 Research
 Open Access
Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling
 Asmat Ara^{1},
 Najeeb Alam Khan^{2}Email author,
 Oyoon Abdul Razzaq^{3},
 Tooba Hameed^{2} and
 Muhammad Asif Zahoor Raja^{4}
https://doi.org/10.1186/s1366201714612
© The Author(s) 2018
 Received: 14 September 2017
 Accepted: 28 December 2017
 Published: 10 January 2018
Abstract
In the present paper, we employ a wavelets optimization method is employed for the elucidations of fractional partial differential equations of pricing European option accompanied by a Lévy model. We apply the Legendre wavelets optimization method (LWOM) to optimize the governing problem. The novelty of the proposed method is the inclusion of differential evolution algorithm (DE) in the Legendre wavelets method for the optimized approximations of the unknown terms of the Legendre wavelets. Sequentially, the functions and components of the pricing models are discretized by utilizing the operational matrix of fractional integration of Legendre wavelets. Illustratively, the implementation of the LWOM is exemplified on a pricing European option Lévy model and successfully depicted the stock paths. Moreover, comparison analysis of the BlackScholes model with a class of Lévy model and LWOM with qhomotopy analysis transform method (qHATM) is also deliberated out. Accordingly, the technique is found to be appropriate for financial models that can be expressed as partial differential equations of integer and fractional orders, subjected to initial or boundary conditions.
Keywords
 pricing models
 Lévy processes
 wavelets approximation
 optimization
1 Introduction
The study of financial theory is a versatile field that connects the assumptions of finance and techniques of mathematics. With the expeditious expansion of financial derivatives like options and futures, it has incited the attention of researchers particularly toward the area of pricing models. In 1973, Black, Scholes, and Merton made an innovative assumption that the stock price and other observable quantities that depend on the volatility of the option price are explicitly related to hedging approaches. The considerable inspiration of the BlackScholesMerton (BS) model defines that by keeping a certain quantity of stock the riskfree rate of the option price, known as delta, is a dynamically hedged option situation. On this key innovation, Scholes and Merton were awarded the Nobel prize in 1997. After the development of the BS model, many endeavors have been organized to sort out the rigorous notions of pricing models. Merton [1] proposed a Merton jump diffusion model (MJD) by considering a symmetric αstable Lévy process in place of an exponential Brownian motion. Many advancements have been contrived on the MJD model by considering the Lévy models with jumps. Bates [2] discovered that the stochastic volatility and elucidated the ‘volatility smile.’ Under the systematic jumps and volatility risk, the Heston model [3] merged a stochastic volatility and a jumpdiffusion process (SVJD). Some stochastic models with infinite activity pure jump processes, such as FMLS (finite moment log stable), CGMY (CarrMadanGemanYor) and KoBol (KoponenBoyarchenkoLevendorskii) can be found in [4–12]. The characteristics of these price models are more flexible and present realistic descriptions of the price process at various time scales.
Fractional derivatives offer a more delicate mechanism for many fields, in comparison with the integerorder derivatives, to confine the characteristics of processes, materials, etc. This special branch has analyzed various problems of different aspects by many researchers and scientists. Numerous papers have been published in this regard to explore and enhance the definitions and properties in order to overcome the inadequacies of previous definitions of fractional calculus, such as conformable derivatives [13], fractional derivatives with smooth [14], nonlocal and nonsingular kernel [15], etc. [16, 17]. Many researchers have proposed different techniques to solve fractional partial differential equations numerically and analytically. A general finite difference scheme is applied on three FPDEs under some infinite activity Lévy models (FMLS, KoBol, and CGMY) using the GrunwaldLetnikov definition [18]. The approach in [19] is based on a combination of the Laplace transformation and homotopy methods for the approximate analytical solution of FPDEs in the LiouvilleCaputo and CaputoFabrizio sense. In [20] a new Adomian decomposition method based on conformable derivatives is utilized to solve FPDEs. The local fractional derivative with homotopy perturbation Sumudu transform technique is studied to solve FDEs in [21]. Hence, by means of different theories of fractional derivatives, the behaviors of many FPDEs have been studied, and various techniques have been developed [22, 23].
