This article proposes a semi-martingale approximation to a fractional Lévy process that is capable of capturing long and short memory in the stochastic process together with fat tails. The authors use the semi-martingale process in option pricing and empirically compare its performance to other option pricing models, including a stochastic volatility Lévy process. They contribute to the empirical literature by being the first to report the implied Hurst index computed from observed option prices using the Lévy process model. Calibrating the implied Hurst index of S&P 500 option prices in a period that covers the 2008 financial crisis, they find that the risk-neutral measure is characterized by a short memory in turbulent markets and a long memory in calm markets.