Student Number : 8804388Y - MSc Dissertation - School of Computational and Applied Mathematics - Faculty of Science / This dissertation considers the pricing and hedging of contingent claims in a general semimartingale market. Initially the focus is on a complete market, where it is possible to price uniquely and hedge perfectly. In this context the two fundamental theorems of asset pricing are explored. The market is then extended to incorporate risk that cannot be hedged fully, thereby making it incomplete. Using quadratic cost criteria, optimal hedging approaches are investigated, leading to the derivations of the minimal martingale measure and the variance-optimal martingale measure. These quadratic approaches are then applied to the problem of minimizing the basis risk that arises when an option on a non-traded asset is hedged with a correlated asset. Closed-form solutions based on the Black-Scholes equation are derived and numerical results are compared with those resulting from a utility maximization approach, with encouraging results.