positions. The position will first become flat, as we exit, before it takes the next long or short.
2. Long-only.Enter on a close above the upper channel(using the entry window). Exit to cash if the price closes below the lower channel(exit window).
3. Long-only with a Moving Average filter. Another variant to filter out intermediate and long term bearish market phases and smoothen the results with a simple, logical trend following filter. Here, we enter and exit a long position with the same set of rules as in the long-only variant, but with an additional check that the price has closed above the 200 MA during entry.

Bias control and realism. We use adjusted close prices for returns, shift signals to avoid look-ahead bias, and apply transaction costs on position changes so the equity curve is not a fantasy.

Benchmarking properly. I put each variant next to a buy-and-hold baseline over a multi-year period. You will see where breakouts shine, where they lag, and why exits matter as much as entries.

What you will learn

• How to compute the Donchian Channelsand wire them into a robust trading strategy.
• Why a one-line shift can save you from hidden look-ahead bias
• How different window choices and filters change the character of the strategy
• How to read equity curves and basic stats like CAGR, Sharpe, and max drawdown without overfitting your choices

Why this matters

Breakout systems are transparent, testable, and easy to extend. Once the plumbing is correct, you can try portfolios, volatility sizing, regime filters, and walk-forward checks. This is the scaffolding for that kind of work.

Download the Code

If you want to replicate everything from the video, download the codes below.

Visit blog to download

Next Steps

• Pressure-test the idea. Change windows, tickers, and date ranges. Check if results hold outside your calibration period. Try a simple volatility position sizing rule and see what it does to drawdowns.
• Portfolio view. Run a small basket of liquid instruments and equal-weight the signals. Breakouts often behave better in a diversified set.
• Walk-forward logic. Split the data into in-sample and out-of-sample, or do a rolling re-fit of windows. You want robustness, not a one-off lucky decade.

Learn and build with QuantInsti

Quantra: hands-on courses you can finish this week
If you want structured, bite-sized learning that complements this video, start with Quantra. Begin with Python and backtesting fundamentals, then move to learning tracks.

Quantra is a Python-based, interactive e-learning platform by QuantInsti for quantitative and algorithmic trading. It provides self-paced courses with a focus on practical, hands-on learning to teach users how to backtest a trading strategy, and implement trading strategies, including those using machine learning.

Explore courses on Quantra

EPAT: a complete path into quant roles and workflows
If you are ready for depth and career outcomes, EPAT gives you a broad, applied curriculum with mentorship and an active alumni network. It connects the dots from research to execution across markets and asset classes.

EPAT (Executive Programme in Algorithmic Trading) by QuantInsti is a comprehensive, online, 6-month certification program designed to equip professionals and aspirants with the skills needed for algorithmic and quantitative trading. It covers a wide curriculum from foundational finance and statistics to advanced topics like AI/ML in trading and is taught by industry experts. The program includes live lectures, hands-on projects, and focuses on practical skills in Python, backtesting, and strategy implementation.

Discover EPAT

Blueshift: take your research toward live
When your research looks solid, move to Blueshift for higher-quality simulations. Blueshift is an all-in-one automated trading platform that brings institutional-class infrastructure for investment research, backtesting, and algorithmic trading to everyone, anywhere and anytime. It is fast, flexible, and reliable. It is also asset-class and trading-style agnostic. Blueshift helps you turn your ideas into investment-worthy opportunities.
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Disclaimer: This blog post is for informational and educational purposes only. It does not constitute financial advice or a recommendation to trade any specific assets or employ any specific strategy. All trading and investment activities involve significant risk. Always conduct your own thorough research, evaluate your personal risk tolerance, and consider seeking advice from a qualified financial professional before making any investment decisions.

TL;DR

Most investors focus on picking stocks, but asset allocation, how you distribute your investments, matters even more. While poor allocation can cause concentrated risks, a methodical approach to allocation would lead to a more balanced portfolio, better aligned with the portfolio objective.

This blog explains why Risk Parity is a powerful strategy. Unlike equal-weighting or mean-variance optimisation, Risk Parity allocates based on each asset’s risk (volatility), aiming to balance the portfolio so that no single asset dominates the risk contribution.

A practical Python implementation shows how to build and compare an Equal-Weighted Portfolio vs. a Risk Parity Portfolio using the Dow Jones 30 stocks.

Key results:

• Risk Parity outperforms with higher annualized return (15.6% vs. 11.5%), lower volatility (9.9% vs. 10.7%), better Sharpe ratio (1.57 vs. 1.07), and smaller max drawdown (-4.8% vs. -5.8%).
• While compelling, Risk Parity depends on historical volatility, it needs frequent rebalancing, and may underperform in certain market conditions.

To get the most out of this blog, it’s helpful to be familiar with a few foundational concepts.

Pre-requisites

First, a solid understanding of Python fundamentals is essential. This includes working with basic programming constructs as well as libraries frequently used in data analysis. You can explore these concepts in-depth through Basics of Python Programming.

Since the blog builds on financial data handling, you’ll also need to be comfortable with stock market data analysis. This involves learning how to obtain market datasets, visualise them effectively, and perform exploratory analysis in Python. For this, check out Stock Market Data: Obtaining Data, Visualization & Analysis in Python.

By covering these prerequisites, you’ll be well-prepared to dive into the concepts discussed in this blog and apply them with confidence.

Table of contents

• What is Asset Allocation?
• How Can We Solve This Allocation Imbalance?
• Risk Parity Allocations Process in Python
• Frequently Asked Questions
• A Note on Limitations

Ever wondered where your portfolio's risk is coming from?

Most investors focus heavily on picking the right stocks or funds, but what if the way you allocate your capital is more important than the assets themselves? Research consistently shows that asset allocation is the key driver of long-term portfolio performance. For example, Vanguard has published multiple papers reinforcing that asset allocation is the dominant factor in portfolio performance.

In this post, we take a closer look at Risk Parity, a smart and systematic approach to portfolio construction that aims to balance risk, not just capital. Instead of letting one asset class dominate your portfolio's risk, Risk Parity spreads exposure more evenly, potentially leading to greater stability across market cycles.

Quantitative Portfolio Management is a 3-step process.

1. Asset selection
2. Asset Allocation
3. Portfolio rebalance and monitoring

In modern portfolio theory, research has shown that “Asset Allocation” has played a major role in portfolio performance. We will understand Asset Allocation in-depth and then move to understanding one of the possible ways to allocate assets, the Hierarchical Risk Parity method.

What is Asset Allocation?

Let us take an example of a novice investor. This investor has a portfolio of 5 stocks and has invested $30,000 in them.

How he/she bought specific proportions of the stocks could depend on subjective analysis or on the funds they have now to buy shares. And this leads to a random exposure of different stocks. As given below, let’s assume that the novice investor is buying stocks, and this is how the allocation looks:

Note: Some of the numbers below could be approximations, for demonstration purposes.

As a result, the proportion of each stock bought would widely vary.

Note: The number of shares is not a whole number. The calculations are approximations only for demonstration purposes.

We clearly see that NVDA has a significantly higher weightage of 39% while APPL has merely a weightage of 6%. There is a great disparity in the allocation of funds across the different stocks.

Case 1: NVDA underperforms; it will have a significant impact on your portfolio. Which could lead to large drawdowns, and this is high idiosyncratic risk.

Case 2: APPL outperforms, due to a much lower weightage of the stock in your portfolio. You won’t benefit from it.

How Can We Solve This Allocation Imbalance?

Quantitative Portfolio Managers do not allocate funds based on subjectivity. It is industry practice to adopt logical, tested, and effective ways to do it.

Uneven fund allocation can expose your portfolio to concentrated risks. To address this, several systematic asset allocation strategies have been developed. Let’s explore the most notable ones:

1. Equal Weighting:

Approach: Assigns equal capital to each asset.

Note: The number of shares is not a whole number. The calculations are approximations only for demonstration purposes.

• Pros: Simple, intuitive, and reduces concentration risk.
• Cons: Ignores differences in volatility or asset correlation. May overexpose to riskier assets.

Real world example: MSCI World Equal Weighted Index

2. Mean-Variance Optimisation (MVO)

Approach: Based on Modern Portfolio Theory, it aims to maximise expected return for a given level of risk. Though it looks simple, this approach is followed by several fund managers; its effectiveness comes with periodically rebalancing the portfolio exposures :

1. Expected returns
2. Asset volatilities
3. Covariances between assets

Note: The number of shares is not a whole number. The calculations are approximations only for demonstration purposes.

Monte Carlo simulation is often used to test portfolio robustness across different market scenarios. To understand this method better, please read Portfolio Optimisation Using Monte Carlo Simulation.

The plot below shows an example of how portfolios with different expected returns and volatilities are created using the Monte Carlo Simulation method. Thousands, if not more, combinations of weights are considered in this process. The portfolio weights with the highest Sharpe ratio (marked as +) are often taken as the portfolio with the most optimal weightages.

Note: This is only for demonstration purposes, not for stocks used for our example.

• Pros: Theoretically optimal: When inputs are accurate, MVO can construct the most efficient portfolio on the risk-return frontier.
• Cons: Highly sensitive to input assumptions, especially expected returns, which are difficult to forecast.

3. Risk-Based Allocation: Risk Parity

Approach: Instead of allocating capital equally or based on returns, Risk Parity allocates based on risk contribution from each asset. The goal is for each asset to contribute equally to the total portfolio volatility. The process to achieve this includes the following steps.

1. Estimate each asset’s volatility
2. Compute the inverse of volatility (i.e., lower volatility → higher weight).
3. Normalise the inverse of volatility to get final weights.

What is volatility?

Volatility refers to the degree of variation in the price of a financial instrument over time. It represents the speed and magnitude of price changes, and is often used as a measure of risk.

In simple terms, higher volatility means greater price fluctuations, which can imply more risk or more opportunity.

Formula for Standard Deviation:

$$\sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (r_i - \bar{r})^2}$$

Inverse of Volatility:

The inverse of volatility is simply the reciprocal of volatility. It is often used as a measure of risk-adjusted exposure or to allocate weights inversely proportional to risk in portfolio construction.

σ=Volatility

Then the Inverse of Volatility is:  1/σ

Normalise the inverse of volatility to get final weights :

To determine the final portfolio weights, we take the inverse of each asset's volatility and then normalise these values so that their sum equals 1. This ensures assets with lower volatility receive higher weights while maintaining a fully allocated portfolio.

Example of Risk Parity weighted approach(applying the above approach):

The number of shares is not a whole number. The calculations are approximations only for demonstration purposes.

Result: No single asset dominates the portfolio risk.

Note:

• Volatility is an example based on an assumed % standard deviation.
• “Risk Parity Weight” is proportional to 1 / volatility, normalised to 100%.
  The exposure is calculated as: Risk Parity Weight × Total Capital.
• Shares = Exposure ÷ Price.

Pros:

• Does not rely on expected returns.
• Simple, robust, and uses observable inputs.
• Reduces portfolio drawdowns during volatile periods.

Cons:

• May overweight low-volatility assets (e.g., bonds), underweight growth assets.
• Ignores correlations between assets (unlike HRP).

Other Allocation Methods to Know:

Risk Parity Allocations Process in Python

Step 1: Let’s start by importing the relevant libraries

Step 2: We fetch the data for 30 stocks using their Yahoo Finance ticker symbols.

• These 30 stocks are the current 30 constituents of the Dow Jones Industrial Average Index.
• We fetch the data from one month before 2024 begins. And target a window of the entire year 2024. This is done because we use a 20-day rolling period to compute volatilities and rebalance the portfolios. 20 trading days roughly translates to one month.
• Only the “Close” prices are extracted, and the data frame is flattened for further analysis.

Step 3: We create a function to compute the returns of portfolios that are either equally weighted or weighted using the Risk Parity approach.

Purpose: To compute a portfolio's cumulative NAV (Net Asset Value) using equal-weighted or risk-parity rebalancing at fixed intervals.

• price_df: DataFrame containing historical price data of multiple assets, indexed by date.
• rebalance_period (default = 20):
  Number of trading days between each portfolio rebalancing.
• method (default = 'equal'):
  Portfolio weighting method - either 'equal' for equal weights or 'risk_parity' for inverse volatility weights.

Step-by-Step Logic

• Daily Returns Calculation: The function starts by computing daily returns using pct_change() on the price data and dropping the first NaN row.

• Rolling Volatility Estimation: A rolling standard deviation is computed over the rebalance window to estimate asset volatility. To avoid look-ahead bias, this is shifted by one day using .shift(1).

• Start Alignment: The earliest date all rolling volatility is available is identified. The returns and volatility DataFrames are trimmed accordingly.

• NAV Initialisation: A new Series is created to store the portfolio NAV, initialised at 1.0 on the first valid date.

• Rebalance Loop: The function loops through the data in windows of rebalance_period days:

• Volatility and Weights on Rebalance Day: On the first day of each window:

• For 'equal': assigns equal weights to all assets.

• For 'risk_parity': assigns weights inversely proportional to asset volatility.

• Cumulative Returns & NAV Computation: The window’s cumulative returns are calculated and combined with weights to compute the NAV path.

• NAV Normalisation: The NAV is normalised to match the last value of the previous window, ensuring smooth continuity.

Final Output: Returns a time series of the portfolio’s NAV, excluding any missing values.

Step 4: Portfolio Construction

We now proceed to construct two portfolios using the historical price data. This involves calling the portfolio construction function defined earlier. Specifically, we generate:

1. An Equal-Weighted Portfolio, where each asset is assigned the same weight at every rebalancing period.
2. A Risk Parity Portfolio, where asset weights are determined based on inverse volatility, aiming to equalise risk contribution across all holdings.

Both portfolios are rebalanced periodically based on the specified frequency.

Step 5: Portfolio Performance Evaluation

In this step, we evaluate the performance of the two constructed portfolios: Equal-Weighted and Risk Parity, by computing key performance metrics:

• Daily Returns: Calculated from the cumulative NAV series to observe day-to-day performance fluctuations.
• Annualised Return: Derived using the compound return over the entire investment period, scaled to reflect yearly performance.
• Annualised Volatility: Estimated from the standard deviation of daily returns and scaled by the square root of 252 trading days to annualise.
• Sharpe Ratio: A measure of risk-adjusted return, computed as the ratio of annualised return to annualised volatility, assuming a risk-free rate of 0.
• Maximum Drawdown: The maximum observed peak-to-trough decline in portfolio value, indicating the worst-case historical loss.

These metrics offer a comprehensive view of how each portfolio performs in terms of both return and risk. We also visualise the cumulative NAVs of both portfolios to observe their performance trends over time.

Frequently Asked Questions

What exactly is Risk Parity?

Risk Parity is a portfolio allocation strategy that assigns weights such that each asset contributes equally to the total portfolio volatility, rather than simply allocating equal capital to each asset. The goal is to prevent any single asset or asset class from dominating the portfolio’s overall risk exposure.

How does it differ from Equal Weighting or Mean-Variance Optimisation?

• Equal Weighting: This method allocates the same amount of capital to each asset. It is simple and intuitive, but does not consider the risk (volatility) of each asset, potentially leading to concentrated risk.
• Mean-Variance Optimisation (MVO): Based on Modern Portfolio Theory, MVO seeks to maximise expected return for a given level of risk by considering expected returns and covariances. However, it is highly sensitive to the accuracy of input forecasts.
• Risk Parity: Instead of focusing on returns or allocating equal capital, Risk Parity adjusts weights based on the volatility of each asset, allocating more capital to lower-volatility assets to equalise their risk contributions.

Why is asset allocation so important?

Research has shown that asset allocation is the primary driver of long-term portfolio returns, far more significant than selecting individual securities. A well-thought-out allocation helps manage risk and enhances the likelihood of meeting investment goals.

How is volatility calculated in Risk Parity?

Volatility is typically measured as the standard deviation of past returns over a rolling window (for example, a 20-day rolling standard deviation). In Risk Parity, assets with lower volatility are assigned higher weights to balance their contribution to total portfolio risk.

Is there Python code to implement this?

Yes. The blog provides complete Python code examples using libraries such as pandas for data handling, yfinance for fetching historical prices, and custom functions to rebalance portfolios either by equal weights or by inverse volatility (Risk Parity).

Does Risk Parity always outperform other strategies?

No. While Risk Parity often leads to more stable performance and better risk-adjusted returns, especially in diversified or volatile markets, it may underperform simpler strategies like Equal-Weighted portfolios during strong bull markets that favour high-risk assets.

What are the limitations of Risk Parity?

• It relies on the historical volatility to set target weights, which may not accurately reflect  the future behaviour of assets, especially during abrupt changes or crises.
• It typically requires frequent rebalancing, which can increase transaction costs and potential slippage.
• It may under-allocate to high-growth assets in trending markets, limiting upside in strong rallies.

Are there more advanced methods beyond standard Risk Parity?

Yes. For example, Hierarchical Risk Parity (HRP) uses clustering to understand asset relationships and aims to allocate risk more efficiently by addressing some of the weaknesses of traditional mean-variance approaches, such as instability due to input sensitivity.

Conclusion

The comparative analysis highlights the clear advantages of using a Risk Parity approach over a traditional Equal-Weighted portfolio. While both portfolios deliver positive returns, Risk Parity stands out with:

• Higher Annualised Return (15.60% vs. 11.47%)
• Lower Volatility (9.90% vs. 10.72%)
• Superior Risk-Adjusted Performance, as seen in the Sharpe Ratio (1.57 vs. 1.07)
• Smaller Max Drawdown (-4.76% vs. -5.83%)

These results demonstrate that by aligning portfolio weights with asset risk (rather than capital), the Risk Parity portfolio may enhance return potential along with better downside protection and smoother performance over time.

The NAV chart further reinforces this conclusion, showing a more consistent and resilient growth trajectory for the Risk Parity strategy.

In summary, for investors prioritising stability over growth, Risk Parity offers a compelling alternative to conventional allocation methods.

A Note on Limitations

Although the Risk Parity portfolio delivered stronger returns during the period taken in our example, its performance advantage is not assured in every market phase. Like any strategy, Risk Parity comes with limitations. It relies heavily on historical volatility estimates, which may not always accurately reflect future market conditions, especially during sudden regime shifts or extreme events.

It tends to shine in portfolios that mix high‑ and low‑volatility assets, like stocks and bonds, where equal capital allocation would otherwise concentrate risk.However, if low‑volatility assets underperform or if all assets have similar risk profiles,

Additionally, the strategy often requires frequent rebalancing, which can increase transaction costs and introduce slippage. In strong directional markets, particularly those favouring higher-risk assets, simpler strategies like Equal-Weighted may outperform due to their greater exposure to momentum.

Hence, while Risk Parity provides a systematic way to balance portfolio risk, it should be used with an understanding of its assumptions and practical limitations.

Next Steps:

After reading this blog, you may want to enhance your understanding of portfolio design and explore techniques that provide more structure to risk-return trade-offs.

A good place to begin is with Portfolio Variance/Covariance Analysis, which explains how asset correlations impact portfolio volatility. This will provide you with the foundation to understand why diversification works and where it doesn’t.

From there, Portfolio Optimisation Using Monte Carlo Simulation introduces a more dynamic approach. By running thousands of simulated outcomes, you can test how different allocations behave under uncertainty and identify combinations that balance risk and reward.

To round it off, Portfolio Optimisation Methods walks through a range of optimisation frameworks, covering classical mean-variance models as well as alternative methods, so you can compare their strengths and apply them in different market conditions.

Working through these next steps will equip you with practical techniques to analyse, simulate, and optimise portfolios, a skill set that’s critical for anyone looking to manage capital with confidence.

You can explore all of these in detail in the Portfolio Management & Position Sizing Learning Track, which includes the Quantitative Portfolio Management course for a comprehensive understanding of portfolio construction and optimisation.

For those looking to expand beyond portfolio theory into the broader realm of systematic trading, check the Executive Programme in  Algorithmic Trading - EPAT. Its comprehensive curriculum, led by top faculty like Dr. Ernest P. Chan, offers a leading Python algorithmic trading course for career growth. EPAT covers core trading strategies that can be adapted and extended to High-Frequency Trading. Get personalised support for specialising in trading strategies with live project mentorship.

Disclaimer: This blog post is for informational and educational purposes only. It does not constitute financial advice or a recommendation to trade any specific assets or employ any specific strategy. All trading and investment activities involve significant risk. Always conduct your own thorough research, evaluate your personal risk tolerance, and consider seeking advice from a qualified financial professional before making any investment decisions.

