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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.

There are many mathematical models [

Some studies in the context of fractional calculus have investigated the use of the Black and Scholes equation for European options. Much of this research has involved the use of the Caputo–Liouville and the Riemann–Liouville derivatives. In [

Modeling the physical phenomena using the fractional-order derivative has many advantages. First of all, it permits us to study the differential models with arbitrary noninteger-order derivatives. Second, it permits us to take into account the memory effect; that is, the next behavior of the dynamic is explained by the past behavior of the dynamic. As we will observe in this paper, the fractional-order derivative can play a regulator role in differential dynamics. In this paper, we investigate and introduce the fractional Black–Scholes equation described by a new fractional-order derivative: the Mittag-Leffler fractional derivative [

This manuscript is structured as follows: In Section

In this section, we recall the definitions of fractional derivatives with nonsingular kernels. Fractional calculus began with the Riemann–Liouville fractional derivative and the Caputo–Liouville fractional derivative; for definitions, refer to [

The Caputo–Fabrizio fractional derivative of the function

The Caputo–Fabrizio integral for a given function

The exponential form is a particular case of the Mittag-Leffler function. 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 [

The Atangana–Baleanu–Caputo derivative for a function

The Atangana–Baleanu integral for a given function

We were motivated to consider these two fractional derivatives because of their successful application in modeling real-life phenomena. In this paper, 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.

In this section, we introduce the Black–Scholes equation in the context of the Atangana–Baleanu fractional derivative. We begin by recalling the classical model proposed by Black and Scholes [

Note that equation (

Taking into account the interest rate

Combining equations (

We made the following assumptions related to equation (

The use of equation (

From which it follows the classical Euler equation given by

In the next section, we try to prove our new model is well defined, admit a unique solution, and use numerical and analytical methods to approach it. For the readers and more understanding of the paper, we summarize the description of the parameters used in this paper in Table

Parameters of the Black–Scholes equation.

Parameters | Description of the parameters. |
---|---|

The value of an option and the expiration time, respectively. | |

The volatility of the underlining stock. | |

The balance between the free interest rate and the volatility of the stock. | |

The risk-less interest rate. | |

The strike price of the underlying stock and the asset price, respectively. |

In this section, we describe the procedure of discretization used in this paper. The method is called the Adams–Bashforth numerical scheme and was introduced in fractional calculus by Atangana in [

The solution of the fractional differential equation described by

The approximation of the function

Using equation (

The above discretization proposed by Atangana in [

We will find a threshold for the function

Using the assumption

Applying the norm to both sides of equation (

From equation (

We can conclude the Adams–Bashforth numerical scheme is unconditionally stable. In the next section, we apply the Adams–Bashforth numerical scheme to the numerical approximation of the fractional Black–Scholes equation described by the fractional derivative with Mittag-Leffler.

In this section, we describe the Adams–Bashforth numerical scheme for the fractional Black–Scholes equation represented by the Atangana–Baleanu fractional derivative. Let us begin the numerical approximation of the fractional Black–Scholes equation. Let

The next step consists of finding the discretization of the functions

Using the central difference approximation for the second-order derivative with respect to the space coordinate and the numerical approximation for the space derivative, we obtain the following discretization at the points

Using the central difference approximation again for the second-order derivative with respect to the space coordinate and the numerical approximation for the space derivative, we obtain the following discretization at the points

Numerical discretization using the Adams–Bashforth method for the Black–Scholes equation is obtained by combining equations (

For the computation of our numerical schemes, we make some changes to the variables, such that the terms depending on

Finally, the numerical scheme using the Adams–Bashforth method for the fractional Black–Scholes equation is given by

To complete the numerical discretization given in equation (

In this section, we use the fractional integrator to propose the analytical solution of the fractional Black–Scholes equation described by the Atangana–Baleanu fractional derivative. The method is described in the following theorem.

The solution of the fractional differential equation described by

Consider the fractional Black–Scholes equation (

Under assumption

We adopt the procedure described in Theorem

We recover the approximate solution of the classical Black–Scholes equation when

Due to space limitation, all term are not written. Finally, we describe the analytical solution for the classical Black–Scholes equation (

The method adopted here gives a solution which is in good agreement with the classical solution of the Black–Scholes equation (

From which, we rewrite equation (

Equation (

Now, we replace

Multiplying the function

We can see that the solution represented in equation (

Note that, in the money “call” is when the price of the underlying asset is higher than the strike price. It is in the interest of the holder of the option to exercise it. He/she has made good anticipations (

In this section, we analyse the volatility of the fractional Black–Scholes equation using an analytical solution. In the market, we can buy the call and put with different strike prices and maturity. The volatility analyses the liquidity of the cost of the call and the put in the market. Many types of volatility can be generated by the Black–Scholes equation. In general, volatility measures risk in the financial markets. Volatility is used to control both upward and downward movements. It is calculated from log returns. The delta is part of the Greek letters of options. It is the derivative of the price of the call or put option in relation to the price of the underlying asset. It is used for trading, arbitrage, or hedging operations on options. It is an important indicator of market risk management. The Basel Committee on International Banking Regulation recommends that banks use delta for exposures to options. In reality delta is a sensitivity factor.

Here, we recall the formula of the volatility delta of the fractional Black–Scholes equation. The delta measures the sensitivity of the option price to a change in the price of the underlying security. Under variable changes, this is expressed as follows:

Using the approximate solution (

Given the conditions in equation (

Earlier, we stated that the volatility studied in this paper does not depend on the order of the fractional derivative or the balance between the free interest rate and the volatility of the stocks

Volatility surface of the Black–Scholes equation.

The volatility surface gives the sensitivity of the option price when the asset price

Volatility when

In this section, we illustrate our results graphically. Specifically, we depict the behavior of the solutions of the fractional Black–Scholes equation obtained with numerical schemes and the recursive method. We begin with the approximate solution generated by the recursive method previously described in Section

Behavior of the Black–Scholes equation with

Figure

Behavior of the Black–Scholes equation with

Figure

Analytical solution vs numerical approximation with

The main question now is how to find the optimal order of

To support the numerical discretization, we describe in Table

Option values when

0.1541 | 0.01 | 0.3240 | 0.3242 | 1.1467 |

0.4677 | 0.03 | 0.8455 | 0.8456 | 1.5381 |

0.2401 | 0.05 | 0.5745 | 0.5747 | 1.1762 |

0.1697 | 0.02 | 0.4924 | 0.4925 | 1.1457 |

We mainly observe that the results in Table

In this paper, we have discussed the numerical scheme and the analytical solution for the fractional Black–Scholes equation described by the Atangana–Baleanu derivative. As observed, the analytical solution of the fractional Black–Scholes equation with the Atangana–Baleanu fractional derivative is not trivial. We have used the Adams–Bashforth numerical scheme to approach the solution, as it is useful and straightforward for proposing the approximate solutions of the fractional Black–Scholes equation. We have also considered the liquidity of the cost of the call and the put in the market, namely, the volatility. The graphical representations have proved the good agreements between the analytical solution and the numerical solutions for the fractional Black–Scholes equation.

No data were used to support this study.

The authors declare that they have no conflicts of interest.