Goodness-of-fit of the Heston model

Abstract

An analytical formula for the probability distribution of stock-market returns, derived from the Heston model assuming a mean-reverting stochastic volatility, was recently proposed by Dragulescu and Yakovenko in Quantitative Finance 2002. While replicating their results, we found two significant weaknesses in their method to pre-process the data, which cast a shadow over the effective goodness-of-fit of the model. We propose a new method, more truly capturing the market, and perform a Kolmogorov-Smirnov test and a Chi Square test on the resulting probability distribution. The results raise some significant questions for large time lags -- 40 to 250 days -- where the smoothness of the data does not require such a complex model; nevertheless, we also provide some statistical evidence in favour of the Heston model for small time lags -- 1 and 5 days -- compared with the traditional Gaussian model assuming constant volatility.