TL;DR:

This blog introduces retrospective simulation, inspired by Taleb’s "Fooled by Randomness," to simulate 1,000 alternate historical price paths using a non-parametric Brownian bridge method. Using SENSEX data (2000–2020) as in-sample data, the author optimises an EMA crossover strategy across the in-sample data first, and then applies it to the out-of-sample data using the optimum parameters obtained from the in-sample backtest. While the strategy outperforms the buy-and-hold approach in in-sample testing, it significantly underperforms in out-of-sample testing (2020–2025), highlighting the risk of overfitting to a single realised path. The author then runs the backtest across all 1,000 simulated paths to identify the most frequently successful SEMA-LEMA parameter combinations.

The author also calculates VaR and CVaR using over 5 million simulated returns and compares return extremes and distributional characteristics, revealing heavy tails and high kurtosis. This framework holds promise for more robust strategy validation by evaluating how strategies might perform across multiple plausible market scenarios.

Introduction

In “Fooled by Randomness”,  Taleb says at one place, “In the beginning, when I knew close to nothing (that is, even less than today), I wondered whether the time series reflecting the activity of people now dead or retired should matter for predicting the future.”

This got me thinking. We often run simulations for the probable paths a time series can take in the future. However, the premise for these simulations is based on historical data. Given the stochastic nature of asset prices (read more), the realised price path had the choice of an infinite number of paths it could have taken, but it traversed through only one of those infinite possibilities. And I thought to myself, why not simulate those alternate paths?

In common practice, this approach is referred to as bootstrap historical simulation. I chose to refer to it as retrospective simulation, as a more intuitive counterpart to the terms ‘look-ahead’ and ‘walk-forward’ used in the context of simulating the future.

Article map

Here’s an outline of how this article is laid out:

• Data Download
• Retrospective Simulation using Brownian Bridge
• Exponential Moving Average Crossover Strategy Development and Backtesting on In-Sample Data, and Parameter Optimisation
• Backtesting on Out-of-Sample Data
• Backtesting on Simulated Paths and Optimising to Extract the Best Parameters
• Discussion on Randomness
• Out-of-Sample Backtesting using Optimised Parameters based on Simulated Data Backtesting
• Discussion on the Results and the Approach
• Evaluation of VaR and C-VaR
• Distribution of Simulated Data
• Where This Approach Can Be Most Useful
• Suggested Next Steps
• In Summary
• Frequently Asked Questions
• Conclusion

Data Download

We import the necessary libraries and download the daily data of the SENSEX index, a broad market index based on the Bombay Stock Exchange of India.

I have downloaded the data from January 2000 to November 2020 as the in-sample data, and from December 2020 to April 2025 as the out-of-sample data. We could have put in a gap (an embargo) between the in-sample and out-of-sample data to minimise, if not eliminate, data leakage (read more). In our case, there’s no direct data leakage. However, since stock levels (prices) are known to bear autocorrelation, like we saw above, the SENSEX index on the first trading day of December 2020 would be highly correlated with its level on the last trading day of November 2020.

Thus, when we train our model on data that includes the last trading day of November 2020, it extracts information from that day’s level and uses it to get trained. Since our testing dataset is from the first trading day of December 2020, some residual information from the training dataset is present in the testing dataset.

As an extension, the training set contains some information that is also present in the testing dataset. However, this information will diminish over time and eventually become insignificant. Having said that, I didn’t maintain a gap between the in-sample and out-of-sample datasets so that we can focus on understanding the core concept of this article.

You can use any yfinance ticker to download data for an asset of your liking. You can also adjust the dates to suit your needs.

Retrospective Simulation using Brownian Bridge

The next part is the main crux of this blog. This is where I simulate the possible paths the asset could have taken from January 2000 to November 2020. I have simulated 1000 paths. You can modify it to make it 100 or 10000, as you like. The higher the value, the greater our confidence in the results, but there is a tradeoff in computational time. I have simulated only the closing prices. I kept the first-day and last-day prices the same as the realised ones, and simulated the in-between prices.

Keeping the price fixed on the first day makes sense. But the last day? If the prices are to follow a random walk (read more), the closing price levels of most, if not all, paths should be different, isn’t it? But I made an assumption here. Given the efficient market hypothesis, the index would have a fair price by the end of November 2020, and after moving on its capricious course, it would converge back to this fair price.

Why only November 2020?

Was the level of the index at its fairest price at that time? No way of knowing. However, one date is as good as any other, and we need to work with a specific date, so I chose this one.

Another consideration here is on what basis we allow the simulated paths to meander. Should it be parametric, where we assume the time series to follow a specific distribution, or non-parametric, where we don’t make any such assumption? I chose the latter. The financial literature discusses prices (and their returns) as belonging approximately to certain underlying distributions. However, when it comes to outlier events, such as highly volatile price jumps, these assumptions begin to break down, and it is these events that a quant (trader, portfolio manager, investor, analyst, or researcher) should be prepared for.

For the non-parametric approach, I have modified the Brownian bridge approach. In a pure Brownian bridge approach, the returns are assumed to follow a Gaussian distribution, which again becomes somewhat parametric (read more). However, in our approach, we calculate the realized returns from the in-sample closing prices and use these returns as a sample for the simulation generator to choose from. We are using bootstrapping with replacement (read more), which means that the realized returns aren’t just being shuffled; some values may be repeated, while some may not be used at all. If the values are simply shuffled, all simulated paths would land at the last closing price of the in-sample data. How do we ensure that the simulated prices converge to the final close price of the in-sample data?  We’ll use geometric smoothing for that.

Another consideration: since we use the realized returns, we are priming the simulated paths to resemble the realized path, correct? Sort of, but if we were to generate pseudo-random numbers for these returns, we would have to make some assumption about their distribution, making the simulation a parametric process.

Here’s the code for the simulations:

Note that I did not use a random seed when generating the simulated paths. I’ll mention the reason at a later stage.

Let’s plot the simulated paths:

The above graph shows that the starting and ending prices are the same for all 1,000 simulated paths. We should note one thing here. Since we are working with data from a broad market index, whose levels depend on many interlinked macroeconomic variables and factors, it's highly unlikely that the index would have traversed most of the paths simulated above, given the same macroeconomic events that occurred during the simulation period. We are making an implicit assumption here that the specified macroeconomic variables and factors differ in each of the simulated paths, and the interactions between these variables and factors result in the simulated levels that we generate. This holds for any other asset class or asset you decide to replace the SENSEX index with, for retrospective simulation purposes.

Exponential Moving Average Crossover Strategy Development and Backtesting on In-Sample Data, and Parameter Optimisation

Next, we develop a simple trading strategy and conduct a backtest using the in-sample data. The strategy is a simple exponential moving average crossover strategy, where we go long when the short-period exponential moving average (SEMA) of the close price goes above the long-period exponential moving average (LEMA), and we go short when the SEMA crosses the LEMA from above (read more).

Through optimisation, we’ll attempt to find the best SEMA and LEMA combination that yields the maximum returns. For the SEMA, I use lookback periods of 5, 10, 15, 20, … up to 100, and for the LEMA, 20, 30, 40, 50, … up to 300.

The condition is that for any given SEMA and LEMA combination, the LEMA lookback period should be greater than the corresponding SEMA lookback period. We would perform backtests on all different combinations of these SEMA and LEMA values and choose the one that yields the best performance.

We’ll plot:

1. the equity curve of the strategy with the best-performing SEMA and LEMA lookback values, plotted against the buy-and-hold equity,
2. the buy and sell signals plotted along with the close prices of the in-sample data and the SEMA and LEMA lines,
3. the underwater plot of the strategy, and,
4. a heatmap of the returns for different LEMA and SEMA calculations.

We’ll calculate:

1. the SEMA and LEMA lookback values for the best-performing combination,
2. the total returns of the strategy,
3. the maximum drawdown of the strategy, and,
4. the Sharpe ratio of the strategy.

We will also review the top 10 SEMA and LEMA combinations and their respective performances.

Here’s the code for all of the above:

And here are the outputs of the above code:

Best SEMA: 5, Best LEMA: 40

Total Return: 873.43%
Maximum Drawdown: -41.28 %
Sharpe Ratio: 0.59

Top 10 Parameter Combinations:

SEMA  LEMA    Return
2       5    40  8.734340
3       5    50  7.301270
62     15    60  6.021219
89     20    50  5.998316
116    25    40  5.665505
31     10    40  5.183363
92     20    80  5.071913
32     10    50  5.022373
58     15    20  4.959147
27      5   290  4.794400

The heatmap shows a gradual change in color from one adjacent cell to the next. This suggests that slight modifications to the EMA values don’t lead to drastic changes in the strategy's performance.  Of course, it would be more gradual if we were to reduce the spacing between the SEMA values from 5 to, say, 2, and between the LEMA values from 10 to, say, 3.

The strategy outperforms the buy-and-hold strategy, as shown in the equity plot. Good news, right? Note here that this was in-sample backtesting. We ran the optimisation on a given dataset, took some information from it, and applied it to the same dataset. It’s like using the prices for the next year (which are unknown to us now, except if you’re time-travelling!) to predict the prices over the next year. However, we can utilise the information gathered from this dataset to apply it to another dataset. That is where we use the out-of-sample data.

Backtesting on Out-of-Sample Data

Let’s run the backtest on the out-of-sample dataset:

Before we see the outputs of the above codes, let’s list what we are doing here.

We are plotting:

1. The equity curve of the strategy plotted alongside that of the buy-and-hold, and,
2. The underwater plot of the strategy.

We are calculating:

1. Strategy returns,
2. Buy-and-hold returns,
3. Strategy maximum drawdown,
4. Strategy Sharpe ratio,
5. Buy-and-hold Sharpe ratio, and,
6. Strategy hit ratio.

For the Sharpe ratio calculations, we assume a risk-free rate of return of 0. Here are the outputs:

Out-of-Sample Strategy Total Return: 15.46%
Out-of-Sample Buy-and-Hold Total Return: 79.41%
Out-of-Sample Strategy Maximum Drawdown: -15.77 %
Out-of-Sample Strategy Sharpe Ratio: 0.30
Out-of-Sample Buy-and-Hold Sharpe Ratio: 0.56
Out-of-Sample Hit Ratio: 53.70%

The strategy underperforms the underlying by a significant margin. But that’s not what we're primarily interested in, as far as this blog is concerned. We need to consider that we ran an optimisation on only one of the many paths that the prices could have taken during the in-sample period, and then extrapolated that to the out-of-sample backtest. This is where we use the simulation we performed at the beginning. Let’s run the backtest on the different simulated paths and check the results.

Backtesting on Simulated Paths and Optimising to Extract the Best Parameters

This would keep printing the corresponding SEMA and LEMA values for the best strategy performance, and the performance itself for the simulated paths:

Completed optimization for column 0: SEMA=65, LEMA=230, Return=1.8905
Completed optimization for column 1: SEMA=45, LEMA=140, Return=4.4721
.....................................................................
Completed optimization for column 998: SEMA=10, LEMA=20, Return=3.6721
Completed optimization for column 999: SEMA=15, LEMA=20, Return=9.8472

Here’s a snap of the output of this code:

Now, we’ll sort the above table so that the SEMA and LEMA combination with the best returns for the most paths is at the top, followed by the second-best combination, and so on.

Let’s check how the table would look:

Here’s a snapshot of the output:

Of the 1000 paths, 47 showed the best returns with a combination of SEMA 5 and LEMA 20. Since I didn’t use a random seed while generating the simulated paths, you can run the code multiple times and obtain different outputs or results. You’ll see that the best SEMA and LEMA combination in the above table would most likely be 5 and 20. The frequencies can change, though.

How do I know?

Because I’ve done so, and have gotten the combination of 5 and 20 in the first place every time (followed by 100 and 300 in the second place). Of course, it’s not that there’s a zero chance of getting some other combination in the top row.

Out-of-Sample Backtesting using Optimised Parameters based on Simulated Data Backtesting

We’ll extract the SEMA and LEMA look-back combination from the previous step that yields the best returns for most of the simulated paths. We’ll use a dynamic approach to automate this selection. Thus, if instead of 5 and 20, we were to obtain, say, 90 and 250 as the optimal combination, the same would be selected, and the backtest would be performed using that.

Let’s use this combination to run an out-of-sample backtest:

Here are the outputs:

Out-of-Sample Strategy Total Return: -7.73%
Out-of-Sample Buy-and-Hold Total Return: 79.41%
Out-of-Sample Strategy Maximum Drawdown: -23.70 %
Out-of-Sample Strategy Sharpe Ratio: -0.05
Out-of-Sample Buy-and-Hold Sharpe Ratio: 0.56
Out-of-Sample Hit Ratio: 52.50%

Discussion on the Results and the Approach

Here, the strategy not only underperforms the underlying but also generates negative returns. So what’s the point of all this effort that we put in? Let’s note that I employed the moving average crossover strategy to illustrate the application of retrospective simulation using a modified Brownian bridge. This approach is more suitable for testing complex strategies with multiple conditions, and machine learning (ML)-based and deep learning (DL)-based strategies.

We have approaches such as walk-forward optimisation and cross-validation to overcome the problem of optimising or fine-tuning a strategy or model on only one of the many possible traversable paths.

However, this approach of retrospective simulation ensures that you don’t have to rely on only one path but can employ multiple retrospective paths. However, since running an ML-based strategy on these simulated paths would be too computationally intensive for most of our readers who don’t have access to GPUs or TPUs, I chose to work with a simple strategy.

Additionally, if you wish to modify the approach, I have included some suggestions at the end.

Evaluation of VaR and C-VaR

Let’s move on to the next part. We’ll utilise the retrospective simulation to calculate the value at risk and the conditional value at risk (read more: 1, 2, 3).

Output:

Value at Risk - 90%: -0.014976172535594811
Value at Risk - 95%: -0.022113806787530325
Value at Risk - 99% -0.04247765359038646
Expected Shortfall - 90%: -0.026779592114352924
Expected Shortfall - 95%: -0.035320511964199504
Expected Shortfall - 99% -0.058565593363193474

Let’s decipher the above output. We first calculated the daily percent returns of all 1000 simulated paths. Every path has 5,155 days of data, which yielded 5,154 returns per path. When multiplied by 1,000 paths, this resulted in 5,154,000 values of daily returns. We used all these values and found the lowest 90th, 95th, and 99th percentile values, respectively.

From the above output, for example, we can say with 95% certainty that if the future prices follow paths similar to these simulated paths, the maximum drawdown that we can face on any given day would be 2.21%. The expected drawdown would be 3.53% if that level gets breached.

Let’s talk about the extremes now. Let’s compare the maximum and minimum daily returns of the simulated paths and the realised in-sample path.

Realized Lowest Daily Return: -0.1315258002691394
Realized Highest Daily Return: 0.17339334818061447

The maximum values from both approaches are close, at around 17.4%. Same for the minimum values, at around -13.2%. This makes a case for using this approach in financial modelling.

Distribution of Simulated Data

Let’s see how the simulated returns are distributed and compare them visually to a normal distribution. We’ll also calculate the skewness and the kurtosis.

Skewness: -0.11595652411010503
Kurtosis: 9.597364213156881

The argument ‘kde’, when set to ‘True’, smooths the histogram curve, as shown in the above plot. Also, if you want a more granular (coarse) visual of the distribution, you can increase (reduce) the value in the ‘bins’ argument.

Though the histogram resembles a bell curve, it’s far from a normal distribution. It exhibits heavy kurtosis, meaning there are significant chances of finding returns that are many standard deviations away from the mean. And this isn’t any surprise, since that’s how equity and equity-index returns are inherently.

Where This Approach Can Be Most Useful

While the strategy I used here is simple and illustrative, this retrospective simulation framework comes into its own when applied to more complex or nuanced strategies. It’s beneficial in cases where:

1. You're testing multi-condition or ML-based models that might overfit on a single realized path.
2. You want to stress test a strategy across alternate historical realities that didn’t happen, but very well could have.
3. Traditional walk-forward or cross-validation techniques don’t seem to be enough, and you want an added lens to evaluate generalisability.
4. You're exploring how a strategy might behave (or might have behaved had the price taken on any alternate price path) under extreme market moves that aren’t present in the actual historical path (but could well have).

In essence, this method enables you to shift your paradigm from “what happened” to “what could have happened,” a subtle yet impactful change in approach.

Suggested Next Steps

If you found this approach interesting, here are a few ways you can extend it:

1. Try more sophisticated strategies: Apply this retrospective simulation to mean-reversion, volatility breakout, or reinforcement learning-based strategies.
2. Introduce macro constraints: Anchor the simulations around known macroeconomic markers or regime changes to test how strategies behave in such environments.
3. Use intermediate anchor points: Instead of just fixing the start and end prices, try anchoring the simulation at quarterly or annual levels to better control drift and convergence.
4. Train ML models on simulated paths: If you’re working with supervised learning or deep learning models, train them on multiple simulated realities instead of one.
5. Portfolio-level testing: Use this framework to evaluate VaR, CVaR, or stress-test an entire portfolio, not just a single strategy.

This is just the beginning, and how you build on it depends on your curiosity, computing resources, and the questions you're trying to answer.

In Summary

1. We introduced a retrospective simulation framework using a non-parametric Brownian bridge approach to simulate alternate historical price paths.
2. We employed a simple EMA crossover strategy to illustrate how this simulation can be integrated into a traditional backtesting loop.
3. We extracted the best SEMA and LEMA combinations after running backtests on the simulated in-sample paths, and then used those for backtesting on the out-of-sample data.
4. This simulation method enables us to test how strategies would behave not only in response to what happened, but also in response to what could have happened, helping us avoid overfitting and uncover robust signals.
5. The same simulated paths can be used to derive distributional insights, such as tail risk (VaR, CVaR) or return extremes, offering a deeper understanding of the strategy’s risk profile.

Frequently Asked Questions

1. Curious why we simulate price paths at all?
Real market data shows only one path the market took, among many possible paths. But what if we want to understand how our strategy would behave across many plausible realities in the future, or would have behaved across such realities in the past? That’s why we use simulations.

2. What exactly is a Brownian bridge, and why was it used?
A Brownian bridge simulates price movements that start and end at specific values, like real historical prices. This helps ensure simulated paths are anchored in reality while still allowing randomness in between. The main question we ask here is “What else could have happened in the past?”.

3. How many simulated paths should I generate to make this analysis meaningful?
We used 1000 paths. As mentioned in the blog, when the number of simulated paths increases, computation time increases, but our confidence in the results grows too.

4. Is this only for simple strategies like moving averages?
Not at all. We used the moving average crossover just as an example. This framework can be (and should be) used when you’re testing complex, ML-based, or multi-condition strategies that may overfit to historical data.

5. How do I find the best parameter settings (like SEMA/LEMA)?
For each simulated path, we backtested different parameter combinations and recorded the one that gave the highest return. By counting which combinations performed best across most simulations, we identified the combination that is most likely to perform well. The idea is not to rely on the combination that works on just one path.

6. How do I know which parameter combo to use in the markets?
The idea is to pick the combo that most frequently yielded the best results across many simulated realities. This helps avoid overfitting to the single historical path and instead focuses on broader adaptability. The principle here is not to let our analysis and backtesting be subject to chance or randomness, but rather to have some statistical significance.

7. What happens after I find that "best" parameter combination?
We run an out-of-sample backtest using that combination on data the model hasn’t seen. This tests whether the strategy works outside of the data on which the model is trained.

8. What if the strategy fails in the out-of-sample test?
That’s okay, and in this example, it did! The point is not to "win" with a basic strategy, but to show how simulation and robust testing reveal weaknesses before real money is involved. Of course, when you backtest an actual alpha-generating strategy using this approach and still get underperformance in the out-of-sample, it likely means that the strategy isn’t robust, and you’ll need to make changes to the strategy.

9. How can I use these simulations to understand potential losses?
We followed the approach of flattening the returns from all simulated paths into one big distribution and calculating risk metrics like Value at Risk (VaR) and Conditional VaR (CVaR). These show how bad things can get, and how often.

10. What’s the difference between VaR and CVaR?

• VaR tells us the worst expected loss at a given confidence level (e.g., “you’ll lose no more than 2.2% on 95% of days”).
• CVaR goes a step further and says, “If you lose more than that, here’s the average of those worst days.”.

11. What did we learn from the VaR/CVaR results in this example?
We saw that 99% of days resulted in losses no worse than ~4.25%. But when losses exceeded that threshold, they averaged ~5.86%. That’s a useful insight into tail risk. These are the rare but severe events that can highly affect our trading accounts if not accounted for.

12. Are the simulated return extremes realistic compared to real markets?
Yes, they matched very closely with the maximum and minimum daily returns from the real in-sample data. This validates that our simulation isn’t just random but is grounded in reality.

