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Stochastic Calculus of Variations in Mathematical Finance

:	Paul Malliavin, Anton Thalmaier
:	Stochastic Calculus of Variations in Mathematical Finance
:	Springer-Verlag
:	9783540307990
:	1
:	CHF 47.50
:

:	Volkswirtschaft
:	English

:	142
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

Malliav n calculus provides an infinite-dimensional differential calculus in the context of continuous paths 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.

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.

	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