Option Pricing with Ambiguous Correlation and Fast Mean-reverting Volatilities

In pricing an option on multiple assets, volatilities of individual assets and the correlations among different assets are necessary inputs. In practice, implied volatility surface for individual underlying asset can be used to calibrate the marginal distribution, but information on correlations is generally difficult to obtain. By regarding the correlation between two assets as an ambiguous parameter, we obtain the worst case price to option seller. The corresponding pricing problem is formulated as a stochastic optimal control problem in which the uncertain correlation takes the role as a control function. Consequently, the Black-Scholes equation is replaced by an HJB equation for deriving of the option price bounds. We solve this problem using the framework of stochastic volatility asymptotics and explain how volatility surfaces of individual assets can be robustly pulled together to estimate the price bounds of an option on multiple assets. Empirical study with foreign exchange option data illustrates the practical use of the proposed approach.