A robust numerical solution to a time-fractional Black–Scholes equation

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

doi:10.1186/s13662-021-03259-2

Cited by 14 publications

(14 citation statements)

References 38 publications

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“…When examining stocks with continuous dividend payments, a pricing equation proposed by Nuugulu et al [174] closely resembles the one (50). To estimate European put option premiums, an implicit finite difference scheme with O(∆t + h) was constructed.…”

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

confidence: 90%

“…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%

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

Zhang,

Liu

et al. 2024

Fractal Fract

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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.

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

confidence: 90%

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

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“…The time fractional Black-Scholes equation is a fractional partial differential equation, which is used for modeling the prices of the options [49][50][51][52][53][54][55].…”

Section: Time Fractional Black-scholes Equationmentioning

confidence: 99%

Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations

Dimitrov,

Georgiev,

Todorov

2023

Fractal Fract

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In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments.

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“…The proposed method follows a front-fixing algorithm under which option premiums and their corresponding optimal option exercise boundaries are computed. The front-fixing method has been applied successfully to a wide range of problems arising in population dynamics [27] and finance [28][29][30][31][32][33][34]. There are numerous other similar techniques used in solving American option problems, for example singularity separating method [32,35] and Penalty methods [9,34].…”

Section: Introductionmentioning

confidence: 99%

A Robust Numerical Simulation of a Fractional Black–Scholes Equation for Pricing American Options

Nuugulu,

Gideon,

Patidar

2024

J Nonlinear Math Phys

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After the discovery of fractal structures of financial markets, fractional partial differential equations (fPDEs) became very popular in studying dynamics of financial markets. Available research results involves two key modelling aspects; firstly, derivation of tractable asset pricing models, those that closely reflects the actual dynamics of financial markets. Secondly, the development of robust numerical solution methods. Often times, most the effective models are of a nonlinear nature, and as such, reliable analytical solution methods are seldomly available. On the other hand, the accurate value of American options strongly lies on the unknown free boundaries associated with these types of derivative contracts. The free boundaries emanates from the flexibility of the early exercise features with American options. To the best of our knowledge, the approach of pricing American options under the fractional calculus framework has not been extensively explored in literature, and an obvious wider research gap still exist on the design of robust solution methods for pricing American option problems formulated under the fractional calculus framework. Therefore, this paper serve to propose a robust numerical scheme for solving time-fractional Black–Scholes PDEs for pricing American put option problems. The proposed scheme is based on the front-fixing algorithm, under which the early exercise boundaries are transformed into fixed boundaries, allowing for a simultaneous computation of optimal exercise boundaries and their corresponding fair premiums. Results herein indicate that, the proposed numerical scheme is consistent, stable, convergent with order $${\mathcal {O}}(h^2,k)$$ O ( h 2 , k ) , and also does guarantee positivity of solutions under all possible market conditions.

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A robust numerical solution to a time-fractional Black–Scholes equation

Cited by 14 publications

References 38 publications

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

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

Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations

A Robust Numerical Simulation of a Fractional Black–Scholes Equation for Pricing American Options

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