Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type

In the finance market, the Black–Scholes equation is used to model the price change of the underlying fractal transmission system. Moreover, the fractional differential equations recently are accepted by researchers that fractional differential equations are a powerful tool in studying fractal geometry and fractal dynamics. Fractional differential equations are used in modeling the various important situations or phenomena in the real world such as fluid flow, acoustics, electromagnetic, electrochemistry and material science.
In this paper, we have discussed the analytic solution of the time-fractional Black-Scholes equation with two assets in sense of the Katugampola fractional derivative in Caputo type. The method, used here to provide analytical solutions, is called the generalized Laplace homotopy perturbation method. The idea of the generalized Laplace homotopy perturbation method comes from combining the technique of both the homotopy perturbation method and generalized Laplace transform. The result shows that the analytic solution of the time-fractional Black–Scholes equations in the sense of Caputo type Katugampola fractional derivative can be carried out in the form of the two-parameters Mittag–Leffler function. Since the Katugampola fractional derivative in Caputo type generalizes both Caputo and Caputo–Hadamard derivatives, in the case of the time-fractional Black–Scholes equations in the Caputo derivative sense, the analytic solution is also obtained. Unfortunately, in the case of the time-fractional Black–Scholes equations in the Caputo-Hadamard derivative sense, there is no solution. Moreover, we can observe that the orders of the Katugampola fractional derivative in Caputo type have an effect on the price of the European option. Moreover, the European option price is overpriced when alpha = 0.25.

Reference: Sirunya Thanompolkrang, Wannika Sawangtong and Panumart Sawangtong