A Numerical Scheme for the Quantile Hedging Problem

Abstract

We consider numerical approximations to the quantile hedging price of a European claim in a nonlinear market with Markovian dynamics. We study an equivalent stochastic target problem with the conditional probability of success as a new state variable, in addition to the asset value process. We propose numerical approximations based on piecewise constant policy time stepping coupled with novel finite difference schemes. We prove convergence in the monotone case combining backward stochastic differential equation arguments with the Barles and Jakobsen and Barles and Souganidis approaches for nonlinear PDEs. The difficulties compared to the classical setting consist in the construction of monotone schemes under degeneracy due to the perfectly correlated joint process, the unboundedness of the control variable, and the effect of the boundaries in the probability variable on the analysis. We extend the method to a class of nonmonotone schemes using higher order interpolation and prove convergence for linear drivers. In a numerical section, we illustrate the performance of our schemes by considering an example in a financial market with imperfections, and show that a standard nonmonotone scheme produces financially counterintuitive solutions.