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Paper text:
standard models of stock - market fluctuations predict a normal ( gaussian ) distribution for stock price log - returns @xcite
. however , the empirical distribution exhibits significant kurtosis with a greater probability mass in its tails and centre of the distribution @xcite . in quantitative finance 2002 ( @xcite ) , drgulescu and yakovenko ( dy ) proposed an analytical formula for the probability density function ( _ pdf _ ) of stock price log - returns , based on the heston model @xcite  a geometric brownian motion for the log - returns time series coupled with a stochastic mean - reverting volatility .
the resulting _ pdf _ is claimed to outperform the gaussian on a large scale of time lags ( @xmath5 and @xmath2 days ) .
+      this model starts with the usual assumption that the price @xmath6 follows a geometric brownian motion described by the following stochastic differential equation @xmath7    [ cols= " < , < " , ]      +
the dy analytical formula is quite exciting , and would validate the heston model , if the resulting expected distribution was similar to the empirical one .
our method of pre - processing the data is different from the one suggested by drgulescu and yakovenko , and , we believe , more truly captures the market .
we find that the dy formula consistently outperforms the gaussian in terms of best fit ( the error between the model and the observed data ) , but a higher complexity is the price to pay : the number of parameters is greater in the heston model , which enables the fitted _ pdf _ to exhibit kurtosis and fat tails when the empirical dataset does , but also penalizes it when it does nt .
hence , in terms of goodness - of - fit ( the trade - off between best fit and complexity ) , the gaussian is preferable for low frequencies ( @xmath8 days ) , since the empirical dataset is then quite smooth and does not exhibit kurtosis . for medium frequencies ( @xmath9 days ) ,
both models are accepted at a .05 confidence level . finally , for high frequencies , although it performs better than the gaussian , the dy formula is still rejected ( @xmath10 and @xmath4 days ) , mainly because of the extremely high kurtosis of the observed data .
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 drgulescu 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 @xmath0 test on the resulting probability distribution . 
 the results raise some significant questions for large time lags  @xmath1 to @xmath2 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  @xmath3 and @xmath4 days  compared with the traditional gaussian model assuming constant volatility .    
 heston model , dy formula , mean - reverting stochastic volatility , goodness - of - fit