13. Do the simulated returns follow a normal distribution?
Not quite. The returns showed high kurtosis (fat tails) and slight negative skewness, meaning extreme moves (both up and down) are more common than a normal distribution would have. This mirrors real market behaviour.

14. Why does this matter for risk management?
If our strategy assumes normal returns, we’re heavily underestimating the probability of significant losses. Simulated returns reveal the true nature of market risk, helping us prepare for the unexpected.

15. Is this just an academic exercise, or can I apply this practically?
This approach is incredibly useful in practice, especially when you’re working with:

• Machine learning models that are  prone to overfitting
• Strategies designed for high-risk environments
• Portfolios where stress testing and tail risk are crucial
• Regime-switching or macro-anchored models

It helps shift our mindset from "What worked before?" to “What would have worked across many alternate market scenarios?”, and that can be one latent source of alpha.

Conclusion

Hope you learned at least one new thing from this blog. If so, do share what it is in the comments section below and let us know if you’d like to read or learn more about it. The key takeaway from the above discussion is the importance of performing simulations retrospectively and applying them to financial modelling. Apply this approach to more complex strategies and share your experiences and findings in the comments section. Happy learning, happy trading 🙂

Credits

José Carlos Gonzáles Tanaka and Vivek Krishnamoorthy, thank you for your meticulous feedback; it helped shape this article!

Chainika Thakar, thanks for rendering and publishing this, and making it available to the world, that too on your birthday!

Disclaimer: All investments and trading in the stock market involve risk. Any decision to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.

TL;DR

Most trading strategies fail because they assume the market behaves the same all the time.
But real markets shift between calm and chaotic, and strategies must adapt accordingly.

This project builds a Python-based adaptive trading strategy that:

• Detects current market regime using a Hidden Markov Model (HMM)
• Trains specialist ML models (Random Forests) for each regime
• Uses the most relevant model based on regime prediction
• Filters weak signals to reduce noise
• Compares performance vs. Buy-and-Hold
• Uses walk-forward backtesting to remain adaptive over time
• Applies this to Bitcoin, but easily extendable to other assets

It’s a modular, beginner-friendly framework that you can customise, extend, and evolve for real-world deployment.

Prerequisites

To get the most out of this blog, it’s helpful to be familiar with a few foundational concepts. First, understanding Python fundamentals and libraries is essential, especially the use of Pandas for handling time-series data. You can explore these in-depth through Python for Trading: A Step-By-Step Guide and Pandas in Python: A Guide to High-Performance Data Analysis.

Since the blog heavily leans on probabilistic modeling, having prior exposure to Markov processes and their extension into Hidden Markov Models is recommended. For that, Markov Model - An Introduction and Intro to Hidden Markov Chains will provide the necessary conceptual grounding.

Additionally, as this strategy aims to adapt to changing market conditions, knowledge of walk-forward optimization can be beneficial. Walk-Forward Optimization (WFO): A Framework for More Reliable Backtesting helps you understand how to evaluate models over shifting regimes.

A common reason trading strategies fail is that they’re too rigid.

Let me unpack that.

They apply the same logic whether the market is calm and trending or volatile and chaotic. A strategy that works well in one environment can easily fall apart in another.

So, what’s the solution? It might not be a "better" rigid strategy, but an adaptive one to these “market regimes”.

So, what are we going to do today?

We're going to build a Python-based trading strategy that first tries to figure out the market's current "mood" (or regime) and then uses a machine learning model trained specifically for that environment. We'll walk through the entire script, function by function, so you can see how it all fits together.

This is a practical framework you can experiment with and build on. Let's get into the code.

Are you ready? Get your popcorn, eat it with the left hand, scroll down with the right!

The Foundation: Imports and Setup

First things first, let's get our imports out of the way. If you've done any quantitative analysis in Python, these libraries should look familiar. They are the standard tools for data handling, machine learning, and finance. For a good summary of the most useful libraries, QuantInsti's Blog on the Best Python Libraries for Algorithmic Trading is a great resource.

Python code:

Step 1: Getting the Data

In algo trading:No data, no strategy!

So, our first function, get_data, is a simple utility to download historical market data using yfinance. We also calculate the daily percentage returns here, as this will be a key input for our regime detection model later.

Python code:

Step 2: Feature Engineering

Raw price data alone isn't very useful for a machine learning model. We need to give it more context. This is where feature engineering comes in.

The engineer_features function does two main things:

1. Calculates Technical Indicators: It uses the ta library to generate dozens of indicators like RSI, MACD, and Bollinger Bands. This gives our model information about momentum, volatility, and trends.
2. Ensures Stationarity: This is a crucial step in time series analysis. We test each indicator to see if it's "stationary." A non-stationary indicator (like a moving average on a trending stock) can mislead a model. If an indicator isn't stationary, we convert it to a percentage change to make it more stable.

Finally, we define our target y_signal: 1 if the price goes up the next day, and -1 if it goes down. This is what our model will try to predict.

Python code:

Step 3: The Backtesting Engine

This is where the core logic of the strategy lives. A backtest shows how a strategy might have performed in the past. We use a "walk-forward" method, which is more realistic than a simple train-test split because it continuously retrains the models on more recent data. This helps the strategy adapt to changing market behavior over time. To learn more about this method, check out QuantInsti's article on Walk-Forward Optimization.

The run_backtest function is doing a lot, so let's break it down.

The Code: run_backtest

Python code:

Breaking Down the Backtest Logic

So, you saw this whole code script and you stopped eating your popcorn, right?

Don’t worry! We got you covered:

On each day of the backtest, the script performs these steps:

1. Slice the Data:

It creates a window_size (4 years) of the most recent historical data to work with.

2. Detect the Market Regime:

• It trains a Hidden Markov Model (HMM) on the daily returns of the historical data. The HMM's job is to find hidden "states" in the data. We've set it to find two states, which often correspond to low-volatility and high-volatility periods.
• The HMM then labels each day in our historical data as belonging to either "Regime 0" or "Regime 1".

3. Train Specialist Models:

• Now, instead of training one general model, we train two specialists using Random Forest Classifiers.
• Model 0 is trained only on data the HMM labeled as "Regime 0." It becomes our low-volatility expert.
• Model 1 is trained only on "Regime 1" data, making it our high-volatility expert.

4. Forecast and Generate a Signal:

• First, the HMM predicts the probability of tomorrow being in Regime 0 vs. Regime 1.
• We then feed today's data to both specialist models. Model 0 gives us its prediction, and Model 1 gives us its prediction. These are probabilities of an upward move.
• Here's the key part: if the HMM is leaning towards Regime 0 for tomorrow, we use the signal from Model 0. If it expects Regime 1, we use the signal from Model 1.

5. Filter Out Weak Signals as a Risk Management Tool:

• We don't want to trade on every minor signal. A 51% probability isn't very convincing. We set a limit threshold.
• We only go long (1) if the chosen model's probability is high enough (e.g., > 0.53).
• Otherwise, we stay neutral (0). This helps filter out noise.

Step 4&5: Visualizing Results and Running the Script

After all that work, we need to see if it paid off. The plot_results function calculates the strategy's cumulative returns and plots them against a simple Buy-and-Hold strategy for comparison.

Python code:

The compute_perf_stats function prints a table with relevant metrics to evaluate the performance of both strategies.

Python code:

Last but not least, the main execution block (if __name__ == '__main__':) is where you set the parameters like the ticker and date range, and run the whole process.

For this exercise, we use Bitcoin as our preferred asset. Import data from 2008 to 2025, show backtesting results from January 2024, and create the prediction feature with the first lead of the close-to-close returns.

Python code:

See the plot:

And the performance stats table:

The results look promising because the strategy returns have lower volatility than the buy-and-hold returns. Although this is just a sample. There are some things you can do to improve the results:

• Add more input features
• Add risk-management thresholds
• Instead of training your ML model in the regime-specific training samples, you can generate multiple paths of synthetic data based on each regime and optimize your ML model based on those synthetic samples. Check out our blog, TGAN for trading.
• You can use more ML models for each regime and create the signal based on a meta learner.

Frequently Asked Questions

1. What is a "market regime"?

A market regime is a broad characterisation of market behaviour, such as high volatility versus low volatility. This framework uses machine learning (HMM) to detect such regimes dynamically.

2. Why train separate models for different regimes?

Because one-size-fits-all models might tend to underperform in some cases. Models trained on specific market conditions might be better at capturing behavior patterns relevant to that regime.

3. What kind of data does this strategy use?

• Price data from Yahoo Finance via yfinance
• Engineered features like RSI, MACD, Bollinger Bands
• Daily returns and their regime-labeled patterns

4. What machine learning models are used?

• Hidden Markov Models (HMMs) to classify regimes
• Random Forest Classifiers for predicting the next move within each regime
• (Optionally) Meta learners or ensemble models can be added later

5. What’s “walk-forward” backtesting?

A realistic evaluation method where the model is retrained over expanding windows of historical data. This simulates how a strategy might behave when deployed live.

6. Why Bitcoin?

Bitcoin offers high volatility, clear regime shifts, and continuous market access, making it ideal for showcasing adaptive strategies. But the framework works for stocks, forex, or futures too.

7. Can I run this without coding?

Some coding knowledge is required, particularly in Python, pandas, and scikit-learn. But the functions are modular, well-commented, and beginner-friendly.

8. How can I improve this strategy?

• Add more engineered features (volume, macro data, sentiment, etc.)
• Use synthetic data to augment training
• Add stop-loss or drawdown thresholds
• Experiment with different ML models (XGBoost, LSTMs, Transformers)
• Add a meta learner to combine model predictions

Conclusion

By identifying the market state first and then applying a specialist model, this strategy builds adaptability into its core logic. It’s less about having a single perfect model and more about having the right model for the right conditions.

What we've built here is a framework for thinking about market dynamics. The best way to learn is by doing, so I encourage you to grab the script and play with it. Try different tickers, adjust the conviction limit, swap out the Random Forest for another model, or add new features. This is a solid foundation for developing your own robust trading strategies.

Next Steps

Once you’ve worked through the blog and understand how regime classification and model selection work in tandem, you might want to build on this framework using more advanced tools.

A natural next step is to explore alternative models like XGBoost for better predictive power. The blog XGBoost for Time Series Forecasting in Trading walks through its implementation. To further expand your modeling horizons, Directional Change in Trading introduces a novel way to detect market shifts that goes beyond time-based segmentation.

At the same time, robust risk management is crucial when using multiple models, and Position Sizing in Trading offers a practical framework for capital allocation based on model confidence and volatility.

For structured learning, the Technical Indicators & Strategies in Python course on Quantra provides a foundation in strategy design using rule-based indicators, helping you contrast them with your machine-learning approach.

If you’re interested in diving deeper into supervised learning, model evaluation, and time-series forecasting, you’ll find the Machine Learning & Deep Learning in Trading learning track on Quantra highly relevant.

Finally, if you are looking for the best algorithmic trading course to take your strategy-building journey further moving from this Python concept to a live deployment, the Executive Programme in Algorithmic Trading (EPAT) offers a comprehensive curriculum, including modules on machine learning, backtesting, and API integration with brokers.

Disclaimer: This blog post is for informational and educational purposes only. It does not constitute financial advice or a recommendation to trade any specific assets or employ any specific strategy. All trading and investment activities involve significant risk. Always conduct your own thorough research, evaluate your personal risk tolerance, and consider seeking advice from a qualified financial professional before making any investment decisions.

TL;DR

Bayesian statistics offers a flexible, adaptive framework for making trading decisions by updating beliefs with new market data. Unlike traditional models, Bayesian methods treat parameters as probabilities, making them ideal for uncertain, fast-changing financial markets. Understanding Bayesian methods is increasingly seen as a core competency in quant trading, where probabilistic thinking drives better decision-making.

They’re used in risk management, model tuning, classification, and incorporating expert views or alternative data. Tools like PyMC and Bayesian optimisation make it accessible for quants and traders aiming to build smarter, data-driven strategies.

This blog covers:

• The Bayesian Basics
• Why Bayesian Statistics Shines in Algorithmic Trading
• Practical Bayesian Applications in Algorithmic Trading
• Recent Industry Buzz in Bayesian Algorithmic Trading
• The Upsides and Downsides: What to Keep in Mind
• Frequently Asked Questions

Want to ditch rigid trading models and truly harness the power of incoming market information? Imagine a system that learns and adapts, just like you do, but with the precision of mathematics. Welcome to the world of Bayesian statistics, a game-changing framework for algorithmic traders. It’s all about making informed decisions by logically blending what you already know with what the market is telling you right now.

Let's explore how this can sharpen your trading edge!

This approach contrasts with the traditional, or "frequentist," view of probability, which often sees probabilities as long-run frequencies of events and parameters as fixed, unknown constants (Neyman, 1937).

Bayesian statistics, on the other hand, treats parameters themselves as random variables about which we can have beliefs and update them as more data comes in (Gelman et al., 2013). Honestly, this feels tailor-made for trading, doesn't it? After all, market conditions and relationships are hardly ever set in stone. So, let's jump in and see how you can use Bayesian stats to get a leg up in the fast-paced world of finance and algorithmic trading.

Prerequisites

To fully grasp the Bayesian methods discussed in this blog, it is important to first establish a foundational understanding of probability, statistics, and algorithmic trading.

For a conceptual introduction to Bayesian statistics, Bayesian Inference Methods and Equation Explained with Examples offers an accessible explanation of Bayes' Theorem and how it applies to uncertainty and decision-making, foundational to applying Bayesian models in markets.

What You'll Learn:

• The core idea behind Bayesian thinking is updating beliefs with new evidence.
• Understanding Bayes' Theorem: your mathematical tool for belief updating.
• Why Bayesian methods are a great fit for the uncertainties of financial markets.
• Practical examples of Bayesian statistics in algorithmic trading:
• Estimating model parameters that adapt to new data.
• Building simple predictive models (like Naive Bayes for market direction).
• Incorporating expert views or alternative data into your models.
• The Pros, Cons, and Recent Trends of Using Bayesian Approaches in Quantitative Finance.

The Bayesian Basics

Prior Beliefs, New Evidence, Updated Beliefs

Okay, let's break down the fundamental magic of Bayesian statistics. At its core, it's built on a wonderfully simple yet incredibly powerful idea: our understanding of the world is not static; it evolves as we gather more information.

Think about it like this: you've got a new trading strategy you're mulling over.

• Prior Belief (Prior Probability): Based on your initial research, backtesting on historical data, or even a hunch, you have some initial belief about how profitable this strategy might be.  Let's say you think there's a 60% chance it will be profitable. This is your prior.
• New Evidence (Likelihood): You then deploy the strategy on a small scale or observe its hypothetical performance over a few weeks of live market data. This new data is your evidence. The likelihood function tells you how probable this new evidence is, given different underlying states of the strategy's true profitability.
• Updated Belief (Posterior Probability): After observing the new evidence, you update your initial belief. If the strategy performed well, your confidence in its profitability might increase from 60% to, say, 75%. If it performed poorly, it might drop to 40%. This updated belief is your posterior.

This whole process of tweaking your beliefs based on new info is neatly wrapped up and formalised by what is called the Bayes' Theorem.

Bayes' Theorem: The Engine of Bayesian Learning

So, Bayes' Theorem is the actual formula that ties all these pieces together. If you have a hypothesis (let’s call it H) and some evidence (E), the theorem looks like this:

Bayes’ Theorem:

\( P(H \mid E) = \frac{P(E \mid H) \cdot P(H)}{P(E)} \)

Where:

• P(H|E) is the Posterior Probability: The probability of your hypothesis (H) being true after observing the evidence (E). This is what you want to calculate; your updated belief.
• P(E|H) is the Likelihood: The probability of observing the evidence (E) if your hypothesis (H) were true. For example, if your hypothesis is “this stock is bullish,” how likely is it to see a 2% price increase today?
• P(H) is the Prior Probability: The probability of your hypothesis (H) being true before observing the new evidence (E). This is your initial belief.
• P(E) is the Probability of the Evidence (also called Marginal Likelihood or Normalising Constant): The overall probability of observing the evidence (E) under all possible hypotheses. It's calculated by summing (or integrating) P(E|H) × P(H) over every possible H. This ensures the posterior probabilities sum up to 1.

Let's try to make this less abstract with a quick trading scenario.

Example: Is a News Event Bullish for a Stock?

Suppose a company is about to release an earnings report.

• Hypothesis (H): The earnings report will be significantly better than expected (a "positive surprise").
• Prior P(H): Based on analyst chatter and recent sector performance, you believe there's a 30% chance of a positive surprise. So, P(H) = 0.30.
• Evidence (E): In the hour before the official announcement, the stock price jumps 1%.
• Likelihood P(E|H): You know from past experience that if there's a genuinely positive surprise brewing, there's a 70% chance of seeing such a pre-announcement price jump due to insider information or some sharp traders catching on early. So, P(E|H) = 0.70.
• Probability of Evidence P(E): This one's a little more involved because the price could jump for other reasons, too, right? Maybe the whole market is rallying, or it's just a false rumour. Let's say:
  • The probability of the price jump if it's a positive surprise (P(E|H)) is 0.70 (as above).
  • The probability of the price jump if it's not a positive surprise (P(E|not H)) is, say, 0.20 (it's less likely, but possible).
  • Since P(H) = 0.30, then P(not H) = 1 - 0.30 = 0.70.
  • So, P(E) = P(E|H)P(H) + P(E|not H)P(not H) = (0.70 * 0.30) + (0.20 * 0.70) = 0.21 + 0.14 = 0.35.

Now we can calculate the Posterior \( P(H \mid E) \):

\( P(H \mid E) = \frac{0.70 \times 0.30}{0.35} = \frac{0.21}{0.35} = 0.60 \)

Boom! After seeing that 1% price jump, your belief that the earnings report will be a positive surprise has shot up from 30% to 60%! This updated probability can then inform your trading decision, perhaps you're now more inclined to buy the stock or adjust an existing position.

Of course, this is a super-simplified illustration. Real financial models are juggling a significantly greater number of variables and much more complex probability distributions. But the beautiful thing is, that core logic of updating your beliefs as new info comes in? That stays exactly the same.

Source

Why Bayesian Statistics Shines in Algorithmic Trading

Financial markets are a wild ride, full of uncertainty, constantly changing relationships (non-stationarity, if you want to get technical), and often, not a lot of data for those really rare, out-of-the-blue events. Bayesian methods offer several advantages in this environment:

1. Handles Uncertainty Like a Pro: Bayesian statistics doesn't just give you a single number; it naturally deals with uncertainty by using probability distributions for parameters, instead of pretending they are fixed, known values (Bernardo & Smith, 2000). This gives you a much more realistic picture of what might happen.
2. Updating Beliefs with New Data: Algorithmic trading systems constantly process new market data. Bayesian updating allows models to adapt dynamically. For instance, the volatility of an asset isn't constant; a Bayesian model can update its volatility estimate as new price ticks arrive.
3. Working with Small Data Sets: Traditional frequentist methods often require large sample sizes for reliable estimates. Bayesian methods, however, can give you pretty sensible insights even with limited data, because they let you bring in "informative priors" – basically, your existing knowledge from experts, similar markets, or financial theories (Ghosh et al., 2006). This is a lifesaver when you're trying to model rare events or new assets that don't have a long history.
4. Model Comparison and Averaging:  Bayesian techniques provide a really solid way (e.g., using Bayes factors or posterior predictive checks) to compare different models and even average out their predictions. This often leads to more robust and reliable results (Hoeting et al., 1999).
5. Lets You Weave in Qualitative Insights: Got a strong economic reason why a certain parameter should probably fall within a specific range? Priors give you a formal way to mix that kind of qualitative hunch or expert opinion with your hard quantitative data.
6. Clearer Interpretation of Probabilities: When a Bayesian model tells you "there's a 70% chance this stock will go up tomorrow," it means exactly what it sounds like: it’s your current degree of belief. This can be a lot more straightforward to act on than trying to interpret p-values or confidence intervals alone (Berger & Berry, 1988).

Practical Bayesian Applications in Algorithmic Trading

Alright, enough theory! Let's get down to brass tacks. How can you actually use Bayesian statistics in your trading algorithms?

1. Adaptive Parameter Estimation: Keeping Your Models Fresh

So many trading models lean heavily on parameters – like the lookback window for your moving average, the speed of mean reversion in a pairs trading setup, or the volatility guess in an options pricing model. But here’s the catch: market conditions are always shifting, so parameters that were golden yesterday might be suboptimal today.

