Soliton wave profiles and dynamical analysis of fractional Ivancevic option pricing model

Abstract

This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions.

Introduction

Since nonlinear partial differential equations (NLPDEs) replicate a variety of real-world physical phenomena, they are extremely important in both science and engineering. The most fundamental models required to study nonlinear processes are these equations. Nonlinear processes are common in many disciplines, including biology, population ecology, fluid mechanics, nonlinear mechanics, aerospace engineering, marine science, and climatology. It is impossible to overestimate the importance of solving the mathematical models underlying these physical processes. On the other hand, the consensus is that there is no systematic way to solve nonlinear partial differential equations in closed form. Even with this difficulty, scientists have produced a number of effective techniques for deriving workable and distinct solutions to NLPDEs. The Black-Scholes (BS) model is a popular mathematical model for option pricing, as seen in¹. Traders can get a better understanding of the risks and rewards involved in trading options by knowing the mathematics underlying the BS model. Another mathematical model for option pricing is the binomial model². This model ascertains the value of an option through a sequence of computations that take into account the price, volatility, and time to expiration of the underlying asset.

The most significant and urgent research question of the present moment is how to monitor and control market price fluctuations in the financial system and markets. To enable the production of more advanced goods, experts and scientists can develop a wide range of technical and scientific instruments. The generalized exponential rational function and modified auxiliary equations techniques utilized and developed diverse types of analytical solutions^3,4. Raheel et al.⁵ constructed soliton solution via three different schemes and demonstrated their propagation graphically. The modulational instability and other dynamical aspects have been investigated in the Ivancevic option pricing model^6,7. Stochastic differential equations are widely used in finance, and their stability is investigated⁸. The optimal consumption and portfolio policies were identified by the creators of a continuous-time financial model^9,10. A global information system is produced through the modelling of an international financial market and its underlying components^11–13. The writers of created the intricate schematics for the resilient economic model¹⁴. The concept of risk factors has been incorporated into the control strategy, and an algorithm has been supplied to determine which economic zone should be targeted. A national economic model consisting of linked nonlinear stochastic multi-dimensional operator equations was created and assessed by Adomian¹⁵ and many more^16,17.

Furthermore, NPDEs seem to be in charge of a wide range of complex biological, chemical, and physical processes^18–20. For NPDEs, closed-form solutions are required in order to understand complex processes. Therefore, it is imperative and critical to find some solutions, particularly wave solutions. Thus, scientists are encouraged to provide new insights and improve tried-and-true strategies. The multiple exp-function method²¹, the Kudryashov approach²², the bifurcation and chaos analysis techniques²³ and so on are just a few examples of important and effective strategies. A plethora of NPDE types and technical techniques exist for solving and analyzing these equations, as the works of^24–28 illustrate.

Employing a variety of mathematical tools and techniques, the pricing model’s mathematical solution, its stability under behaviour-dependent physical parameters, and the fractional Ivancevic option pricing model’s sensitivity analysis are explained. This nonlinear model (which supersedes the more conventional Black-Scholes option-pricing model) is best defined mathematically by adaptive nonlinear Schrödinger (NLS) equations that characterize the option pricing wave function in terms of time and stock price. The conformable fractional derivative Ivancevic pricing model and the BS model are two of the most widely used financial models for option pricing. Although the theoretical price of an option can be determined using both models, their methods and presumptions are different. Early in the new millennium, the conformable fractional derivative Ivancevic pricing model was created^29–31. It is used to price options with non-constant volatility and is based on fractional calculus. Because it accounts for the underlying asset’s fluctuating volatility, this model is more accurate than the BS model when pricing options with non-constant volatility. The more conventional BS model is predicated on the idea that the underlying asset’s volatility is constant. While both the BS model and the conformable fractional derivative Ivancevic pricing model are helpful instruments for pricing options, their optimal applications differ.