Motivated by the worthmentioning research works found in the literature, in this endeavor, exploring the applications of FPDEs in financial mathematics, we consider the BS model and a class of Lévy models (FMLS), which is used to model stock price [5, 18]. The proposed method, the Legendre wavelets optimization method (LWOM), is successfully implemented on BS and FMLS. The technique is an amalgamation of the Legendre wavelets approximation and differential evolution algorithm. The Legendre wavelets method has been extensively used to approximate the unknown functions of integer and fractionalorder differential equations [24, 25]. Besides, the differential evolution (DE) algorithm, the famous metaheuristic scheme, has nowadays gained popularity for its global optimization attribute [26, 27]. Here, after discretizing the functions using the Legendre wavelets, the equations are optimized using the DE scheme to find the unknown parameters. The transparent numerical comparison of the BS and FMLS models with qhomotopy analysis transform method (qHATM) [18] also expounded.
The outline of the paper is as follows. In Section 2, the fractional definition and properties are outlined. In Section 3, financial and mathematical backgrounds of pricing options models are acquainted. Key features of LWOM are explained in Section 4, whereas the implementation of the method on European call option model and discussion of results is given in Section 5. Section 6 contains the conclusive annotations observed from the facts and figures of the whole study.
2 Fractional prerequisites
In this section, we give a few basic results and definitions from fractional calculus, which are helpful for the advance evolution.
Definition 2.1
 (i)
\(\mathrm{\mathrm{I}}_{\tau}^{\nu} \mathrm{\mathrm{I}}_{\tau}^{\mu} \phi ( z,\tau ) = \mathrm{\mathrm{I}}_{\tau}^{\nu+ \mu} \phi ( z,\tau ) = \mathrm{\mathrm{I}}_{\tau}^{\mu} \mathrm{\mathrm{I}}_{\tau}^{\nu} \phi ( z,\tau )\).
 (ii)
\(\mathrm{D}_{\tau}^{\nu} \mathrm{\mathrm{I}}_{\tau}^{\mu} \phi ( z,\tau ) = \mathrm{\mathrm{I}}_{\tau}^{\mu \nu} \phi ( z,\tau )\).
 (iii)
\(I_{\tau}^{\nu} \mathrm{D}^{\mu} \phi ( z,\tau ) = \phi ( z,\tau )  \sum_{k = 0}^{n  1} \frac{\tau^{k}}{k!} \frac{ \partial^{k}\phi ( z,\tau ) \vert _{\tau= 0}}{\partial\tau^{k}}\).
 (iv)The Caputo fractional derivative of order \(\nu> 0\) for \(q ( \tau ) = \tau^{\alpha} \) is$$\mathrm{D}_{\tau}^{\nu} q ( \tau ) = \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{\Gamma ( \alpha+ 1 )}{\Gamma ( \alpha \nu+ 1 )}\tau^{\alpha \alpha} & \mbox{if } m > \nu\ge m  1, \\ 0& \mbox{if } \nu\in \{ 0,1,2, \ldots,m  1 \}. \end{array}\displaystyle \right . $$
Lemma 2.2
Let \(m  1 < \nu\le m\) and \(\phi (\cdot,\tau ) \in C^{m} ( [ 0,T ] )\). Then \(\mathrm{\mathrm{I}}_{\tau}^{\nu} \ \mathrm{D}_{\tau}^{\mu} \phi ( z,\tau ) = \phi ( z,\tau )  \sum_{i = 0}^{m  1} \eta_{i} ( z ) \tau^{i}\), where \(\eta_{i} ( z ) = \frac{1}{i!} \frac{\partial^{i}\phi ( z,\tau )}{\partial\tau^{i}}\).
Definition 2.3
Definition 2.4
3 Background of option pricing model
The mathematical and financial background of pricing options is described here in order to identify the problem. The Brownian motion is a famous stochastic process and is a part of the BlackScholes model. It is profoundly recognized as a Wiener process, named after Norbert Wiener, who was the first to explain the process mathematically [29].
The more standard type of stochastic process is a Lévy process in which the Brownian motion is an example. It consists of three terms, drift, diffusion, and a jump, which play a crucial role in constructing different market models.
4 Features of Legendre wavelets optimization method
4.1 The Legendre wavelets approximation
Theorem 4.1
4.2 The differential evolution
For the optimization purpose, the differential evolution (DE) algorithm is utilized in this endeavor. This effective heuristic optimizing technique was proposed by Storn and Price [26]. Among many other global optimizers, DE is considered to be more significant for its simplicity and strong populationbased stochastic search technique over a continuous domain. The key features of DE are three control parameters, that is, the population size NP, crossover constant CR, and scaling factor Sf. These parameters may extensively affect the optimization performance of the DE; therefore, in [26, 27] some simple rules are defined for the selection of these parameters. Thus, in the DE algorithm, the solutions are easily obtained by just specifying the population set, an approximate solution, and the objective function.
5 Implementation of LWOM
Global optimum residual errors \(\pmb{\boldsymbol{R} ( v_{ij} )}\) for \(\pmb{NP = 20}\)
γ  BS  FMLS  

