Option pricing and hedging with temporal correlations
We consider the problem of option pricing and hedging when stock returns are correlated in time.
Within a quadratic-risk minimisation scheme, we obtain a general formula, valid for weakly correlated non-Gaussian processes.
We show that for Gaussian price increments, the correlations are irrelevant, and the Black-Scholes formula holds with the volatility of the price increments on the scale of the re-hedging.
For non-Gaussian processes, further non trivial corrections to the 'smile' are brought about by the correlations, even when the hedge is the Black-Scholes Delta-hedge.
We introduce a compact notation which eases the computations and could be of use to deal with more complicated models.