"Empirical and theoretical studies have attempted to establish the U-shape of the log-ratio of conditional risk-neutral and physical probability density functions. The main subject of this paper is to question the use of such a U-shaped pricing kernel to improve option pricing performances in a non-Gaussian setting. Starting from the so-called Inverse Gaussian GARCH model (IG-GARCH), known to provide semi-closed form formulas for classical European derivatives when an exponential affine pricing kernel is used, we build a new pricing kernel that is non-monotonic and that still has this remarkable property. Using a daily dataset of call options written on the S &P500 index, we compare the pricing performances of these two IG-GARCH models proving, in this framework, that the new exponential U-shaped stochastic discount factor clearly outperforms the classical exponential affine one. What is more, several estimation strategies including options or VIX information are evaluated taking advantage of the analytical tractability of these models. We prove that the parsimonious estimation approach using returns and VIX historical data remains competitive without having to work with the cross-section of options."
"This project attempts to fill several gaps in the GARCH option pricing literature, in particular, from an empirical point of view. Firstly, in the spirit of Christoffersen et al. (2004) the aim of our study is to provide an intensive comparison analysis of empirical performances, in VIX index or options valuation, between different GARCH type models using Gaussian or non-Gaussian distributions under different classes of risk neutral measures. Furthermore, particular attention is granted on the choice of the information set (VIX, options, returns) in the estimation process. As a natural non-Gaussian alternative we favor the so-called NIG distribution not only because it is known to fit the statistical properties of asset returns remarkably but also because, combined with the Esscher and EGP SDF, the pricing equations may be solved explicitly. What is more, monotonic and non-monotonic pricing kernels performances, but few of them consider all these factors at the same time. Our study is a mean of making a contribution to understand the global impact of these complementary aspects (24 combinations of GARCH/distribution/SDF/estimation are tested). Secondly, inspired by the work of Hao & Zhang (2013) that proposes to explain the poor pricing performances of Gaussian GARCH models by their inefficiency to capture the variance risk premium, we also explore in this paper if it is possible to partly classify GARCH option pricing models by their ability to simply reproduce the VIX index. From purely numerical aspects, such a conclusion would be very interesting to backtest these models in an efficient way only using VIX information, when available, instead of complex option datasets."
"This working-paper derives from a very simple finding: under Gaussian hypotheses, some GARCH-type models have outstanding properties (closed-form expressions for the VIX and/or option prices) that fail when NIG innovations are involved. Nevertheless, it is now well documented that Gaussian GARCH option pricing models produce poor pricing errors when compared with skewed and fat-tailed counterparts. Thus, inspired by the so-called quasi-maximum likelihood estimator, a new two-steps approach is provided to both take benefit of these remarkable features in Gaussian environment and work with more realistic distributions. This strategy estimates separately the volatility and the distribution parameters supposing Gaussian innovations in the first step to incorporate VIX or options information in the estimation process. In a second step, the NIG distribution is fitted from the residuals obtained in the previous stage. What is more, we provide an empirical test for our new estimation methodology on a large dataset of options written on the S&P500."