A Closed-Form Solution for Options with Ambiguity about Stochastic Volatility∗

We derive a closed-form solution for the price of a European call option in the presence of ambiguity about the stochastic process that determines the variance of the underlying asset's return. The option pricing formula of Heston (1993) is a particular case of ours, corresponding to the case in which there is no ambiguity (uncertainty is exclusively risk). In the presence of ambiguity, the variance uncertainty price becomes either a convex or a concave function of the instantaneous variance, depending on whether the variance ambiguity price is negative or positive. We nd that if the variance ambiguity price is positive, the option price is decreasing in the level of ambiguity (across all moneyness levels). The opposite happens if the variance ambiguity price is negative. Consistently, in the former (and more natural) scenario, ambiguity aversion decreases the option's implied volatility, which helps to explain the variance premium puzzle.