Uncertain Volatility Model: A Monte-Carlo Approach
Bloomberg L.P.; Columbia University - Department of Mathematics; New York University - Courant Institute of Mathematical Sciences
Société Générale - Paris, France
January 21, 2010
The uncertain volatility model has long ago attracted the attention of practitioners as it provides worst-case pricing scenario for the sell-side. The valuation of a financial derivative based on this model requires solving a fully non-linear PDE. One can rely on finite difference schemes only when the number of variables (that is, underlyings and path-dependent variables) is small - in practice no more than three. In all other cases, numerical valuation seems out of reach. In this paper, we outline two accurate, easy-to-implement Monte-Carlo-like methods which hardly depend on dimensionality. The first method requires a parameterization of the optimal covariance matrix and consists in 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 non-linear second-order parabolic PDEs. Both methods are illustrated by numerical experiments.
Number of Pages in PDF File: 25
Keywords: Uncertain volatility model, optimization of non-smooth function, backward stochastic differential equation, Monte-Carlo simulation, regression, Malliavin
Date posted: January 21, 2010