A fractional Heston model with

We present a modification of the classical Heston model, where the volatility process is defined by means of a fractional integration of a diffusion process. Our construction allows us to easily compute a martingale representation for the volatility process. From this representation, and using Itô calculus, we develop approximation formulas for the corresponding option prices and implied volatilities, and we prove their accuracy. These formulas give us a tool to study the main properties of the implied volatility. In particular, we study the evolution of the at-the-money skew slope as a function of time to maturity. We see that this model preserves the short-time behaviour of the Heston model, at the same time it explains the slow decrease of the smile amplitude when time to maturity increases. Numerical examples are given.