Parameter estimation for time-fractional Black-Scholes equation with S &P 500 index option

Abstract

This paper aims to estimate the parameters of the time-fractional Black-Scholes (TFBS) partial differential equation with the Caputo fractional derivative by using the real option prices of the S &P 500 index options. First, the numerical solution is obtained by developing a high-order scheme with order (3 - α) for the time discretisation. Some theoretical analyses such as stability and convergence are presented in order to verify the efficiency and accuracy of the proposed scheme. Secondly, we employ a modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) to identify the fractional order α and implied volatility σ of the TFBS equation, and explore the financial meanings of α under extreme stock market conditions such as the Covid-19 and the 2008 global financial crisis. We analyse the values of α and compare the mean squared errors of both the TFBS model and the BS model. Our empirical results show that α may be regarded as a market fluctuation indicator for classifying financial environments, and the TFBS model is more capable of fitting real option data compared with the BS model, especially for put options during the economic downturn. In addition, we find and discuss an interesting relation between α and σ from both the TFBS model and the BS model in three expressions, which could be an open problem for further research.