This is where Bayesian methods are super handy. They let you treat these parameters not as fixed numbers, but as distributions that get updated as new data rolls in. Imagine you're estimating the average daily return of a stock.

• Prior: You might start with a vague prior idea(e.g., a normal distribution centred around 0 with a wide spread (standard deviation)) or a more educated guess based on how similar stocks in the sector have performed historically.
• Likelihood: As each new trading day provides a return, you calculate the likelihood of observing that return given different possible values of the true average daily return.
• Posterior: Bayes' theorem combines the prior and likelihood to give you an updated distribution for the average daily return. This posterior becomes the prior for the next day's update.It's a continuous learning loop!

Hot Trend Alert: Techniques like Kalman Filters (which are inherently Bayesian) are widely used for dynamically estimating unobserved variables, like the "true" underlying price or volatility, in noisy market data (Welch & Bishop, 2006). Another area is Bayesian regression, where the regression coefficients (e.g., the beta of a stock) are not fixed points but distributions that can evolve.
For more on regression in trading, you might want to check out how Regression is Used in Trading.

Simplified Python Example: Updating Your Belief about a Coin's Fairness (Think Market Ups and Downs)

Let's say we want to get a handle on the probability of a stock price going up (we'll call it 'Heads') on any given day. This is a bit like trying to figure out if a coin is fair or biased.

Python Code:

Output:

Initial Prior: Alpha=1, Beta=1
Observed Data: 6 'up' days, 4 'down' days
Posterior Belief: Alpha=7, Beta=5
Updated Estimated Probability of an 'Up' Day: 0.58
95% Credible Interval for p_up: (0.31, 0.83)

In this code:

• We start off with a Beta(1,1) prior, which is uniform and suggests any probability of an 'up' day is equally likely.
• Then, we observe 10 days of market data with 6 'up' days.
• The posterior distribution becomes Beta(1+6, 1+4) = Beta(7, 5).
• Our new point estimate for the probability of an 'up' day is 7 / (7+5) = 0.58, or 58%.
• The credible interval gives us a range of plausible values.

The graph provides a clear visual for this belief-updating process. The flat blue line represents our initial, uninformative prior, where any probability for an 'up' day was considered equally likely. In contrast, the orange curve is the posterior belief, which has been sharpened and informed by the observed market data. The peak of this new curve, centered around 0.58, represents our updated, most probable estimate, while its more concentrated shape signifies our reduced uncertainty now that we have evidence to guide us.

This is a toy example, but it shows the mechanics of how beliefs get updated. In algorithmic trading, this could be applied to the probability of a profitable trade for a given signal or the probability of a market regime persisting.

2. Naive Bayes Classifiers for Market Prediction: Simple but Surprisingly Smart!

Next up, let's talk about Naive Bayes. It's a straightforward probabilistic classifier that uses Bayes' theorem, but with a "naive" (or let's say, optimistic) assumption that all your input features are independent of each other. Despite its simplicity, it can be surprisingly effective for tasks like classifying whether the next day's market movement will be 'Up', 'Down', or 'Sideways' based on current indicators. (Rish, 2001)

Here’s how it works (conceptually):

• Define Features: These could be technical indicators (e.g., RSI < 30, MACD crossover), price patterns (e.g., yesterday was an engulfing candle), or even sentiment scores from financial news.
• Collect Training Data: Gather historical data where you have these features and the actual outcome (Up/Down/Sideways).
• Calculate Probabilities from Training Data:
  • Prior Probabilities of Outcomes: P(Up), P(Down), P(Sideways) – simply the frequency of these outcomes in your training set.
  • Likelihood of Features given Outcomes: P(Feature_A | Up), P(Feature_B | Up), etc. For instance, "What's the probability RSI < 30, given the market went Up the next day?"
• Make a Prediction: For new data (today's features):
  • Calculate the posterior probability for each outcome:
  • P(Up | Features) ∝ P(Up) * P(Feature_A | Up) * P(Feature_B | Up) * ...
  • P(Down | Features) ∝ P(Down) * P(Feature_A | Down) * P(Feature_B | Down) * ...
  • (And similarly for Sideways)
  • The outcome with the highest posterior probability is your prediction.

Python Snippet Idea (Just a concept, you'd need sklearn for this):

Python Code:

Output:

Naive Bayes Classifier Accuracy (on dummy data): 0.43

This accuracy score of 0.43 indicates the model correctly predicted the market's direction 43% of the time on the unseen test data. Since this result is below 50% (the equivalent of random chance), it suggests that, with the current dummy data and features, the model does not demonstrate predictive power. In a real-world application, such a score would signal that the chosen features or the model itself may not be suitable, prompting a re-evaluation of the approach or further feature engineering.

This little snippet gives you the basic flow. Building a real Naive Bayes classifier for trading takes careful thought about which features to use (that's "feature engineering") and rigorous testing (validation). That "naive" assumption that all features are independent might not be perfectly true in the messy, interconnected world of markets, but it often gives you a surprisingly good starting point or baseline model.
Curious about where to learn all this? Don’t worry, friend, we’ve got you covered! Check out this course.

3. Bayesian Risk Management (e.g., Value at Risk - VaR)

You've probably heard of Value at Risk (VaR), it's a common way to estimate potential losses. But traditional VaR calculations can sometimes be a bit static or rely on simplistic assumptions. Bayesian VaR allows for the incorporation of prior beliefs about market volatility and tail risk, and these beliefs can be updated as new market shocks occur. This can lead to risk estimates that are more responsive and robust, especially when markets get choppy.

For instance, if a "black swan" event occurs, a Bayesian VaR model can adapt its parameters much more quickly to reflect this new, higher-risk reality. A purely historical VaR, on the other hand, might take a lot longer to catch up.

4. Bayesian Optimisation for Finding Goldilocks Strategy Parameters

Finding those "just right" parameters for your trading strategy (like the perfect entry/exit points or the ideal lookback period) can feel like searching for a needle in a haystack. Bayesian optimisation is a seriously powerful technique that can help here. It cleverly uses a probabilistic model (often a Gaussian Process) to model the objective function (like how profitable your strategy is for different parameters) and selects new parameter sets to test in a way that balances exploration (trying new areas) and exploitation (refining known good areas) (Snoek et al., 2012). This can be much more efficient than just trying every combination (grid search) or picking parameters at random.

Hot Trend Alert:Bayesian optimisation is a rising star in the broader machine learning world and is incredibly well-suited for fine-tuning complex algorithmic trading strategies, especially when running each backtest takes a lot of computational horsepower.

5. Weaving in Alternative Data and Expert Hunches (Opinions)

These days, quants are increasingly looking at "alternative data" sources, things like satellite images, the general mood on social media, or credit card transaction trends. Bayesian methods give you a really natural way to integrate such diverse and often unstructured data with traditional financial data. You can set your priors based on how reliable or strong you think the signal from an alternative data source is.

And it's not just about new data types. What if a seasoned portfolio manager has a strong conviction about a particular sector because of some geopolitical development that's difficult to quantify? That "expert opinion" can actually be formalised into a prior distribution, allowing it to influence the model's output right alongside the purely data-driven signals.

Recent Industry Buzz in Bayesian Algorithmic Trading

While Bayesian methods have been around in finance for a while, a few areas are really heating up and getting a lot of attention lately:

1. Bayesian Deep Learning (BDL): You know how traditional deep learning models give you a single prediction but don't really tell you how "sure" they are? BDL is here to change that! It combines the power of deep neural networks with Bayesian principles to produce predictions with associated uncertainty estimates (Neal, 1995; Gal & Ghahramani, 2016). This is crucial for financial applications where knowing the model's confidence is as important as the prediction itself. For example, imagine a BDL model not just predicting a stock price, but also saying it's "80% confident the price will land between X and Y".
2. Probabilistic Programming Languages (PPLs): Languages like Stan, PyMC3 (Salvatier et al., 2016), and TensorFlow Probability are making it easier for quants to build and estimate complex Bayesian models without getting bogged down in the low-level mathematical details of inference algorithms like Markov Chain Monte Carlo (MCMC). This easier access is really democratising the use of sophisticated Bayesian techniques across the board (Carpenter et al., 2017).
3. Sophisticated MCMC and Variational Inference: As our models get more ambitious, the computational grunt work needed to fit them also grows. Thankfully, researchers are constantly cooking up more efficient MCMC algorithms (like Hamiltonian Monte Carlo) and speedier approximate methods like Variational Inference (VI) (Blei et al., 2017), making larger Bayesian models tractable for real-world trading.
   If you want to learn more about MCMC, QuantInsti has an excellent blog on Introduction to Monte Carlo Analysis.
4. Dynamic Bayesian Networks for Spotting Market Regimes: Financial markets often seem to flip between different "moods" or "regimes", think high-volatility vs. low-volatility periods, or bull vs. bear markets. Dynamic Bayesian Networks (DBNs) can model these hidden market states and the probabilities of transitioning between them, allowing strategies to adapt their behavior accordingly (Murphy, 2002).

The Upsides and Downsides: What to Keep in Mind

Like any powerful tool, Bayesian methods come with their own set of pros and cons.

Advantages:

• Intuitive framework for updating beliefs.
• Quantifies uncertainty directly.
• Works well with limited data by using priors.
• Allows incorporation of expert knowledge.
• Provides a coherent way to compare and combine models.

Limitations:

• Choice of Prior: The selection of a prior can be subjective and can significantly influence the posterior, especially with small datasets. A poorly chosen prior can lead to poor results. While techniques for "objective" or "uninformative" priors exist, their appropriateness is often debated.
• Computational Cost: For complex models, estimating the posterior distribution (especially using MCMC methods) can be computationally intensive and time-consuming, which might be a constraint for high-frequency trading applications.
• Mathematical Complexity: While PPLs are helpful, a solid understanding of probability theory and Bayesian concepts is still needed to apply these methods correctly and interpret results.

Frequently Asked Questions

Q. What makes Bayesian statistics different from traditional (frequentist) methods in trading?
Bayesian statistics treats model parameters as random variables with a and allows beliefs to be updated with new data. In contrast, frequentist methods assume parameters are fixed and require large data samples. Bayesian thinking is more dynamic and well-suited to the non-stationary, uncertain nature of financial markets.

Q. How does Bayes’ Theorem help in trading decisions? Can you give an example?
Bayes’ Theorem is used to update probabilities based on new market information. For example, if a stock price jumps 1% before earnings, and past data suggests this often precedes a positive surprise, Bayes’ Theorem helps revise your confidence in that hypothesis, turning a 30% belief into 60%, which can directly influence your trade.

Q. What are priors and posteriors in Bayesian models, and why do they matter in finance?
A prior reflects your initial belief (from past data, theory, or expert views), while a posterior is the updated belief after considering new evidence. Priors help improve performance in low-data or high-uncertainty situations and allow integration of alternative data or human intuition in financial modelling.

Q. What types of trading problems are best suited for Bayesian methods?
Bayesian techniques are ideal for:

• Parameter estimation that adapts (example, volatility, beta, moving average lengths)
• Market regime detection using dynamic Bayesian networks
• Risk management (example, Bayesian VaR)
• Strategy optimisation using Bayesian Optimisation
• Classification tasks with Naive Bayes models
  These approaches help build more responsive and robust strategies.

Q. Can Bayesian methods work with limited or noisy market data?
Yes! Bayesian methods shine in low-data environments by incorporating informative priors. They also handle uncertainty naturally, representing beliefs as distributions rather than fixed values, crucial when modelling rare market events or new assets.

Q. How is Bayesian optimisation used in trading strategy design?
Bayesian optimisation is used to tune strategy parameters (like entry/exit thresholds) efficiently. Instead of brute-force grid search, it balances exploration and exploitation using a probabilistic model (example, Gaussian Processes), making it perfect for costly backtesting environments.

Q. Are simple models like Naive Bayes really useful in trading?
Yes, Naive Bayes classifiers can serve as lightweight baseline models to predict market direction using indicators like RSI, MACD, or sentiment scores. While the assumption of independent features is simplistic, these models can offer fast and surprisingly solid predictions, especially with well-engineered features.

Q. How does Bayesian thinking enhance risk management?
Bayesian models, like Bayesian VaR (a, update risk estimates dynamically as new data (or shocks) arrive, unlike static historical models. This makes them more adaptive to volatile conditions, especially during rare or extreme events.

Q. What tools or libraries are used to build Bayesian trading models?
Popular tools include:

• PyMC and PyMC3 (Python)
• Stan (via R or Python)
• TensorFlow Probability
  These support techniques like MCMC and variational inference, enabling the development of everything from simple Bayesian regressions to Bayesian deep learning models.

Q. How can I get started with Bayesian methods in trading?
Start with small projects:

• Test a Naive Bayes classifier on market direction.
• Use Bayesian updating for a strategy’s win rate estimation.
• Try parameter tuning with Bayesian optimisation.
• Then explore more advanced applications and consider learning resources such as Quantra’s courses on machine learning in trading and EPAT for a comprehensive algo trading program with Bayesian techniques.

Conclusion: Embrace the Bayesian Mindset for Smarter Trading!

So, there you have it! Bayesian statistics offers an incredibly powerful and flexible way to navigate the unavoidable uncertainties that come with financial markets. By giving you a formal way to blend your prior knowledge with new evidence as it streams in, it helps traders and quants build algorithmic strategies that are more adaptive, robust, and insightful.

While it's not a magic bullet, understanding and applying Bayesian principles can help you move beyond rigid assumptions and make more nuanced, probability-weighted decisions. Whether you're tweaking parameters, classifying market conditions, keeping an eye on risk, or optimising your overall strategy, the Bayesian approach encourages a mindset of continuous learning, and that’s absolutely vital for long-term success in the constantly shifting landscape of algorithmic trading.

Start small, perhaps by experimenting with how priors impact a simple estimation, or by trying out a Naive Bayes classifier. As you grow more comfortable, the rich world of Bayesian modeling will open up new avenues for enhancing your trading edge.

If you're serious about taking your quantitative trading skills to the next level, consider Quantra's specialised courses like "Machine Learning & Deep Learning for Trading" to enhance Bayesian techniques, or EPAT for comprehensive, industry-leading algorithmic trading certification. These equip you to tackle complex markets with a significant edge.

Keep learning, keep experimenting!

Further Reading

For a structured and applied learning path with Quantra, start with Python for Trading: Basic, then move to Technical Indicators Strategies in Python.

For machine learning, explore the Machine Learning & Deep Learning in Trading: Beginners learning track, which provides practical hands-on insights into implementing models like Bayesian classifiers in financial markets.

If you're a serious learner, you can take the Executive Programme in Algorithmic Trading (EPAT), which covers statistical modelling, machine learning, and advanced trading strategies with Python.

References

• Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 236(767), 333-380.
  https://royalsocietypublishing.org/doi/10.1098/rsta.1937.0005
• Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.
  https://www.routledge.com/Bayesian-Data-Analysis/Gelman-Carlin-Stern-Dunson-Vehtari-Rubin/p/book/9781439840955
• Bernardo, J. M., & Smith, A. F. M. (2000). Bayesian Theory. John Wiley & Sons.
  https://onlinelibrary.wiley.com/doi/book/10.1002/9780470316870
• Ghosh, J. K., Delampady, M., & Samanta, T. (2006). An Introduction to Bayesian Analysis: Theory and Methods. Springer.
  http://ndl.ethernet.edu.et/bitstream/123456789/58197/1/41%20pdf.pdf
• Hoeting, J. A., Madigan, D., Raftery, A. E., & Volinsky, C. T. (1999). Bayesian model averaging: A tutorial. Statistical Science, 14(4), 382-417.
  https://www.stat.colostate.edu/~jah/papers/statsci.pdf
• Welch, G., & Bishop, G. (2006). An introduction to the Kalman filter. TR 95-041, University of North Carolina at Chapel Hill, Department of Computer Science.
  https://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf
• Rish, I. (2001, August). An empirical study of the naive Bayes classifier. In IJCAI 2001 workshop on empirical methods in artificial intelligence (Vol. 3, No. 22, pp. 41-46).
  https://www.researchgate.net/publication/228845263_An_Empirical_Study_of_the_Naive_Bayes_Classifier
• Snoek, J., Larochelle, H., & Adams, R. P. (2012). Practical Bayesian optimization of machine learning algorithms. Advances in neural information processing systems, 25.
  https://papers.nips.cc/paper_files/paper/2012/hash/05311655a15b75fab86956663e1819cd-Abstract.html
• Neal, R. M. (1995). Bayesian learning for neural networks. (Doctoral dissertation, University of Toronto).
  https://glizen.com/radfordneal/ftp/thesis.pdf
• Gal, Y., & Ghahramani, Z. (2016). Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In the International Conference on machine learning (pp. 1050-1059). PMLR.
  https://proceedings.mlr.press/v48/gal16.html
• Salvatier, J., Wiecki, T. V., & Fonnesbeck, C. (2016). Probabilistic programming in Python using PyMC3. PeerJ Computer Science, 2, e55.
  https://peerj.com/articles/cs-55/
• Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., ... & Riddell, A. (2017). Stan: A probabilistic programming language. Journal of Statistical Software, 76(1), 1-32.
  https://www.jstatsoft.org/article/view/v076i01
• Blei, D. M., Kucukelbir, A., & McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518), 859-877.
  https://www.tandfonline.com/doi/full/10.1080/01621459.2017.1285773
• Murphy, K. P. (2002). Dynamic Bayesian Networks: Representation, Inference and Learning. (Doctoral dissertation, University of California, Berkeley).
  https://www.cs.ubc.ca/~murphyk/Thesis/thesis.html

Disclaimer: This blog post is for informational and educational purposes only. It does not constitute financial advice or a recommendation to trade any specific assets or employ any specific strategy. All trading and investment activities involve significant risk. Always conduct your own thorough research, evaluate your personal risk tolerance, and consider seeking advice from a qualified financial professional before making any investment decisions.

TL;DR

Linear regression fits a straight‑line equation between a target (dependent) variable and one or more predictors, with Simple LR using a single factor and Multiple LR handling many. Coefficients show direction and strength (beta), and Ordinary Least Squares finds the “best” line by minimising squared errors. Traders still rely on it because it’s fast, interpretable, and forms the base for risk models, factor analysis, pairs trades, stat‑arb, and hedge‑ratio calculations. To work well, key assumptions—linearity, independent and homoscedastic errors, normal residuals, and low multicollinearity—must hold; otherwise results mislead. Model quality is gauged via R², adjusted R², coefficient p‑values, RMSE, and, above all, out‑of‑sample tests. Limits include sensitivity to outliers, purely linear vision, and shifting market regimes, so practitioners often extend it with regularised variants, rolling windows, or non‑linear and ML methods when relationships get more complex. In the context of quantitative trading, linear regression helps traders identify price relationships and build predictive models based on historical data.

This blog covers:

• What Exactly is Linear Regression? Unveiling the Basics
• Why Do Algorithmic Traders Still Swear By Linear Regression?
• The Ground Rules: Assumptions of Linear Regression
• How Does Linear Regression Actually Work Its Magic? Meet Ordinary Least Squares (OLS)
• Linear Regression in Action: Algorithmic Trading Examples & Case Studies
• Let's Get Practical: Linear Regression with Python
• How Good is Your Model? Evaluating Linear Regression Performance
• Limitations and Pitfalls of Linear Regression in Trading
• Beyond the Basics: What's Next?
• Conclusion: Your Journey with Linear Regression
• Frequently Asked Questions

Hey there, my friend, you are the real trader! Ever wonder how you can systematically get a handle on market movements or figure out the connections between different financial bits and pieces? Well, you're in the right spot! One of the absolute foundational tools in any quant trader's toolkit is Linear Regression.

Now, it might give you flashbacks to your college stats class, but trust me, its power and how much it's used in today's speedy algorithmic trading world are a big deal.

This guide is all about walking you through what linear regression really is, why it's such a big deal in quantitative finance, and how you can start using it—all from a practical, "get-your-hands-dirty" angle for algorithmic trading. We'll keep the math talk intuitive, show you some Python code examples, and check out how it's used in real-world trading.

Prerequisites

Before diving into the practical applications of linear regression in algorithmic trading, it's essential to have a foundational understanding of a few key areas. Start with Algorithmic Trading Basics to get acquainted with how automated strategies function in financial markets. Follow that with Machine Learning Basics, which lays the groundwork for supervised and unsupervised learning models used in financial prediction.

A strong foundation in statistics is critical as well, and Mathematics for Algorithmic Trading provides the necessary background on concepts like mean, variance, correlation, and probability distributions. Since Python is a standard tool for implementing regression models in trading, Python Trading Library offers practical guidance on using Python for handling market data and building strategies.