It is one of the most significant and intriguing economic models that is being studied. Several precise solutions have been found in the literature by using this model in various ways. For instance, new solutions to this model have been obtained in³² by using the fractional reduced differential transform technique. In³³, solutions for complex, travelling, periodic, trigonometric, hyperbolic, dark, bright, and dark-bright functions were obtained using the modified exponential method and the rational sine-Gordon expansion approach. Rogue wave and dark wave solitons for the Ivancevic option pricing equation are constructed using the trial function technique, as described in³⁴. To the best of our knowledge, bifurcation and chaos theories have not been taken into consideration for the qualitative analysis of the truncated conformable fractional derivative Ivancevic option pricing model. We also use a modified version of the Kudryashov approach³⁵ to construct meaningful wave solutions for the model under study. The results are applicable to two different fields because of the two unique features of the models under study. The computed solutions provide a dependable and accurate method for valuing options for financial practitioners. The dark soliton solutions, obtained by the modified Kudryashov method, have been presented in 3D, 2D, and density plots; people with a background in physics will find them appealing. This opens up new avenues for investigation and promotes more study in the area of mathematical finance.

The paper is divided into several sections, including Section “Description of modified Kudryashov analytical method”, which discusses the truncated conformable fractional derivative and its properties; Section “Investigation of the mathematical Ivancevic option pricing system”, which outlines a description of the model and exact solutions to the model; Section “Dynamical study of the Ivancevic option pricing model” contains the phase portraits, chaos and sensitivity analysis using dynamical systems. Finally, a conclusion is presented.

Conformal fractional derivative

Definition

Assume that a $h:{\mathbb{R}}^{+}\to \mathbb{R}$ is differential function. The conformal derivative with fractional order $0<\alpha \le 1$ as³⁶:

$$\begin{array}{c}\hfill D_t^{\alpha }(h(t))=\underset{\epsilon \to 0}{lim}\frac{h(t+\epsilon t^{1-\alpha })-h(t)}{\epsilon },t>0.\end{array}$$ 1

Description of modified Kudryashov analytical method

We will go over how to solve travelling wave problems in nonlinear partial differential equations by applying the modified Kudryashov analytical method in this section²².

Suppose we have a general non-linear partial differential equation:

$$\begin{array}{c}\hfill \mathrm{\Lambda }(F,F_s,F_t,F_s,\dots )=0.\end{array}$$ 2

Where F denote a function of option price over time. The following sections outline the subsequent steps of the technique.

Step 1. Utilizing wave transformation.

$$\begin{array}{c}\hfill F(x,t)=Q(\zeta ),\zeta =k(x+\tau t).\end{array}$$ 3

Introduce a novel independent variable, denoted as $\zeta$, with real parameters k and $\tau$ for Eq. (2). Upon substituting Eq. (3) into Eq. (2), we obtain the resulting ordinary differential equation.

$$\begin{array}{c}\hfill \mathrm{\Pi }(Q,Q_{}^{_{}^{\prime }},Q_{}^{_{}^{\prime \prime }},Q_{}^{_{}^{\prime \prime \prime }},\dots )=0.\end{array}$$ 4

The powers of Q denote differentiation with respect to $\zeta$.

Step 2. Suppose the solution for Eq. (4) takes the following form:

$$\begin{array}{c}\hfill Q(\zeta )=A_0+\sum _{i=1}^N{A}_i{W}^i(\zeta ),\end{array}$$ 5

where, the coefficients $A_i$ are real constants,and N show the balanced number .Further, the function $W(\zeta )$ must satisfy the following equation:

$$\begin{array}{c}\hfill \frac{\mathit{dW}}{d\zeta }=(W^2(\zeta )-W(\zeta ))ln(b),\end{array}$$ 6

Step 3. Determine $N$ in Eq. (5) by balancing the highest-order derivative with the highest-degree non-linear term. Then, use values from Eqs. (5) and (6) in Eq. (2) to find the coefficients of $W(\zeta )$, simplify the equations, and solve them, typically using software like Mathematica. Obtain solutions for Eq. (6) with:

$$\begin{array}{c}\hfill W(\zeta )=\frac{1}{1+C{b}^{\zeta }}.\end{array}$$ 7

Investigation of the mathematical Ivancevic option pricing system

Ivancevic presented an excellent nonlinear option pricing model based on the Elliott wave market theory³⁷, the modern adaptive markets hypothesis derived from Lo³⁸, and the quantum neural computation technique³⁹. This method was later dubbed the Ivancevic option pricing model by him. The following is the conformable fractional derivative Ivancevic option pricing model^29–31:

$$\begin{array}{c}\hfill \iota D_{M,t}^{\alpha ,\nu }F+\frac{\lambda }{2}{D}_{M,2s}^{2\alpha ,\nu }F+\mu F{|F|}^2=0,\iota =\sqrt{-1},\end{array}$$ 8

• The adaptive nonlinear Schrödinger equation models option pricing as $F=F(s,t)$ and describes financial market behavior^29–31.

