Positivity Preserving Schemes for Black-Scholes Equation

Abstract

Mathematical finance is a field of applied mathematics, concerned with financial markets. In the market of financial derivatives the most important problem is the so called option valuation problem, i.e. to compute a fair value for the option. The solution of the Black-Scholes equation determines the option price, respectively according to the used initial conditions. In this paper, first we show that the positivity is not ensured with classical finite difference schemes when applied to the Black-Scholes equation for very small time steps. Next, by reforming the discretization of the reaction term of equation, a family of efficient explicit schemes are derived that is free of spurious oscillations around discontinuities and preserving positivity.

Keywords: Positivity, Nonstandard discretization, Black-Scholes equation