The modified homotopy perturbation method and its application to the dynamics of price evolution in Caputo-fractional order Black Scholes model

Abstract

BackgroundFollowing a financial loss in trades due to lack of risk management in previous models from market practitioners, Fisher Black and Myron Scholes visited the academic setting and were able to mathematically develop an option pricing equation named the Black–Scholes model. In this study, we address the solution of a Caputo fractional-order Black–Scholes model using an analytic method named the modified initial guess homotopy perturbation method.MethodologyForemost, the classical Black Scholes model relaxed for European option style is generalized to be of Caputo derivative. The introduced method is established by coupling a power series function of arbitrary order with the renown He’s homotopy perturbation method. The convergence of the method is demonstrated using the fixed point theorem, and its methodology is illustrated by solving a generalized theoretical form of the fractional order Black Scholes model. The applicability of the method is proven by solving three different fractional order Black–Scholes equations derived from different market scenarios and its effectiveness is confirmed as feasible series of arbitrary orders that accelerate fast to the exact solution at an integer order were obtained. The computation of these results was carried out using Mathematica 12 software. Subsequently, the obtained outcomes were utilized in Maple 18 software to conduct a series of numerical simulations. These simulations aimed to analyze the influence of the fractional order on the dynamics of payoff functions regarding the share value as the option approached its expiration date under varying market constraints. In all three scenarios, the results showed that option values decrease as the expiration date approaches the integer order. Furthermore, the comparative outcomes reveal that Caputo fractional order derivatives control the flexibility of the classical Black–Scholes model because its payoff curve exhibits more sensitivity to changes associated with market characteristic parameters, such as volatility and interest rates.RecommendationsWe propose that the results of this work should be examined and implemented by mathematicians and economists to better comprehend the influence of Caputo-fractional order derivatives in understanding the dynamics of option price evolution of financial assets.