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Derivation and Application of Some Fractional Black–Scholes Equations Driven by Fractional G-Brownian Motion

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Abstract

In this paper, a new concept for some stochastic process called fractional G-Brownian motion (fGBm) is developed and applied to the financial markets. Compared to the standard Brownian motion, fractional Brownian motion and G-Brownian motion, the fGBm can consider the long-range dependence and uncertain volatility simultaneously. Thus it generalizes the concepts of the former three processes, and can be a better alternative in real applications. Driven by the fGBm, a generalized fractional Black–Scholes equation (FBSE) for some European call option and put option is derived with the help of Taylor’s series of fractional order and the theory of absence of arbitrage. Meanwhile, some explicit option pricing formulas for the derived FBSE are also obtained, which generalize the classical Black–Scholes formulas for the prices of European options given by Black and Scholes in 1973.

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Acknowledgements

This research was supported by National Natural Science Foundation of China (Nos. 72101061 and 71974038), and Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515011749).

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Author notes

  1. Changhong Guo and Yong He contributed equally to this work.

Authors and Affiliations

  1. School of Management, Guangdong University of Technology, Yinglong Road, Guangzhou, 510520, Guangdong, People’s Republic of China

    Changhong Guo & Yong He

  2. Department of Mathematics, South China Agricultural University, Wushan Street, Guangzhou, 510640, Guangdong, People’s Republic of China

    Shaomei Fang

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  1. Changhong Guo
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  2. Shaomei Fang
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  3. Yong He
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Correspondence to Shaomei Fang.

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Guo, C., Fang, S. & He, Y. Derivation and Application of Some Fractional Black–Scholes Equations Driven by Fractional G-Brownian Motion. Comput Econ 61, 1681–1705 (2023). https://doi.org/10.1007/s10614-022-10263-5

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