Analytical and semi-analytical pricing for European vanilla and barrier options. Pure Python, no black-box dependencies — every formula derives directly from the mathematics.

Closed-form pricing, first-order Greeks, and implied volatility inversion for European calls and puts.

where $d_1 = \frac{\ln(S/K) + (r + \sigma^2/2),T}{\sigma\sqrt{T}}$ and $d_2 = d_1 - \sigma\sqrt{T}$.

Implied volatility is recovered via Brent's method with arbitrage-bound checks.

import black_scholes as bs

result = bs.greeks(S=100, K=105, T=0.5, r=0.05, sigma=0.20, option_type="call")
# BSResult(price=5.14, delta=0.398, gamma=0.019, vega=0.271, theta=-0.017, rho=0.089)

iv = bs.implied_vol(market_price=5.14, S=100, K=105, T=0.5, r=0.05)
# 0.19999...

Semi-analytical pricing via Gil-Pelaez Fourier inversion of the Heston (1993) characteristic function.

$$dS = r,S,dt + \sqrt{v},S,dW_1$$ $$dv = \kappa(\theta - v),dt + \sigma_v\sqrt{v},dW_2, \quad d W_1 d W_2 = \rho,dt$$

The characteristic function is evaluated using the Albrecher et al. (2007) formulation to avoid branch-cut discontinuities. The option price is:

$$C = S,P_1 - K e^{-rT} P_2, \quad P_j = \frac{1}{2} + \frac{1}{\pi}\int_0^\infty \text{Re}!\left[\frac{e^{-iu\ln K},\phi_j(u)}{iu}\right]du$$

Calibration to a strip of market prices via L-BFGS-B (minimises sum of squared relative errors).

from heston import HestonParams, price, calibrate

params = HestonParams(v0=0.04, kappa=2.0, theta=0.04, sigma_v=0.30, rho=-0.70)
c = price(S=100, K=105, T=0.5, r=0.05, params=params)

fitted = calibrate(market_prices, strikes, S=100, T=0.5, r=0.05)

All eight standard single-barrier types under Black-Scholes dynamics (Reiner & Rubinstein, 1991).

In-Out parity $V_{\text{KI}} + V_{\text{KO}} = V_{\text{vanilla}}$ is verifiable via check_in_out_parity. Optional cash rebate on knock-out.

import barrier_options as barrier

p = barrier.price(S=100, K=100, H=90, T=0.5, r=0.05, sigma=0.20,
                  barrier_type="down-and-out", option_type="call")

assert barrier.check_in_out_parity(S=100, K=100, H=90, T=0.5, r=0.05, sigma=0.20)

pip install -r requirements.txt
python examples/demo.py

• Black, F. & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy.
• Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility. Review of Financial Studies.
• Reiner, E. & Rubinstein, M. (1991). Breaking Down the Barriers. Risk Magazine.
• Albrecher, H., Mayer, P., Schachermayer, W. & Teichmann, J. (2007). The Little Heston Trap. Wilmott Magazine.