Empirical studies have shown that GARCH models can be successfully used to describe
option prices. Pricing such option contracts requires the risk neutral return dynamics of
underlying asset. Since under the GARCH framework the market is incomplete, there is
more than one risk neutral measure. In this thesis, we study the locally risk neutral valuation relationship, the mean correcting martingale measure, the conditional Esscher transform and the second order Esscher transform as martingale measure candidates. All these methods lead to the respective risk neutral return dynamics. We empirically examine in-sample and out-ofsample performance of Gaussian-TGARCH and Normal inverse Gaussian (NIG)-TGARCH models under these risk neutral measures.