What Exactly is Linear Regression? Unveiling the Basics

At its heart, linear regression is a statistical method used to model the relationship between a dependent variable (the one you want to predict) and one or more independent variables (the factors you believe influence the dependent variable) by fitting a linear equation to observed data (Draper & Smith, 1998). Think of it as drawing the "best-fitting" straight line through a scatter plot of data points.

So, what is linear regression in trading? Can I use linear regression to predict stock prices?

Let's see!

Picture this: You're eyeing a stock's price (your dependent variable) and you're curious how it's swayed by, let's say, how the overall market index is doing (that's your independent variable). Linear regression helps you quantify this relationship.

• Simple Linear Regression (SLR): This is the most basic form, involving one dependent variable and one independent variable. The relationship is drawn out  as:
  Y = β₀ + β₁X + ε

Where:

• Y is the dependent variable (e.g., stock return).
• X is the independent variable (e.g., market return).
• β₀ is the intercept – the value of Y when X is 0. It represents the expected value of the dependent variable when all independent variables are zero. (Fabozzi, Focardi & Rachev, 2007).
• β₁ is the slope coefficient – it measures how much Y changes for a one-unit change in X. This is the classic "beta" in finance, indicating a stock's sensitivity to market movements.
• ε is the error term, representing the part of Y that the model can't explain.

Source

• Multiple Linear Regression (MLR): Now, let's be real – markets are rarely driven by just one factor. MLR beefs up the concept by letting us use multiple independent variables. This is where it gets super interesting for us traders because we can incorporate various factors like interest rates, volatility indices, commodity prices, or even custom indicators.
  The formula looks like this:
  
  Y = β₀ + β₁X₁ + β₂X₂ + ... + βₚXₚ + ε

Where:

• X₁, X₂, ..., Xₚ are the different independent variables (e.g., Fed interest rates, oil prices)
• β₁, β₂, ..., βₚ are their respective slope coefficients.

Source

Understanding these coefficients is key:

• According to Gujarati & Porter (2009), a positive β means that as your independent variable goes up, your dependent variable tends to go up too, and vice versa for a negative β.
• The magnitude tells you the strength of the influence.

Why Do Algorithmic Traders Still Swear By Linear Regression?

Even with all the fancy machine learning models out there, good old linear regression is still a favorite in algorithmic trading for some solid reasons:

1. It's Simple and You Can Explain It: It's relatively easy to understand and explain. The coefficients (β values) directly tell you the direction and strength of relationships, which is crucial for building conviction in a trading strategy. You can clearly spell out why your model is making the calls it's making.
   E.g., a β of 1.2 implies 20% higher sensitivity to market movements in case the independent variable is a market index.
2. It's Quick on its Feet: Linear regression models are generally fast to train and run. This makes them great for strategies that need to make decisions quickly, especially if you're in the high-frequency or medium-frequency trading game  (Tsay, 2010).
3. Foundational for Complex Models: Many advanced quantitative strategies use linear regression as a starting point or something to compare against (Hastie, Tibshirani & Friedman, 2009). Understanding it well provides a solid foundation for exploring more sophisticated techniques.
4. Risk Management: It's widely used in risk models. For instance, calculating a portfolio's beta to the market is a direct application. It helps you understand and hedge out market exposure (Jorion, 2007).
5. Identifying Potential Arbitrage/Relative Value: By running a regression of one asset's price against another (or a whole basket of them), traders can spot deviations from their usual historical relationship. This could flag a pairs trading or relative value opportunity.

What’s Trending Now? While traditional linear regression is a stalwart, a recent trend involves enhancing it with machine learning techniques. For example, regularization methods like Ridge and Lasso regression (which are variants of linear regression) are increasingly used to prevent overfitting and to make them more robust. This is especially handy when you've got lots of predictors that are all kind of related, which happens a lot in finance (Abu-Mostafa, Magdon-Ismail & Lin, 2012). The industry is also seeing more use of linear regression in factors-based investing and “smart beta” strategies (Ang, 2014).

The Ground Rules: Assumptions of Linear Regression

Linear regression is a powerful beast, but to use it right, you've got to play by its rules—these are its key assumptions. If you ignore them, you could end up with some really misleading results, so it pays to know them (Berry, 1993). Think of these as the "terms and conditions" before you click "I agree" in your house mortgage contract: 1

1. Linearity: The relationship between the independent variables and the mean of the dependent variable is linear.

• What’s linearity in coefficients: Linearity means that the effect of each independent variable on the dependent variable is constant — in other words, the model assumes a straight-line relationship. For every one-unit change in an independent variable, the dependent variable changes by a fixed amount (determined by the coefficient), regardless of the level of the variable. This doesn't mean the data itself must be linear, but that the relationship the model fits is a straight line in the parameter space.
• Why it matters in trading: If you're trying to model a non-linear relationship (e.g., the impact of volatility, which can have diminishing or accelerating effects) with a simple linear model, your predictions will be off.
• How to check & fix it: Scatter plots are your first friend here—they can give you a visual hint. If it looks curvy/non-linear, you might need to transform your variables (e.g., log transformation for prices) or consider non-linear models.

2. Independence of Errors (No Autocorrelation): The error terms (ε) are independent of each other. In time-series data, this means the error in one period is not correlated with the error in the next.

• Why it matters in trading: Financial time series often exhibit autocorrelation (e.g., momentum or mean reversion). If your residuals (the errors) are autocorrelated, your coefficient estimates might still be unbiased, but their standard errors will be wrong, leading to unreliable significance tests.
• How to check & fix it: Use the Durbin-Watson test or plot residuals over time to see if there's a pattern. If you find autocorrelation, you might try adding lagged variables (like yesterday's return) as a predictor, or use models built specifically for time series data, like ARIMA.

3. Homoscedasticity (Constant Variance of Errors): The variance of the error terms is constant across all levels of the independent variables.

• Why it matters in trading:  In time series analysis, markets are famous for having "mood swings"—periods of high volatility followed by periods of higher volatility, and calm periods followed by calmer. This is called heteroskedasticity (the opposite of homoscedasticity). If it's present, your OLS estimates are still unbiased, but they are not the "best" (Minimum Variance Unbiased Estimator or BLUE – Best Linear Unbiased Estimator), and standard errors will be biased (Engle, 1982).
• How to check & fix it: Plot residuals against predicted values or, in case of cross-sectional data, use tests like Breusch-Pagan or White. Using "robust standard errors" (like White's heteroskedasticity-consistent standard errors) or specialized models like GARCH can help with time series analysis.

Source
Caption: The left plot shows homoscedasticity, where the spread of residuals remains consistently tight across all levels of the independent variable — indicating a satisfactory model. In contrast, the right plot shows heteroscedasticity, where residuals spread out unevenly (wider at higher values), signaling a violation of constant variance and an unsatisfactory model fit for linear regression.

4. Normality of Errors: The error terms are normally distributed.

• Why it matters in trading: While linear regression can handle some deviation from this, especially with larger sample sizes (Central Limit Theorem), normality is important for valid hypothesis testing and building reliable confidence intervals. The catch? Financial returns often have "fat tails" (this is called leptokurtosis), which means extreme events are more common than a perfect normal distribution would predict.
• How to check & fix it: You can look at histograms or Q-Q plots of your residuals, or use statistical tests like Shapiro-Wilk or Jarque-Bera. If things look non-normal, transformations or robust regression techniques might be needed. Another solution is to normalize the independent variables with scalers like the z-score or the min-max scaler.

5. No Perfect Multicollinearity: The independent variables are not perfectly correlated with each other.

• Why it matters in trading: If two independent variables are perfectly (or highly) correlated (e.g., using both a 5-day moving average and a 7-day moving average of the same price series), the model gets confused and can't figure out the individual effect of each one on the dependent variable. This leads to unstable and unreliable coefficient estimates.
• How to check & fix it: Calculate a correlation matrix of independent variables or use a metric called Variance Inflation Factor (VIF). If high multicollinearity exists, consider removing one of the correlated variables or combining them (e.g., into an index).

Being diligent about checking these assumptions is a hallmark of a good quantitative analyst. It's not just about hitting "run" on the model; it's about making sure you're running it the right way.

How Does Linear Regression Actually Work Its Magic? Meet Ordinary Least Squares (OLS)

The most common way to find the "best-fitting" line in linear regression is the Ordinary Least Squares (OLS) method. The intuition is simple: OLS tries to draw a line that minimizes the sum of the squared differences between the observed values of Y and the values of Y predicted by the linear model (Ŷ). These differences are called residuals ( e = Y - Ŷ ) (Jarantow, 2023).

Mathematically, it's trying to:

Minimize Σ(Yᵢ - Ŷᵢ)² which is Minimize Σ(Yᵢ - (β₀ + β₁Xᵢ))²

Why squared differences?

• Squaring ensures that negative and positive residuals don't cancel each other out.
• It penalizes larger errors more heavily.
• The math works out nicely to give a unique solution for β₀ and β₁ (Gauss, 1809).

While the calculus required to derive the formulas for β₀ and β₁ is beyond our conversational scope, statistical software and Python libraries effortlessly handle these calculations.

Linear Regression in Action: Algorithmic Trading Examples & Case Studies

Alright, enough theory! Let's see how linear regression is actually used to build trading strategies.

1. Pairs Trading:

• The Idea: Identify two historically correlated assets (e.g., two stocks in the same sector, or a stock and an ETF). When the price ratio or spread between them deviates significantly from its historical mean, you bet on them reverting to the mean.
• Linear Regression's Role: You can regress the price of Stock A against the price of Stock B: Price_A = β₀ + β₁ * Price_B + ε.
• The residuals (ε) of this regression represent the deviation from the historical relationship. When the residual becomes unusually large (positive or negative), it might be a signal to trade (Vidyamurthy, 2004).
• Example: Let's say Shell (RDS.A) and BP (BP) historically move together. We regress RDS.A prices on BP prices. If the current residual is significantly positive, it suggests RDS.A is overpriced relative to BP (or BP is underpriced relative to RDS.A). A strategy could be to short RDS.A and go long BP, expecting them to revert.
• Recent Trend: Traders are increasingly using rolling regressions to dynamically adjust the hedge ratio (β₁). This helps the strategy adapt as the correlation between the assets naturally changes over time.

Want to explore pairs trading in more detail? QuantInsti has a great primer on Pairs Trading Basics.

2. Statistical Arbitrage with ETFs and Futures:

• The Idea: Exploiting temporary price differences between an ETF and its underlying basket of assets it's supposed to track, or between a stock index future and the underlying cash index (Zhao et al., 2024).
• Linear Regression's Role: You regress the ETF's price (or futures price) against the value of its underlying components (or the cash index).The model's parameters, such as the intercept (β₀) and slope (β₁), along with the residuals (ε), are analyzed to identify mispricing. If an ETF is perfectly tracking its underlying assets, financial theory tells us that  β₁ should be very close to 1.
• ETF_Price = β₀ + β₁ * Underlying_Basket_Value + ε
• Example: If an S&P 500 ETF is trading at a price significantly different from what the regression against the actual S&P 500 index value predicts (once you've accounted for things like costs and dividends), an arbitrage opportunity might exist. This requires fast execution and careful transaction cost management.

For a deeper dive into these kinds of strategies, check out QuantInsti's article on Statistical Arbitrage Trading Strategies

3. Factor Modeling (e.g., Predicting Stock Returns):

• The Idea: Explain or predict stock returns using various market or fundamental factors (e.g., market risk (Beta), size (SMB), value (HML) from the Fama-French models).
• Linear Regression's Role: It looks something like this:
  Stock_Return = β₀ + β₁ * Market_Factor + β₂ * Size_Factor + β₃ * Value_Factor + ... + ε
• Example: A quant might build a model to predict next month's returns for a universe of stocks based on factors like past 12-month momentum, book-to-price ratio, and earnings yield. The regression coefficients help understand which factors the market is currently rewarding (Fama & French, 1993).
• Industry Trend: The quant world is always on the hunt for new factors (it's sometimes called the "factor zoo"!) and uses Multiple Linear Regression (MLR) to see if they work and to build multi-factor models. There's also a significant focus on "factor timing" – trying to predict when certain factors will outperform.

4. Hedging Strategies:

• The Idea: Minimize the risk of a portfolio by taking an offsetting (opposite) position in a related asset.
• Linear Regression's Role: It's key for figuring out the optimal hedge ratio. Let's say you've got a portfolio of tech stocks and want to hedge against market downturns using a Nasdaq 100 futures contract. You'd run a regression:: Portfolio_Returns = β₀ + β₁ * Nasdaq_Futures_Returns + ε
  The β₁ (beta) tells you how many units of Nasdaq futures you need to short for every unit of your portfolio to minimize its sensitivity to Nasdaq movements (Hull, 2018).
• Practical Note: Hedge ratios are often dynamic and re-estimated regularly using rolling regressions mentioned earlier.

Understanding risk is crucial. QuantInsti has resources on Risk Management in Trading that touch upon concepts like beta.

These are just a few examples. The versatility of linear regression means it can be adapted to many other scenarios, such as volatility forecasting, transaction cost analysis, and optimising order execution.

Let's Get Practical: Linear Regression with Python

Python, with its powerful libraries like statsmodels and scikit-learn, makes implementing linear regression straightforward.

And, how to apply linear regression in python? Let's look at a simple example.

Suppose we want to model the relationship between the daily returns of a specific stock (e.g., AAPL) and the daily returns of the S&P 500 index (e.g., SPY).

Explanation of Code:

1. Import necessary libraries
   Load the Python libraries needed to fetch financial data, manipulate data frames, run regression models, and plot results.
2. Download close price data for AAPL and SPY
   Fetch historical daily closing prices for both AAPL (Apple Inc.) and SPY (S&P 500 ETF) over a defined time period (e.g., 2023).
3. Drop any missing data
   Eliminate any rows with missing values to ensure clean data for regression.
4. Calculate daily returns
   Convert the price series into daily percentage returns for both AAPL and SPY, which will be used for the regression.
5. Set the dependent and independent variables
   Define AAPL returns as the dependent variable (the one we’re trying to predict) and SPY returns as the independent variable (the market driver).
6. Add a constant term to the independent variable
   This ensures the regression model includes an intercept (β₀), representing the return of AAPL when SPY’s return is zero.
7. Fit the linear regression model using OLS
   Run the Ordinary Least Squares (OLS) regression to estimate the intercept and slope (β₁) of the model.
8. Print and interpret the regression results
   Display the regression output, including coefficients, p-values, and the R-squared value, which shows how well SPY returns explain AAPL returns.
9. Visualize the regression line
   Create a scatter plot of SPY vs AAPL returns and overlay the regression line to visualise how closely the model fits the data.

Python Code:

Output:

This basic setup is the launchpad for many quantitative analyses. If you want to do multiple linear regression, you could easily expand this by adding more independent variables.

How Good is Your Model? Evaluating Linear Regression Performance

You might ask us now: Once I've built my linear regression model, how do I tell if it's actually any good, or just spitting out random numbers? How to check performance of linear regression model in Python?

We got you covered, my friend!

Here are some of the key things to look at, many of which you'll find in that model.summary() output we just generated:

1. R-squared (R²):

• What it is: This tells you what proportion of the change in your dependent variable can be explained by your independent variable(s). It's a percentage, running from 0 to 1 (or 0% to 100%).
• How to read it: Generally, a higher R-squared generally indicates a better fit. An R² of 0.65 means that 65% of the variation in Y can be explained by X(s).
• The Catch in Trading: In finance, especially for return prediction, R-squared values are often quite low (e.g., < 0.10 or even < 0.05). This doesn't necessarily mean the model is useless. A small but consistent predictive edge can be super valuable. Be wary of extremely high R-squared values in financial time series, as they might indicate overfitting or a relationship that's just a fluke or a spurious regression.

2. Adjusted R-squared:

• What it is: This is a slightly tweaked version of R-squared that adjusts for the number of predictors in the model. It only increases if the new predictor improves the model more than would be expected by chance.
• How to read it: It's really handy when comparing models with different numbers of independent variables.

3. Coefficients (β):

• You might have asked up to this point: How to get coefficients of linear regression in Python​?
• What they are: The estimated intercept and slopes.
• How to read them: Their sign (+ or -) tells you the direction of the relationship, and their magnitude indicates the strength. In trading, it's not just about whether a coefficient is statistically significant; you also need to think about economic significance. Does the size of the coefficient make real-world sense, and is it big enough that you could actually make money from it after accounting for trading costs?

4. P-values (for coefficients):

• What they are: The p-value is the probability of observing the current (or more extreme) data if the null hypothesis (that the coefficient is actually zero, meaning no relationship) were true.
• How to read it: A small p-value (typically < 0.05) suggests that you can reject the null hypothesis. In plain English, it means your independent variable has a real effect on your dependent variable – it's statistically significant.
• Caution: Statistical significance doesn't automatically imply economic significance or predictive power out-of-sample (i.e., on new data) (Ioannidis, 2005).

5. Standard Error of the Regression (or Root Mean Squared Error - RMSE):

• What it is: Measures the typical distance between the observed values and the regression line. It's in the same units as the dependent variable.
• How to read it: A smaller RMSE generally means a better fit – your model's predictions are closer to reality.

6. Out-of-Sample Testing (Crucial for Trading):

• What it is: Evaluating your model on data it hasn't seen during training. This is the true test of a trading model's predictive power.
• How to do it: Split your data into a training set (to build the model) and a test set (to evaluate it). Metrics like Mean Squared Error (MSE), Mean Absolute Error (MAE), or actual simulated trading performance (P&L, Sharpe ratio) on the test set are vital (Aronson, 2006).
• Why it's King: Due to overfitting, a model might look great on in-sample data (the data used to build it) but fail miserably on new data. This is a common pitfall in quant trading.

Recent Trend in Evaluation: There's a big push these days for using really robust out-of-sample validation methods like walk-forward optimization and k-fold cross-validation, especially in algo trading. These help make sure your models aren't just perfectly curve-fitted to old data but actually have some genuine predictive ability (Pardo, 2008).

Limitations and Pitfalls of Linear Regression in Trading

Now, while linear regression is super useful, it's not a magic crystal ball. You've got to be aware of its limitations, or you could get into trouble:

1. It only sees straight lines (Linearity Assumption): It only captures linear relationships. If the true relationship is non-linear, the model will be misspecified.
2. Sensitivity to outliers: OLS can be heavily influenced by extreme values (outliers) because it minimizes squared errors. One massive outlier can skew your entire regression line. Robust regression techniques can mitigate this.
3. Correlation vs. Causation: Regression can show that X and Y move together, but it cannot prove that X causes Y. A lurking variable might cause both, or the relationship could be coincidental. This is a fundamental concept in statistics and is critical for strategy development.
   Wanna learn more about this? Check out this blog!
4. Relationships change (Stationarity of relationships):  Financial markets are alive and always changing. Relationships that held in the past may not hold in the future (concept drift). So, a model you build on historical data needs to be constantly watched and recalibrated from time to time. For example, a stock's beta isn't necessarily constant forever.
5. Danger of Overfitting: Including too many independent variables (especially if they are not truly predictive) can lead to a model that fits the training data perfectly but performs poorly on new data.
6. Ignoring Non-Normal Errors / Fat Tails: As mentioned, financial returns often have "fat tails." Standard OLS might underestimate risk if this isn't accounted for.

Beyond the Basics: What's Next?

Linear regression is a fantastic launchpad. When you're ready to explore further, you might want to check out:

• Non-linear Regression Models: For when those relationships just aren't straight lines. QuantInsti has an overview of Advanced Regression Models in Finance and also discusses Types of Regression in Finance which can touch on non-linear aspects.
• Logistic Regression: This is your go-to when what you're trying to predict is categorical – for example, will the market go up or down? (Yes/No). Check out Logistic Regression for Trading with Python.
• Time Series Models (like ARIMA, GARCH): These are specifically built to handle the kinds of patterns and changing volatility you often see in financial data that unfolds over time.
  For a general intro, see Time Series Analysis in Python.
  Dive into volatility with the GARCH Model in Python for Alpha Trading.
  And for forecasting, there's the ARIMA Model in Python for Trading.
• Machine Learning Ensemble Methods: Think Random Forests or Gradient Boosting. These clever methods combine lots of simpler models (like decision trees, which are themselves related to regression) to make predictions that are often more robust and accurate. QuantInsti covers these in broader machine learning topics:
  Decision Tree Algorithm for Trading in Python
  Random Forest Algorithm in Python for Trading
  XGBoost Algorithm for Trading with Python Examples

Frequently Asked Questions

What is linear regression in trading?

