Finance is only the second field of human endeavor (after physics) that draws on essentially every branch of mathematics: calculus, linear algebra, probability, differential equations, measure theory, stochastic calculus. This outstanding reference surveys it all, detailing results that are important for financial professionals and illustrating them with examples. This isn't a book to learn from. Although it does develop concepts, starting with the basics and building from there, explanations are brief and to-the-point. The book is a reference for quantitative professionals. Can't remember the formula for a Taylor series expansion? Perhaps you are wondering exactly what a Laplace transform is ... or a Hankel matrix ... or a filtration. Would you like a quick two-page explanation of the HJM framework for modeling interest rates? It is all here! What is more, it is mathematically precise. This isn't some vague, watered-down, hand-waving excuse for mathematics. It is for readers who plan on using the mathematics.

The authors illustrate how the mathematics is used in finance, with discussions of fixed income markets, portfolio theory, futures and forwards, options pricing and more. They offer insights for even the most experienced practitioners. A particular example of this is Chapter 3, which details historical milestones in finance, starting in the 1800s with the Lausanne School founded by Pareto and Walras.

The book isn't perfect. Its definition of duration is missing a minus sign. Value-at-risk would have been an excellent context to illustrate much of the math, but its treatment is simplistic. A few topics, such as Copulas or the Monte Carlo method, are mentioned but really not explained. For a book of this magnitude, these are trivial criticisms.