High-accuracy finite-difference methods for the valuation of options

Abstract

A finite-difference scheme often employed for the valuation of options from the Black–Scholes equation is the Crank–Nicolson (CN) scheme. The CN scheme is second order in both time and asset. For a rapid valuation with a reasonable resolution of the option price curve, it requires extremely small steps in both time and asset. In this paper, we present high-accuracy finite-difference methods for the Black–Scholes equation in which we employ the fourth-order L-stable Simpson-type (LSIMP) time integration schemes developed earlier and the well-known Numerov method for discretization in the asset direction. The resulting schemes, called LSIMP–NUM, are fourth order in both time and asset. The LSIMP–NUM schemes obtained can provide a rapid, stable and accurate resolution of option prices, allowing for relatively large steps in both time and asset. We compare the computational efficiency of the LSIMP–NUM schemes with the CN and Douglas schemes by considering valuation of European options and American options via the linear complementarity approach.

Keywords:

• Valuation of options
• Black–Scholes equation
• Crank–Nicolson method
• Douglas method
• L-stable Simpson-type rules
• Numerov method
• High-accuracy finite-difference methods
• European options
• American options
• Linear complementarity approach