Stochastic Calculus of Variations in Mathematical Finance / Edition 1

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Overview

Malliavin calculus provides an infinite-dimensional differential calculus in the context of continuous path stochastic processes. The calculus includes formulae of integration by parts and Sobolev spaces of differentiable functions defined on a probability space. This new book, demonstrating the relevance of Malliavin calculus for Mathematical Finance, starts with an exposition from scratch of this theory. 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. Finite-dimensional projections of infinite-dimensional Sobolev spaces lead to Monte Carlo computations of conditional expectations useful for computing American options. The discretization error of the Euler scheme for a stochastic differential equation is expressed as a generalized Watanabe distribution on the Wiener space. 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.

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Editorial Reviews

From the Publisher
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 shastic calculus of variations in mathematical finance. ... In conclusion, this book aims to explain the role played by the shastic 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)

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Product Details

  • ISBN-13: 9783540434313
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 12/19/2005
  • Series: Springer Finance Series
  • Edition description: 2006
  • Edition number: 1
  • Pages: 142
  • Product dimensions: 0.44 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

1 Gaussian Stochastic Calculus of Variations 1
1.1 Finite-Dimensional Gaussian Spaces, Hermite Expansion 1
1.2 Wiener Space as Limit of its Dyadic Filtration 5
1.3 Stroock-Sobolev Spaces of Functionals on Wiener Space 7
1.4 Divergence of Vector Fields, Integration by Parts 10
1.5 Ito's Theory of Stochastic Integrals 15
1.6 Differential and Integral Calculus in Chaos Expansion 17
1.7 Monte-Carlo Computation of Divergence 21
2 Computation of Greeks and Integration by Parts Formulae 25
2.1 PDE Option Pricing; PDEs Governing the Evolution of Greeks 25
2.2 Stochastic Flow of Diffeomorphisms; Ocone-Karatzas Hedging 30
2.3 Principle of Equivalence of Instantaneous Derivatives 33
2.4 Pathwise Smearing for European Options 33
2.5 Examples of Computing Pathwise Weights 35
2.6 Pathwise Smearing for Barrier Option 37
3 Market Equilibrium and Price-Volatility Feedback Rate 41
3.1 Natural Metric Associated to Pathwise Smearing 41
3.2 Price-Volatility Feedback Rate 42
3.3 Measurement of the Price-Volatility Feedback Rate 45
3.4 Market Ergodicity and Price-Volatility Feedback Rate 46
4 Multivariate Conditioning and Regularity of Law 49
4.1 Non-Degenerate Maps 49
4.2 Divergences 51
4.3 Regularity of the Law of a Non-Degenerate Map 53
4.4 Multivariate Conditioning 55
4.5 Riesz Transform and Multivariate Conditioning 59
4.6 Example of the Univariate Conditioning 61
5 Non-Elliptic Markets and Instability in HJM Models 65
5.1 Notation for Diffusions on R[superscript N] 66
5.2 The Malliavin Covariance Matrix of a Hypoelliptic Diffusion 67
5.3 Malliavin Covariance Matrix and Hormander Bracket Conditions 70
5.4 Regularity by Predictable Smearing 70
5.5 Forward Regularity by an Infinite-Dimensional Heat Equation 72
5.6 Instability of Hedging Digital Options in HJM Models 73
5.7 Econometric Observation of an Interest Rate Market 75
6 Insider Trading 77
6.1 A Toy Model: the Brownian Bridge 77
6.2 Information Drift and Stochastic Calculus of Variations 79
6.3 Integral Representation of Measure-Valued Martingales 81
6.4 Insider Additional Utility 83
6.5 An Example of an Insider Getting Free Lunches 84
7 Asymptotic Expansion and Weak Convergence 87
7.1 Asymptotic Expansion of SDEs Depending on a Parameter 88
7.2 Watanabe Distributions and Descent Principle 89
7.3 Strong Functional Convergence of the Euler Scheme 90
7.4 Weak Convergence of the Euler Scheme 93
8 Stochastic Calculus of Variations for Markets with Jumps 97
8.1 Probability Spaces of Finite Type Jump Processes 98
8.2 Stochastic Calculus of Variations for Exponential Variables 100
8.3 Stochastic Calculus of Variations for Poisson Processes 102
8.4 Mean-Variance Minimal Hedging and Clark-Ocone Formula 104
A Volatility Estimation by Fourier Expansion 107
A.1 Fourier Transform of the Volatility Functor 109
A.2 Numerical Implementation of the Method 112
B Strong Monte-Carlo Approximation of an Elliptic Market 115
B.1 Definition of the Scheme [characters not reproducible] 116
B.2 The Milstein Scheme 117
B.3 Horizontal Parametrization 118
B.4 Reconstruction of the Scheme [characters not reproducible] 120
C Numerical Implementation of the Price-Volatility Feedback Rate 123
References 127
Index 139
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