An obstacle for mathematical scientists learning about applications to finance is the abundance of new terminology (e.g. derivatives, American knock-in barrier options), acronyms (e.g. CDS, CDO, CBOE) and conventions (e.g. some years have 360 days, the default order for foreign exchange quotations). An obstacle for people in finance learning mathematical finance is the deep mathematics underneath the formulas (e.g. stochastic differential equations and martingales). The field is still advancing with extensions of "classic" theory based on Brownian motion to more general stochastic processes. New structured products aim for more general risk and reward profiles than simple profit (or loss). This book aims to quickly and succinctly overcome these obstacles and to introduce the new ideas.

The book is in four parts, the first being an overview of financial instruments and markets using precalculus mathematics. The emphasis of this section is carefully defining the terms and conditions of standard financial instruments along with the markets they trade in. Most chapters have formulas giving instrument values or payoffs, usually algebraic consequences of the definitions. The second and longest part is a brisk overview of the mathematics behind financial mathematics, starting with basic probability theory and proceeding to stochastic processes leading to the Black-Scholes-Merton formula, derived both from the partial differential equation and the risk-neutral martingale approach. Measure theory appears implicitly with Radon-Nikodym derivatives, conditional expectations and martingales but is not in the foreground and is usually illustrated with either finite fields or distributions with a density. Numerical methods and Monte Carlo simulation are also developed and used as necessary. The third section extends the methods and solutions to stochastic volatility and jump-diffusion models along with interest rate modeling. The valuation methods here are stochastic differential equations and the related PDEs and integral-differential equations. The final section is a survey of modern structured financial products which are combinations of financial instruments. Large banks typically provide structured products to insurance or pension funds having a need for exposure in desirable markets but with a need for limited potential loss. The overall treatment is brisk. Most theorems and formulas get a short proof or derivation. Longer proofs get deferred to the references. Some proofs are heuristic, rather than formal or rigorous. Each chapter ends with 3 to 5 problems with a range of difficulty levels.

The book is carefully prepared, has comprehensive coverage of financial mathematics with a helpful glossary of acronyms, references and an index. The authors state that the book intends to unify derivatives modeling and financial engineering practice. Selecting sections appropriately based on the audience, this book would be a natural choice in advanced undergraduate courses and master's level courses in financial mathematics, financial engineering, applied stochastic processes, and finance. The book would also serve as a useful reference for academics and practicing financial engineers.

Steven Dunbar is Professor Emeritus at the University of Nebraska-Lincoln, with an interest in the applications of mathematics to biology, economics, and political science.