This book presents the mathematics that underpins pricing models for derivative securities, such as options, futures, and swaps, in modern financial markets. The mathematical concepts used in idealised continuous-time models are sophisticated, relying for the most part on the modern stochastic calculus and its ramifications. In the discrete-time framework, however, many of the underlying ideas can be explained much more simply. The treatment is careful and detailed rather than comprehensive, aiming in particular to provide a clear understanding of pricing and hedging for call and put options. From here the reader can progress to the use of similar methods for more exotic instruments and further research. The text should prove useful to graduates with a sound mathematical background, ideally including a first course on measure-theoretic probability, who wish to understand the mathematical models on which the multitude of current financial instruments used in derivative markets is based. It is well suited to the needs of the rapidly increasing range of quantitatively oriented Master's programmes that provide an entry into this burgeoning field of research and practice, and should equally be useful to risk managers and other practitioners looking for the mathematical tools with which to understand modern pricing and hedging models and their application.