• Referred to as the Ivancevic option pricing model, it uses $t$ for time, $s$ for asset price, and $\lambda$ for volatility.

• The Landau coefficient $\mu =\mu (r,l)$ represents market potential, with interest rate and market temperature relating to it.

• The potential field is represented by ${|F|}^2$.

The computation of wave profiles for pricing model

Consider the travelling wave transform is given as follows:

$$\begin{array}{c}\hfill F(s,t)=Q(\zeta )\times exp\left(\iota ,\left(\frac{k_1}{\alpha },s^{\alpha },+,\frac{k_2}{\alpha },t^{\alpha }\right)\right),\zeta =\left(\frac{k_3}{\alpha },s^{\alpha },+,\frac{k_4}{\alpha },t^{\alpha }\right),\end{array}$$ 9

where $Q(\zeta )$ represents the amplitude of the financial wave function, while $k_2$ and $k_4$ represent the time velocity. The product’s asset price yields the parameters $k_1$ and $k_3$. Plugging in Eq. (9) yields real and imaginary components, respectively, when compared to Eq. (8),

$$\begin{array}{c}\hfill 2\mu Q^3+\lambda {k_3}^2{Q}^{″}-(\lambda {k_1}^2+2k_2)Q=0,\end{array}$$ 10

$$\begin{array}{c}\hfill (\lambda k_1{k}_3+k_4)Q^{\prime }=0.\end{array}$$ 11

One can achieve the velocity of wave function from Eq. (11) and also can develop following relation:

$$\begin{array}{c}\hfill k_4=-\lambda k_1{k}_3.\end{array}$$ 12

Now, we will gain the exact analytical soliton solution of Eq. (10) by using the modified Kudryashov method. Balancing the terms $Q^{″}$ and $Q^3$ of Eq. (10), we get $N=1$. Therefore, the solution Eq. (5) takes the linear form:

$$\begin{array}{c}\hfill Q(\zeta )=A_0+A_{1}W(\zeta ).\end{array}$$ 13

The Eq. (13) is substituted into Eq. (10) and get a system of algebraic equations by equating the distinct powers of $W(\zeta )$.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill W{(\zeta )}^0=& -A_0\lambda k_1^2-2A_0{k}_2+2A_0^3\mu =0,\hfill \\ \hfill W{(\zeta )}^1=& A_1\lambda k_3{log}^2(b)-A_1\lambda k_1^2-2A_1{k}_2+6A_0^2{A}_1\mu =0,\hfill \\ \hfill W{(\zeta )}^2=& 6A_0{A}_1^2\mu -3A_1\lambda k_3{log}^2(b)=0,\hfill \\ \hfill W{(\zeta )}^3=& 2A_1\lambda k_3{log}^2(b)+2A_1^3\mu =0.\hfill \\ \hfill \end{array}\end{array}$$ 14

Now, the system mentioned above (14) is solved by utilizing the symbolic computation software Mathematica and getting the following two nontrivial sets of solutions:

Family-I

$$\begin{array}{c}\hfill [A_0=\pm \frac{\sqrt{\lambda k_1^2+2k_2}}{\sqrt{2}\sqrt{\mu }},A_1=\mp \frac{\sqrt{2}\sqrt{\lambda k_1^2+2k_2}}{\sqrt{\mu }},k_3=-\frac{2\left(\lambda ,k_1^2,+,2,k_2\right)}{\lambda {log}^2(b)}],\end{array}$$ 15

the exact solution of Eq. (9) yield the following form:

