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Interest Rate Models : An Introduction - Andrew J.G. Cairns

Interest Rate Models

An Introduction

Paperback Published: 25th January 2004
ISBN: 9780691118949
Number Of Pages: 288

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The field of financial mathematics has developed tremendously over the past thirty years, and the underlying models that have taken shape in interest rate markets and bond markets, being much richer in structure than equity-derivative models, are particularly fascinating and complex. This book introduces the tools required for the arbitrage-free modelling of the dynamics of these markets. Andrew Cairns addresses not only seminal works but also modern developments. Refreshingly broad in scope, covering numerical methods, credit risk, and descriptive models, and with an approachable sequence of opening chapters, Interest Rate Models will make readers--be they graduate students, academics, or practitioners--confident enough to develop their own interest rate models or to price nonstandard derivatives using existing models.

The mathematical chapters begin with the simple binomial model that introduces many core ideas. But the main chapters work their way systematically through all of the main developments in continuous-time interest rate modelling. The book describes fully the broad range of approaches to interest rate modelling: short-rate models, no-arbitrage models, the Heath-Jarrow-Morton framework, multifactor models, forward measures, positive-interest models, and market models. Later chapters cover some related topics, including numerical methods, credit risk, and model calibration. Significantly, the book develops the martingale approach to bond pricing in detail, concentrating on risk-neutral pricing, before later exploring recent advances in interest rate modelling where different pricing measures are important.

Industry Reviews

"This book provides an excellent introduction to the field of interest-rate modeling for readers at the graduate level with a background in mathematics. It covers all key models and topics in the field and provides first glances at practical issues (calibration) and important related fields (credit risk). The mathematics is structured very well."--R diger Kiesel, University of Ulm, coauthor of Risk-Neutral Valuation
"A very useful book that provides clear and comprehensive discussions of the topic that are not easily available elsewhere."--Edwin J. Elton, New York University, author of Modern Portfolio Theory and Investment Analysis

Preface ixAcknowledgements xiii1. Introduction to Bond Markets 11.1 Bonds 11.2 Fixed-Interest Bonds 21.3 STRIPS 101.4 Bonds with Built-in Options 101.5 Index-Linked Bonds 101.6 General Theories of Interest Rates 111.7 Exercises 132. Arbitrage-Free Pricing 152.1 Example of Arbitrage: Parallel Yield Curve Shifts 162.2 Fundamental Theorem of Asset Pricing 182.3 The Long-Term Spot Rate 192.4 Factors 232.5 A Bond Is a Derivative 232.6 Put-Call Parity 232.7 Types of Model 242.8 Exercises 253. Discrete-Time Binomial Models 293.1 A Simple No-Arbitrage Model 293.2 The Ho and Lee No-Arbitrage Model 303.3 Recombining Binomial Model 323.4 Models for the Risk-Free Rate of Interest 373.5 Futures Contracts 453.6 Exercises 484. Continuous-Time Interest Rate Models 534.1 One-Factor Models for the Risk-Free Rate 534.2 The Martingale Approach 554.3 The PDE Approach to Pricing 604.4 Further Comment on the General Results 644.5 The Vasicek Model 644.6 The Cox-Ingersoll-Ross Model 664.7 A Comparison of the Vasicek and Cox-Ingersoll-Ross Models 704.8 Affine Short-Rate Models 744.9 Other Short-Rate Models 774.10 Options on Coupon-Paying Securities 774.11 Exercises 785. No-Arbitrage Models 855.1 Introduction 855.2 Markov Models 865.3 The Heath-Jarrow-Morton (HJM) Framework 915.4 Relationship between HJM and Markov Models 965.5 Exercises 976. Multifactor Models 1016.1 Introduction 1016.2 Affine Models 1026.3 Consols Models 1126.4 Multifactor Heath-Jarrow-Morton Models 1156.5 Options on Coupon-Paying Securities 1166.6 Quadratic Term-Structure Models (QTSMs) 1186.7 Other Multifactor Models 1186.8 Exercises 1197. The Forward-Measure Approach 1217.1 A New Numeraire 1217.2 Change of Measure 1227.3 Derivative Payments 1227.4 A Replicating Strategy 1237.5 Evaluation of a Derivative Price 1247.6 Equity Options with Stochastic Interest 1267.7 Exercises 1288. Positive Interest 1318.1 Introduction 1318.2 Mathematical Development 1318.3 The Flesaker and Hughston Approach 1348.4 Derivative Pricing 1358.5 Examples 1368.6 Exercises 1429. Market Models 1439.1 Market Rates of Interest 1439.2 LIBOR Market Models: the BGM Approach 1449.3 Simulation of LIBOR Market Models 1529.4 Swap Market Models 1539.5 Exercises 15510 Numerical Methods 15910.1 Choice of Measure 15910.2 Lattice Methods 16010.3 Finite-Difference Methods 16810.4 Numerical Examples 17810.5 Simulation Methods 18410.6 Exercise 19611 Credit Risk 19711.1 Introduction 19711.2 Structural Models 19911.3 A Discrete-Time Model 20111.4 Reduced-Form Models 20611.5 Derivative Contracts with Credit Risk 21811.6 Exercises 22212 Model Calibration 22712.1 Descriptive Models for the Yield Curve 22712.2 A General Parametric Model 22812.3 Estimation 22912.4 Splines 23412.5 Volatility Calibration 23812.6 Exercises 239Appendix A: Summary of Key Probability and SDE Theory 241A.1 The Multivariate Normal Distribution 241A.2 Brownian Motion 241A.3 Ito Integrals 242A.4 One-Dimensional Ito and Diffusion Processes 243A.5 Multi-Dimensional Diffusion Processes 244A.6 The Feynman-Kac Formula 245A.7 The Martingale Representation Theorem 246A.8 Change of Probability Measure 246Appendix B: The Vasicek and CIR Models: Proofs 249B.1 The Vasicek Model 249B.2 The Cox-Ingersoll-Ross Model 253References 265Index 271

ISBN: 9780691118949
ISBN-10: 0691118949
Series: International Studen
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 288
Published: 25th January 2004
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 22.86 x 15.24  x 1.91
Weight (kg): 0.41

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