Bounds and Asymptotic Approximations for Utility Prices when Volatility is Random

This paper is a contribution to the valuation of derivative securities in a stochastic volatility framework, which is a central problem in financial mathematics. The derivatives to be priced are of European type with the payoff depending on both the stock and the volatility. The valuation approach uses utility-based criteria under the assumption of exponential risk preferences. This methodology yields the indifference prices as solutions to second order quasilinear PDEs. Two sets of price bounds are derived that highlight the important ingredients of the utility approach, namely, nonlinear pricing rules with dynamic certainty equivalent characteristics, and pricing measures depending on correlation and the Sharpe ratio of the traded asset. The problem is further analyzed by asymptotic methods in the limit of the volatility being a fast mean-reverting process. The analysis relates the traditional market-selected volatility risk premium approach and the preference-based valuation techniques.