Interest Rate Derivatives: Valuation, Calibration and Sensitivity Analysis

The class of interest rate models introduced by O. Cheyette in 1994 is a subclass of the general HJM framework with a time dependent volatility parameterization. This book addresses the above mentioned class of interest rate models and concentrates on the calibration, valuation and sensitivity analysis in multifactor models. It derives analytical pricing formulas for bonds and caplets and applies several numerical valuation techniques in the class of Cheyette model, i.e. Monte Carlo simulation, characteristic functions and PDE valuation based on sparse grids. Finally it focuses on the sensitivity analysis of Cheyette models and derives Model- and Market Greeks. To the best of our knowledge, this sensitivity analysis of interest rate derivatives in the class of Cheyette models is unique in the literature. Up to now the valuation of interest rate derivatives using PDEs has been restricted to 3 dimensions only, since the computational effort was too great. The author picks up the sparse grid technique, adjusts it slightly and can solve high-dimensional PDEs (four dimensions plus time) accurately in reasonable time. Many topics investigated in this book are new areas of research and make a significant contribution to the scientific community of financial engineers. They also represent a valuable development for practitioners.
Table of Contents
Cover
Interest Rate Derivatives - Valuation, Calibration and Sensitivity Analysis
ISBN 9783642349249 ISBN 9783642349256
Foreword
Preface
Acknowledgment
Contents
Literature Review
The Cheyette Model Class
2.1 The Heath-Jarrow-Morton Framework
2.2 Derivation of the Cheyette Model Class
2.3 Particular Models in the Cheyette Model Class
2.3.1 Ho-Lee Model
2.3.2 Hull-White Model
2.3.3 The Three Factor Exponential Model
2.4 Remarks on the Cheyette Model Class
Analytical Pricing Formulas
3.1 Bonds 3.1.1 Multifactor Cheyette Model
3.1.2 One-Factor Cheyette Model
3.1.3 The Three Factor Exponential Model
3.2 Caplets/Floorlets
3.2.1 The Three Factor Exponential Model
3.3 Swaptions
Calibration
4.1 Literature Review
4.2 The Calibration Problem 4.2.1 Formulation
4.2.2 Constraints
4.2.3 Characterization of the Optimization Space
4.2.4 Quality Check
4.3 Optimization Methods
4.3.1 Newton Algorithm
4.3.2 Powell Algorithm
4.3.3 Downhill Simplex Algorithm
4.3.4 Simulated Annealing
4.3.5 Genetic Optimization
4.4 Numerical Results
Monte Carlo Methods
5.1 Literature Review
5.2 Simulations in the Cheyette Model Class
5.3 Quasi-Monte Carlo Simulation
5.4 Pricing Bonds and European Options 5.4.1 Pricing Under the Forward Measure
5.4.2 Distribution of the State Variables
5.4.3 Covariance
5.4.4 Numerical Results
5.5 Pricing Bermudan Swaptions
5.5.1 Problem Formulation
5.5.2 Random Tree Methods
5.5.3 Numerical Results
5.6 Snowballs
5.6.1 Numerical Results
Characteristic Function Method
6.1 Literature Review
6.2 Affin Diffusion Setup 6.2.1 Fundamentals
6.2.2 Classifcation of the Cheyette Model Class
6.3 Characteristic Functions 6.3.1 Fundamentals
6.3.2 Characteristic Functions in the Affin Diffusion Setup
6.3.3 Characteristic Functions in the Cheyette Model Class
6.4 Pricing with Characteristic Functions 6.4.1 Fundamentals
6.4.2 Caplets
6.4.3 Numerical Results
6.5 Numerical Analysis
PDE Valuation
7.1 Literature Review
7.2 Derivation of the Valuation PDE
7.2.1 Some Specifi Examples
7.3 Boundary Conditions
7.3.1 Terminal Condition
7.3.2 Spatial Domain
7.4 Remarks on the Valuation PDE
7.5 Numerical Method for PDE Valuation
7.5.1 Finite Difference with PSOR
7.5.2 Sparse Grid Implementation
7.6 Numerical Results
7.6.1 Bonds
7.6.2 Caplets
7.6.3 European Swaptions
7.6.4 Bermudan Swaptions
Comparison of Valuation Techniques for Interest Rate Derivatives
8.1 Plain Vanillas 8.1.1 Bonds
8.1.2 Caplets
8.1.3 European Swaptions
8.2 Exotics 8.2.1 Bermudan Swaptions
8.2.2 Snowballs
Greeks
9.1 Literature Review
9.2 Bonds
9.2.1 Model-Greeks
9.2.2 Market-Greeks
9.3 Caplets
9.3.1 Model-Greeks
9.3.2 Market-Greeks
9.3.3 Stability of Greeks
9.4 Swaptions 9.4.1 European Swaptions
9.4.2 Bermudan Swaptions
Conclusion
Additional Calculus in the Class of Cheyette Models
Mathematical Tools
Market Data
References
Index