A closed-form option pricing approximation formula for a fractional Heston model

We present a method to develop simple option pricing approximation formulas for a fractional Heston model, where the volatility process is defined by means of a fractional integration of a diffusion process. 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. Then, by means of classical Itô's calculus we decompose option prices as the sum of the classical Black-Scholes formula with volatility parameter equal to the root-mean-square future average volatility plus a term due to correlation and a term due to the volatility of the volatility. This decomposition procedure does not need the volatility process to be Markovian and allows us to develop easy-to-apply approximation formulas for option prices and implied volatilities, as well as to study their accuracy. Numerical examples are given.