Numerical pricing based on fractional Black–Scholes equation with time-dependent parameters under the CEV model: Double barrier options

The empirical observations show that the constant elasticity of variance (CEV) model is a practical approach to capture the implied volatility smile phenomenon. Also, due to the appearance of long-range dependence (or long memory effect) in stock returns or volatility, the fractional Black–Scholes models became particularly important in finance. We will generalize the CEV model to a fractional-CEV model with Jumarie’s fractional model (Jumarie, 2008) to capture the long memory effect in addition to show the negative relationship between stock price and its return volatility. The asset price dynamics of this model follows from a fractional stochastic differential equation, and its volatility is a function of the underlying asset price. We derive the fractional Black–Scholes equation by using the Itoˆ Lemma and fractional Taylor’s series. Previously, Jumarie’s model was used to determine the value of European and American options. In this study, given the importance of the barrier option, we use the proposed model for pricing a European double barrier option. Since most fractional PDEs do not have closed-form analytical solutions, we solve the option pricing problem numerically. Then, we investigate the stability and convergence of the proposed scheme by applying the Fourier analysis. Finally, some numerical results are given in the last section by computing the European double barrier option.