||In this thesis, various new methodologies for pricing multivariate path dependent options in closed form are presented. A splitting direction technique is developed to find the transition density functions of multi-asset barrier options and occupation time derivatives with one state variable having the barrier feature. Based on the lognormal assumption of the asset price movement, our formulation has been successfully applied to derive the analytical formulas of multi-asset options with external two-sided barriers and sequential barriers, multi-asset step options and delayed barrier options. Analytic price formulas for European quanto options are also derived in this thesis. The success of the analytic tractability of these quanto lookback options depends on the availability of a succinct analytic representation of the joint density function of the extreme value and terminal value of the stock price and exchange rate. A new perspective of understanding lookback options, a sub-replication and replenishing premium approach, is motivated by the existence of the closed form solution in the quanto lookback case. By choosing an appropriate sub-replication of the option, we are able to represent the option value by the sub-replication and adding a suitable amount of replenishing premium to obtain the price of lookback options. Without limitation to any form of the asset price process, we are able to provide an analytic price presentation for single asset lookbacks, semi-lookbacks and double lookbacks. In practice, such a representation provides nice analytical tractability if the asset price movement follows a Geometric Brownian motion. The contingent claim approach is also useful for modeling risky debts. It allows us to model the option of the issuer to default on the debt contract. We employ the option pricing techniques to examine the fix-float differentials of credit spreads with imposing systematic jump risk. Some characteristics of risky fixed and floating rate debts are reported.