Numerical Methods for Non-Linear Black–Scholes Equations

In recent years non-linear Black–Scholes models have been used to build transaction costs, market liquidity or volatility uncertainty into the classical Black–Scholes concept. In this article we discuss the applicability of implicit numerical schemes for the valuation of contingent claims in these models. It is possible to derive sufficient conditions, which are required to ensure the convergence of the backward differentiation formula (BDF) and Crank–Nicolson scheme (CN) scheme to the unique viscosity solution. These stability conditions can be checked a priori and convergent schemes can be constructed for a large class of non-linear models and payoff profiles. However, if these conditions are not satisfied we show that the schemes are not convergent or produce spurious solutions. We study the practical implications of the derived stability criterions on relevant numerical examples.