Local and implied volatilities with the mixed-modified-fractional-Dupire model

In this paper, we use the Mellin transform to obtain the analytical formulas of European option (call or put) values, when the evolution of the underlying asset return is governed by a mixed modified fractional stochastic process. As an extension of the Dupire model Dupire (1994)[12], we also introduce the so-called "Mixed-Modified-Fractional-Dupire model", by giving the expression of it's local volatility and it's sensitivity in relation to the Hurst coefficient $H$. Finally, in the same vein, we highlight an analytical relationship between local volatility and implied volatility.