Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Sample path Large Deviation Principles (LDP) of the Freidlin-Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model. Using the sample path LDP for the Heston model, the problem is considered of selecting an importance sampling change of drift, for both the price and volatility, which minimize the variance of Monte Carlo estimators for path-dependent option prices. An asymptotically optimal change of drift is identified as a solution to a two-dimensional variational problem. The case of the arithmetic average Asian put option is solved in detail.