Fractional Functional with two Occurrences of Integrals and Asymptotic Optimal Change of Drift in the Black-Scholes Model

The fractional action-like variational approach for the case of functional with two occurrences of fractional action integrals is constructed. Our motivation is based on the fact that a functional with two occurrences of integrals has many advantages in optimization problems and financial engineering issues. After deriving the corresponding fractional Euler-Lagrange equation and discussing some examples, we addressed the problem of Asian call option. We derived the asymptotically optimal change of fractional drift for the geometric average Asian call option and gave some new consequences.