This book presents the modern mathematical approach to continuous-time asset pricing theory. The author’s previously noted contributions include a very well-known text on stochastic differential calculus, a technique which is naturally heavily used in the present book. The main aspects of mathematical finance covered are optimum portfolio selection, options and contingent claims pricing, and the consumption based Capital Asset Pricing Model. Topics are presented in the context of complete and incomplete markets and the implications of the introduction of transactions costs and of constraints on short-sales are analysed.