In this thesis we search for optimal hedging strategy in stock index futures markets by providing a comprehensive comparison of variety types of models in the related literature. We concentrate on the strategy that minimizes portfolio risk, i.e., minimum variance hedge ratio (MVHR) estimated from a range of time series models with different assumptions of market volatility. There are linear regression models assuming time-invariant volatility; GARCH-type models capturing time-varying volatility, Markov regime switching (MRS) regression models assuming state-varying volatility, and MRS-GARCH models capturing both time-varying and state-varying volatility. We use both Maximum Likelihood Estimation (MLE) and Bayesian Gibbs-Sampling approach to estimate the models with four commonly used index futures contracts: S&P 500, FTSE 100, Nikkei 225 and Hang Seng index futures. We apply risk reduction and utility maximization criterions to evaluate hedging performance of MVHRs estimated from these models. The in-sample results show that the optimal hedging strategy for the S&P 500 and the Hang Seng index futures contracts is the MVHR estimated using the MRS-OLS model, while the optimal hedging strategy for the Nikkei 225 and the FTSE 100 futures contracts is the MVHR estimated using the Asymmetric-Diagonal-BEKK-GARCH and the Asymmetric-DCC-GARCH model, respectively. As in the out-of sample investigation, the time-varying models such as the BEKK-GARCH models especially the Scalar-BEKK model outperform those state-varying MRS models in majority of futures contracts in both one-step- and multiple-step-ahead forecast cases. Overall the evidence suggests that there is no single model that can consistently produce the best strategy across different index futures contracts. Moreover, using more sophisticated models such as MRS-GARCH models provide some benefits compared with their corresponding single-state GARCH models in the in-sample case but not in the out-of-sample case. While comparing with other types of models MRS-GARCH models do not necessarily improve hedging efficiency. Furthermore, there is evidence that using Bayesian Gibbs-sampling approach to estimate the MRS models provides investors more efficient hedging strategy compared with the MLE method.