Two high-order compact difference schemes with temporal graded meshes for time-fractional Black-Scholes equation

Gu, Jie; Nong, Lijuan; Yi, Qian; Chen, An

doi:10.3934/nhm.2023074

Cited by 3 publications

(1 citation statement)

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“…On the other hand, Gu et al [148] developed a temporal non-uniform mesh combined with the compact difference scheme, which exhibits a convergent order of O(∆t 2 + h 4 ). Soon afterwards, a improved work was presented in [149]. The authors proposed the sixth-order/eighth-order compact difference scheme, incorporating a trapezoidal formulation based on temporally graded meshes for the temporal integral term.…”

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%