The book "Martingale Methods in Financial Modelling" presents a comprehensive overview of financial modelling techniques, focusing on martingale methods in financial mathematics. It covers a wide range of topics including financial derivatives, options, futures, forwards, interest rate models, and exotic options. The book is divided into two parts: Part I deals with discrete-time security markets and basic financial derivatives, while Part II focuses on continuous-time models and fixed-income markets. The first edition was written based on courses given at several universities, aiming to provide a textbook for both undergraduate and graduate students. The second edition, published in 2005, incorporates recent developments in financial modelling and pricing of derivatives, while also simplifying some sections to maintain the book's size. The second printing in 2006 includes additional proofs, expanded explanations, and updated references. Part I covers discrete-time security markets, including the Cox-Ross-Rubinstein model, the Black-Scholes model, and the valuation of American options. It also discusses futures markets, game contingent claims, and the properties of financial derivatives. Part II focuses on continuous-time models, including the Black-Scholes model, stochastic volatility models, and interest rate models. It also covers exotic options, volatility risk, and alternative approaches to financial modelling. The book provides a detailed analysis of various financial models, including the Black-Scholes model, the Bachelier model, the Black model, and stochastic volatility models. It also discusses the valuation and hedging of financial derivatives, the properties of financial markets, and the use of martingale measures in financial modelling. The book is accompanied by an appendix that provides an overview of Itô stochastic calculus, which is essential for understanding the mathematical foundations of financial modelling. The book is intended for students and professionals in finance and mathematics who are interested in financial modelling and the application of martingale methods in financial mathematics.The book "Martingale Methods in Financial Modelling" presents a comprehensive overview of financial modelling techniques, focusing on martingale methods in financial mathematics. It covers a wide range of topics including financial derivatives, options, futures, forwards, interest rate models, and exotic options. The book is divided into two parts: Part I deals with discrete-time security markets and basic financial derivatives, while Part II focuses on continuous-time models and fixed-income markets. The first edition was written based on courses given at several universities, aiming to provide a textbook for both undergraduate and graduate students. The second edition, published in 2005, incorporates recent developments in financial modelling and pricing of derivatives, while also simplifying some sections to maintain the book's size. The second printing in 2006 includes additional proofs, expanded explanations, and updated references. Part I covers discrete-time security markets, including the Cox-Ross-Rubinstein model, the Black-Scholes model, and the valuation of American options. It also discusses futures markets, game contingent claims, and the properties of financial derivatives. Part II focuses on continuous-time models, including the Black-Scholes model, stochastic volatility models, and interest rate models. It also covers exotic options, volatility risk, and alternative approaches to financial modelling. The book provides a detailed analysis of various financial models, including the Black-Scholes model, the Bachelier model, the Black model, and stochastic volatility models. It also discusses the valuation and hedging of financial derivatives, the properties of financial markets, and the use of martingale measures in financial modelling. The book is accompanied by an appendix that provides an overview of Itô stochastic calculus, which is essential for understanding the mathematical foundations of financial modelling. The book is intended for students and professionals in finance and mathematics who are interested in financial modelling and the application of martingale methods in financial mathematics.