A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics

Nuugulu, Samuel Megameno; Gideon, Frednard; Patidar, Kailash C.

doi:10.1016/j.chaos.2021.110753

Cited by 14 publications

(6 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To estimate European put option premiums, an implicit finite difference scheme with O(∆t + h) was constructed. Meanwhile, Nuugulu et al [175] improved upon the results in [174] by introducing a robust numerical method based on extending a C-N finite difference approach with O(∆t 2 + h 2 ). Considering the time-varying dynamics of asset prices in the market, Rezaei et al [118] proposed a more complicated equation, which incorporates time-varying interest rates and dividend parameters.…”

Section: Equations Derived By Fractional Taylor Series and Their Solu...mentioning

confidence: 99%

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Zhang,

Liu

et al. 2024

Fractal Fract

View full text Add to dashboard Cite

The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.

show abstract

Section: Equations Derived By Fractional Taylor Series and Their Solu...mentioning

confidence: 99%

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Zhang,

Liu

et al. 2024

Fractal Fract

View full text Add to dashboard Cite

show abstract

“…For example, Samuel M Nuugulu derives the time-fractional B-S partial differential equation for option pricing, who proposes a robust numerical method which arises from the extension of a Crank Nicholson finite difference method. [1] Ruifang Yan uses a difference method of parallel nature to solve time-space fractional B-S model. The auther proposes a kind of difference format of parallel nature which uses an implicit format and an explicit format instead of the Saul'yev asymmetric format on the inner boundary.…”

Section: Introductionmentioning

confidence: 99%

Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics

Sun

2023

Chinese Phys. B

View full text Add to dashboard Cite

This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics. The traditional reproducing kernel (RK) method which deals with this problem is very troublesome. This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel (AMPIRK) method for the first time. This method has three obvious advantages which are as follows. Firstly, the piecewise number is reduced. Secondly, the calculation accuracy is improved. At last, the waste time caused by too many fragments is avoided. Then four numerical examples show that this new method has a higher precision and is a more timesaving numerical method than the others. The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.

show abstract

“…Fractional calculus, and specifically fractional differential equations, are useful mathematical tools for modelling the dynamics of systems and phenomena in very diverse fields in the applied sciences. Some applications can be found in [7,[15][16][17][18][19][20] among others. The discovery of the fractal nature of financial markets, and the subsequent development of fractal-based asset-pricing models, has intensified the search for accurate and stable numerical methods for solving these somewhat involved yet useful asset-pricing models.…”

Section: Introductionmentioning

confidence: 99%

“…At present, several numerical methods for fractional Black-Scholes models have been suggested. The existing methods can be categorised into three classes: methods based on finite difference [16,[19][20][21][22][23][24][25], finite elements [26][27][28] and those based on the spectral approach [29][30][31]. Compared to the other two classes, the finite difference-based methods are proven to be more robust, efficient and tractable in solving fractional Black-Scholes equations.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Method for Pricing Double-Barrier Options on an Underlying Stock Governed by a Fractal Stochastic Process

2023

Self Cite

View full text Add to dashboard Cite

After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2+k2).

show abstract

A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics

Cited by 14 publications

References 14 publications

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics

An Efficient Numerical Method for Pricing Double-Barrier Options on an Underlying Stock Governed by a Fractal Stochastic Process

Contact Info

Product

Resources

About