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Interest rate models : an infinite dimensional stochastic analysis perspective

Author: R Carmona; Michael Tehranchi
Publisher: Berlin ; New York : Springer, ©2006.
Series: Springer finance.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"Interest Rate Models: an Infinite Dimensional Stochastic Analysis Perspective studies the mathematical issues that arise in modeling the interest rate term structure. These issues are approached by casting the interest rate models as stochastic evolution equations in infinite dimension."--Jacket.
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: R Carmona; Michael Tehranchi
ISBN: 3540270655 9783540270652 3642066003 9783642066009 3540270671 9783540270676
OCLC Number: 63905352
Description: xiv, 235 pages : illustrations ; 24 cm.
Contents: Part I. The Term Structure of Interest Rates : --
1. Data and Instruments of the Term Structure of Interest Rates --
2. Term Structure Factor Models --
Part II. Infinite Dimensional Stochastic Analysis : --
3. Infinite Dimensional Integration Theory --
4. Stochastic Analysis in Infinite Dimensions --
5. The Malliavin Calculus --
Part III. Generalized Models for the Term Structure : --
6. General Models --
7. Specific Models. Part I. The Term Structure of Interest Rates : --
1. Data and Instruments of the Term Structure of Interest Rates --
1.1. Time Value of Money and Zero Coupon Bonds --
1.1.1. Treasury Bills --
1.1.2. Discount Factors and Interest Rates --
1.2. Coupon Bearing Bonds --
1.2.1. Treasury Notes and Treasury Bonds --
1.2.2. The STRIPS Program --
1.2.3. Clean Prices --
1.3. Term Structure as Given by Curves --
1.3.1. The Spot (Zero Coupon) Yield Curve --
1.3.2. The Forward Rate Curve and Duration --
1.3.3. Swap Rate Curves --
1.4. Continuous Compounding and Market Conventions --
1.4.1. Day Count Conventions --
1.4.2. Compounding Conventions --
1.4.3. Summary --
1.5. Related Markets --
1.5.1. Municipal Bonds --
1.5.2. Index Linked Bonds --
1.5.3. Corporate Bonds and Credit Markets --
1.5.4. Tax Issues --
1.5.5. Asset Backed Securities --
1.6. Statistical Estimation of the Term Structure --
1.6.1. Yield Curve Estimation --
1.6.2. Parametric Estimation Procedures --
1.6.3. Nonparametric Estimation Procedures --
1.7. Principal Component Analysis --
1.7.1. Principal Components of a Random Vector --
1.7.2. Multivariate Data PCA --
1.7.3. PCA of the Yield Curve --
1.7.4. PCA of the Swap Rate Curve --
2. Term Structure Factor Models --
2.1. Factor Models for the Term Structure --
2.2. Affine Models --
2.3. Short Rate Models as One-Factor Models --
2.3.1. Incompleteness and Pricing --
2.3.2. Specific Models --
2.3.3. A PDE for Numerical Purposes --
2.3.4. Explicit Pricing Formulae --
2.3.5. Rigid Term Structures for Calibration --
2.4. Term Structure Dynamics --
2.4.1. The Heath-Jarrow-Morton Framework --
2.4.2. Hedging Contingent Claims --
2.4.3. A Shortcoming of the Finite-Rank Models --
2.4.4. The Musiela Notation --
2.4.5. Random Field Formulation --
2.5. Appendices --
Part II. Infinite Dimensional Stochastic Analysis : --
3. Infinite Dimensional Integration Theory --
3.1. Introduction --
3.1.1. The Setting --
3.1.2. Distributions of Gaussian Processes --
3.2. Gaussian Measures in Banach Spaces and Examples --
3.2.1. Integrability Properties --
3.2.2. Isonormal Processes --
3.3. Reproducing Kernel Hilbert Space --
3.3.1. RKHS of Gaussian Processes --
3.3.2. The RKHS of the Classical Wiener Measure --
