Stochastic Calculus of Variations in Mathematical Finance (eBook)

Paul Malliavin, Anton Thalmaier (Autoren)

eBook Download: PDF

2006 | 2006
XII, 142 Seiten
Springer Berlin (Verlag)
978-3-540-30799-0 (ISBN)

Infos

Highly esteemed author

Topics covered are relevant and timely

Inhalt

Preface 7
Contents 9
1Gaussian Stochastic Calculus of Variations 12
1.1 Finite-Dimensional Gaussian Spaces,Hermite Expansion 12
1.2 Wiener Space as Limit of its Dyadic Filtration 16
1.3 Stroock–Sobolev Spaces of Functionals on Wiener Space 18
1.4 Divergence of Vector Fields, Integration by Parts 21
1.5 Itö’s Theory of Stochastic Integrals 26
1.6 Differential and Integral Calculus in Chaos Expansion 28
1.7 Monte-Carlo Computation of Divergence 32
2 Computation of Greeks and Integration by Parts Formulae 36
2.1 PDE Option Pricing PDEs Governing the Evolution of Greeks36
2.2 Stochastic Flow of Diffeomorphisms Ocone-Karatzas Hedging41
2.3 Principle of Equivalence of Instantaneous Derivatives 44
2.4 Pathwise Smearing for European Options 44
2.5 Examples of Computing Pathwise Weights 46
2.6 Pathwise Smearing for Barrier Option 48
3 Market Equilibrium and Price-Volatility Feedback Rate 52
3.1 Natural Metric Associated to Pathwise Smearing 52
3.2 Price-Volatility Feedback Rate 53
3.3 Measurement of the Price-Volatility Feedback Rate 56
3.4 Market Ergodicity and Price-Volatility Feedback Rate 57
4 Multivariate Conditioning and Regularity of Law 60
4.1 Non-Degenerate Maps 60
4.2 Divergences 62
4.3 Regularity of the Law of a Non-Degenerate Map 64
4.4 Multivariate Conditioning 66
4.5 Riesz Transform and Multivariate Conditioning 70
4.6 Example of the Univariate Conditioning 72
5 Non-Elliptic Markets and Instability in HJM Models 76
5.1 Notation for Diffusions on 77
5.2 The Malliavin Covariance Matrix of a Hypoelliptic Di.usion 78
5.3 Malliavin Covariance Matrix and Hörmander Bracket Conditions 81
5.4 Regularity by Predictable Smearing 81
5.5 Forward Regularity by an Infnite-Dimensional Heat Equation 83
5.6 Instability of Hedging Digital Options 84
5.7 Econometric Observation of an Interest Rate Market 86
6 Insider Trading 88
6.1 A Toy Model: the Brownian Bridge 88
6.2 Information Drift and Stochastic Calculus of Variations 90
6.3 Integral Representation 92
of Measure-Valued Martingales 92
6.4 Insider Additional Utility 94
6.5 An Example of an Insider Getting Free Lunches 95
7 Asymptotic Expansion and Weak Convergence 98
7.1 Asymptotic Expansion of SDEs Depending on a Parameter 99
7.2 Watanabe Distributions and Descent Principle 100
7.3 Strong Functional Convergence of the Euler Scheme 101
7.4 Weak Convergence of the Euler Scheme 104
8 Stochastic Calculus of Variations for Markets with Jumps 108
8.1 Probability Spaces of Finite Type Jump Processes 109
8.2 Stochastic Calculus of Variations for Exponential Variables 111
8.3 Stochastic Calculus of Variations for Poisson Processes 113
8.4 Mean-Variance Minimal Hedging and Clark–Ocone Formula 115
A Volatility Estimation by Fourier Expansion 118
A.1 Fourier Transform of the Volatility Functor 120
A.2 Numerical Implementation of the Method 123
B Strong Monte-Carlo Approximation of an Elliptic Market 126
B.1 De.nition of the Scheme 127
B.2 The Milstein Scheme 128
B.3 Horizontal Parametrization 129
B.4 Reconstruction of the Scheme 131
C Numerical Implementation of the Price-Volatility Feedback Rate 134
References 138
Index 150

Textauszug

5 Non-Elliptic Markets and Instability in HJM Models (p.65)

In this chapter we drop the ellipticity assumption which served as a basic hypothesis in Chap. 3 and in Chap. 2, except in Sect. 2.2.

We give up ellipticity in order to be able to deal with models with random interest rates driven by Brownian motion (see [61] and [104]). The empirical market of interest rates satis.es the following two facts which rule out the ellipticity paradigm:

1) high dimensionality of the state space constituted by the values of bonds at a large numbers of distinct maturities;

2) low dimensionality variance which, by empirical variance analysis, within experimental error of 98/100, leads to not more than 4 independent scalar-valued Brownian motions, describing the noise driving this highdimensional system (see [41]).

Elliptic models are therefore ruled out and hypoelliptic models are then the most regular models still available. We shall show that these models display structural instability in smearing instantaneous derivatives which implies an unstable hedging of digital options.

Practitioners hedging a contingent claim on a single asset try to use all trading opportunities inside the market. In interest rate models practitioners will be reluctant to hedge a contingent claim written under bounds having a maturity less than .ve years by trading contingent claims written under bounds of maturity 20 years and more. This quite di.erent behaviour has been pointed out by R. Cont [52] and R. Carmona [48].

R. Carmona and M. Tehranchi [49] have shown that this empirical fact can be explained through models driven by an in.nite number of Brownian motions. We shall propose in Sect. 5.6 another explanation based on the progressive smoothing e.ect of the heat semigroup associated to a hypoelliptic operator, an e.ect which we call compartmentation.

This in.nite dimensionality phenomena is at the root of modelling the interest curve process: indeed it has been shown in [72] that the interest rate model process has very few .nite-dimensional realizations.

Section 5.7 develops for the interest rate curve a method similar to the methodology of the price-volatility feedback rate (see Chap. 3). We start by stating the possibility of measuring in real time, in a highly traded market, the full historical volatility matrix: indeed cross-volatility between the prices of bonds at two di.erent maturities has an economic meaning (see [93, 94]). As the market is highly non-elliptic, the multivariate price-volatility feedback rate constructed in [19] cannot be used. We substitute a pathwise econometric computation of the bracket of the driving vector of the di.usion. The question of e.ciency of these mathematical objects to decipher the state of the market requires numerical simulation on intra-day ephemerides leading to stable results at a properly chosen time scale.

Produkt-Details

• Erscheint lt. Verlag: 25.2.2006
• Reihe/Serie: Springer Finance
• Reihe/Serie: Springer Finance
• Zusatzinfo: XII, 142 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: American option • Calculus • Insider information • Malliavin calculus • Market equilibrium • Monte Carlo weight • MSC (2000): 60H30, 60H07, 60G44, 62P20, 91B24 • Price sensitivity • Quantitative Finance • Stochastic Calculus • Stochastic Processes • Volatility measurement
• ISBN-10: 3-540-30799-0 / 3540307990
• ISBN-13: 978-3-540-30799-0 / 9783540307990