Abstract
In this article, the analytical solutions of economically important model named as the Ivancevic option pricing model (IOPM) along new definition of derivative have been explored. For this purpose, function, extended sinh-Gordon equation expansion (EShGEE) and extended -expansion methods have been utilized. The resulting solutions are dark, bright, dark-bright, periodic, singular, and other kinds of solutions. These solutions are obtained and also verified by a Mathematica tool. Some of the gained results are explained by 2-D, 3-D, and contour plots.
1. Introduction
Many mathematical models have been developed in many areas of sciences in the form of nonlinear partial differential equations (NLPDEs). Numerous techniques are made to gain exact solutions of NLPDEs such as generalized exponential rational function scheme (GERFS) [1–4], -expansion and Adomian decomposition schemes [5], new generalized expansion method [6], simplest equation and Kudryashov’s new function techniques [7], modified simple equation scheme [8], modified Kudryashov simple equation technique [9], first integral technique [10], Bäcklund transformation scheme [11], extended jacobi elliptic function expansion technique [12], and extended -expansion and improved -expansion schemes [13].
In modern century, one of the most studied fields from all over the world is the economy or finance. Therefore, such problems were studied to be explained and investigated by using scientific norms. Thus, such works introduce more intellectual ways for the user. Therefore, to observe financial market is highly important. Deeper properties of the modeling of a global financial market produce global informative systems. Especially, these dynamical systems can be used for deep investigation of the productions. The first step is to treatment its mathematical models being either complex-valued or real values with wave function. Therefore, many models were developed by experts in extracting their wave distributions in today and future direction. In our study, we use three methods function, extended sinh-Gordon equation expansion (EShGEE), and extended -expansion methods. These methods have various applications. Likely, some new kind of analytical results of perturbed Gerdjikov–Ivanov model (pGIM) have been achieved by using function and extended function expansion methods in [14]. By applying function and hyperbolic function methods, various types of wave solutions of two nonlinear Schrödinger equations are gained in [15]. New trigonometry and hyperbolic function type soliton solutions of (2 + 1)-dimensional hyperbolic and cubic-quintic nonlinear Schrödinger equations are achieved by applying extended sinh-Gordon equation expansion technique in [16]. Bright, dark, and bright-dark soliton solutions of generalized nonlinear Schrödinger equation have been determined by utilizing extended sinh-Gordon equation expansion approach in [17]. Some exact solitons of (2 + 1)-dimensional that improved the Eckhaus equation are calculated by using extended -expansion technique in [18]. Different types of exact solitons of time-fractional parabolic equations are obtained by using extended -expansion scheme in [19].
Our considering model is one of the important and interesting economical models, namely, the Ivancevic option pricing model (IOPM). It can be possible to derive the Ivancevic option pricing model by using the Brownian movement like the Black–Scholes option pricing model. The Ivancevic option pricing model is an adaptive-wave model which is a nonlinear wave alternative for the standard Black–Scholes option-pricing model, representing controlled Brownian behavior of financial markets, which is formally defined by adaptive nonlinear Schrodinger (NLS) equations, defining the option-pricing wave function in terms of the stock price and time. In the literature, few techniques have been used on this model to get different exact solutions. For example, new solutions have been achieved in this model by applying the fractionally reduced differential transform technique in [20]. Dark, bright, dark-bright, complex, travelling, periodic, trigonometric, and hyperbolic function solutions have been achieved by applying rational sine-Gordon expansion scheme and modified exponential method in [21]. Rogue wave and dark wave solitons of the Ivancevic option pricing equation have been obtained by using the trial function method in [22].
The fundamental purpose of the work is to explore analytical solutions of the truncated M-fractional Ivancevic option pricing model based on function, extended sinh-Gordon equation expansion, and extended -expansion methods.
The paper is structured as follows: The brief introduction of model has been given in Section 2, together with other useful properties and characterizations. Section 3 contains the description of methodologies. The mathematical analysis of model and its analytical solutions have been provided in Section 4. Section 5 some solutions have been represented through different types of graphics. Finally, Section 6 contains some discussion about the graphs and conclusion of our research.
