Option pricing in a Garch model with tempered stable innovations

The key problem for option pricing in Garch models is that the risk-neutral distribution of the underlying at maturity is unknown. Heston and Nandi solved this problem by computing the characteristic function of the underlying by a recursive procedure. Following the same idea, Christoffersen, Heston and Jacobs proposed a Garch-like model with inverse Gaussian innovations and recently Bellini and Mercuri obtained a similar procedure in a model with Gamma innovations. We present a model with tempered stable innovations that encompasses both the CHJ and the BM models as special cases. The proposed model is calibrated on S&P500 closing option prices and its performance is compared with the CHJ, the BM and the Heston-Nandi models.