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An Elementary Introduction to Mathematical Finance

Sheldon M. Ross
Cambridge University Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Tom Schulte
, on

This new, third edition further bolsters Ross’ text as an excellent introduction to mathematical finance. With plentiful preliminary material, the book can work for self-study, assuming that the reader has a solid background in calculus and statistics fundamentals. It would also be helpful to understand the fundamentals of LP models and their duals, as this is an expectation made on the reader for grasping the proof of the key Arbitrage Theorem. Of course, the text can work well as an introductory course for undergraduates. Use as a textbook on the basics of option pricing better fits the structure of the text, which includes exercises at the end of each chapter without any solutions. The examples, while detailed, are less numerous than the proofs, lemmas, and propositions the reader will need in order to understand in order to progress.

The mathematically inclined reader or ardent student can find here the proofs and theorems that build up to arbitrage-defeating options pricing. The starting point is the most preliminary probabilistic notions: tossing a fair coin and considering fair dice. However, with fewer than three hundred pages, the derivation of the Black-Scholes option pricing formula comes up so quickly that it cannot be said that this is a self-contained book. A one semester course using this text would take such probability fundamentals as prerequisites and assuredly start with Chapter 3, a mathematical description of Brownian motion necessary for the models that follow.

The topics covered include utility functions, optimal portfolio selections, and the capital assets pricing model. The key new features of this third edition are new chapters on stochastic order relations and stochastic dynamic programming.

The derivation and application of Black-Scholes remains the culminating and central concept of this work. Nassim Nicholas Taleb’s book The Black Swan and news reports such as “…The combined value of the options is about $6 million to $6.5 million, according to a calculation of their value using the Black-Scholes model” (Published May 18, 2011 on bring this algorithm of the “quants” into common parlance. This Ross text can demystify the mathematics behind Black-Scholes and similar models.

Tom Schulte teaches mathematics to some of Oakland County’s future MBAs in Michigan.

1. Probability
2. Normal random variables
3. Geometric Brownian motion
4. Interest rates and present value analysis
5. Pricing contracts via arbitrage
6. The Arbitrage Theorem
7. The Black–Scholes formula
8. Additional results on options
9. Valuing by expected utility
10. Stochastic order relations
11. Optimization models
12. Stochastic dynamic programming
13. Exotic options
14. Beyond geometric motion models
15. Autoregressive models and mean reversion.