Abstract
The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.
Highlights
Fractional Operators without Singular KernelsWe recall the definitions of fractional derivatives with nonsingular kernels
We use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. e graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation
Fractional derivatives occupy an important place in fractional calculus, so this paper investigates the use of fractional derivatives for modeling financial and economic models. e Black–Scholes model is an important tool used in finance to predict the value of an option [2]. ere are many styles of options: European options [3, 4], American options [5, 6], and Asian options. e pricing of options is a subject that has been very intensely debated in economics
Summary
We recall the definitions of fractional derivatives with nonsingular kernels. E Caputo–Fabrizio fractional derivative of the function u: R × [0, +∞[ ⟶ R of order α is defined in the form. E Caputo–Fabrizio integral for a given function u: R × [0, +∞[ ⟶ R, of order α ∈ Motivated by the fact that the Cauchy problem with the Caputo–Fabrizio derivative generates a solution with an exponential function, Atangana and Baleanu proposed another fractional derivative with a Mittag-Leffler kernel in 2016 [21]. E Atangana–Baleanu–Caputo derivative for a function u: R × [0, +∞[ ⟶ R, of order α ∈ E Atangana–Baleanu integral for a given function u: R × [0, +∞[ ⟶ R, of order α ∈ We apply the Atangana–Baleanu fractional derivative in modeling the value of options and investigate the fractional Black–Scholes equation described by the Atangana–Baleanu–Caputo fractional derivative
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