Customer Reviews

Option Valuation Under Stochastic Volatility: With Mathematica Code

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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.

Other reviewers discussed the virtues of this book as a first book devoted to option pricing under stochastic volatility. And, indeed, the book provides a detailed exposition of stochastic volatility models. What I want to add to the other reviews is that this book is more than just about stochastic volatility. The book gives a careful exposition of the application of the two important mathematical methods to contingent claim valuation: the method of integral transforms (Fourier and Laplace in particular) and the method of eigenfunction expansions. Long the core tools in mathematical physics, these important methods now find more and more applications in financial economics. They can be applied to option pricing, interest rate modeling, and, more generally, any problems in economics that involve calculations with diffusion processes. The author clearly demonstrates how to use these powerful tools for calculations in finance. Researchers working in the area of derivatives pricing, both in academia and on the Street, will not want to miss this point.

This book is a must have. Things that I found particularly useful+interesting:

* The section on deriving a series expansion for the implied volatility in powers of the moneyness for a general stochastic volatility model in the small-time limit. I have used this in my own work.
* The derivation of the large-time asymptotics for the Heston model using the saddle point method.
* Easy to use Mathematica code to compute Inverse Fourier transforms.
* Dicussion on how to compute the Fourier transform of a call/put price as opposed to just the log Stock price density.

The book introduces the reader to the whole new world of pricing derivatives under stochastic volatility , which had been restricted to the academics for some time now . For finance professionals not satisfied with the performance of the constant volatility Black Scholes models and it's other variants , this is a must read . The exposition is clear and the author makes a good attempt to keep the material intutive , inspite of the nature of the subject to get mathematically complex quite fast .

A worthwhile addition to the collection of books on derivative pricing .

It's rather frustrated to see the book with this title only values plain call options. It didn't give a general approach for valuing general payoff securities. The transform formula only works for plain call option, and even this need to use put call parity once to convert the payoff function so that the fundamental solution can apply.