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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-Ublenbeck (or AR1) process. We then apply our results to two related problems in the finance literature: (1) options pricing in a world of stochastic volatility, and (2) the relationship between stochastic volatility and the nature of "fat tailes" in stock price distributions. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.
Stein et al. (Tue,) studied this question.