J.-P. Fouque, G. Papanicolaou and K. R. Sircar, 2000

x + 202 pp., \pounds34.75
ISBN 0-521-79163-4

It has become apparent over many years that the time-honoured
Black-Scholes model for pricing financial derivatives is not adequate
for describing modern market phenomena. One such phenomenon is the
so-called `smile' curve. The implied volatility is the value of the
volatility parameter that when entered in the Black-Scholes formula
for a financial derivative yields the price that is observed in the
market. If the Black-Scholes assumptions were true, this would be
constant for all derivatives. However, for European call options, the
implied volatility for high or low strike prices is higher than for
middle prices (a smile curve). Given that practitioners use implied
(calibrated) parameter values rather than values estimated from past
history, this is very serious and it is extremely surprising that the
Black-Scholes model has survived for so long.

This book looks at models where the volatility is itself a stochastic
process which is oscillating around a constant value (`mean
reverting'). The reader should always keep in mind that, in contrast
with the price process, the value of the volatility process is not
directly observable. The structure is preserved under the equivalent
martingale measures used for the pricing of derivatives. Such models
are seen to explain the smile phenomenon. A large amount of research
has been published on the topic. One should mention the early work of
Hull and White and many others mentioned by the authors. A further
assumption that the authors make is that the rate of mean reversion is
large (`volatility clustering'). This is not a standard assumption in
the literature. However, the authors assure us with evidence that this
is reasonable in practice. The pricing of derivatives is done by using
the partial differential equations method but the martingale approach
is also provided as an alternative.

The book is directed towards practitioners as well as researchers in
the field. It could also provide the basis for a graduate course on
stochastic volatility.

Chapter 1 provides an introduction to stochastic finance, focusing on
the Black-Scholes model. In Chapter 2 the authors demonstrate the need
for stochastic volatility models and provide a survey of them. The
case where the volatility process and the price process are
uncorrelated is, not surprisingly, easier to handle. It is shown, with
reference to Renault and Touzi who came up with the argument, how such
a model explains smiles. Of course the correlated case is more
difficult and more important in practice. Usually the two processes
are negatively correlated with the exception of foreign exchange
models where they could be uncorrelated. In Chapter 3 empirical
evidence for the mean reverting nature of stochastic volatility is
provided as well as some stochastic simulations. It is shown that the
Black-Scholes formula can be used as a first approximation; the
process converges to the usual Black-Scholes log-normal model as the
rate of mean reversion becomes larger. The main purpose of the rest of
the book is to obtain correction terms to this first approximation.

Chapter 4 explains how to estimate the rate of mean reversion. Chapter
5 is the core of the book; it is on the pricing of European
derivatives. Asymptotic results are obtained for a fairly general
class of diffusion models. These lead to correction terms for the
asymptotic approximation in Chapter 3. It is shown that the error of
the new approximation is of an order that is inversely proportional to
the rate of mean reversion. Chapter 6 lays out the implementation
procedure for pricing (also in the introduction of the book) as well
as investigating the stability of the calibration of the model over
time; both subjects are extremely useful to the practitioner. There is
also a section about adjustments in the presence of dividend payments
and a discussion on whether obtaining one more term in the asymptotic
expansion is worthwhile. One complication is that any further terms
would depend on the current state of the volatility process which is
not directly observable. Chapter 7 discusses hedging issues, obtaining
correction terms for the hedging parameters; this is not a
straightforward extension of previous results. Stochastic volatility
models imply incomplete markets, so there is no perfect hedge.

Chapter 8 has results for binary, barrier and Asian options and
Chapter 9 for American options. Chapter 10 is about portfolio
optimization and some generalizations. Examples are models with jumps,
non-Markovian models and multidimensional models. Also the martingale
approach is offered as an alternative derivation to earlier
results. Chapter 11 applies the techniques developed in the book to
interest rate models.

This is an excellent book that succeeds admirably in all its aims. It
can satisfy both practitioners and researchers at the same time. It is
very well written and it is concise and informative at the same
time. The main strength of the book is the methodology that it
suggests. It is robust, as it does not depend on the exact form of the
model, and it is fairly easy to understand and to apply.  Angelos
Dassios