Fixed Income Modelling

by Munk, Claus

• ISBN13: 9780199575084
• ISBN10: 0199575088
• Format: Hardcover
• Copyright: 2011-09-05
• Publisher: Oxford University Press

Summary

Fixed Income Modelling offers a unified presentation of dynamic term structure models and their applications to the pricing and risk management of fixed income securities. It explains the basic fixed income securities and their properties and uses as well as the relations between those securities. The book presents and compares the classical affine models, Heath-Jarrow-Morton models, and LIBOR market models, and demonstrates how to apply those models for the pricing of various widely traded fixed income securities. It offers a balanced presentation with both formal mathematical modelling and economic intuition and understanding.

The book has a number of distinctive features including a thorough and accessible introduction to stochastic processes and the stochastic calculus needed for the modern financial modelling approach used in the book, as well as a separate chapter that explains how the term structure of interest rates relates to macro-economic variables and to what extent the concrete interest rate models are founded in general economic theory. The book focuses on the most widely used models and the main fixed income securities, instead of trying to cover all the many specialized models and the countless exotic real-life products. The in-depth explanation of the main pricing principles, techniques, and models as well as their application to the most important types of securities will enable the reader to understand and apply other models and price other securities. The book includes chapters on interest rate risk management, credit risk, mortgage-backed securities, and relevant numerical techniques. Each chapter concludes with a number of exercises of varying complexity.

Suitable for MSc students specializing in finance and economics, quantitatively oriented MBA students, and first- or second-year PhD students, this book will also be a useful reference for researchers and finance professionals and can be used in specialized courses on fixed income or broader courses on derivatives.

Table of Contents

List of Figures	p. xiii
List of Tables	p. xv
Introduction and Overview	p. 1
What is fixed income analysis?	p. 1
Basic bond market terminology	p. 2
Bond markets and money markets	p. 12
Fixed income derivatives	p. 18
An overview of the book	p. 21
Exercises	p. 23
Extracting Yield Curves from Bond Prices	p. 25
Introduction	p. 25
Bootstrapping	p. 26
Cubic splines	p. 29
The Nelson-Siegel parametrization	p. 33
Additional remarks on yield curve estimation	p. 36
Exercises	p. 37
Stochastic Processes and Stochastic Calculus	p. 38
Introduction	p. 38
What is a stochastic process?	p. 39
Brownian motions	p. 47
Diffusion processes	p. 52
Itô processes	p. 53
Stochastic integrals	p. 54
Itô's Lemma	p. 58
Important diffusion processes	p. 59
Multi-dimensional processes	p. 68
Change of probability measures	p. 76
Exercises	p. 80
A Review of General Asset Pricing Theory	p. 82
Introduction	p. 82
Assets, trading strategies, and arbitrage	p. 84
State-price deflators, risk-neutral probabilities, and marker prices of risk	p. 88
Other useful probability measures	p. 97
Complete vs. Incomplete markets	p. 100
Equilibrium and representative agents in complete markets	p. 102
Extension to intermediate dividends	p. 104
Diffusion models and the fundamental partial differential equation	p. 106
Concluding remarks	p. 114
Exercises	p. 115
The Economics of the Term Structure of Interest Rates	p. 116
Introduction	p. 116
Real interest rates and aggregate consumption	p. 117
Real interest rates and aggregate production	p. 122
Equilibrium term stricture models	p. 125
Real and nominal interest rates and term structures	p. 128
The expectation hypothesis	p. 140
Liquidity preference, market segmentation, and preferred habitats	p. 145
Concluding remarks	p. 146
Exercises	p. 147
Fixed Incomes Securities	p. 150
Introduction	p. 150
Forwards and futures	p. 151
European options	p. 158
Caps, floors, and collars	p. 163
Swaps and swaptions	p. 169
American-style derivatives	p. 178
An overview of term structure models	p. 180
Exercises	p. 182
One-Factor Diffusion Models	p. 184
Introduction	p. 184
Affine models	p. 185
Merton's model	p. 196
Vasicek's model	p. 200
The Cox-Ingersoll-Ross model	p. 214
Generalized affine models	p. 220
Non-affine models	p. 221
Parameter estimation and empirical tests	p. 224
Concluding remarks	p. 228
Exercises	p. 229
Multi-Factor Diffusion Models	p. 231
Introduction	p. 231
The general multi-factor setting	p. 234
Affine multi-factor models	p. 236
Two-factor affine diffusion models	p. 243
Three-factor affine models	p. 252
Generalized affine models	p. 255
Other multi-factor diffusion models	p. 257
Final remarks	p. 261
Exercises	p. 262
Calibration of Diffusion Models	p. 264
Introduction	p. 264
Time-inhomogeneous affine models	p. 265
The Ho-Lee model (extended Merton)	p. 267
The Hull-White model (extended Vasicek)	p. 269
The extended CIR model	p. 273
Calibration to other market data	p. 274
Initial and future term structures in calibrated models	p. 275
Calibrated non-affine models	p. 277
Is a calibrated one-factor model just as good as a multi-factor model?	p. 278
Final remarks	p. 280
Exercises	p. 280
Health-Jarrow-Morton Models	p. 281
Introduction	p. 281
Basic assumption	p. 281
Bond price dynamics and the drift restriction	p. 283
Three well-known special cases	p. 285
Gaussian HJM models	p. 289
Diffusion representation of HJM models	p. 292
HJM models with forward-rate dependent volatilities	p. 298
HJM models with unspanned stochastic volatility	p. 299
Concluding remarks	p. 299
Exercises	p. 300
Market Models	p. 303
Introduction	p. 303
General LIBOR market models	p. 304
The lognormal LIBOR market model	p. 312
Alternative KIBOR market models	p. 316
Swap market models	p. 318
Further remarks	p. 321
Exercises	p. 321
The Measurement and Management of Interest Rate Risk	p. 323
Introduction	p. 323
Traditional measures of interest rate risk	p. 323
Risk measures in one-factor diffusion models	p. 328
Immunization	p. 335
Risk measures in multi-factor diffusion models	p. 342
Duration-based pricing of options on bonds	p. 346
Alternative measures of interest rate risk	p. 351
Exercises	p. 354
Defaultable Bonds and Credit Derivatives	p. 355
Introduction	p. 355
Some basic concepts, relations and practical issues	p. 357
Structural models	p. 368
Reduced-form models	p. 384
Hybrid models	p. 401
Copulas	p. 402
Markets for credit derivatives	p. 407
Credit default swaps (CDSs)	p. 410
Collateralized debt obligations (CDOs)	p. 415
Concluding remarks	p. 418
Exercises	p. 420
Mortagages and Mortgage-backed Securities	p. 422
Introduction	p. 422
Mortgages	p. 423
Mortgage-backed bonds	p. 429
The prepayment option	p. 430
Rational prepayment models	p. 433
Empirical prepayment models	p. 445
Risk measures for mortgage-backed bonds	p. 448
Other mortgage-backed securities	p. 448
The subprime crisis	p. 450
Concluding remarks	p. 452
Exercises	p. 453
Stock and Currency Derivatives When Interest Rates are Stochastic	p. 454
Introduction	p. 454
Stock options	p. 454
Options on forwards and futures	p. 460
Currency derivatives	p. 464
Final remarks	p. 470
Exercises	p. 471
Numerical Techniques	p. 472
Introduction	p. 472
Numerical solution of PDEs	p. 474
Monte Carlo simulation	p. 492
Approximating trees	p. 517
Concluding remarks	p. 527
Exercises	p. 528
Appendix A: Results on the Lognormal Distribution	p. 532
References	p. 535
Index	p. 553
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