Linear regression fits a straight-line equation between a dependent variable (target) and one or more independent variables (predictors), helping traders model relationships and forecast outcomes.

What’s the difference between simple and multiple linear regression?

Simple linear regression uses one predictor , while multiple linear regression involves two or more predictors to model more complex relationships.

What do linear regression coefficients represent?

Coefficients (often called betas) indicate the direction and strength of each predictor’s influence on the target variable.

How does Ordinary Least Squares (OLS) work?

OLS finds the “best-fit” line by minimizing the sum of squared differences between actual and predicted values.

Why is linear regression still used in trading?

It’s fast, interpretable, and foundational for strategies like risk models, factor analysis, pairs trading, statistical arbitrage, and hedge ratio estimation.

What are the key assumptions of linear regression?

Linear regression assumes linear relationships in coefficients, independent and homoscedastic errors, normally distributed residuals, and nolow multicollinearity among predictors.

How do traders assess the quality of a regression model?

They use metrics like R², adjusted R², p-values for coefficients, F test to test model significance, Root Mean Square Error (RMSE), and out-of-sample performance tests.

What are the limitations of linear regression in finance?

Limitations include sensitivity to outliers, an inability to capture non-linear relationships, and performance issues during changing market regimes.

How can traders improve or extend linear regression models?

Extensions include regularised regression (e.g., Lasso, Ridge), rolling-window models for adapting to market shifts, and non-linear or machine learning methods for complex dynamics.

Conclusion: Your Journey with Linear Regression

Linear regression is way more than just another statistical technique you learned once; it's a really versatile and understandable tool that plays a huge role in the world of quantitative trading. From sniffing out pairs trading opportunities and building factor models to keeping a lid on risk, its uses are all over the place.

The real key to using linear regression successfully in trading is to really get its assumptions, be super careful when you're evaluating how well it's performing (especially on that out-of-sample data!), and always keep its limitations in mind. If you can combine that statistical discipline with a good understanding of how financial markets actually work, you'll be in a great position to develop smarter and, hopefully, more profitable algorithmic trading strategies.

Ready?

Set?

Happy (quantitative) trading!

‌Next Steps

Once you are familiar with the fundamentals, the next logical step is to deepen your understanding of model assumptions. Linear Regression: Assumptions and Limitations is a must-read, as it covers the statistical assumptions required for regression to work effectively and highlights common pitfalls in model interpretation.

For readers interested in extending their regression models to more advanced techniques, Advanced Regression Models in Finance and Types of Regression in Finance explore variations such as Ridge, Lasso, and logistic regression, each with unique strengths in handling financial data complexities. If you are looking to apply regression to time-series data, Time Series Analysis offers a broader view of forecasting techniques and stationarity, which are often necessary for reliable predictive modeling.

To see how linear regression ties into real trading strategies, explore Pairs Trading Basics and Statistical Arbitrage. These strategies rely heavily on identifying mean-reverting relationships and co-movement of asset prices scenarios where regression can be highly effective.

A good strategy is only as strong as its evaluation, so Backtesting Trading Strategies becomes critical in testing your model under historical data and market conditions. You'll learn how to validate results, refine your strategy, and avoid common overfitting pitfalls.

For a more structured and in-depth learning experience, the best algorithmic trading course, the Executive Programme in Algorithmic Trading (EPAT), is highly recommended. It offers comprehensive coverage of machine learning, regression techniques, statistical modelling, and Python-based strategy implementation, making it ideal for those looking to turn theoretical knowledge into practical, real-world trading systems.

References

• Draper, N.R., & Smith, H. (1998). Applied Regression Analysis. Wiley.(Fundamental text on regression) https://www.wiley.com/en-us/Applied+Regression+Analysis%2C+3rd+Edition-p-9780471170822
• Fabozzi, F. J., Focardi, S. M., & Rachev, S. T. (2007). The basics of financial econometrics: Tools, concepts, and asset management applications. John Wiley & Sons. https://nibmehub.com/opac-service/pdf/read/The%20Basics%20of%20Financial%20Econometrics%20_%20tools-%20concepts-%20and%20asset%20management%20applications.pdf
• Gujarati, D. N., & Porter, D. C. (2009). Basic Econometrics (5th ed.). McGraw-Hill(General econometrics, coefficient interpretation). https://archive.org/details/basic-econometric-by-damodar-n.-gujarati-and-dawn-c.-porter
• Tsay, R. S. (2010). Analysis of financial time series (3rd ed.), Wiley.
  https://cpb-us-w2.wpmucdn.com/blog.nus.edu.sg/dist/0/6796/files/2017/03/analysis-of-financial-time-series-copy-2ffgm3v.pdf
• Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning: Data mining, inference, and prediction (2nd ed.). Springer. (Section on Foundation for complex models).
  https://link.springer.com/book/10.1007/978-0-387-84858-7
• Jorion, P. (2007). Value at risk: The new benchmark for managing financial risk (3rd ed.). McGraw-Hill.
  https://www.academia.edu/8519246/Philippe_Jorion_Value_at_Risk_The_New_Benchmark_for_Managing_Financial_Risk_3rd_Ed_2007
• Abu-Mostafa, Y. S., Magdon-Ismail, M., & Lin, H. T. (2012). Learning from data. AMLBook. (General machine learning concepts, including regularization).
  http://amlbook.com/
• Ang, A. (2014). Asset management: A systematic approach to factor investing. Oxford University Press. (Factor investing insights).
  https://global.oup.com/academic/product/asset-management-9780199959327
• Berry, W. D. (1993). Understanding regression assumptions. Sage Publications. (Series: Quantitative Applications in the Social Sciences, general discussion throughout).
  https://wrlc-gm.primo.exlibrisgroup.com/discovery/fulldisplay?docid=sagesrmob10.4135%2F9781412986427&context=PC&vid=01WRLC_GML:01WRLC_GML&lang=en&search_scope=MyInst_and_CI&adaptor=Primo%20Central&tab=Everything&query=any,contains,Understanding%20Regression%20Assumptions&offset=0
• Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987-1007. (Seminal paper on ARCH/GARCH for heteroskedasticity).
  https://doi.org/10.2307/1912773
• Montgomery, D. C., Peck, E. A., & Vining, G. G. (2021). Introduction to linear regression analysis (6th ed.). Wiley.
  https://www.wiley.com/en-us/Introduction+to+Linear+Regression+Analysis%2C+6th+Edition-p-9781119578727
• Jarantow, S.W. (2023). Introduction to the Use of Linear and Nonlinear Regression Analysis. Current Protocols, 3(7), e801. (Section: Linear Least Squares/Ordinary Least Squares).
  https://currentprotocols.onlinelibrary.wiley.com/doi/full/10.1002/cpz1.801
• Gauss, C. F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Perthes et Besser. (Method of least squares discussed in Book 2, Section 3, Article 186, approx. p. 221 in some reprints like the Davis 1857 English translation).
  https://www.researchgate.net/publication/364785738_Theoria_Motus_Corporum_Coelestium_in_Sectionibus_Conicis_Solem_Ambientium
• Vidyamurthy, G. (2004). Pairs trading: Quantitative methods and analysis. John Wiley & Sons. (Pairs trading mechanics, Chapter 2).
  https://download.e-bookshelf.de/download/0000/5844/79/L-G-0000584479-0002384386.pdf
• Zhao, K., Li, Z., & Zhang, W. (2024). Design and Analysis of an Innovative Arbitrage Strategy: Bridging Stock Index Futures and Cross-border ETFs.
  https://www.researchgate.net/publication/382599502_Design_and_Analysis_of_an_Innovative_Arbitrage_Strategy_Bridging_Stock_Index_Futures_and_Cross-border_ETFs
• Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56.
  https://www.bauer.uh.edu/rsusmel/phd/Fama-French_JFE93.pdf
• Hull, J. C. (2018). Options, futures, and other derivatives (10th ed.). Pearson Education. (Section on Hedging strategies).
  https://tfal.in/wp-content/uploads/2023/09/5_6091323572117045477.pdf
• Ioannidis, J. P. (2005). Why most published research findings are false. PLoS Medicine, 2(8), e124. (Discussion on p-values and statistical significance, relevant section: "Why Most Published Research Findings Are False").
  https://doi.org/10.1371/journal.pmed.0020124
• Aronson, D. R. (2006). Evidence-based technical analysis: Applying the scientific method and statistical inference to trading signals. John Wiley & Sons. (Chapter on backtesting and out-of-sample validation).
  https://www.researchgate.net/publication/286014244_Evidence-Based_Technical_Analysis_Applying_the_Scientific_Method_and_Statistical_Inference_to_Trading_Signals
• Pardo, R. (2008). The evaluation and optimization of trading strategies (2nd ed.). John Wiley & Sons. (Walk-forward optimization, Part III of the book).
  https://download.e-bookshelf.de/download/0000/5709/82/L-G-0000570982-0002382554.pdf

Disclaimer: This blog post is for informational and educational purposes only. It does not constitute financial advice or a recommendation to trade any specific assets or employ any specific strategy. All trading and investment activities involve significant risk. Always conduct your own thorough research, evaluate your personal risk tolerance, and consider seeking advice from a qualified financial professional before making any investment decisions.

TL;DR

Understanding probability, independence, correlation, and cointegration is key to building robust trading strategies. While correlation shows short-term co-movements, cointegration captures long-term ties, and independence means no influence between variables. Visual tools and Python-based analysis help identify these relationships, supporting smarter diversification and precise hedging. Algorithms and AI further apply these ideas across strategies, but real-world shifts and human biases remind us that market relationships evolve. Mastering these concepts enables more adaptive, data-driven trading.

This blog covers:

• The Building Blocks
• What is Independence, Statistically?
• Understanding the Concepts: Independence, Correlation, and Cointegration Defined
• Seeing is Believing: Visual and Quantitative Tools
• From Brain Food to Real Action: Leveraging Independence in Your Trading Arsenal
• The Human Factor: Data Science Tools and Our Own Brain Quirks
• Reality Check: Limitations and Caveats
• Frequently Asked Questions

Ever look at the stock market and feel like it’s just a blur of randomness—spikes, dips, and noise with no clear rhyme or reason? You’re not alone. But here’s the thing: beneath the chaos, there are patterns. And one of the most powerful tools for spotting them is a statistical gem called independent events.

Forget the dry textbook stuff for a moment. This concept isn’t just academic—it’s practical. It’s the key to recognising signals that truly stand apart from the usual market noise. It’s how you start building a portfolio where one bad day doesn’t wreck your entire plan. And it’s the secret behind smarter, sharper strategies that don’t just ride the market’s mood—they cut through it.

Prerequisites

To grasp the concepts of statistical independence, correlation, and cointegration in trading, it's important to start with foundational knowledge in probability and statistics. Begin with Probability in Trading, which introduces the role of probabilistic thinking in financial markets. Follow it with Statistics & Probability Distribution, where you’ll learn about key statistical measures and how they apply to market data. These concepts are critical for interpreting market relationships and designing robust trading strategies. You can further reinforce your foundation with the Statistics & Probability for Trading Quantra course, which offers interactive content tailored for market practitioners.

Complement this understanding with Stock Market Data: Analysis in Python, which walks through acquiring and processing real market data—a vital step before running statistical models. For coding fluency, Basics of Python Programming and the Python for Trading (Basic) course offer hands-on experience with Python, ensuring you're equipped to analyze time series and build models effectively.

So, in this guide, we're going to take a journey together. Not just to define these terms, but to truly internalize them. We'll explore:

• The core idea of independence and what it means in trading
• A little bit of simple math to keep us grounded (I promise, not too scary!).
• Clear examples from everyday life and, of course, the financial battleground.
• A good look at what independence, correlation, and cointegration actually are, and critically, how they’re different.
• Actionable ways to weave this knowledge into robust trading strategies and risk management.
• Expanded, real-world algorithmic trading examples, showing these concepts in action.
• The essential caveats – because no concept is a magic bullet.

Ready to move past just scratching the surface and get a real handle on this?

Let's dive in!

The Building Blocks

Alright, before we dive deeper, let's make sure we're speaking the same language. Here are a few foundational concepts:

• Probability: Simply put, this is the measure of how likely an event is to occur. It’s expressed on a scale from 0 (impossible) to 1 (it’s a sure thing!).
  Mathematically, if A is any event, then P(A) is the probability that event A occurs.
• Random Variable: Think of this as a variable whose value is determined by the outcome of a random phenomenon.The daily price wiggle of a stock? A classic example.
• Conditional Probability: This is the chance of something happening given that something else has already happened. We write it as P(A|B) – "the probability of A, if B has occurred." This is super important for understanding events that aren't independent (dependent events). If A and B are dependent, then:

P(A and B) = P(A) × P(B|A)

What is Independence, Statistically?

Two events are independent if one happens without changing the odds of the other happening. They're effectively in their own lanes.

Think: Event A is "Stock X goes up," and Event B is "It rains today." If they're independent, Stock X's rise (or fall) has zero impact on whether it rains, and the rain isn't bothered by what Stock X is doing.

Mathematically, this means knowing A happened doesn't change B's odds, so the probability of B given A (P(B|A)) is just the same as B's original probability (P(B)). Remember our conditional probability rule for any two events: P(A and B) = P(A) × P(B|A)? Well, for independent events, since P(B|A) simply equals P(B), the formula simplifies nicely to:

P(A and B) = P(A) × P(B)

Essentially, you just multiply their individual chances.

Spotting Independence: From Daily Life to Market Dynamics

It’s always easier to grasp these ideas when you see them in action. In everyday life, independent events show up in things like flipping two coins or rolling a pair of dice—where one outcome doesn’t affect the other.

Source

Extending this idea to Financial Markets and Trading:

• Super Diversified Global Assets: Think about assets from totally different parts of the world and the economy. Say, bonds from a city in California and shares in a tech startup in Bangalore, India. They're likely operating under very different economic pressures and business drivers. Now, in our super-connected global market, are any two assets perfectly, 100% statistically independent? Probably not. But this kind of diversification aims to get them as close as possible, with low correlation (Markowitz, 1952). A crisis hitting one is much less likely to wallop the other in the same way directly. True statistical independence is more of an ideal we shoot for.
• Unrelated Industry Performance (Usually): The stuff that makes cocoa bean prices jump (like weather in West Africa or crop diseases) is generally pretty separate from what drives the stock price of a big aerospace defense company (think government contracts or global political tensions).

A Quick Heads-Up on a Common Mix-Up:

Sometimes you'll see two things react to the same event but in totally opposite ways.

Take the early days of the COVID-19 pandemic, for instance. E-commerce giants like Amazon saw demand skyrocket as we all started shopping online from our couches. Meanwhile, airline companies like Delta watched their revenues nosedive because no one was flying.
It's super tempting to look at that and think, "Aha! Independent events!" because their fortunes went in completely different directions. But hold on – this isn't actually statistical independence.
It’s a classic case of strong negative correlation. Both were reacting to the same global event (the pandemic), just in opposite ways because of how it hit their specific businesses. For example, Baker et al. (2020) reported a very strong negative correlation-around -0.82 between Amazon and Delta in mid-2020.

So, just because things move in polar opposite directions doesn't mean they're truly independent of each other. It's a subtle but important difference to keep in mind!

Understanding the Concepts: Independence, Correlation, and Cointegration Defined

Let's break down these crucial terms individually before we compare them.

What is Statistical Independence?
Independence, in a statistical sense, signifies a complete lack of predictive power between two events or variables. Variable X gives you no clues about Variable Y, and Y offers no hints about X. There's no hidden string connecting them, no shared underlying reason that would make them move together or apart in any predictable way.

What is Correlation?
Correlation is a number that tells us how much and in what direction the returns (like the daily percentage change) of two assets tend to move together. It’s a score from -1 to +1:

• +1 (Perfect Positive Correlation): This means that the assets' returns move perfectly in the same direction. When one goes up, the other goes up by a proportional amount, and vice versa.
• -1 (Perfect Negative Correlation): This indicates that the assets' returns move perfectly in opposite directions.When one goes up, the other goes down by a proportional amount.
• 0 (Zero Correlation): This shows there's no clear linear connection in how their returns change.

Correlation is usually about how things co-move in the shorter term.
Craving the full scoop? This blog’s got you covered.

What is Cointegration?
This one's a bit more nuanced and thinks long-term. It’s about when two or more time series (like the prices of assets) are individually wandering around without a clear anchor (we call this non-stationary – they have trends and don't snap back to an average). BUT, if you combine them in a certain linear way, that combination is stationary – meaning it tends to hang around a stable average over time. So, even if individual prices drift, cointegration means they're tethered together by some deep, long-run economic relationship (Engle & Granger, 1987).

Classic Example: Think crude oil and gasoline prices. Both might trend up or down over long stretches due to inflation or significant economic shifts. However, the spread (the difference) between their prices, which is related to refinery profits, often hovers around a historical average. They can't stray too far from each other for too long.

Comparing these terms:

Now, let's see how these concepts stand apart – a critical distinction for any serious trader.

Super Important Point:  Zero Correlation ≠ Independence!
This is a classic trip-up! Two assets can have zero linear correlation but still be dependent. Imagine Asset A does great when Asset B is either doing really well or really badly (picture a U-shape if you plotted them). The linear correlation might be near zero, but they're clearly not independent; knowing Asset B's extreme performance tells you something about Asset A.

Recap: Independence means no relationship; correlation is about short-term linear return patterns; cointegration points to long-term price relationships. Understanding these nuances is vital for building robust strategies.

Seeing is Believing: Visual and Quantitative Tools

Visualizing data and quantifying relationships can transform abstract concepts into actionable insights.

Price Charts & Scatter Plots:

As mentioned, overlaying price charts (like the AMZN vs. DAL example) or creating scatter plots of returns can offer initial clues. A scatter plot of returns for two truly independent assets would look like a random cloud with no discernible pattern.

Left: Random scatter indicating no correlation (independent variables), Right: Pattern showing a non-linear relationship (non-linear dependent variables)
Source

Beware! For reliable analysis, always use high-quality historical data from reputable providers like Yahoo Finance, Bloomberg, Refinitiv, or directly from the exchanges. Garbage in, garbage out!

Calculating Correlation with Python:

Don't worry if you're not a coder, but for those who are, a simple Python script can quickly show you the linear relationship

Python code snippet:

Output:

yf.download() has changed argument auto_adjust default to True
Ticker       CVX       XOM
Ticker
CVX     1.000000  0.837492
XOM     0.837492  1.000000
Ticker      AAPL      MSFT
Ticker
AAPL    1.000000  0.547987
MSFT    0.547987  1.000000
Ticker       GLD       SPY
Ticker
GLD     1.000000  0.004044
SPY     0.004044  1.000000 

The correlation matrix for XOM/CVX shows a high 0.837492, meaning these oil stocks’ returns move closely together, driven by similar market factors. AAPL/MSFT (0.547987, moderate) and GLD/SPY (0.004044, near-zero) indicate tech stocks have some co-movement, while gold and the S&P 500 are, possibly, nearly independent, otherwise, they have a non-linear correlation.

From Brain Food to Real Action: Leveraging Independence in Your Trading Arsenal

This isn't just interesting theory; it's about giving you a real strategic advantage.

• Next-Level Diversification: True diversification isn't just about owning many different assets; it's about owning assets whose price movements are, as much as possible, driven by independent factors. This is your best shield against unexpected shocks in one part of your portfolio.Want to learn more ? Check out this blog !
• Precision Hedging: Hedging is about taking positions to protect against potential losses. Understanding independence (or the lack of it!) helps you pick better hedges – assets that are likely to move predictably (often negatively correlated) against your primary holdings under specific conditions, or assets that offer a safe haven due to their independent nature.
• Building Resilient Portfolios: By thoughtfully mixing asset classes (stocks, bonds, commodities, real estate, alternative stuff) that have historically shown low correlation and are affected by different big-picture economic drivers, you can build portfolios that are designed to handle a wider variety of market storms.
  
• Navigating Volatility Storms: When markets freak out, correlations often spike—everyone panics and does the same thing (herd behaviour). Knowing this and which assets might keep some independence (or even become negatively correlated, like some "safe-haven" assets) is key for quick-thinking risk management.

Modern Tools That Amp Up These Ideas:

• Risk Parity Models: These are smart allocation strategies that try to make sure each asset class in your portfolio contributes an equal amount of risk, not just an equal amount of money. This relies heavily on good estimates of volatility and, you guessed it, correlations between assets.
  Keen to learn more ? This blog has you covered!
• AI and Machine Learning: Yep, AI can sift through massive piles of data to find complex, non-linear connections and fleeting moments of independence that a human might totally miss. This can lead to more dynamic and quick-to-adapt portfolio changes.
• The Rise of Alternative Data: We're talking info from unusual places—satellite pics of oil tankers, credit card spending data, real-time supply chain info, what people are saying on social media. This can give unique, potentially independent clues about what's happening with the economy or specific companies, giving you an edge if you know how to read it.