$$\begin{array}{c}\hfill F(s,t)=\pm \frac{\sqrt{\lambda k_1^2+2k_2}}{\sqrt{2}\sqrt{\mu }}\mp \frac{\sqrt{2}\sqrt{\lambda k_1^2+2k_2}}{\sqrt{\mu }}\times \frac{exp(\iota (k_1\frac{\mathrm{\Gamma }(\beta +1)}{\alpha }{s}^{\alpha }+k_2\frac{\mathrm{\Gamma }(\beta +1)}{\alpha }{t}^{\alpha }))}{1+C{b}^{\left(k_3,\frac{\mathrm{\Gamma }(\beta +1)}{\alpha },s^{\alpha },+,k_4,\frac{\mathrm{\Gamma }(\beta +1)}{\alpha },t^{\alpha }\right)}}.\end{array}$$ 16

The above-depicted figures present the travelling wave solutions and the impact of fractional order on a solution. Figures 1 and 2 present the influence of fractional order parameters on the prorogation and demonstrate how fractional order controls the behaviour of the travelling wave solution. Figure 3 presents the 2D impact of fractional parameters and a graphical comparison with classical order.

Fig. 1.

Three, Contour, and two-dimensional propagation for the solution F(s, t).

Fig. 2.

Three, Contour, and three-dimensional propagation for the solution F(s, t).

Fig. 3.

Two-dimensional propagation of influence of fractional order for the solution F(s, t).

Dynamical study of the Ivancevic option pricing model

This section will present phase portraits via bifurcation analysis, periodic, quasi-periodic and chaotic patterns by chaos analysis and sensitive investigation of the considered model.

Bifurcation analysis

In this section, the dynamical structure approach will be used to examine the governing model. Here, we will apply the phase portraits analysis found in Eq. (10), and each phase portrait will be shown graphically^40,41. Next, the planar dynamical system of the following form is obtained by applying the Galilean transformation to Eq. (10):

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\frac{\mathit{dQ}}{d\zeta }=P,\hfill & \\ \frac{\mathit{dP}}{d\zeta }=Y_1^{\star }Q-Y_2^{\star }{Q}^3,\hfill \end{array}\right)\end{array}$$ 17

where $Y_1^{\star }=\frac{\lambda {k_1}^2+2k_2}{\lambda {k_3}^2}$ and $Y_2^{\star }=\frac{2\mu }{\lambda {k_3}^2}$. There are the following equilibrium points in the system (17):

$$\begin{array}{c}\hfill E_1^{\star }=(0,0),E_2^{\star }=\left(\sqrt{\frac{Y_1^{\star }}{Y_2^{\star }}},,,0\right),E_3^{\star }=\left(-,\sqrt{\frac{Y_1^{\star }}{Y_2^{\star }}},,,0\right).\end{array}$$ 18

Moreover, the Jacobian of (17) is:

$$\begin{array}{c}\hfill J(Q,P)=\left|\begin{array}{cc}0& 1\\ Y_1^{\star }-3Y_2^{\star }{Q}^2& 0\end{array}\right|=3Y_2^{\star }{Q}^2-Y_1^{\star }.\end{array}$$ 19

Since $J(Q,P)<0$, (Q, P) is the saddle point; additionally, it would be a centre for $J(Q,P)>0$ and a cusp for $J(Q,P)=0$. You can get the following outcomes by varying the ranges of the parameters.

Case 1: $Y_1^{\star }>0$ and $Y_2^{\star }>0$

Three equilibrium points, as shown in Fig. 4a, have been obtained from the system (17) by applying $Y_1^{\star }=1$ and $Y_2^{\star }=1$. These are $E_1^{\star }=(0,0),E_2^{\star }=(1,0),E_3^{\star }=(-1,0)$. It is evident that $E_1^{\star }$ is a saddle point and $E_2^{\star }$ and $E_3^{\star }$ are center points.

Fig. 4.

Phase portraits for dynamical system (17).

Case 2: $Y_1^{\star }<0$ and $Y_2^{\star }<0$

The equilibrium points $E_1=(0,0),E_2=(1,0),E_3=(-1,0)$ have been retrieved from system (17) displayed in Fig. 4b by employing $Y_1^{\star }=-1$ and $Y_2^{\star }=-1$. The phase portrait indicates that while $E_1$ is a centre point, $E_2$ and $E_3$ display saddle points.

