Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation

Abstract

• HAM and its applications in the valuation of ECO with TFBSE in a Caputo sense are presented. • By means of the marked point process, the classical Black-Scholes equation turns to a time-fractional Black-Scholes equation. • The valuation formula for the price of a European call option with fractional order is obtained via HAM. • Illustrative examples were presented to measure the performance of HAM in terms of accuracy, effectiveness and suitability. • The physical behaviour of the option prices has been shown in terms of plots for different values of fractional order. This paper presents the applications of Homotopy Analysis Method (HAM) in the valuation of a European Call Option (ECO) with Time-Fractional Black-Scholes Equation (TFBSE). The fractional derivative is considered in the sense of Caputo. Also, it is assumed that the stock price pays no dividend and follows the geometric Brownian motion. Based on HAM, a series solution for TFBSE has been obtained successfully. The valuation formula for the price of ECO with fractional order is also obtained. The accuracy, effectiveness and suitability of HAM were tested on two illustrative examples in the context of the Crank Nicolson Method (CRN), Binomial Model (BM) and the Black-Scholes Model (BSM). The comparative study of the results obtained via HAM, CRN, BM and BSM is presented. Furthermore, the physical behaviour of the option prices obtained via HAM has been shown in terms of plots for diverse fractional order. Moreover, HAM is found to be accurate, effective and suitable for the solution of TFBSE. Hence, it can be concluded that HAM converges faster to the analytical solution and is a good alternative tool to determine the price of ECO with fractional order.