This comprehensive course on financial mathematics is aimed at beginning graduate students in any field with a good quantitative background, and is appropriate for advanced undergraduates in mathematics and statistics. It is also invaluable as a reference for practitioners in financial engineering.Via an accessible presentation of the theory of probability and stochastic processes needed to construct and employ most models commonly used in investment finance, including binomial trees, Brownian motion, martingales, Markov processes, and Lévy processes, the book covers no-arbitrage option pricing, hedging, and portfolio optimization in the Black–Scholes–Merton framework, exotic and American options, and fixed-income securities under stochastic interest rates.Rather than presenting mathematical topics as a succession of theorems and proofs, the book adopts a compact and to-the-point character, justifying formulas in a way that motivates their usage, thus covering a wide array of important models and quantitative tools in a didactic fashion. Special topics not easily found in textbooks at this level are included: volatility estimation and calibration, incomplete market such as stochastic volatility models, energy and weather derivatives, credit risk and credit derivatives, jump diffusions. The text provides specific numerical techniques for financial algorithms, from multinomial and finite-difference methods, to variance reduction for Monte–Carlo methods, to special algorithms for stochastic interest rates and American options.A large number of exercises illustrate the theory and practice of each financial topic, some of which encourage the reader to engage in high-level programming. This is facilitated by an extensive online companion to the text, in the form of a primer on programming, tailored to the text's exercises, covering languages popular in the financial industry, including C++, C#, Matlab, and Excel/VBA.