Comparison between the probability distribution of returns in the Heston model and empirical data for stock indexes

Abstract

We compare the probability distribution of returns for the three major stock-market indices (Nasdaq, S&P500, and Dow-Jones) with an analytical formula recently derived by Drȃgulescu and Yakovenko for the Heston model with stochastic variance. For the period of 1982-1999, we find a very good agreement between the theory and the data for a wide range of time lags from 1 to 250 days. On the other hand, deviations start to appear when the data for 2000-2002 are included. We interpret this as a statistical evidence of the major change in the market from a positive growth rate in 1980s and 1990s to a negative rate in 2000s.

Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact

“…The intraday time lags were chosen at multiples of 5 minutes, which divide exactly the 6.5 hours (390 minutes) of the trading day. The interday returns are as described in [4,5] for time lags from 1 day to 1 month = 20 trading days.…”

Fitting the Heston Stochastic Volatility Model to Chinese Stocks

“…Without trimming, Daniel et al (2005) have shown that the Heston model does not provide a good fit to the Dow Jones stock market index. Meanwhile, Silva and Yakovenko (2003) have verified the goodness-of-fit of the Heston model by explaining the NASDAQ, DowJones and S&P 500 indices and by documenting different results for different dataset periods. Our results show that without trimming the dataset the Heston model fits the empirical distribution of index and stock returns in the Chinese stock market well, especially for the daily log-returns.…”

“…The goodness-of-fit of the Heston model to the historical data of log-returns has been tested in studies by Dragulescu and Yakovenko (2002), Silva and Yakovenko (2003) and Daniel et al (2005), which together show mixed results regarding the performance of the Heston model. Dragulescu and Yakovenko (2002) have derived the closed form of the probability distribution of log-returns in the Heston model to show that the Heston model provides a good fit to the DowJones index returns at different time intervals.…”

Volatility Effects on the Escape Time in Financial Market Models

“…It received a great attention in the financial literature specially in connection with option pricing [Fouque et al, 2000]. The Heston model was verified empirically with both stocks [Silva & Yakovenko, 2003, Drǎgulescu & Yakovenko, 2002 and options [Hull & White, 1987;Hull, 2004], and good agreement with the data has been found. It was also recently investigated by econophysicists [Miccichè et al, 2002;Drǎgulescu & Yakovenko, 2002;Silva et al, 2004;Bonanno & Spagnolo, 2005].…”