Volatility Tracking

Volatility can be thought of as an inverse function of liquidity.  While liquidity is quantitatively vague, volatility is not.  It can be measured as the scale factor of the distribution of market returns.  Volatility is traditionally measured by taking standard deviation of a series of logarithmic returns.  Implicit in such a measure is that the variance of the distribution is finite.  Such an assumption may not be correct for financial markets so we present methods of measuring the scale factor which will work if the variance of a distribution is infinite.

Below is the plot of the SPX (the S&P 500 index as calculated by the CME Group).  The plot below shows the closing prices since mid-February 2009, capturing the market bottom in March.

For our analysis we will scale each day's one minute logarithmic return data.  In the plot below three different methods are used.  In red is a method based on the empirical characteristic function, the unit scale factor is chosen to be identical to that of a stable distribution.  The calculation is made at the one point on the empirical characteristic function that would be independent of stable distributional shape parameter, α.  The method is otherwise distributionally independent.  In blue the stable scale factor, γ, is calculated from each day's one minute log returns by maximum likelihood method.  This method assumes a stable distribution.  In green is a calculation made by L-moments.  The assumption with this method is that the expectation of the distribution is finite.  To be on the same scale with the other fits this calculation is adjusted by a factor chosen to make it parallel to the stable distribution fit.  Each of these methods is completely independent of one another, but the L-moment method would have slightly less amplitude than the others if it were not rescaled.

The highest amplitude of volatility coincided with the market bottom, perhaps slightly preceding it.  We plan to track this volatility decay.  Our expectation is that volatility will slowly fall until some event drives it up again.  Volatility typically increases during the quarterly reporting seasons, but the current plot does not show too much disturbance of the trend in the reporting months of April and July.  The plot below shows the decay since the recent peak in volatility along with a logarithmic decay model and the model 95% confidence bands in green.

We expect that at some point we will see another volatility spike, but we believe that the timing of this is probably not predictable.  However, if volatility crosses significantly above the upper confidence band, we then predict market prices will fall and probably fall significantly.  

Keep in mind that the calculations above are sort of an average volatility for the day.  In fact the volatility varies throughout the day.  The autocorrelation plot below of absolute value of log returns picks up the daily cycle which is consistent with higher volatility at the market open and close and the lowest volatility occuring at mid day.

Intra-day samples of several hours may also be evaluated by the methods above.

We have developed a simpler measure of volatility that can track the same information from intra-day high and low prices: HighLowVolatility.

© Copyright 2010 mathestate    Wed 10 Mar 2010