It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The analytical solution of the proposed model is investigated by the Laplace transform homotopy perturbation method.
Abstract:The Black Scholes model is a well-known and useful mathematical model in financial markets. In this paper, the two-dimensional Black Scholes equation with European call option is studied. The explicit solution of this problem is carried out in the form of a Mellin-Ross function by using Laplace transform homotopy perturbation method. The solution example demonstrates that the proposed scheme is effective.
Abstract:The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We prove the existence and uniqueness of a solution for the FAFDEs. Furthermore, the stability analysis of the model is investigated and the numerical simulation is accordingly performed to support the proposed model.
This paper aims to study the quenching problem in a fractional heat equation with the Riemann-Liouville fractional derivative. The existence and uniqueness of a solution for the problem are obtained by transforming the problem to an equivalent integral equation. The condition for the quenching occurrence in a finite time is given. Furthermore, the quenching point set is shown.
In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.
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. There is an important question in finance: “Can the fractional differential equation be applied in the financial market?”. The answer is “Yes”. Due to the self-similar property of the fractional derivative, it can reply to the long-range dependence better than the integer-order derivative. Thus, these advantages are beneficial to manage the fractal structure in the financial market. In this article, the classical Black–Scholes equation with two assets for the European call option is modified by replacing the order of ordinary derivative with the fractional derivative order in the Caputo type Katugampola fractional derivative sense. The analytic solution of time-fractional Black–Scholes European call option pricing equation with two assets is derived by using the generalized Laplace homotopy perturbation method. The used method is the combination of the homotopy perturbation method and generalized Laplace transform. The analytic solution of the time-fractional Black–Scholes equation is carried out in the form of a Mittag–Leffler function. Finally, the effects of the fractional-order in the Caputo type Katugampola fractional derivative to change of a European call option price are shown.
This article deals with the novel method for finding solutions for the initial-boundary value problems (IBVPs), which is called the Sawangtong's Green function homotopy perturbation method, shortly called SGHPM. The SGHPM is a method which combines the homotopy perturbation method with Green's function method. The convergence analysis for the SGHPM is shown. Furthermore, some examples are presented to illustrate the validity of the proposed method and to ensure that SGHPM is a technique which is powerful and efficient for finding approximate analytic solutions of IBVPs.
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