σ = 0.01  σ = 0.1  σ = 0.01  σ = 0.1  
2.0  9.4463 × 10^{−8}  7.3315 × 10^{−15}  2.0887 × 10^{−15}  7.3314 × 10^{−15} 
1.2  8.7099 × 10^{−15}  7.3312 × 10^{−15}  6.9499 × 10^{−15}  4.2934 × 10^{−15} 
Comparison of LWOM with qHATM [ 22 ] at \(\pmb{\sigma= 0.1}\) , \(\pmb{x = 1}\) , and \(\pmb{n = 3}\)
t  α = 2  

qHATM ħ = 0.00085  LWOM  
0.2  0.383141  0.361403 
0.4  0.759879  0.771825 
0.6  1.223750  1.242120 
0.8  1.796510  1.785540 
1.0  2.475530  2.418280 
Comparison of LWOM with qHATM [ 22 ] at \(\pmb{\sigma= 0.1}\) , \(\pmb{t = 1}\) , and \(\pmb{n = 3}\)
x  α = 2  

qHATM ħ = −0.00083  LWOM  
0.2  2.432330  2.432340 
0.4  2.428880  2.428890 
0.6  2.425380  2.425400 
0.8  2.421850  2.421860 
1.0  2.418270  2.418280 
6 Conclusions

The Legendre wavelets optimization method is successfully implemented in the fractional partial differential models that occur in financial modeling.

Legendre wavelets produce a good approximation of arbitrary functions.

LWOM enables to interpret the effects of the parameters smoothly.

The numerical solutions of the model enable us to locate the pricing variations and the parameters affecting the stock market.

From the simulated stock paths of the BS and FMLS models it can be easily depicted that the European call options of both the models converge as \(\gamma\to2\).

Since Lévy models consider the jumps of the market, FMLS being a class of Lévy models shows an incremental path of the stock prices at \(t = 0\), which differs from the path of the BS model.

By comparison of the tables the solutions obtained from LWOM are found to be in a good agreement with qHATM at particular ħ.

Since efficiency of LWOM does not depend on any auxiliary parameter as that of qHATM, it converges toward an accurate approximation more competently with less time consumption.

The proposed technique is appropriate for different financial models that can be expressed as partial or ordinary differential equations of integer and fractional orders, subjected to initial/boundary conditions.
Declarations
Acknowledgements
Author Najeeb Alam Khan would like to thank the Dean faculty of science of the University of Karachi for supporting this research.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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