2. Model Description
Scholes and Robert Merton published their now-well-known option pricing formula which would have an important effect on the development of quantitative finance. In their model, typically known as Black–Scholes, the value of an option depends on the future volatility of a stock rather than on its expected return. The Ivancevic option pricing model is an adaptive-wave model which is a nonlinear wave alternative for the standard Black–Scholes option-pricing model, representing controlled Brownian behavior of financial markets, which is formally defined by adaptive nonlinear Schrodinger (NLS) equations, defining the option-pricing wave.
Let’s assume the M-fractional Ivancevic option pricing model (IOPM) [22] given as follows:where .
This model was first developed by Ivancevic [23] to fulfill both behavioral and efficient markets. Here, describes the option price wave profile. While t is time variable and s is asset price of the model. Parameter represents the volatility which shows either stochastic process itself or only a constant. Where is called Landau coefficient which describes adaptive market potential. In nonadaptive simplest case, and r become equal which shows the interest rate while in adaptive case, may be connected to market temperature and it depends on the set of tractable parameters . In third term, shows the probability density function which denotes the potential field.
3. Description of Methodologies
3.1. Summary of Function Scheme
Here, we will give complete concept of this scheme.
Assuming the nonlinear partial differential equation (PDE),
Equation (2) transformed into nonlinear partial differential equation as follows:
By using following transformations,
Considering root of equation (3) is shown in [24–27].where and are undetermined. Positive integral value of is calculated by utilizing homogenous balance technique in equation (3). Putting equation (5) into equation (3) gives
Taking in equation (6) equal to 0, a set of algebraic equations is gained which is given as
By using the got roots, we attain analytical results of equation (2).
3.2. Detail of Extended Sinh-Gordon Equation Expansion Method (EShGEEM)
Here, we will describe main steps of this technique. Step 1: Let a nonlinear partial differential equation be given as where denotes the wave function. Assuming the travelling wave transformation, Inserting equation (9) into equation (8), we attain the nonlinear ODE given as Step 2: Assuming the results of equation (9) in the series form, Here, , , and are unknowns. Consider a function p of that satisfies the following equation: Natural number m can be attained with the use of homogenous balance approach. Equation (12) is gained from sinh-Gordon equation as shown as By what being present in [28], we get the results of equation (13) given as follows: In addition, where . Step 3: Using equation (11) with equation (13) into equation (10), we get the algebraic equations involving . We take the every coefficient of equal to 0, to attain system of algebraic equations having and . Step 4: By solving the obtained system of algebraic equations, one may obtain value of and . Step 5: By achieved solutions, equations (14) and (15), we get the wave solitons of equation (10) shown as
3.3. Explanation of Extended Expansion Method
Here, we will represent some main steps of method given in [13]. Step 1: Considering the nonlinear PDE, where denotes the wave profile. Considering the travelling wave transformations: Step 2: Using equation (19) along equation (18), we attain the nonlinear ODE as follows: Step 3: Assuming the results of equation (20) in the series form given as In equation (21), and are undetermined and . Applying homogenous balance technique into equation (20), natural number m can be obtained. Function satisfies the Riccati differential equation. where a, b, c, and d are constants. Step 4: Considering equation (22) has roots in the form: When and , then When and , then When and , then When and , then When and , then where a, b, c, d, , and are constants. Step 5: Putting equation (21) with equation (22) into equation (20) and collecting coefficients of each power of . Taking every coefficient equal to 0, we attain a set of algebraic equations involving , , , and other parameters. Step 6: Finding the gain system of equations with the use of the tool. Step 7: Putting the attained solutions into the equation (21) and we gain analytical solutions of equation (18).
4. Mathematical Treatment of Model
Suppose the travelling wave transform given aswhere shows the amplitude of wave function while and represent the time velocity. Parameters and are obtaining from asset price of the product.
