1 Introduction -- 2 The Valuation of Contingent Claims: A Survey -- 3 Existence of Consistent Price Systems -- 3.1 The basic model -- 3.2 Arbitrage and equivalent martingale measures -- 3.3 Examples -- 4 The Continuous-time Trading Model -- 4.1 Continuous-time self-financing trading strategies -- 4.2 A characterization of P*-attainable contingent claims -- 4.3 Classes of P*-attainable contingent claims for specific security price processes -- 4.4 The relationship between P*-attainable contingent claims and solutions of associated differencedifferential equations -- 4.5 Complete securities market models -- 4.6 Counterexamples -- 5 Extensions of the BLACK/SCHOLES Model -- 5.1 Determination of an equivalent martingale measure -- 5.2 The relationship between the original and the discounted model -- 5.3 Completeness and the determination of self-financing trading strategies in the case of a European call option -- 5.4 Completeness and the relaxation of assumptions -- 5.5 Incompleteness caused by variations of assumptions -- 6 From Preference-free to Preference-dependent Valuations of Contingent Claims: the Hedge Approach in Incomplete Models -- 7 Conclusion -- References -- A Appendix -- A 1 Notation -- A 2 Mathematical Tools -- A 2.1 Miscellany -- A 2.2 Measure theory -- A 2.3 Stochastic calculus.