Numerical Approximations of a Nonlinear Volatility Model with European Options

Abstract

Black-Scholes model plays a very significant role in the world of quantitative finance. In this paper, the focus are on both nonlinear and linear Black-Scholes (BS) equations with numerical approximations. We aim to find an effective numerical approximations for Black-Scholes model. Several models from the most relevant class of nonlinear Black-Scholes equations with European option are analyzed in this study. The problem is approached by transforming the problem into a convection-diffusion equation and later it is approximated with the help of finite difference method (Crank-Nicolson). The result of finite difference schemes (Crank-Nicolson) for several volatility models are presented, including the Risk Adjusted Pricing Methodology (RAPM), Leland’s model and the Barles’-Soner’s Model. At the same time, it is attempted to illustrate a comparison of different volatility models. In the case of linear Black-Scholes model, we approximate the model with finite difference method (FDM) and finite element method (FEM) and compare the results. All the numerical schemes are implemented in MATLAB and corresponding graphs are also presented here.

GANIT J. Bangladesh Math. Soc. 42.1 (2022) 050- 068