Numerical Schemes for Black-Scholes Equation with Error Dynamics

Abstract

. This paper focuses on the numerical solution of the Black-Scholes equation (BSE), which is used in finance to price options. The modified version of BSE to heat equation is subjected to two-time level finite difference method such as the Crank-Nicolson method and three-time level finite difference method such as the DuFort-Frankel method. The error dynamics is represented by the Global Spectral Analysis (GSA) method, which contradict the error dynamics of the von Neumann method, where the signal and error follow the same difference equation. For different maturities, volatilities and interest rates, both techniques are tested for accuracy. For the converted heat equation of BSE, the three-time level method is determined to be more accurate than the two-time level method. Finally, we conclude that risk can be reduced by short-term investment in a low interest, high-volatility market with a good approximation using the three-time level finite difference method for European call option for converted BSE to heat equation.