Using equation (28) into equation (1), we gain real part and imaginary part given as
Real part:
Imaginary part:
From equation (30), we get the velocity of wave function given as follows:
Utilizing the homogenous balance method into equation (29), we achieve
Now, we will gain the exact solutions of equation (29) by using three abovementioned methods.
4.1. Analytical Solutions via Function Technique
Equation (5) changes into the following for :
Inserting equation (32) into equation (29), a system of equations is achieved. By solving the system, we obtain different solution sets given as follows: Set 1: From equations (28), (32) and (33), we get Set 2: From equations (28), (32) and (35), we get Where .
4.2. Exact Solutions through EShGEEM
For , equations (9), (16) and (17) and becomeHere, are undetermined. Utilizing equation (39) into equation (29), we attain algebraic equations containing and other parameters. By using the Mathematica tool, we get different solution sets given as Set 1: From equations (28), (37) and (40), we get From equations (28), (38), and (40), we get Set 2: From equations (28), (37), and (43), we get From equations (28), (38), and (43), we get Set 3: From equations (28), (37), and (46) we get From equations (28), (38), and (46), we get Set 4: From equations (28), (37), and (49), we get From equations (28), (38), and (49), we get Set 5: From equations (28), (37), and (52), we get From equations (28), (38), and (52), we get Set 6: From equations (28), (37), and (55), we get From equations (28), (38), and (55), we get Set 7: By using equations (28), (37), and (58), we obtain From equations (28), (38) and (58), we get Set 8: From equations (28), (37) and (61), we get From equations (28), (38), and (61), we get where .
4.3. Analytical Solutions via Extended -Expansion Technique
Equation (21) changes into following form for :where and are undetermined.
Inserting equation (64) along equation (22) into equation (29) and manipulating the set having and other parameters, we gain different sets of solutions given as Set 1: From equations (23), (28), (64), and (65), we get From equations (24), (28), (64), and (65), we get From equations (26), (28), (64), and (65), we get From equations (27), (28), (64), and (65), we get Set 2: From equations (23), (28), (64), and (70), we get From equations (24), (28), (64), and (70), we get From equations (26), (28), (64), and (70), we get From equations (27), (28), (64), and (70), we get Set 3: From equations (23), (28), (64), and (75), we get From equations (24), (28), (64), and (75), we get From equations (26), (28), (64), and (75), we get From equations (27), (28), (64), and (75), we get Set 4: From equations (23), (28), (64), and (80), we get From equations (24), (28), (64), and (80), we get From equations (26), (28), (64), and (80), we get From equations (27), (28), (64), and (80), we get Here, for all abovementioned solutions.
5. Illustrations with Graphics
In this portion, we will represent some 2-D, 3-D, and contour structures that help us to classify the type of results. Figures 1–3 show some of the analytical solutions. In Figure 1, we apply our technique to represent the plot of (34) for . Furthermore, Figure 2 denotes the plot of (41) . Finally, the plot of (66) for is presented in Figure 3. We see that the wave retains its shape over time, moves to the right, and breaks by changing the value of .
Through our analysis of the forms presented in the previous section, we can reach important results as follows: First, in Figure 1, we apply the function technique to represent the plot of (34) at . Further, Figure 2 denotes the plot of (41) at using EShGEE technique. Finally, the plot of (66) for presented in Figure 3 using the extended -expansion technique.
6. Conclusion
In this article, we obtain modernistic analytical solutions to the Ivancevic option pricing model along M-fractional derivative by utilizing function, extended sinh-Gordon equation expansion, and extended -expansion methods. The achieved results are also verified and demonstrated with different plots by Mathematica tool. The obtained results are also explained graphically by 2-dimensional, 3-dimensional, and contour plots. Finally, it is suggested that to deal with the other fractional nonlinear PDEs, the function, extended sinh-Gordon equation expansion, and extended -expansion methods are very helpful, reliable, and straight forward. The results achieved in this paper may be useful for the progress in the supplementary analyzing of this model.
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors have contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Acknowledgments
The work in this study was supported, in part, by the Open Access Program from the American University of Sharjah.