Numerical Investigation of Fractional Step-Down ELS Option

In this paper, we use the finite difference methods to explore step-down Equity Linked Securities (ELS) options under the fractional Black-Scholes model. We establish Crank-Nicolson scheme under one asset and study the impact of Hurst exponent (H) on return of repayment under fixed stock price. We also explore the impact of stock price on return of repayment under different H. Through numerical experiments, it is found that the return of repayment of options is related to H, and the result of difference scheme will increase with the increase of H. In the case of two assets, we establish implicit scheme, and in the case of three assets, we use operator splitting method (OSM) method to establish semi-implicit scheme. We get the result that the H also influences the return of repayment in two and three assets. We also conduct Greeks analysis. Through Greeks analysis, we find that the long-term correlation of stocks has a huge impact on investment gains or losses. Therefore, we take historical volatility (fractal exponents) into account which can significantly reduce risk and increase revenue for investors.