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 is known in explicit form only one day ahead and not at maturity. This problem was solved in the HestonNandi model (1997), where it is possible to compute the characteristic function of the underlying by a recursive procedure and options can be priced by Inverse Fourier Transform, see Heston (1993). Following the same idea, Christo¤ersen Heston and Jacobs (2004) proposed a Garchlike model with Inverse Gaussian Innovations and recently Bellini and Mercuri (2007) obtained a similar recursive procedure in a model with Gamma innovations. In this work, we present a new Garch-like model with Tempered Stable innovations that encompasses both the CJH and the BM models as special cases. As it is costumary for this class of models, the pricing measure is chosen by means of the Conditional Esscher Transform (Siu et al. 2004). The TS model is calibrated on SP500 closing option prices and its performance is compared with the CJH, the BM and the Heston Nandi models. Keywords Option Pricing, Garch model, Tempered Stable distribution, Semi-analytical valuation 1 Introduction It is well known that, from an empirical point of view, the Black-Scholes model is not able to capture some "stylized facts" such as skewness, heavy tails and volatility clustering that are observed in real …nancial time series. To model these stylized phenomena, Mandelbrot (1963) proposed the use of -Stable distributions. Unfortunately this class of distributions has in…nite variance and hence the tails might be too heavy to model real …nancial data. In order to gain more adaptability, a new family of distributions, called Tempered Stable (TS), was obtained by multiplying the density of the positively Dipartimento Metodi Quantitativi per le Scienze Economiche ed Aziendali, Università degli Studi di Milano-Bicocca. E-mail: [email protected]