Category Archives: volatility

As mentioned in a previous post, I intend to model quadratic variation in terms of multiple pairings of intensity (duration) and return level processes.   At a minimum want a pairing for “non-jump” related returns and a pairing for “jump” related returns.

To do this it is necessary to partition returns into the categories based on threshold.   We may further want to disregard price movements below a certain level unless they cumulatively add up to a return with significance within a period.   Towards this end my duration measurement function uses a threshold to determine whether a return is to be considered as an event or not.  In pseudocode:

r ← {0} ∪ diff(log(series))
t ← times (series)
durations ← {}
for (i in 2:length(r))
{
    # determine cumulative return since last acceptance
    cumr ← <cummulative return since last event or max cum window>

    # determine whether qualifying event has occurred
    if (|cumr| ≥ threshold or |r[i]| ≥ threshold)
        durations ← durations ∪ {t[i] - <Tlastevent>}
}

For the diffusion portion of the process, in this 2 second sampled data set (EUR/USD low-liquidity period), a threshold of 3e-5 (equivalent of about 1/2 pip), seems to work well:

The jump portion of the process should be set so as to capture desired jump features and not much more, here I show with a threshold of 2e-4 (equivalent to about 3 pips):

Spikes manifest in intra-day markets frequently.   These are often short-lived and associated with buying/selling programs more often than change in fundamental factors, particularly in low-liquidity periods.   In evaluating duration and variance measures was trying to determine reasonable jump thresholds.

Below is a price series demonstrating a variety of spiking behavior:

Taking a look at the region around the jump at high frequency, we note that the jump did not occur with one trade rather with multiple within a short space of time:

From a duration perspective, if we want to capture the spike as one event of a given magnitude we either need to consider the cumulative return over a given window or sample with a longer period.   Here is the same with a 2 second sample:

Here is the 2 second sampling of the original range.   With the longer sample period, the spikes in return are more evident (compare to the first graph in the post):

I’ve been looking at a number of different volatility models for both daily and intra-day variance.   In the process have looked at GARCH(1,1) for daily and have developed a number of duration based variance (DV) models for intra-day.

Daily Returns
I had concluded early on that GARCH(1,1) generally fits daily data well but fails miserably with intra-day data.    Unfortunately, for some markets, GARCH performs miserably as well.

I was looking at Canadian 2Y CMT yields over the last 10 years.   There are significant spikes in return for any given year.    For example in 2003, on the raw daily returns, GARCH does poorly (squared returns: black, sigma squared: red):

As an experiment I used (very) lightly smoothed prices as a basis for GARCH testing.  The smoothing gave continuity to the price function and to the 1st derivatives (returns).    GARCH(1,1) calibrated on the same data set, but with minimal smoothing yields a much higher degree of fit (25 x):

Of course we are now projecting sigma^2 on a lightly smoothed price series and not the original series.   Note that the magnitude of returns has reduced as well, as the smoothing allows for more gradual transitions.

Let’s look at a smaller section of returns on the raw price series:

Visually there appears to be a autoregressive component in squared returns, but involves jumps followed by periods of near zero return.   Subsequent jumps appear to follow a near linear decay in magnitude.    You will note that the last set appears to be a poor fit, however, there is a cluster of low jumps, that when considered in intensity space would have a higher combined value (ie, consider the cumulative return over a small window).

This behavior would point to modeling this as a intensity function (equivalent to duration).

Intra-day Squared Returns
Autocorrelation plots of liquid periods and illiquid periods both show significant autocorrelation out to 10+ seconds, indicating that we should be looking at an AR process at some level with long memory:

Thus far the most successful model is a duration based hawkes process which is “infinitely” autoregressive with exponential decay.

GARCH has been shown to be a reasonable estimator of variance for daily or longer period returns. Some have adapted GARCH to use intraday returns to improve daily returns. GARCH does very poorly in estimating intra-day variance, however.

The GARCH model is based on the empirical observation that there is strong autocorrelation in the square of returns for lower frequencies (such as daily). This can be easily seen by observing clustering and “smooth” decay of squared returns on daily returns for many assets.

where the second equation is the ML optimization for the parameters.

Here is an example for daily Canadian 2Y CMT yield.  The red is the GARCH(1,1) variance, the black is the series, and the grey is the log return, and the green circle is the predicted variance for the next period:

Contact me if you would like to get the R source code for the above.

Intra-day squared returns, however, have many jumps, with little in the way of autocorrelated decay pattern. Looking at the EUR/USD series, the squared returns have jumps that reduce the ML to the point where GARCH parameterization does not converge.  There does appear to be a longer-term pattern, though, allowing for a model, though not GARCH.

With expanded processing power and general access to tick data, research has begun to focus on intra-day variance estimation. In particular, expressing variance in terms of price duration has become an emergent theme. Andersen, Dobrev, and Schaumburg are among a growing community developing this in a new direction.

At this point have disqualified GARCH as a useful measure for my intra-day strategies but am planning to use for a daily strategy. I am investigating a formulation of a duration based measure for intra-day volatility.