Since its introduction in the early 80s, the risk-neutral valuation principle has proved to be an important tool in the pricing and hedging of financial derivatives. This book provides a self-contained treatment of the probabilistic theory behind the risk-neutral valuation principle and its application to the pricing and hedging of financial derivatives. On the probabilistic side, both discrete- and continuous-time stochastic processes are treated. These foundation and general principles of modern stochastic finance are established in the first six chapters, which are essentially revised and updated versions of the corresponding chapters in the first edition. Chapters 7 and 8, which treat incomplete markets and interest rate theory, are completely restructured and greatly expanded in order to include recent developments. A new ninth chapter discusses aspects of modelling credit risk.