3.4. Topological Supports, Carriers, Equivalence and Singularity --
3.4.1. Topological Supports of Gaussian Measures --
3.4.2. Equivalence and Singularity of Gaussian Measures --
3.5. Series Expansions --
3.6. Cylindrical Measures --
3.6.1. The Canonical (Gaussian) Cylindrical Measure of a Hilbert Space --
3.6.2. Integration with Respect to a Cylindrical Measure --
3.6.3. Characteristic Functions and Bochner's Theorem --
3.6.4. Radonification of Cylindrical Measures --
3.7. Appendices --
4. Stochastic Analysis in Infinite Dimensions --
4.1. Infinite Dimensional Wiener Processes --
4.1.1. Revisiting some Known Two-Parameter Processes --
4.1.2. Banach Space Valued Wiener Process --
4.1.3. Sample Path Regularity --
4.1.4. Absolute Continuity Issues --
4.1.5. Series Expansions --
4.2. Stochastic Integral and Ito Processes --
4.2.1. The Case of E*- and H*-Valued Integrands --
4.2.2. The Case of Operator Valued Integrands --
4.2.3. Stochastic Convolutions --
4.3. Martingale Representation Theorems --
4.4. Girsanov's Theorem and Changes of Measures --
4.5. Infinite Dimensional Ornstein-Uhlenbeck Processes --
4.5.1. Finite Dimensional OU Processes --
4.5.2. Infinite Dimensional OU Processes --
4.5.3. The SDE Approach in Infinite Dimensions --
4.6. Stochastic Differential Equations --
5. The Malliavin Calculus --
5.1. The Malliavin Derivative --
5.1.1. Various Notions of Differentiability --
5.1.2. The Definition of the Malliavin Derivative --
5.2. The Chain Rule --
5.3. The Skorohod Integral --
5.4. The Clark-Ocone Formula --
5.4.1. Sobolev and Logarithmic Sobolev Inequalities --
5.5. Malliavin Derivatives and SDEs --
5.5.1. Random Operators --
5.5.2. A Useful Formula --
5.6. Applications in Numerical Finance --
5.6.1. Computation of the Delta --
5.6.2. Computation of Conditional Expectations. Part III. Generalized Models for the Term Structure : --
6. General Models --
6.1. Existence of a Bond Market --
6.2. The HJM Evolution Equation --
6.2.1. Function Spaces for Forward Curves --
6.3. The Abstract HJM Model --
6.3.1. Drift Condition and Absence of Arbitrage --
6.3.2. Long Rates Never Fall --
6.3.3. A Concrete Example --
6.4. Geometry of the Term Structure Dynamics --
6.4.1. The Consistency Problem --
6.4.2. Finite Dimensional Realizations --
6.5. Generalized Bond Portfolios --
6.5.1. Models of the Discounted Bond Price Curve --
6.5.2. Trading Strategies --
6.5.3. Uniqueness of Hedging Strategies --
6.5.4. Approximate Completeness of the Bond Market --
6.5.5. Hedging Strategies for Lipschitz Claims --
7. Specific Models --
7.1. Markovian HJM Models --
7.1.1. Gaussian Markov Models --
7.1.2. Assumptions on the State Space --
7.1.3. Invariant Measures for Gauss-Markov HJM Models --
7.1.4. Non-Uniqueness of the Invariant Measure --
7.1.5. Asymptotic Behavior --
7.1.6. The Short Rate is a Maximum on Average --
7.2. SPDEs and Term Structure Models --
7.2.1. The Deformation Process --
7.2.2. A Model of the Deformation Process --
7.2.3. Analysis of the SPDE --
7.2.4. Regularity of the Solutions --
7.3. Market Models --
7.3.1. The Forward Measure --
7.3.2. LIBOR Rates Revisited.
Series Title: Springer finance.
Responsibility: René A. Carmona, Michael R. Tehranchi.
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Abstract:

This book presents the mathematical issues that arise in modeling the interest rate term structure by casting the interest-rate models as stochastic evolution equations in infinite dimensions.  Read more...

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