Empirical studies have shown that financial time series exhibit negative skewness and excess kurtosis. GARCH models can be successfully used to model security returns. This thesis utilizes NGARCH and Normal Mixture NGARCH models to price options. The Gibbs sampling method is explained and implemented for parametric inference, and Bayesian inference results are compared with those obtained with Maximum Likelihood Estimates. Pricing option contracts requires the derivation of risk neutral return dynamics of the underlying asset. There is an infinite number of risk neutral measures under the incomplete market GARCH framework. In this thesis, we study the conditional Esscher transform and the Extended Girsanov Principle as the martingale measure candidates. We use the Radon Nikodym derivatives from both risk neutral measures to derive and compare the option prices for GARCH models based on Gaussian and Mixture of Gaussian innovations.