The Black-Scholes model for the pricing of derivatives whilst very effective is also known to be imperfect. A number of ways of generalizing the model to cope with these imperfections have been developed. One popular method is to allow the instantaneous volatility parameter to follow a stochastic process. This allows the possibility of observed volatilities in the market to evolve from day to day and also to produce market type "smiles" that is graphs of volatility against strike which are smile shaped rather than the horizontal line implied by the Black-Scholes model.
In this book, Lewis develops pricing formula for options under stochastic volatility models. This is mainly via the use of transform methods, that is a closed form solution is developed for the Fourier transform of the price as a function of log of the spot. The actual price is then obtained via a numerical inverse Fourier transform.
The strengths of this book are that it covers an important area that heretofore has been restricted to research papers and that it provides a large number of careful derivations and formulas.
The principal weakness is that the approach is too formula-based. The reader does not gain many conceptual insights from the author. Indeed one gains the impression that the author is technically strong but does not have a good conceptual understanding of the subject. The author does not really make a case for stochastic volatility models as opposed to other generalizations of the Black-Scholes model.
The book is restricted to vanilla options with no discussion of how using a stochastic volatility model impacts on the price of exotic options.
In conclusion, this book is not bad but it is also not great. If you are involved in studying or implementing stochastic volatility models you will certainly want to buy a copy. However the definitive book on stochastic volatility remains to be written.
