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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–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.