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Published: 30th November 2010This 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 Variations | p. 1 |
| Finite-Dimensional Gaussian Spaces, Hermite Expansion | p. 1 |
| Wiener Space as Limit of its Dyadic Filtration | p. 5 |
| Stroock-Sobolev Spaces of Functionals on Wiener Space | p. 7 |
| Divergence of Vector Fields, Integration by Parts | p. 10 |
| Ito's Theory of Stochastic Integrals | p. 15 |
| Differential and Integral Calculus in Chaos Expansion | p. 17 |
| Monte-Carlo Computation of Divergence | p. 21 |
| Computation of Greeks and Integration by Parts Formulae | p. 25 |
| PDE Option Pricing; PDEs Governing the Evolution of Greeks | p. 25 |
| Stochastic Flow of Diffeomorphisms; Ocone-Karatzas Hedging | p. 30 |
| Principle of Equivalence of Instantaneous Derivatives | p. 33 |
| Pathwise Smearing for European Options | p. 33 |
| Examples of Computing Pathwise Weights | p. 35 |
| Pathwise Smearing for Barrier Option | p. 37 |
| Market Equilibrium and Price-Volatility Feedback Rate | p. 41 |
| Natural Metric Associated to Pathwise Smearing | p. 41 |
| Price-Volatility Feedback Rate | p. 42 |
| Measurement of the Price-Volatility Feedback Rate | p. 45 |
| Market Ergodicity and Price-Volatility Feedback Rate | p. 46 |
| Multivariate Conditioning and Regularity of Law | p. 49 |
| Non-Degenerate Maps | p. 49 |
| Divergences | p. 51 |
| Regularity of the Law of a Non-Degenerate Map | p. 53 |
| Multivariate Conditioning | p. 55 |
| Riesz Transform and Multivariate Conditioning | p. 59 |
| Example of the Univariate Conditioning | p. 61 |
| Non-Elliptic Markets and Instability in HJM Models | p. 65 |
| Notation for Diffusions on R[superscript N] | p. 66 |
| The Malliavin Covariance Matrix of a Hypoelliptic Diffusion | p. 67 |
| Malliavin Covariance Matrix and Hormander Bracket Conditions | p. 70 |
| Regularity by Predictable Smearing | p. 70 |
| Forward Regularity by an Infinite-Dimensional Heat Equation | p. 72 |
| Instability of Hedging Digital Options in HJM Models | p. 73 |
| Econometric Observation of an Interest Rate Market | p. 75 |
| Insider Trading | p. 77 |
| A Toy Model: the Brownian Bridge | p. 77 |
| Information Drift and Stochastic Calculus of Variations | p. 79 |
| Integral Representation of Measure-Valued Martingales | p. 81 |
| Insider Additional Utility | p. 83 |
| An Example of an Insider Getting Free Lunches | p. 84 |
| Asymptotic Expansion and Weak Convergence | p. 87 |
| Asymptotic Expansion of SDEs Depending on a Parameter | p. 88 |
| Watanabe Distributions and Descent Principle | p. 89 |
| Strong Functional Convergence of the Euler Scheme | p. 90 |
| Weak Convergence of the Euler Scheme | p. 93 |
| Stochastic Calculus of Variations for Markets with Jumps | p. 97 |
| Probability Spaces of Finite Type Jump Processes | p. 98 |
| Stochastic Calculus of Variations for Exponential Variables | p. 100 |
| Stochastic Calculus of Variations for Poisson Processes | p. 102 |
| Mean-Variance Minimal Hedging and Clark-Ocone Formula | p. 104 |
| Volatility Estimation by Fourier Expansion | p. 107 |
| Fourier Transform of the Volatility Functor | p. 109 |
| Numerical Implementation of the Method | p. 112 |
| Strong Monte-Carlo Approximation of an Elliptic Market | p. 115 |
| Definition of the Scheme [characters not reproducible] | p. 116 |
| The Milstein Scheme | p. 117 |
| Horizontal Parametrization | p. 118 |
| Reconstruction of the Scheme [characters not reproducible] | p. 120 |
| Numerical Implementation of the Price-Volatility Feedback Rate | p. 123 |
| References | p. 127 |
| Index | p. 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