Algorithmic Trading in Action: Selected Examples of Independence at Play

The ideas of independence, dependence, correlation, and cointegration are the secret sauce in many fancy trading algorithms. Here’s a peek at some key examples, especially how they relate to these concepts:

Cross-Asset & Global Diversification Algorithms:

• How it works: These algorithms constantly juggle portfolios across diverse asset classes (stocks, bonds, commodities, currencies, real estate) and geographies. They continuously monitor correlations and volatility, trying to keep diversification at a target level.
• Relevance of Independence: The whole point is to mix assets with low, or ideally zero, correlation that comes from independent economic drivers. For example, an algo might buy more Japanese stocks if it thinks their performance is, for the moment, independent of what's happening in the US market due to Japan's specific local policies. The dream is that a dip in one area (say, US tech stocks) is balanced out or barely felt by others (like emerging market bonds or gold).

Factor-Based Investing Algorithms:

• How it works: These algorithms construct portfolios by targeting specific, well-studied "factors" that have historically driven returns– things like Value (cheap stocks), Momentum (stocks on a roll), Quality (solid companies), Low Volatility (less jumpy stocks), or Size (smaller companies). These factors were popularized in foundational work like Fama and French (1993), which identified common risk factors influencing stock and bond returns.
• Relevance of Independence: The idea is that these different factors produce streams of returns that are, to some degree, independent of each other and of the overall market's general movement (beta) over the long haul. An algo might lean a portfolio towards factors expected to do well in the current economic climate or that offer diversification because they don't correlate much with other factors already in the portfolio.
  Want to dig deeper? Check out the full breakdown in this blog.

Event-Driven Strategies (Focusing on Specific News):

• How it works: Algos are built to trade around specific, known corporate or economic events – earnings calls, merger announcements, FDA drug approvals, key economic data releases (like inflation or job numbers).
• Relevance of Independence: The strategy often banks on the market's immediate reaction to the specific news being somewhat independent of the broader market noise at that precise moment. For example, if Company A has a great earnings surprise, its stock might pop even if the overall market is blah or down, all thanks to info specific to Company A.

AI-Driven Sentiment Analysis & Alternative Data Integration:

• How it works: Machine learning models chew through tons of text from news, social media, and financial reports to gauge sentiment (positive, negative, neutral) towards specific assets or the market. Alternative data (like satellite pics of store parking lots, web scraping of job ads, geolocation data) is also used to find non-traditional trading signals.
• Relevance of Independence: The big idea here is that these data sources can offer insights or signals that are independent of traditional financial data (price, volume, company financials). For example, a sudden burst of negative online chatter about a product, spotted before any official sales numbers are out, could be an independent early warning sign for the company's stock.

Want to dive deeper? Two more strategies that lean heavily on the principles of independence and correlation are Market-Neutral & Statistical Arbitrage (StatArb) and Pairs Trading (based on Cointegration). Check out how they work in these quick reads:
https://blog.quantinsti.com/statistical-arbitrage/
https://blog.quantinsti.com/pairs-trading-basics/

Recap: Sophisticated algorithms leverage a deep understanding of independence, correlation, and cointegration to try and find that extra bit of profit (alpha), manage risk, and diversify effectively across all sorts of global markets and assets.

The Human Factor: Data Science Tools and Our Own Brain Quirks

Even though these concepts are statistical, it's humans doing the trading, and humans are, well, human – full of biases!

• Data Science: Your Quantitative Lens: Spotting genuine independence in all the market noise is tough. Data scientists have a whole toolkit:
• Rigorous Statistical Tests: Formal tests like the Pearson correlation coefficient, Spearman rank correlation (for non-linear monotonic relationships), and specific tests for cointegration (e.g., Engle-Granger, Johansen) are must-haves.
• Advanced Time Series Analysis: Techniques like ARIMA, VAR, and GARCH models help to understand dependencies within and between time series data, separating real patterns from random noise.
• Machine Learning Power: AI algorithms can dig up subtle, non-linear patterns of dependence or conditional independence that simpler linear models would completely miss.
• Behavioral Finance: Mind Traps to Avoid:

Source

Our brains are wired to find patterns, sometimes even where none exist. Here are a few common mental traps that can mess up a trader's judgment about independence:

• The Gambler's Fallacy: Wrongly believing that if an independent event (like a stock closing up) has happened a few times in a row, the opposite is now "due" to happen (Nope, each day is a new roll of the dice if they're truly independent
• Representative Bias: Judging how likely something is based on how much it looks like a pattern or stereotype you already have in your head, while ignoring the actual underlying stats. For example, assuming oil stocks XOM and CVX are independent in Jan 2024 because they’re different companies, despite a high 0.84 correlation in 2023 returns showing strong dependence.
• Confirmation Bias: We all do this – looking for, interpreting, and remembering information that confirms what we already believe about how assets are connected, and tuning out evidence that says otherwise. For instance, a trader might focus on a brief period of near-zero correlation (e.g., 0.05 between GLD and SPY in mid-2023) to assume independence, ignoring a longer-term 0.4 correlation indicating dependence.

Just knowing these biases exist is the first huge step towards making more objective, data-driven trading decisions.

Reality Check: Limitations and Caveats

As incredibly useful as all this is, we need to apply the idea of statistical independence with a good dose of realism:

• The Myth of Perfect Independence:  In our super-connected global financial world, finding assets that are perfectly, always independent is like finding a unicorn. Big systemic shocks – a global pandemic, a major financial meltdown, a widespread geopolitical crisis – can make correlations between seemingly unrelated assets suddenly shoot towards 1 (all move together) or -1 (all move opposite) as everyone rushes for (or away from) perceived safety at the same time.
• Models are Guides, Not Crystal Balls: All statistical models, including those used to check for independence or correlation, are simplifications of a far more complex reality. They rely on historical data and assumptions that may not hold true in the future. Market regimes shift, and relationships evolve.
• Dynamic, Not Static, Relationships: How independent or correlated assets are isn't set in stone. It's a moving target that changes over time thanks to evolving economies, tech breakthroughs, new rules, and what investors are feeling. What looks independent today might be strongly correlated tomorrow.

Conclusion

Understanding independent events – and how this concept relates to yet differs from correlation and cointegration – is vital for enhancing your market perspective, portfolio building, and risk management. Consider it an ongoing journey of refinement.

By truly grasping these principles, you can:

• Forge Resilient Portfolios: Move beyond simple diversification to build portfolios designed to handle a wider array of market shocks by seeking genuinely independent return sources.
• Execute Precise Hedging: Gain a clearer understanding of asset relationships to hedge unwanted risks more effectively.
• Uncover Hidden Opportunities: Recognize that many strategies are built on exploiting temporary deviations from statistical relationships or capitalizing on true independencies.
• Cultivate Adaptability: Acknowledge that market relationships are not static, encouraging continuous learning and strategy adjustments.

Financial markets are vast, interconnected, and constantly evolving. While perfect prediction remains elusive, a solid grasp of concepts like statistical independence provides a better compass to navigate, distinguish signals from noise, and identify opportunities.

For those seeking a practical, hands-on learning experience, Quantra by QuantInsti offers excellent courses. The Quantitative Portfolio Management Course covers techniques like Factor Investing and Risk Parity, while the Executive Programme in Algorithmic Trading (EPAT) provides a comprehensive path to mastering trading strategies.

Embracing this learning, questioning assumptions, and letting data guide you will significantly boost your ability to thrive in this ever-changing environment. The effort invested in understanding these concepts is a powerful independent variable in your journey to trading mastery.

References

• Baker, S. R., Bloom, N., Davis, S. J., & Terry, S. J. (2020). COVID-Induced Economic Uncertainty. NBER Working Paper No. 26983.
  https://www.nber.org/papers/w26983
• Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77–91.
  https://onlinelibrary.wiley.com/doi/10.1111/j.1540-6261.1952.tb01525.x
• Engle, R. F., & Granger, C. W. J. (1987). Co-Integration and Error Correction: Representation, Estimation, and Testing. Econometrica, 55(2), 251–276.
  https://www.jstor.org/stable/1913236?origin=crossref
• Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33(1), 3–56. https://doi.org/10.1016/0304-405X(93)90023-5

Next Steps

Once the basics are in place, the next step is to understand how statistical relationships between assets can inform strategy design. Factor Investing helps you recognise systematic return drivers and portfolio construction techniques based on factor exposure. Building on this, Covariance vs Correlation offers a deeper dive into how asset movements relate—fundamental for diversification and hedging.

You can then progress to Johansen Test & Cointegration to understand how long-term equilibrium relationships can signal profitable trading opportunities. This blog pairs well with Stationarity in Time Series and Hurst Exponent, both essential for assessing the stability and memory of financial data.

To apply these concepts practically, explore Statistical Arbitrage, which uses cointegration and mean reversion principles to build pair-based trading strategies. The Pairs Trading with Statistical Arbitrage course teaches you how to develop and test such strategies using Python. For those interested in broader strategy implementation, Backtesting Trading Strategies provides the tools to evaluate historical performance.

Quantitative traders can also benefit from Portfolio Optimization, which builds on correlation insights to construct efficient portfolios. For deeper modeling and predictive techniques, the Machine Learning & Deep Learning in Trading track offers extensive coverage of ML algorithms for forecasting and classification.

Finally, if you're looking to tie all of this together into a comprehensive career-ready framework, the Executive Programme in Algorithmic Trading (EPAT) provides in-depth training in statistical methods, machine learning, Python coding, portfolio theory, and real-world trading systems, making it ideal for serious professionals aiming to lead in quantitative finance.

Frequently Asked Questions

What is the difference between correlation and cointegration?

Correlation measures short-term co-movement between two variables, while cointegration identifies a long-term equilibrium relationship despite short-term deviations between two ore more non-stationary time series.

Why is independence important in trading?

Independence implies no influence between variables. Recognizing independent assets helps avoid false diversification and ensures that combined strategies aren't secretly overlapping.

How does cointegration help in building trading strategies?

Cointegration allows you to build pairs or mean-reversion strategies by identifying asset combinations that revert to a stable long-term relationship, even if each asset is volatile on its own.

Can correlation be used for portfolio diversification?

Yes, but with caution. Correlation is dynamic and can break down during market stress. The conclusion is the following: the lower the correlation, the better for diversification in asset allocation.

How can Python be used to identify these relationships?

Python libraries like statsmodels, scipy, and pandas provide tools to test for correlation, cointegration (e.g., Engle-Granger test), and independence, helping quants validate strategy assumptions.

How do AI and algorithms leverage these concepts?

AI models can automatically detect relationships like cointegration or conditional independence, improving strategy development, regime detection, and risk modeling.

What are the risks of ignoring these concepts?

Ignoring them can lead to overfitting, poor or wrong diversification, or failed hedges—ultimately resulting in unexpected drawdowns during market shifts.

Are these relationships stable over time?

Not always. Market regimes, macro events, and structural shifts can alter statistical relationships. Continuous monitoring and model updates are essential.

Acknowledgements

This blog post draws heavily from the information and insights presented in the following texts:

Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer. https://link.springer.com/book/10.1007/978-0-387-21736-9

1. Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury. https://www.cengage.com/c/statistical-inference-2e-casella-berger/9780534243128/

2. Ross, S. M. (2014). A First Course in Probability (9th ed.). Pearson.
https://www.pearson.com/en-us/subject-catalog/p/first-course-in-probability-a/P200000006334/9780134753119

3. Rodgers, J. L., & Nicewander, W. A. (1988). Thirteen Ways to Look at the Correlation Coefficient. The American Statistician, 42(1), 59–66 https://www.tandfonline.com/doi/abs/10.1080/00031305.1988.10475524

Disclaimer: This blog post is for informational and educational purposes only. It does not constitute financial advice or a recommendation to trade any specific assets or employ any specific strategy. All trading and investment activities involve significant risk. Always conduct your own thorough research, evaluate your personal risk tolerance, and consider seeking advice from a qualified financial professional before making any investment decisions.

Prerequisites

To get the most out of this blog, it helps to start with an overview of machine learning principles. Begin with Machine Learning Basics: Components, Application, Resources and More, which provides a solid introduction to how ML works, key components of ML workflows, and its growing role in financial markets.

Since the blog uses real-world stock data, familiarity with working in Python and handling market datasets is important. The blog Stock Market Data: Obtaining Data, Visualization & Analysis in Python is a great starting point to understand how to download, visualize, and prepare stock price data for modeling.

For a more structured path, the Python for Trading: Basic course on Quantra will help beginners build essential Python skills in a trading context, while Python for Trading dives deeper into data handling and analytics for financial applications.

Table of Contents

• Introduction
• What is Machine Learning?
• Understanding Classification in Machine Learning
• Types of Classification Problems
• Building a Classification Model in Python: Step-by-Step
• Conclusion

Introduction

Have you ever wondered how Netflix recommends shows you might like, or how Tesla cars can recognise objects on the road? These technologies have something important in common – they both use the "first-principles" approach to solve complex problems.

This approach means breaking down complicated issues into smaller, manageable parts and building solutions from the ground up. Today, we'll use this same approach to understand machine learning classification in Python, starting with the basics.

In this beginner-friendly guide, we'll learn how to build a machine learning model that can predict whether to buy or sell a stock. Don't worry if you're new to this – we'll explain everything step by step!

What is Machine Learning?

In simple terms, machine learning gives computers the ability to learn from experience without someone explicitly programming every possible scenario.

Think about how you learned to recognise animals as a child. Your parents might have pointed to a dog and said, "That's a dog." After seeing many dogs, you learned to identify them by yourself. Machine learning works similarly – we show the computer many examples, and it learns patterns from those examples.

Traditional programming tells a computer exactly what to do in every situation:

IF steering wheel turns right

THEN turn the wheels right

Machine learning, however, shows the computer many examples so it can figure out the patterns by itself:

• Here are 1000 images of roads with obstacles
• Here are 1000 images of clear roads

Now, tell me if this new image shows a clear road or has obstacles

This approach is being used in everything from self-driving cars to stock market trading.

Understanding Classification in Machine Learning

Classification is one of the most common tasks in machine learning. It's about putting things into categories based on their features.

Imagine teaching a child about animals:

• You show them a picture of a cat and say, "This is a cat"
• You show them a picture of a dog and say, "This is a dog"

After showing many examples, you test them by showing a new picture and asking, "What animal is this?"

Machine learning classification works the same way:

• We give the model examples with known categories (training data)
• The model learns patterns from these examples
• We test the model by asking it to classify new examples it hasn't seen before

In trading, we might use classification to predict whether a stock price will go up or down tomorrow based on today's market information.

Types of Classification Problems

Before diving into our Python example, let's quickly understand the main types of classification problems:

Binary Classification: Only two possible categories

• Example: Will the stock price go up or down?
• Example: Is this email spam or not?

Multi-class Classification: More than two categories

• Example: Should we buy, hold, or sell this stock?
• Example: Is this image a cat, dog, or bird?

Imbalanced Classification: When one category appears much more frequently than the others

• Example: Predicting rare events like market crashes
• Example: Detecting fraud in banking transactions (most transactions are legitimate)

Our example below will focus on binary classification (predicting whether the S&P 500 index will go up or down the next day).

Building a Classification Model in Python: Step-by-Step

Let's build a simple classification model to predict whether the S&P 500 price will increase or decrease the next trading day.

Step 1: Import the Required Libraries

First, we need to import the Python libraries that will help us build our model:

These libraries give us the tools we need without having to code everything from scratch.

Step 2: Get Your Data

We'll download S&P 500 data using the yfinance library:

This code downloads five years of S&P 500 ETF (SPY) data and plots the closing price.

Figure: Close Prices Plot for SPY

Step 3: Define What You Want to Predict

This is our "target variable" - what we're asking the model to predict. In this case, we want to predict whether tomorrow's closing price will be higher or lower than today's:

Step 4: Choose Your Prediction Features

These are the clues we give our model to make predictions. While we could use many different indicators, we'll keep it simple with two basic features:

Step 5: Split Data into Training and Testing Sets

We need to divide our data into two parts:

Training data: Used to teach the model

Testing data: Used to evaluate how well the model learned

This is like studying for a test: you learn from your study materials (training data), then test your knowledge with new questions (testing data).

Step 6: Train Your Model

Now we'll create and train our model using the Support Vector Classifier (SVC):

This single line of code does a lot of work behind the scenes! It creates a Support Vector Classifier and trains it on our training data.

Step 7: Check How Well Your Model Performs

We need to check if our model has learned effectively:

Output:

Train Accuracy: 54.98%
Test Accuracy: 58.33%

Fig: Accuracy Scores for Train and Test Period

An accuracy above 50% on test data suggests our model is better than random guessing.

Step 8: Make Predictions

Now let's use our model to make predictions and calculate potential returns:

This calculates how much money we would make or lose by following our model's predictions.

Step 9: Visualise Your Results

Finally, let's plot the cumulative returns of our strategy to see how it performs:

This shows the total percentage return of our strategy over time.

Conclusion

Congratulations! You've just built a simple machine learning classification model that predicts stock market movements. While this example used the S&P 500, you could apply the same approach to any tradable asset.

Remember, this is just a starting point. To improve your model, you could:

• Add more features (like technical indicators)
• Try different classification algorithms
• Use more data or different time periods
• Add risk management rules

The key to success in machine learning is experimentation and refinement. Try changing different parts of the code to see how it affects your model's performance.

Happy learning and trading!

Note: All investments and trading in the stock market involve risk. This article is for educational purposes only and should not be considered financial advice. Always do your own research and consider consulting with a financial professional before making investment decisions.

Next Steps

After building your first classification model, you can expand your skills by exploring more advanced ML techniques and integrating them into end-to-end trading workflows.

Start with Machine Learning Classification: Concepts, Models, Algorithms and More, which explores decision trees, logistic regression, k-nearest neighbors (KNN), and other core algorithms that can be applied to classification tasks in trading.

To test your strategies effectively, learning how to backtest is crucial. The blog Backtesting: How to Backtest, Strategy, Analysis, and More introduces key concepts like historical data testing, performance metrics, and risk evaluation—vital for assessing any machine learning-based strategy.

To further integrate ML with trading, the blog Machine Learning for Algorithmic Trading in Python: A Complete Guide offers a full walkthrough of building trading systems powered by machine learning, including feature engineering and model selection.

For a hands-on learning experience, you can explore the Trading with Machine Learning: Classification and SVM course on Quantra, which takes your classification knowledge further and teaches how to apply models in live financial scenarios.

If you're aiming for a comprehensive, career-oriented learning path, the Executive Programme in Algorithmic Trading (EPAT) is highly recommended. EPAT covers Python programming, machine learning, backtesting, and model evaluation, with real-world trading applications and industry mentorship—ideal for professionals serious about algorithmic trading.

File in the download:

ML Classification- Python Notebook

Visit blog to download

Note: The original post has been revamped on 27^th May 2025 for recentness, and accuracy.

Disclaimer: All investments and trading in the stock market involve risk. Any decision to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.

Welcome to the second part of this two-part blog series on the bias-variance tradeoff and its application to trading in financial markets.

In the first part, we attempted to develop an intuition for the bias-variance decomposition. In this part, we’ll extend our learnings from the first part and develop a trading strategy.

Prerequisites

If you have some basic knowledge of Python and ML, you should be able to read and comprehend the article. These are some pre-requisites:

• https://blog.quantinsti.com/bias-variance-machine-learning-trading/
• Linear algebra (basic to intermediate)
• Python programming (basic to intermediate)
• Machine learning (working knowledge of regression and regressor models)
• Time series analysis (basic to intermediate)
• Experience in working with market data and creating, backtesting, and evaluating trading strategies

Also, I have added some links for further reading at relevant places throughout the blog.

If you're new to Python or need a refresher on it, you can start with Basics of Python Programming and then move to Python for Trading: Basic on Quantra for trading-specific applications.

To familiarize yourself with machine learning, and with the concept of linear regression, you can go through Machine Learning for Trading and Predicting Stock Prices Using Regression.

Because the article also covers time series transformations and stationarity, you can familiarize yourself with Time Series Analysis. Knowledge of handling financial market data and practical experience in strategy creation, backtesting, and evaluation will help you apply the article's learnings to your strategies.

