Fast Explicit Positivity-preserving Schemes for the Black-Scholes Equation

In this paper we propose a fast explicit positivity preserving finite difference scheme for finding an accurate solution of the famous Black-Scholes partial differential equation. We apply a non-traditional approximation of the spacial derivatives using values at different time levels. We explore the convergence, consistency and positivity of the iteration matrix of the obtained numerical method for the Black-Scholes equation with non-constant coefficients. The main advantages of the constructed method are monotonicity and positivity preserving properties of the numerical solution that in addition satisfies a discrete maximum principle. The latter is an important property for obtaining a smooth numerical solution in case of convection-diffusion equations with discontinuous initial conditions such as pricing digital or discrete options. The presented explicit method is characterized by a fast computational speed, it has a straightforward implementation and is very efficient for low volatility option valuation problems.