Pricing European Options under Fractional Black–Scholes Model with a Weak Payoff Function

The purpose of this paper is to obtain an explicit solutions of the fractional Black–Scholes model with a weak payoff function. To do this, we derive fractional Black–Scholes equation by creating a self-financing portfolio strategy under Leland’s strategy. Then, we use the Mellin transform method for solving this equation and obtain the price of a European option as a particular case of the proposed solution. A sensitivity analysis is carried out through numerical experiments which shows the differences between Black–Scholes model and the fractional Black–Scholes model. Moreover, an empirical analysis shows that the fractional Black–Scholes model with Hurst exponent greater than one-half is more precise to predict the real market prices than the classical Black–Sholes model.