Application of Fractional Partial Differential Equations in Computational Finance

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (31 August 2023) | Viewed by 7694

Special Issue Editors

Faculty of Science and Technology, University of Macau, Macau SAR, China
Interests: stochastic differential equations; stochastic optimal control theory; financial mathematics; probability; statistics and their applications

E-Mail Website
Guest Editor
Faculty of Science and Technology, University of Macau, Macau SAR, China
Interests: numerical linear algebra; numerical methods for differential equations; preconditioners

Special Issue Information

Dear Colleagues,

Fractional partial differential equations (FPDEs) are widely used for pricing financial derivatives, solving issues in financial models involving fractional diffusions, fractional Brownian motions, etc. Since most FPDEs remain largely unsolved in the closed form, corresponding numerical methods are of particular interest for researchers seeking solutions.

This Special Issue aims to further advance research on the application and computation of FPDEs for the valuation of various financial derivatives and related theoretical analysis. Topics of interest include (but are not limited to):

  • Application of FPDEs in the valuation of complex financial derivatives, and related issues;
  • New and fast numerical methods for FPDEs in financial derivative pricing;
  • Efficient computation of total value adjustment of financial derivatives with counterparty credit risk via FPDEs solutions;
  • Efficient computation of volatility or risks in financial markets via FPDE solutions;
  • Fractional diffusion simulation.

Dr. Deng Ding
Dr. Siu-Long Lei
Guest Editors

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Keywords

  • fractional partial differential equations
  • financial derivatives
  • fractional diffusions
  • fractional Brownian motions
  • total value adjustments

Published Papers (6 papers)

