The Concepts and Practice of Mathematical Finance [Hardcover]

If you want to get an inexpensive book then go for this.

Finding the right level of mathematical sophistication is a difficult balancing act in which it is impossible to please all readers. Here, the author has had a clear vision that the principal audience is the practising or potential quantitative analyst (or quant) and writes accordingly; it is impossible to do better than taking an approach of this sort. Such a quant must have a certain minimum level of mathematical background (a good degree in a numerate discipline). By definition, this has to be assumed for a decent understanding of the material, but the author always has an eye on what a quant really needs to know. Integrated into this mathematical work is a good deal of information about how markets, banks and other corporations operate in practice, not found in more academically-oriented books.

The first half of the book includes the core material found in any decent first course on the subject including basic stochastic calculus, pricing of European options through discounted expectation under a risk-neutral measure, the Black-Scholes differential equation and so forth. Where this book really stands out, however, is the exceptional clarity with which the key concepts are separated. Not only are three different ways for deriving the Black-Scholes formula presented (through PDEs, expectation, and the limit of discrete tree-models) ; much more significantly, the different roles played by hedging, replication and equivalent martingale measures in enforcing a price are made crystal clear. In whatever way you already think about this material, you will almost certainly come away with something new from reading this treatment. In my case, for example, I gained a much greater understanding of why "risk-neutral" pricing is so called.

The second half of the book, roughly speaking, covers a selection of more sophisticated material. The major areas covered include interest-rate derivatives and models; and more complicated models for stock price evolution (such as stochastic-volatility, jump-diffusion and variance-gamma) that have been proposed to correct inadequacies in the Black-Scholes model such as its failure to explain market smiles. Once the core ideas have been so thoroughly explained in the first half, a great deal of interesting and diverse material can be covered rapidly yet with a great deal of clarity and coherence, relating the new models to core ideas such as uniqueness of prices and hedging issues.

Those with quantitative finance experience are still likely to find a good deal that is new and worthwhile in this book. And if you a thinking about becoming a quant, I cannot think of a better book to read first.