A fractional Black-Scholes model with stochastic volatility and European option pricing

• Published in: Expert systems with applications 2021-09, Vol.178, p.114983, Article 114983
• Main Authors: He, Xin-Jiang, Lin, Sha
• Format: Article
• Language: English
• Subjects: Cosine series; Exact solutions; Explicit and analytical; FMLS model; Fourier series; Fractional partial differential equation; Partial differential equations; Pricing; Securities prices; Stochastic models; Stochastic volatility; Volatility
• ISSN: 0957-4174; 1873-6793
• DOI: 10.1016/j.eswa.2021.114983

•European option pricing is studied under a FMLS model with stochastic volatility.•We have successfully solved a fractional partial differential equation system.•The derived pricing formula is truly explicit, involving no Fourier inversion. In this paper, we introduce the stochastic volatility into the FMLS (finite moment log-stable) model to capture the effect of both jumps and stochastic volatility. However, this additional stochastic source adds another degree of complexity in seeking for analytical formula when pricing European options, as the involved FPDE (fractional partial differential equation) system governing option prices is now of three dimensions. Albeit difficult, we have still managed to present an analytical solution expressed in terms of Fourier cosine series, after a two-step solution procedure is developed for the target FPDE system. This solution is different from the most literature as it is truly explicit, involving no Fourier inversion. It is also shown through the numerical experiments that it converges very rapidly and has potential to be applied in practice.