In this blog, we’ll cover the complete pipeline for using machine learning to build and backtest a trading strategy while utilising the bias-variance decomposition to select the appropriate prediction model. So, here goes…

The flow of this article is as follows:

• Importing Libraries
• Downloading Data
• Defining Technical Indicators as Predictor Variables
• Defining the Target Variable
• Walk Forward Optimization with PCA and VIF
• Trading Strategy Formulation, Backtesting, and Evaluation
• Calling the Functions Defined Previously
• Bias-Variance Decomposition
• Stationarizing the Inputs
• Running a Prediction using the Chosen Model
• Checking the Trade Logs
• Equity Curves, Sharpe, Drawdown, Hit Ratio, Returns Distribution, Average Returns per Trade, and CAGR
• Some Realistic Considerations
• A Note on the Downloadable Python Notebook

As a ritual, the first step is to import the necessary libraries.

Importing Libraries

If you don’t have any of these installed, a ‘!pip install’ command should do the trick (if you don’t want to leave the Jupyter Notebook environment, or if you want to work on Google Colab).

Downloading Data

Next, we define a function for downloading the data. We’ll use the yfinance API here.

Notice the argument ‘multi_level_index’. Recently (I’m writing this in April 2025), there have been some changes in the yfinance API. When downloading price level and volume data for any security through the specified API, the ticker name of the security gets added as a heading.

It looks like this when downloaded:

For people (like me!) who are accustomed to not seeing this extra level of heading, removing it while downloading the data is a good idea. So we set the ‘multi_level_index’ argument to ‘False’.

Defining Technical Indicators as Predictor Variables

Next, since we are using machine learning to build a trading strategy, we must include some features (sometimes called predictor variables) on which we train the machine learning model. Using technical indicators as predictor variables is a good idea when trading in the financial markets. Let’s do it now.

Eventually, we’ll see the list of indicators when we call this function on the asset dataframe.

Defining the Target Variable

The next chronological step is to define the target variable/s. In our case, we’ll define a single target variable, the close-to-close 5-day percent return. Let’s see what this means. Suppose today is a Monday, and there are no market holidays, barring the weekends, this week. Consider the percent change in tomorrow’s (Tuesday’s) closing price over today’s closing price, which would be a close-to-close 1-day percent return. At Wednesday’s close, it would be the 2-day percent return, and so on, till the following Monday, when it would be the 5-day percent return. Here’s the Python implementation for the same:

Why do we use the shift(-5) here? Suppose the 5-day percent return based on the closing price of the following Monday over today’s closing price is 1.2%. By using shift(-5), we are placing this value of 1.2% in the row for today’s OHLC price levels, volume, and other technical indicators. Thus, when we feed the data to the ML model for training, it learns by considering the technical indicators as predictors and the value of 1.2% in the same row as the target variable.

Walk Forward Optimization with PCA and VIF

One essential consideration while training ML models is to ensure that they display robust generalization. This means that the model should be able to extrapolate its performance on the training dataset (sometimes called in-sample data) to the test dataset (sometimes called out-of-sample data), and its good (or otherwise) performance should be attributed primarily to the inherent nature of the data and the model, rather than to chance.

One approach towards this is combinatorial purged cross-validation with embargoing. You can read this to learn more.

Another approach is walk-forward optimisation, which we will use (read more: 1 2).

Another essential consideration while building an ML pipeline is feature extraction. In our case, the total predictors we have is 21. We need to extract the most important ones from these, and for this, we will use Principal Component Analysis and the Variance Inflation Factor. The former extracts the top 4 (a value that I chose to work with; you can change it and see how the backtest changes) combinations of features that explain the most variance within the dataset, while the latter addresses mutual information, also known as multicollinearity.

Here’s the Python implementation of building a function that does the above:

Trading Strategy Formulation, Backtesting, and Evaluation

We now come to the meaty part: the strategy formulation. Here are the strategy outlines:

Initial capital: ₹10,000.

Capital to be deployed per trade: 20% of initial capital (₹2,000 in our case).

Long condition: when the 5-day close-to-close percent return prediction is positive.

Short condition: when the 5-day close-to-close percent return prediction is negative.

Entry point: open of day (N+1). Thus, if today is a Monday, and the prediction for the 5-day close-to-close percent returns is positive today, I’ll go long at Tuesday’s open, else I’ll go short at Tuesday’s open.

Exit point: close of day (N+5). Thus, after I get a positive (negative) prediction today and go long (short) during Tuesday’s open, I’ll square off at the closing price of the following Monday (provided there are no market holidays in between).

Capital compounding: no. This means that our profits (losses) from every trade are not getting added (subtracted) to (from) the tradable capital, which remains fixed at ₹10,000.

Here’s the Python code for this strategy:

Next, we define the functions to evaluate the Sharpe ratio and maximum drawdowns of the strategy and a buy-and-hold approach.

Calling the Functions Defined Previously

Now, we begin calling some of the functions mentioned above.

We’ll start with downloading the data using the yfinance API. The ticker and period are user-driven. When running this code, you’ll be prompted to enter the same. I chose to work with the 10-year daily data of the NIFTY-50, the broad market index based on the National Stock Exchange (NSE) of India. You can choose a smaller timeframe; the longer the time frame, the longer it will take for the subsequent codes to run. After downloading the data, we’ll create the technical indicators by calling the ‘create_technical_indicators’ function we defined previously.

Here’s the output of the above code:

Enter a valid yfinance API ticker: ^NSEI
Enter the number of years for downloading data (e.g., 1y, 2y, 5y, 10y): 10y
YF.download() has changed argument auto_adjust default to True
[*********************100%***********************]  1 of 1 completed

Next, we align the data:

Let’s check the two dataframes ‘indicators’ and ‘data_merged’.

RangeIndex: 2443 entries, 0 to 2442
Data columns (total 21 columns):
 #   Column           Non-Null Count  Dtype  
---  ------           --------------  -----  
 0   sma_5            2443 non-null   float64
 1   sma_10           2443 non-null   float64
 2   ema_5            2443 non-null   float64
 3   ema_10           2443 non-null   float64
 4   momentum_5       2443 non-null   float64
 5   momentum_10      2443 non-null   float64
 6   roc_5            2443 non-null   float64
 7   roc_10           2443 non-null   float64
 8   std_5            2443 non-null   float64
 9   std_10           2443 non-null   float64
 10  rsi_14           2443 non-null   float64
 11  vwap             2443 non-null   float64
 12  obv              2443 non-null   int64  
 13  adx_14           2443 non-null   float64
 14  atr_14           2443 non-null   float64
 15  bollinger_upper  2443 non-null   float64
 16  bollinger_lower  2443 non-null   float64
 17  macd             2443 non-null   float64
 18  cci_20           2443 non-null   float64
 19  williams_r       2443 non-null   float64
 20  stochastic_k     2443 non-null   float64
dtypes: float64(20), int64(1)
memory usage: 400.9 KB 

Index: 2438 entries, 0 to 2437
Data columns (total 28 columns):
#   Column           Non-Null Count  Dtype
---  ------           --------------  -----
0   Date             2438 non-null   datetime64[ns]
1   Close            2438 non-null   float64
2   High             2438 non-null   float64
3   Low              2438 non-null   float64
4   Open             2438 non-null   float64
5   Volume           2438 non-null   int64
6   sma_5            2438 non-null   float64
7   sma_10           2438 non-null   float64
8   ema_5            2438 non-null   float64
9   ema_10           2438 non-null   float64
10  momentum_5       2438 non-null   float64
11  momentum_10      2438 non-null   float64
12  roc_5            2438 non-null   float64
13  roc_10           2438 non-null   float64
14  std_5            2438 non-null   float64
15  std_10           2438 non-null   float64
16  rsi_14           2438 non-null   float64
17  vwap             2438 non-null   float64
18  obv              2438 non-null   int64
19  adx_14           2438 non-null   float64
20  atr_14           2438 non-null   float64
21  bollinger_upper  2438 non-null   float64
22  bollinger_lower  2438 non-null   float64
23  macd             2438 non-null   float64
24  cci_20           2438 non-null   float64
25  williams_r       2438 non-null   float64
26  stochastic_k     2438 non-null   float64
27  Target           2438 non-null   float64
dtypes: datetime64[ns](1), float64(25), int64(2)
memory usage: 552.4 KB

The dataframe ‘indicators’ contains all 21 technical indicators mentioned earlier.

Bias-Variance Decomposition

Now, the primary purpose of this blog is to demonstrate how the bias-variance decomposition can aid in developing an ML-based trading strategy. Of course, we aren’t just limiting ourselves to it; we are also learning the complete pipeline of creating and backtesting an ML-based strategy with robustness. But let’s talk about the bias-variance decomposition now.

We begin by defining six different regression models:

You can add more or subtract a couple from the above list. The more regressor models there are, the longer the subsequent codes will take to run. Reducing the number of estimators in the relevant models will also result in faster execution of the subsequent codes.

In case you’re wondering why I chose regressor models, it’s because the nature of our target variable is continuous, not discrete. Although our trading strategy is based on the direction of the prediction (bullish or bearish), we are training the model to predict the 5-day return, a continuous random variable, rather than the market movement, which is a categorical variable.

After defining the models, we define a function for the bias-variance decomposition:

You can decrease the value of num_rounds to, say, 10, to make the following code run faster. However, a higher value gives a more robust estimate.

This is a good repository to look up the above code:

https://rasbt.github.io/mlxtend/user_guide/evaluate/bias_variance_decomp/

Finally, we run the bias-variance decomposition:

The output of this code is:

Bias-Variance Decomposition for All Models:
                  Total Error      Bias  Variance  Irreducible Error
LinearRegression     0.000773  0.000749  0.000024      -2.270048e-19
Ridge                0.000763  0.000743  0.000021       1.016440e-19
DecisionTree         0.000953  0.000585  0.000368      -2.710505e-19
Bagging              0.000605  0.000580  0.000025       7.792703e-20
RandomForest         0.000605  0.000580  0.000025       1.287490e-19
GradientBoosting     0.000536  0.000459  0.000077       9.486769e-20

Let’s analyse the above table. We’ll need to choose a model that balances bias and variance, meaning it neither underfits nor overfits. The decision tree regressor best balances bias and variance among all six models.

However, its total error is the highest. Bagging and RandomForest display similar total errors. GradientBoosting displays not just the lowest total error but also a higher degree of variance compared to Bagging and RandomForest; thus, its ability to generalise to unseen data should be better than the other two, since it would capture more complex patterns..

You might be compelled to think that with such proximity of values, such in-depth analysis isn’t apt owing to a high noise-to-signal ratio. However, since we are running 100 rounds of the bias-variance decomposition, we can be confident in the noise mitigation that results.

Long story cut short, we’ll choose to train the GradientBoosting regressor, and use it to predict the target variable. You can, of course, change the model and see how the strategy performs under the new model. Please note that we are treating the ML models as black boxes here, as exploring their underlying mechanisms is outside the scope of this blog. However, when using ML models for any use case, we should always be aware of their inner workings and choose accordingly.

Having discussed all the above, is there a way of reducing the errors of one or more of the above regressor models? Yes, and it’s not a mere technique, but an integral part of working with time series. Let’s discuss this.

Stationarizing the Inputs

We are working with time series data (read more), and when performing financial modeling tasks, we need to check for stationarity (read more). In our case, we should check our input variables (the predictors) for stationarity.

Let’s define a function to check the order of integration of the predictor variables. This function would check whether we need to difference the time series (in our case, the predictor variables) and if so, how many times (read more).

Let’s check the predictor variables for stationarity and apply differencing to the required predictors.

Here’s the code:

Here’s a snapshot of the output of the above code:

The above output indicates that 13 predictor variables don’t require stationarisation, while 8 do. Let’s stationarise them.

Let’s verify whether the stationarising got done as expected or not:

Yup, done!

Let’s align the data again:

Let’s check the bias-variance decomposition of the models with the stationarised predictors:

Here’s the output:

Bias-Variance Decomposition for All Models with Stationarised Predictors:


                  Total Error      Bias  Variance  Irreducible Error
LinearRegression     0.000384  0.000369  0.000015       5.421011e-20
Ridge                0.000386  0.000373  0.000013      -3.726945e-20
DecisionTree         0.000888  0.000341  0.000546       2.168404e-19
Bagging              0.000362  0.000338  0.000024      -1.151965e-19
RandomForest         0.000363  0.000338  0.000024       7.453890e-20
GradientBoosting     0.000358  0.000324  0.000034      -3.388132e-20

There you go. Just by following Time Series 101, we could reduce the errors of all the models. For the same reason that we discussed earlier, we’ll choose to run the prediction and backtesting using the GradientBoosting regressor.

Running a Prediction using the Chosen Model

Next, we run a walk-forward prediction using the chosen model:

Now, we create a dataframe, ‘final_data’, that contains only the open prices, close prices, actual/realised 5-day returns, and 5-day returns predicted by the model. We need the open and close prices for entering and exiting trades, and the predicted 5-day returns, to determine the direction in which we take trades. We then call the ‘backtest_strategy’ function on this dataframe.

Checking the Trade Logs

The dataframe ‘trades_df_differenced’ contains the trade logs.

We’ll convert the decimals of the values in the dataframe for better visibility:

Let’s check the dataframe ‘trades_df_differenced’ now:

Here’s a snapshot of the output of this code:

From the table above, it’s apparent that we take a new trade daily and deploy 20% of our tradeable capital on each trade.

Equity Curves, Sharpe, Drawdown, Hit Ratio, Returns Distribution, Average Returns per Trade, and CAGR

Let’s calculate the equity for the strategy and the buy-and-hold approach:

Next, we calculate the Sharpe and the maximum drawdowns:

The above code requires you to input the risk-free rate of your choice. It is typically the government treasury yield. You can look it up online for your geography. I chose to work with a value of 6.6:

Enter the risk-free rate (e.g., for 5.3%, enter only 5.3): 6.6

Now, we’ll reindex the dataframes to a datetime index.

We’ll plot the equity curves next:

This is how the strategy and buy-and-hold equity curves look when plotted on the same chart:

The strategy equity and the underlying move almost in tandem, with the strategy underperforming before the COVID-19 pandemic and outperforming afterward. Toward the end, we’ll discuss some realistic considerations about this relative performance.

Let’s have a look at the drawdowns of the strategy and the buy-and-hold approach:

Let’s take a look at the Sharpe ratios and the maximum drawdown by calling the respective functions that we defined earlier:

Output:

Sharpe Ratio (Strategy with Stationarised Predictors):    0.89
Sharpe Ratio (Buy & Hold):  0.42
Max Drawdown (Strategy with Stationarised Predictors):    -11.28%
Max Drawdown (Buy & Hold):  -38.44%

Here’s the hit ratio:

Hit Ratio of Strategy with Stationarised Predictors: 54.09%

This is how the distribution of the strategy returns looks like:

Finally, let’s calculate the average profits (losses) per winning (losing) trade:

Average Profit for Profitable Trades with Stationarised Predictors: 0.0171
Average Loss Loss-Making Trades with Stationarised Predictors: -0.0146

Based on the above trade metrics, we profit more on average in each trade than we lose. Also, the number of positive trades exceeds the number of negative trades. Therefore, our strategy is safe on both fronts. The maximum drawdown of the strategy is limited to 10.48%.

The reason: The holding period for any trade is 5 days, using only 20% of our available capital per trade. This also reduces the upside potential per trade. However, since the average profit per profitable trade is higher than the average loss per loss-making trade and the number of profitable trades is higher than the number of loss-making trades, the chances of capturing more upsides are higher than those of capturing more downsides.

Let’s calculate the compounded annual growth rate (CAGR):

CAGR (Buy & Hold): 13.0078%
CAGR (Strategy with Stationarised Predictors): 13.3382%

Finally, we’ll evaluate the regressor model's accuracy, precision, recall, and f1 scores (read more).

Confusion Matrix (Stationarised Predictors):
 [[387 508]
 [453 834]]
Accuracy (Stationarised Predictors): 0.5596
Recall (Stationarised Predictors): 0.6480
Precision (Stationarised Predictors): 0.6215
F1-Score (Stationarised Predictors): 0.6345

Some Realistic Considerations

Our strategy outperformed the underlying index during the post-COVID-19 crash period and marginally outperformed the overall market. However, if you are thinking of using the skeleton of this strategy to generate alphas, you’ll need to peel off some assumptions and take into account some realistic considerations:

Transaction Costs: We enter and exit trades daily, as we saw earlier. This incurs transaction costs.

Asset Selection: We backtested using the broad market index, which isn’t directly tradable. We’ll need to choose ETFs or derivatives with this index as the underlying. The strategy's performance would also be subject to the inherent characteristics of the asset that we decide to trade as a proxy for the broad market index.

Slippages: We enter our trades at the market's opening and exit at its close. Trading activity can be high during these periods, and we may encounter considerable slippages.

Availability of Partially Tradable Securities: Our backtest implicitly assumes the availability of fractional assets. For example, if our capital is  ₹2,000 and the entry price is ₹20,000, we’ll be able to buy or sell 0.1 units of the underlying, ignoring all other costs.

Taxes: Since we are entering and exiting trades within very short time frames, apart from transaction charges, we would incur a significant amount of short-term capital gains tax (STCG) on the profits earned. This, of course, would depend on your local regulations.

Risk Management: In the backtest, we omitted stop-losses and take-profits. You are encouraged to include them and let us know your findings on how the strategy's performance gets modified.

Event-driven Backtesting: The backtesting we performed above is vectorized. However, in real life, tomorrow comes only after today, and we must consider this when performing a backtest. You can explore the Blueshift at https://blueshift.quantinsti.com/ and try backtesting the above strategy using an event-driven approach (read more). An event-driven backtest would also account for slippage, transaction costs, implementation shortfalls, and risk management.

Strategy Performance: The hit ratio of the strategy and the model's accuracy are approximately 54% and 56%, respectively. These values are marginally better than those of a coin toss. You should try this strategy with other asset classes and only select those assets on which these values are at least 60% (or higher if you wanna be more conservative). Only after that should you perform an event-driven backtesting using this strategy outline.

A Note on the Downloadable Python Notebook

The downloadable notebook comprises backtesting the strategy and evaluating its performance and the model's performance parameters in a scenario where the predictors are not stationarised and after stationarising them (as we saw above). In the former, the strategy significantly outperforms the underlying model, and the model displays greater accuracy in its predictions despite its higher errors displayed during the bias-variance decomposition. Thus, a well-performing model need not necessarily translate into a good trading strategy, and vice versa.

The Sharpe of the strategy without the predictors stationarised is 2.56, and the CAGR is almost 27% (as opposed to 0.94 and 14% respectively when the predictors are stationarised). Since we used GradientBoosting, a tree-based model that doesn't necessarily need the predictor variables to be stationarised, we can work without stationarising the predictors and reap the benefits of the model's high performance with non-stationarised predictors.

Note that running the notebook will take some time. Also, the performances you obtain will differ a bit from what I’ve shown throughout the article.

There’s no ‘Good’ in Goodbye…

…yet, I’ll have to say so now 🙂. Try out the backtest with different assets by changing some of the parameters mentioned in the blog, and let us know your findings. Also, as we always say, since we aren’t a registered investment advisory, any strategy demonstrated as part of our content is for demonstrative, educational, and informational purposes only, and should not be construed as trading or investment advice. However, if you’re able to incorporate all the aforementioned realistic factors, extensively backtest and forward test the strategy (with or without some tweaks), generate significant alpha, and make substantial returns by deploying it in the markets, do share the good news with us as a comment below. We’ll be happy for your success 🙂. Until next time…

Credits

José Carlos Gonzáles Tanaka and Vivek Krishnamoorthy, thank you for your meticulous feedback; it helped shape this article!
Chainika Thakar, thanks for rendering this and making it available to the world!

Next Steps

After going through the above, you can follow a few structured learning paths if you want to broaden and/or deepen your understanding of trading model performance, ML strategy development, and backtesting workflows.

To master each component of this strategy — from Python and PCA to stationarity and backtesting — explore topic-specific Quantra courses like:

• Data & Feature Engineering for Trading
• Trading with Machine Learning: Regression
• Backtesting Trading Strategies
• Statistics & Machine Learning in Quant Trading

For those aiming to consolidate all of this knowledge into a structured, mentor-led format, the Executive Programme in Algorithmic Trading (EPAT) offers an ideal next step. EPAT covers everything from Python and statistics to machine learning, time series modeling, backtesting, and performance metrics evaluation — equipping you to build and deploy robust, data-driven strategies at scale.

File in the download:

• Bias Variance Decomposition - Python notebook

Feel free to make changes to the code as per your comfort.

Visit blog to download

All investments and trading in the stock market involve risk. Any decision to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.