An accurate solution for the generalized Black-Scholes equations governing option pricing

Abstract

Today industries related to finance are essentially implementing advanced mathematical tools. In 1973, Fisher Black and Myron Scholes developed an eminent stochastic model which later coined as Black-Scholes differential equations for option pricing. This paper illustrates a convenient time integration scheme based on the generalized trapezoidal formulas (GTF $[\alpha = \frac{1}{3}]$) introduced by Chawla et al. in 1996. GTF is applied for the temporal discretization along with the classical finite difference schemes in space direction. The proposed scheme yields the (uniform) stability employing the uniform bound of the inverse operator, as well as second-order spatial accuracy and third-order temporal accuracy under reasonable conditions. Finally, the numerical illustrations and comparison with existing schemes demonstrate the stability and accuracy of the method.