Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2003
2002g:91082
91B28
60G99 60H10 60J60
Fouque, Jean-Pierre
(
1-NCS
)
; Papanicolaou, George
(
1-STF
)
;
Sircar, K. Ronnie
(
1-MI
)
Derivatives in nancial markets with stochastic volatility.
(English. English summary) Cambridge University Press, Cambridge, 2000. xiv+201 pp. $49.95.
ISBN 0-521-79163-4
The book is entirely devoted to a specic, but highly relevant and
nontrivial, problem of pricing and hedging of derivative securities
under the assumption that the so-called volatility of the price of the
underlying process is random, rather than deterministic. Numerous
attempts to extend the classic models of nancial markets to the
case of a stochastic volatility were made by several researchers in
the nineties. It should thus be mentioned that the authors put the
emphasis on a particular, though fairly general, method of dealing
with uncertain volatility developed by themselves in a series of papers. The book is organized as follows: Chapter 1 provides a brief overview of basic notions related to arbitrage pricing and hedging of deriva-
tive securities in the classic Black-Scholes options pricing model.
Subsequently, in Chapter 2 the authors rst justify the need for mo-
delling volatility as a stochastic process. In particular, they discuss
here the so-called volatility smile and they present various alterna-
tive approaches to this issue developed in the literature. Most of these
traditional models are based on the diusion-type specication of the
dynamics of the process modelling volatility. They rightly point out
that there is no canonical stochastic volatility model that is generally
accepted, and the relevance of explicit formulas for particular models
is not obvious. To overcome these drawbacks, they propose to com-
bine the mean-reverting property of volatility with the concept of the
intrinsic time scale, which is introduced and analyzed in Chapter 3.
The main statistical tools used to estimate the rate of mean reversion
of volatility from historical data are presented in Chapter 4. Chapters
5 and 6 discuss asymptotic methods related to the valuation of deriv-
ative securities in the present set-up, as well as the important issue
of model calibration. In Chapter 7, the issue of hedging under mar-
ket incompleteness by the stochastic volatility is briey discussed. In
Chapters 8 and 9, the case of exotic options and American contingent
claims is examined, while various generalizations (in particular, the
portfolio optimization problem) are discussed in Chapter 10. Finally,
in Chapter 11 the authors show how to cover through their original
approach the case of xed-income securities. Though the topic discussed in the book is conceptually rather Results from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2003 dicult, the book itself is highly readable. Since the book starts from
scratch and the style is user friendly, it is in my opinion accessible
to graduate students specializing in the eld of nancial mathematics
and probability theory. Marek Rutkowski (PL-WASWT)
(English. English summary) Cambridge University Press, Cambridge, 2000. xiv+201 pp. $49.95.
ISBN 0-521-79163-4
The book is entirely devoted to a specic, but highly relevant and
nontrivial, problem of pricing and hedging of derivative securities
under the assumption that the so-called volatility of the price of the
underlying process is random, rather than deterministic. Numerous
attempts to extend the classic models of nancial markets to the
case of a stochastic volatility were made by several researchers in
the nineties. It should thus be mentioned that the authors put the
emphasis on a particular, though fairly general, method of dealing
with uncertain volatility developed by themselves in a series of papers. The book is organized as follows: Chapter 1 provides a brief overview of basic notions related to arbitrage pricing and hedging of deriva-
tive securities in the classic Black-Scholes options pricing model.
Subsequently, in Chapter 2 the authors rst justify the need for mo-
delling volatility as a stochastic process. In particular, they discuss
here the so-called volatility smile and they present various alterna-
tive approaches to this issue developed in the literature. Most of these
traditional models are based on the diusion-type specication of the
dynamics of the process modelling volatility. They rightly point out
that there is no canonical stochastic volatility model that is generally
accepted, and the relevance of explicit formulas for particular models
is not obvious. To overcome these drawbacks, they propose to com-
bine the mean-reverting property of volatility with the concept of the
intrinsic time scale, which is introduced and analyzed in Chapter 3.
The main statistical tools used to estimate the rate of mean reversion
of volatility from historical data are presented in Chapter 4. Chapters
5 and 6 discuss asymptotic methods related to the valuation of deriv-
ative securities in the present set-up, as well as the important issue
of model calibration. In Chapter 7, the issue of hedging under mar-
ket incompleteness by the stochastic volatility is briey discussed. In
Chapters 8 and 9, the case of exotic options and American contingent
claims is examined, while various generalizations (in particular, the
portfolio optimization problem) are discussed in Chapter 10. Finally,
in Chapter 11 the authors show how to cover through their original
approach the case of xed-income securities. Though the topic discussed in the book is conceptually rather Results from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2003 dicult, the book itself is highly readable. Since the book starts from
scratch and the style is user friendly, it is in my opinion accessible
to graduate students specializing in the eld of nancial mathematics
and probability theory. Marek Rutkowski (PL-WASWT)
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