An Accurate and Efficient Numerical Method for Black-Scholes Equations

Abstract

Abstract. We present an efficient and accurate finite-difference method for computing Black-Scholes partial differential equations with multiunderlying assets. We directly solve Black-Scholes equations without transformations of variables. We provide computational results showing the performance of the method for two underlying asset option pricing problems.

Numerical Solution for 2D European Option Pricing Using Quarter-Sweep Modified Gauss-Seidel Method

“…The parameters considered are T = 0.1, K=100, r = 0.03, σ 1 = 0.5, σ 2 = 0.5, ρ = 0.5, s 1 = [0,300] and s 2 = [0,300] which are also used by Jeong et al (2009). Nonetheless in this study, we have 100 time steps with the tested matrix sizes, 50,100,150,200,250,300 and 350. Furthermore, will be the error tolerance.…”

“…Furthermore, will be the error tolerance. In computational finance, Root Mean Squared Relative Error (RMSE) is widely used to assess the accuracy of the iterative solutions such as in Zhao et al (2007), Jeong et al (2009) and Koh et al (2010a). The RMSE is defined by:…”

“…In this study, we focus on a two-dimensional Black-Scholes PDE as follows Eq. 1 (Stulz, 1982;Jeong et al, 2009) …”

Valuing option on the maximum of two assets using improving modified Gauss-Seidel method

“…W denotes the cumulative bivariate normal distribution function [6,7]. Jeong et al [2] applied a numerical approach, namely the Multigrid method to solve a two asset option pricing, but on different style of option which is the two assets cash or nothing call option. Nevertheless, we would like to study on the preconditioned iterative method established by Kohno et al [8] that is the Improving Modified GaussSeidel (IMGS) iterative method, in solving the option on the maximum of two assets.…”

“…Option on the maximum of two-asset is a style of exotic option initiated by Stulz [1]. This option can be governed by a two-dimensional Black-Scholes PDE [ (2) where T is the time at maturity and K is the strike price. The strike price is the price to trade the assets at maturity.…”