[We study the stock price distributions that arise when prices follow a diffusion process with a stochastically varying volatility parameter. We use analytic techniques to derive an explicit closed-form solution for the case where volatility is driven by an arithmetic Ornstein-Uhlenbeck (or AR1) process. We then apply our results to two related problems in the finance literature: (i) options pricing in a world of stochastic volatility, and (ii) the relationship between stochastic volatility and the nature of "fat tails" in stock price distributions.]

We study the stock price distributions that arise when prices follow a diffusion process with a stochastically varying volatility parameter. We use analytic techniques to derive an explicit closed-form solution for the case where volatility is driven by an arithmetic Ornstein–Uhlenbeck (or AR1) process. We then apply our results to two related problems in the finance literature: (i) options pricing in a world of stochastic volatility, and (ii) the relationship between stochastic volatility and the nature of “fat tails” in stock price distributions.

[We examine the intraday relationship between returns and returns volatility in the stock index and stock index futures markets. Our results indicate a strong intermarket dependence in the volatility of the cash and futures returns. Price innovations that originate in either the stock or futures markets can predict the future volatility in the other market. We show that this relationship persists even during periods in which the dependence in the returns themselves appears to weaken. The findings are robust to controlling for potential market frictions such as asynchronous trading in the stock index. Our results have implications for understanding the pattern of information flows between the two markets.]

We examine the intraday relationship between returns and returns volatility in the stock index and stock index futures markets. Our results indicate a strong intermarket dependence in the volatility of the cash and futures returns. Price innovations that originate in either the stock or futures markets can predict the future volatility in the other market. We show that this relationship persists even during periods in which the dependence in the returns themselves appears to weaken. The findings are robust to controlling for potential market frictions such as asynchronous trading in the stock index. Our results have implications for understanding the pattern of information flows between the two markets.