Calibrating parametric exponential Lévy models to option market data by incorporating statistical moments priors

We investigate a parametric method for calibrating European option pricing using the state-of-art exponential Lévy models. We propose a derivative-free calibration method constrained by four observable statistical moments (mean, variance, skewness and kurtosis) from underlying time series to conquer the ill-posed inverse problem and to incorporate priors on observable statistical moments. We present a numerical implementation scheme for calibrating the exponential Lévy models and show that it can resolve the instability of the inverse problems empirically and can produce good calibration results. In particular, we apply our approach to real market data sets of S&P 500 call options with significantly better performance.