Case 3: $Y_1^{\star }<0$ and $Y_2^{\star }>0$

As shown in Fig. 4c, there is only one equilibrium point $E_1=(0,0)$ when $Y_1^{\star }=-1$ and $Y_2^{\star }=1$ are used. We have determined that $E_1$ is a centre point by examining the Jacobian.

Case 4: $Y_1^{\star }>0$ and $Y_2^{\star }<0$

As seen in Fig. 4d, one equilibrium point exists $E_1^{\star }=(0,0)$ by using $Y_1^{\star }=1$ and $Y_2^{\star }=-1$. We can determine that $E_1^{\star }$ is a saddle point by examining the Jacobian.

Chaotic analysis

Through the application of an external force, we have created a perturbation term in the dynamical system (17) in this analysis. As a result, the perturbation term system (17) can be expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{\mathit{dQ}}{d\zeta }=P,\hfill \\ \frac{\mathit{dP}}{d\zeta }=Y_1^{\star }Q-Y_2^{\star }{Q}^3+\nu cos(\omega \zeta ).\hfill \end{array}\right)\end{array}$$ 20

This system has been named the perturbed dynamical system with perturbation term $\nu cos(\omega \zeta )$. All other system physical parameters in this study will be set to (20). The different phase portraits of the perturbed dynamical system for different $\nu$ and $\omega$ values will then be shown. The Runge-Kutta method’s fourth order integration method (RK4) is applied to the (20) solutions to create these portraits. The findings demonstrate that, for various values of $\nu$ and $\omega$, the solution of system (20) becomes chaotic at $Y_1^{\star }=-2,Y_2^{\star }=-1$. In Fig. 5, the dynamical behaviour is examined for $\nu =0.1$ and $\omega =1$, both with and without the perturbation term. The same experiment is then repeated for various values of $\nu$ and $\omega$ with the perturbation term included, revealing signs of quasi-chaos, as depicted in Fig. 6. For a more in-depth analysis, it is important to employ a different tool to substantiate these claims.

Fig. 5.

Chaotic analysis for the dynamical system (20) at different values of $\nu$ and $\omega$.

Fig. 6.

Chaotic analysis for the dynamical system (20) at different values of $\nu$ and $\omega$.

Sensitivity analysis

Sensitivity analysis is crucial in understanding how small changes in initial conditions impact the dynamical system’s behaviour. In this section, we investigated the system’s sensitivity under two different initial conditions, varying parameters $Y_1^{\star }$ and $Y_2^{\star }$, and initial values (Q, P). The results, depicted in Fig. 7a–c, demonstrate that even minor adjustments to initial conditions cause noticeable shifts in the solution curves, with no overlap between them. This highlights the system’s sensitivity, emphasizing the need for precise initial conditions to model and predict its behavior⁴² accurately.

Fig. 7.

Sensitive analysis of dynamical system (17) for different initial conditions.

Conclusion

The conformable fractional Ivancevic option pricing model is investigated and exact solutions have been obtained. This model has two distinct aspects, allowing practitioners from both financial and physics backgrounds to benefit from the obtained results. The computed solutions are of great importance to financial practitioners, as they provide a reliable and accurate way to value options. Furthermore, using the modified Kudryashov method, the model has produced soliton solutions that have been represented on 3D, 2D, and contour plots in Figs. 1, 2 and 3. The dynamical system of the studied model has undergone a qualitative analysis, and the phase portraits of the bifurcations analysis are displayed in Fig. 4a–d. It is possible to observe market fluctuations close to the equilibrium points of the corresponding dynamical system; Fig. 4a–d shows how the market price behaves at these equilibrium points. As illustrated in Figs. 5 and 6 tiny changes in the physical parameters of the related perturbed dynamical system also indicate chaotic fluctuations. Additionally, the sensitivity analysis of the pricing model has been conducted at different initial values, as given in Fig. 7a–c, with the outcomes showing that the investigated model is not highly sensitive. It has been reported that the results are novel and can be used after further investigation to understand the behaviour of global financial markets in mathematical finance problems.

Data availability

The datasets used and analysed during the current study are available from the corresponding author upon reasonable request.

Competing interests

The authors declare that they have no competing interests.