This book provides an introduction to the mathematical theory of finance for readers familiar with probability and stochastic processes but with little or no background in finance. It is written in a rigorous mathematical style, aiming to explain the financial motivations and terminology. The book is based on two major revolutions in finance: the development of portfolio theory and the creation of derivative securities pricing models. The first revolution, initiated by Harry Markowitz's 1952 "Portfolio Selection," shifted the focus from selecting the best stock to understanding the trade-offs between risk and return in a portfolio. This led to the development of mean-variance analysis and the use of linear regression. The second revolution, involving the pricing of derivative securities, was pioneered by Fischer Black, Myron Scholes, and Robert Merton, who developed the Black-Scholes model for valuing European call options. This model is based on the principle of no arbitrage and has become a cornerstone of modern finance. The book covers the theory of pricing and hedging contingent claims in complete markets, the problem of optimal consumption and investment for a single agent, and the equilibrium in complete markets. It also addresses the more complex issue of pricing and hedging in incomplete markets and with constraints on portfolio choices. The text includes a detailed discussion of stochastic calculus, partial differential equations, and their applications in finance. The book is structured into six chapters, each covering different aspects of financial mathematics. It includes an extensive bibliography and notes at the end of each chapter, pointing readers to related topics. The authors emphasize the importance of stochastic models in understanding financial markets and the role of financial institutions in facilitating the flow of capital. The book also addresses common misconceptions about the mathematical theory of finance, such as the belief that it is solely about beating the market or that finance is a zero-sum game. The authors acknowledge the contributions of many collaborators and colleagues, and express their gratitude to the audiences and institutions that supported their work. The book is intended for advanced students and professionals in finance and mathematics, providing a comprehensive overview of the mathematical foundations of financial markets.This book provides an introduction to the mathematical theory of finance for readers familiar with probability and stochastic processes but with little or no background in finance. It is written in a rigorous mathematical style, aiming to explain the financial motivations and terminology. The book is based on two major revolutions in finance: the development of portfolio theory and the creation of derivative securities pricing models. The first revolution, initiated by Harry Markowitz's 1952 "Portfolio Selection," shifted the focus from selecting the best stock to understanding the trade-offs between risk and return in a portfolio. This led to the development of mean-variance analysis and the use of linear regression. The second revolution, involving the pricing of derivative securities, was pioneered by Fischer Black, Myron Scholes, and Robert Merton, who developed the Black-Scholes model for valuing European call options. This model is based on the principle of no arbitrage and has become a cornerstone of modern finance. The book covers the theory of pricing and hedging contingent claims in complete markets, the problem of optimal consumption and investment for a single agent, and the equilibrium in complete markets. It also addresses the more complex issue of pricing and hedging in incomplete markets and with constraints on portfolio choices. The text includes a detailed discussion of stochastic calculus, partial differential equations, and their applications in finance. The book is structured into six chapters, each covering different aspects of financial mathematics. It includes an extensive bibliography and notes at the end of each chapter, pointing readers to related topics. The authors emphasize the importance of stochastic models in understanding financial markets and the role of financial institutions in facilitating the flow of capital. The book also addresses common misconceptions about the mathematical theory of finance, such as the belief that it is solely about beating the market or that finance is a zero-sum game. The authors acknowledge the contributions of many collaborators and colleagues, and express their gratitude to the audiences and institutions that supported their work. The book is intended for advanced students and professionals in finance and mathematics, providing a comprehensive overview of the mathematical foundations of financial markets.