Finite difference schemes for a nonlinear Black-Scholes model with transaction cost and volatility risk

Abstract

Several nonlinear Black-Scholes models have been proposed to taketransaction cost, large investor performance and illiquid markets into account. Oneof the most comprehensive models was introduced by Barles and Soner in [4] andconsiders transaction cost in the hedging strategy and risk from an illiquid market.In this paper we compare several finite difference methods for the solution of thismodel with respect to precision and order of convergence. We conclude that standard explicit Euler comes out as the preferred explicit method and standard CrankNicolson with Rannacher time stepping as the preferred implicit method.