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Research

0 pages, 482 KiB  
Article
Pricing European Options under a Fuzzy Mixed Weighted Fractional Brownian Motion Model with Jumps
by Feng Xu and Xiao-Jun Yang
Fractal Fract. 2023, 7(12), 859; https://doi.org/10.3390/fractalfract7120859 - 30 Nov 2023
Cited by 1 | Viewed by 918
Abstract
This study investigates the pricing formula for European options when the underlying asset follows a fuzzy mixed weighted fractional Brownian motion within a jump environment. We construct a pricing model for European options driven by fuzzy mixed weighted fractional Brownian motion with jumps. [...] Read more.
This study investigates the pricing formula for European options when the underlying asset follows a fuzzy mixed weighted fractional Brownian motion within a jump environment. We construct a pricing model for European options driven by fuzzy mixed weighted fractional Brownian motion with jumps. By converting the partial differential equation (PDE) into a Cauchy problem, we derive explicit solutions for both European call options and European put options. The figures and tables demonstrating the effectiveness of the results highlight the suitability of the fuzzy mixed weighted fractional Brownian motion with jump model for option pricing. Full article
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13 pages, 617 KiB  
Article
Option Pricing with Fractional Stochastic Volatilities and Jumps
by Sumei Zhang, Hongquan Yong and Haiyang Xiao
Fractal Fract. 2023, 7(9), 680; https://doi.org/10.3390/fractalfract7090680 - 11 Sep 2023
Viewed by 972
Abstract
Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities [...] Read more.
Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models. Full article
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12 pages, 1017 KiB  
Article
On Finite-Time Blow-Up Problem for Nonlinear Fractional Reaction Diffusion Equation: Analytical Results and Numerical Simulations
by Tareq Hamadneh, Zainouba Chebana, Ibraheem Abu Falahah, Yazan Alaya AL-Khassawneh, Abdallah Al-Husban, Taki-Eddine Oussaeif, Adel Ouannas and Abderrahmane Abbes
Fractal Fract. 2023, 7(8), 589; https://doi.org/10.3390/fractalfract7080589 - 30 Jul 2023
Cited by 1 | Viewed by 1198
Abstract
The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand [...] Read more.
The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up solution is also considered. Finally, some illustrative examples and comparisons with the classical problem with integer order are presented, and the validity of the results is demonstrated. Full article
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15 pages, 585 KiB  
Article
A Preconditioned Iterative Method for a Multi-State Time-Fractional Linear Complementary Problem in Option Pricing
by Xu Chen, Xinxin Gong, Siu-Long Lei and Youfa Sun
Fractal Fract. 2023, 7(4), 334; https://doi.org/10.3390/fractalfract7040334 - 17 Apr 2023
Viewed by 967
Abstract
Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used [...] Read more.
Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used in European option pricing; however, to our knowledge, American option pricing based on such models and their numerical methods is yet to be studied. Hence, a fast algorithm for solving the multi-state time-fractional linear complementary problem arising from the regime-switching time-fractional American option pricing models is developed in this paper. To construct the solution strategy, the original problem has been converted into a Hamilton–Jacobi–Bellman equation, and a nonlinear finite difference scheme has been proposed to discretize the problem with stability analysis. A policy-Krylov subspace method is developed to solve the nonlinear scheme. Further, to accelerate the convergence rate of the Krylov method, a tri-diagonal preconditioner is proposed with condition number analysis. Numerical experiments are presented to demonstrate the validity of the proposed nonlinear scheme and the efficiency of the proposed preconditioned policy-Krylov subspace method. Full article
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19 pages, 1145 KiB  
Article
Total Value Adjustment of Multi-Asset Derivatives under Multivariate CGMY Processes
by Fengyan Wu, Deng Ding, Juliang Yin, Weiguo Lu and Gangnan Yuan
Fractal Fract. 2023, 7(4), 308; https://doi.org/10.3390/fractalfract7040308 - 02 Apr 2023
Cited by 3 | Viewed by 1181
Abstract
Counterparty credit risk (CCR) is a significant risk factor that financial institutions have to consider in today’s context, and the COVID-19 pandemic and military conflicts worldwide have heightened concerns about potential default risk. In this work, we investigate the changes in the value [...] Read more.
Counterparty credit risk (CCR) is a significant risk factor that financial institutions have to consider in today’s context, and the COVID-19 pandemic and military conflicts worldwide have heightened concerns about potential default risk. In this work, we investigate the changes in the value of financial derivatives due to counterparty default risk, i.e., total value adjustment (XVA). We perform the XVA for multi-asset option based on the multivariate Carr–Geman–Madan–Yor (CGMY) processes, which can be applied to a wider range of financial derivatives, such as basket options, rainbow options, and index options. For the numerical methods, we use the Monte Carlo method in combination with the alternating direction implicit method (MC-ADI) and the two-dimensional Fourier cosine expansion method (MC-CC) to find the risk exposure and make value adjustments for multi-asset derivatives. Full article
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17 pages, 647 KiB  
Article
A Convolution Method for Numerical Solution of Backward Stochastic Differential Equations Based on the Fractional FFT
by Kexin Fu, Xiaoxiao Zeng, Xiaofei Li and Junjie Du
Fractal Fract. 2023, 7(1), 44; https://doi.org/10.3390/fractalfract7010044 - 30 Dec 2022
Cited by 2 | Viewed by 1372
Abstract
BSDEs are applied in many areas, particularly in finance and economics. In this paper, we extended the convolution method to numerically solve FBSDEs. First, a generalized θ-scheme is applied to discretize the backwards component. Second, the convolution method is used to solve [...] Read more.
BSDEs are applied in many areas, particularly in finance and economics. In this paper, we extended the convolution method to numerically solve FBSDEs. First, a generalized θ-scheme is applied to discretize the backwards component. Second, the convolution method is used to solve the conditional expectation. Third, the resulting convolution is dealt with numerically by the Fourier transform. Therefore, the fractional FFT algorithm is applied to compute the Fourier and inverse the transforms. Then, we prove some error estimates. Finally, a numerical example is implemented to test the efficiency and stability of the proposed method. Full article
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