  We present an option pricing formula for European options in a stochastic
volatility model. In particular, the volatility process is defined using a
fractional integral of a diffusion process and both the stock price and the
volatility processes have jumps in order to capture the market effect known as
leverage effect. We show how to compute a martingale representation for the
volatility process. Finally, using It\^o calculus for processes with
discontinuous trajectories, we develop a first order approximation formula for
option prices. There are two main advantages in the usage of such approximating
formulas to traditional pricing methods. First, to improve computational
effciency, and second, to have a deeper understanding of the option price
changes in terms of changes in the model parameters.
