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Stochastic Calculus of Variations in Mathematical Finance : Springer Finance - Paul Malliavin

Stochastic Calculus of Variations in Mathematical Finance

Springer Finance

Hardcover Published: 3rd November 2005
ISBN: 9783540434313
Number Of Pages: 142

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This extensive text demonstrates the relevance of Malliavin calculus for Mathematical Finance. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formulae provide stable Monte Carlo schemes for numerical valuation of digital options. Insider information is expressed as an infinite-dimensional drift. The last chapter gives an introduction to the same objects in the context of jump processes where incomplete markets appear.

From the reviews:

"This short book introduces Malliavin calculus and illustrates important applications in finance. ... For readers with the necessary mathematical skills, this is a valuable introduction to the mathematics and financial applications of Malliavin calculus. ... it provides a direct gateway to the relevant literature." (www.riskbook.com, November, 2006)

"The book under review is on the applications of the Malliavin calculus to financial mathematics. ... The authors have written a short book introducing the reader efficiently to the key points of the Malliavin calculus in mathematical finance. ... Also the list of references is comprehensive and updated, and gives a clear picture of the activity and relevance of this approach to many financial problems. ... This book is recommended to all researchers in mathematical finance." (MathSciNet, February, 2007)

"The book under review is on the applications of the Malliavin calculus to financial mathematics. ... The compact form is to the advantage of the reader, who is led to the applications rather quickly. ... This book is recommended to all researchers in mathematical finance. It shows how advanced mathematics can play an important role in solving practical financial problems as well as developing new understanding and concepts." (Fred Espen Benth, Mathematical Reviews, Issue 2007 b)

"The book under review demonstrates the power and versatility of the Malliavin calculus in a variety of problems arising in Mathematical Finance. Despite being mathematically demanding, it is directed not only towards researchers in mathematics, but also to practitioners ... . The book will certainly address in the first place researchers in mathematical finance. It can however be recommended to a much wider public in mathematics beyond probability ... ." (Peter Imkeller, Zentralblatt MATH, Vol. 1124 (1), 2008)

"This monograph is devoted to an updated presentation, in a rigorous mathematical framework, of the applications of the stochastic calculus of variations in mathematical finance. ... In conclusion, this book aims to explain the role played by the stochastic calculus of variations in mathematical finance, and it will be useful for researchers working in these fields." (David Nualart, Bulletin of the American Mathematical Society, Vol. 44 (3), July, 2007)

Gaussian Stochastic Calculus of Variationsp. 1
Finite-Dimensional Gaussian Spaces, Hermite Expansionp. 1
Wiener Space as Limit of its Dyadic Filtrationp. 5
Stroock-Sobolev Spaces of Functionals on Wiener Spacep. 7
Divergence of Vector Fields, Integration by Partsp. 10
Ito's Theory of Stochastic Integralsp. 15
Differential and Integral Calculus in Chaos Expansionp. 17
Monte-Carlo Computation of Divergencep. 21
Computation of Greeks and Integration by Parts Formulaep. 25
PDE Option Pricing; PDEs Governing the Evolution of Greeksp. 25
Stochastic Flow of Diffeomorphisms; Ocone-Karatzas Hedgingp. 30
Principle of Equivalence of Instantaneous Derivativesp. 33
Pathwise Smearing for European Optionsp. 33
Examples of Computing Pathwise Weightsp. 35
Pathwise Smearing for Barrier Optionp. 37
Market Equilibrium and Price-Volatility Feedback Ratep. 41
Natural Metric Associated to Pathwise Smearingp. 41
Price-Volatility Feedback Ratep. 42
Measurement of the Price-Volatility Feedback Ratep. 45
Market Ergodicity and Price-Volatility Feedback Ratep. 46
Multivariate Conditioning and Regularity of Lawp. 49
Non-Degenerate Mapsp. 49
Divergencesp. 51
Regularity of the Law of a Non-Degenerate Mapp. 53
Multivariate Conditioningp. 55
Riesz Transform and Multivariate Conditioningp. 59
Example of the Univariate Conditioningp. 61
Non-Elliptic Markets and Instability in HJM Modelsp. 65
Notation for Diffusions on R[superscript N]p. 66
The Malliavin Covariance Matrix of a Hypoelliptic Diffusionp. 67
Malliavin Covariance Matrix and Hormander Bracket Conditionsp. 70
Regularity by Predictable Smearingp. 70
Forward Regularity by an Infinite-Dimensional Heat Equationp. 72
Instability of Hedging Digital Options in HJM Modelsp. 73
Econometric Observation of an Interest Rate Marketp. 75
Insider Tradingp. 77
A Toy Model: the Brownian Bridgep. 77
Information Drift and Stochastic Calculus of Variationsp. 79
Integral Representation of Measure-Valued Martingalesp. 81
Insider Additional Utilityp. 83
An Example of an Insider Getting Free Lunchesp. 84
Asymptotic Expansion and Weak Convergencep. 87
Asymptotic Expansion of SDEs Depending on a Parameterp. 88
Watanabe Distributions and Descent Principlep. 89
Strong Functional Convergence of the Euler Schemep. 90
Weak Convergence of the Euler Schemep. 93
Stochastic Calculus of Variations for Markets with Jumpsp. 97
Probability Spaces of Finite Type Jump Processesp. 98
Stochastic Calculus of Variations for Exponential Variablesp. 100
Stochastic Calculus of Variations for Poisson Processesp. 102
Mean-Variance Minimal Hedging and Clark-Ocone Formulap. 104
Volatility Estimation by Fourier Expansionp. 107
Fourier Transform of the Volatility Functorp. 109
Numerical Implementation of the Methodp. 112
Strong Monte-Carlo Approximation of an Elliptic Marketp. 115
Definition of the Scheme [characters not reproducible]p. 116
The Milstein Schemep. 117
Horizontal Parametrizationp. 118
Reconstruction of the Scheme [characters not reproducible]p. 120
Numerical Implementation of the Price-Volatility Feedback Ratep. 123
Referencesp. 127
Indexp. 139
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783540434313
ISBN-10: 3540434313
Series: Springer Finance
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 142
Published: 3rd November 2005
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.89