The seminal paper of Black and Scholes (1973) led to the explosive growth of option pricing and hedging theory. However, the assumptions of the Black-Scholes model contradict reality. In the past three decades, a large volume ofresearch has been conducted on the problem of pricing and hedging contingent claims under more realistic assumptions. In particular, two streams of the . literature are directly related to this thesis. One is the development of stochastic volatility jump diffusion models and their option pricing formulas. The other is optimal hedging under market frictions. This thesis consists of four essays. The first two essays propose an affine stochastic volatility jump diffusion model for equity index. This is a rich model motivated by other empirical work. It includes two stochastic volatility factors, jumps in volatility process, and leverage effects. An option pricing formula is obtained by using the integral transform approach. Empirically, the model is nicely calibrated to the FTSEIOO index options data. Once the structural parameters are obtained, we examine the performance of several different calibration schemes as well as the dynamics ofthe state variables. The third and the fourth essays study the problem of optimal hedging of contingent claims in the presence of transactions costs. In the third essay, the market is described by pure diffusion. We introduce a local time analysis approach to this class of problem. This approach is new to the literature. It provides solutions that are consistent to the literature. More importantly, it is capable of providing deeper insights. The approach should stimulate further research. The fourth essay studies the optimal hedging problem with a jump diffusion market in the presence of transactions costs. The local time analysis is no longer appropriate because of the discontinuity in the stock price process, so we use a dynamic programming approach instead. The numerical results in particular extend our knowledge beyond the scope of the current literature. The essay is focused particularly on the impact ofjumps on the optimal hedging policy. Keywords: Option pricing, incomplete markets, stochastic volatility, jump diffusion, transactions costs, local time.