Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs

Rezaei, Mohammad; Yazdanian, A. R.; Ашрафи, Али Реза; Mahmoudi, Seyed Mahdi

doi:10.1007/s10614-021-10148-z

Cited by 3 publications

(3 citation statements)

References 45 publications

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“…Similar techniques were utilized in [136,137] to investigate the same problem incorporating different FDs, namely the C-F derivative and Caputo derivative, respectively. In recent study, Rezaei et al [138] extended Equation (48) by incorporating transaction costs. The new equation considers volatility as a function of spatial derivatives, asset prices and time, with the dividend and interest rate being time dependent.…”

Section: The Finite Difference Methodsmentioning

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 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: The Finite Difference Methodsmentioning

confidence: 99%

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

Zhang,

Liu

et al. 2024

Fractal Fract

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“…Hence, we use fractional order derivatives to describe the trend memory effect, which is another effective tool for describing the memory effect. Therefore, we replace the time derivative ∂U ∂t by the fractional derivative ∂ α U ∂t α in the equation ( 3) under the assumption that the change in the option price with time follows a fractal transmission system (see, e.g., [37,40])…”

Section: The Fractional Model With Transaction Costsmentioning

confidence: 99%

“…Amster et al [36] presented a nonlinear Black-Scholes equation which arises in an option pricing model with transaction costs and assumed that the transaction costs behaved as a nonincreasing linear function. Rezaei et al [37] numerically evaluated the European option with different transaction cost models based on fractional Black-Scholes equation. Using Leland's delta hedging strategy, Wang et al [38] deduced the Black-Scholes equation with transaction costs in a continuous time setting under the fractional Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

The valuation of double barrier options under mixed fractional Brownian motion model with transaction costs

Rezaei

2023

Preprint

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One of the phenomena observed in the real market is long-range dependence. Mixed fractional Brownian motion with Hurst parameter can be a suitable tool to capture it. The main objective of this work is to numerically evaluate the double barrier options base on the fractional Black-Scholes equation under mixed fractional Brownian motion model with transaction costs and time-dependent parameters in a discrete-time setting when the change in the option price with time follows a fractal transmission system. We solve the fractional Black-Scholes equation by using an implicit difference scheme. Also, we numerically investigate the influence of model parameters on the double barrier option price. Numerical results are presented to confirm the theoretical findings. Mathematics Subject Classification. 60G22, 91G20, 91B25.

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Pricing European Double Barrier Option with Moving Barriers Under a Fractional Black–Scholes Model

Rezaei

Yazdanian

2022

Mediterr. J. Math.

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Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs

Cited by 3 publications

References 45 publications

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

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

The valuation of double barrier options under mixed fractional Brownian motion model with transaction costs

Pricing European Double Barrier Option with Moving Barriers Under a Fractional Black–Scholes Model

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