Financial Mathematics: From Discrete to Continuous Time is a study of the mathematical ideas and techniques that are important to the two main arms of the area of financial mathematics: portfolio optimization and derivative valuation. The text is authored for courses taken by advanced undergraduates, MBA, or other students in quantitative finance programs. The approach will be mathematically correct but informal, sometimes omitting proofs of the more difficult results and stressing practical results and interpretation. The text will not be dependent on any particular technology, but it will be laced with examples requiring the numerical and graphical power of the machine. The text illustrates simulation techniques to stand in for analytical techniques when the latter are impractical. There will be an electronic version of the text that integrates Mathematica functionality into the development, making full use of the computational and simulation tools that this program provides. Prerequisites are good courses in mathematical probability, acquaintance with statistical estimation, and a grounding in matrix algebra. The highlights of the text are: A thorough presentation of the problem of portfolio optimization, leading in a natural way to the Capital Market Theory Dynamic programming and the optimal portfolio selection-consumption problem through time An intuitive approach to Brownian motion and stochastic integral models for continuous time problems The Black-Scholes equation for simple European option values, derived in several different ways A chapter on several types of exotic options Material on the management of risk in several contexts