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

Changhong Guo (), Shaomei Fang () and Yong He ()
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Changhong Guo: Guangdong University of Technology
Shaomei Fang: South China Agricultural University
Yong He: Guangdong University of Technology

Computational Economics, 2023, vol. 61, issue 4, No 14, 1705 pages

Abstract: 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.

Keywords: Fractional G-Brownian motion; Fractional Black–Scholes equation; Taylor’s series of fractional order; Uncertain volatility; European option pricing (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10614-022-10263-5

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