The uncertain volatility model: A monte carlo approach

The uncertain volatility model has long attracted the attention of practitioners since it provides a worst-case pricing scenario for the sell side. The valuation of a financial derivative based on this model requires the solution of a fully nonlinear partial differential equation. One can only rely on finite-difference schemes when the number of variables (that is, underlyings and path-dependent variables) is small (no more than three in practice). In all other cases, numerical valuation seems out of reach. In this paper we outline two accurate, easy-to-implement Monte Carlo-like methods that only depend minimally on dimensionality. The first method requires a parameterization of the optimal covariance matrix and involves a series of backward low-dimensional optimizations. The second method relies heavily on a recently established connection between second-order backward stochastic differential equations and nonlinear second-order parabolic partial differential equations. Both methods are illustrated by numerical experiments.