Option pricing via maximization over uncertainty and correction of volatility smile

Abstract

The paper presents a pricing rule for market models with stochastic volatility and with an uncertainty in its evolution law. It is shown that the most common stochastic volatility models allow a possibility that the option price calculated for random volatility with an error in volatility forecasts is lower than the price for the market with zero error of volatility forecast. To eliminate this possibility, we suggest a pricing rule based on maximization of the price via a class of possible equivalent risk-neutral measures. It shown that, in a Markovian setting, this pricing rule requires to solve a parabolic Bellman equation. Some existence results and a priory estimates are obtained for this equation.