Term-Structure Models

Changing interest rates constitute one of the major risk sources for banks, insurance companies, and other financial institutions. Modeling the term-structure movements of interest rates is a challenging task. This volume gives an introduction to the mathematics of term-structure models in continuous time. It includes practical aspects for fixed-income markets such as day-count conventions, duration of coupon-paying bonds and yield curve construction; arbitrage theory; short-rate models; the Heath-Jarrow-Morton methodology; consistent term-structure parametrizations; affine diffusion processes and option pricing with Fourier transform; LIBOR market models; and credit risk.

The focus is on a mathematically straightforward but rigorous development of the theory. Students, researchers and practitioners will find this volume very useful. Each chapter ends with a set of exercises, that provides source for homework and exam questions. Readers are expected to be familiar with elementary Itô calculus, basic probability theory, and real and complex analysis.

Auteur

Damir Filipovic is head of the Vienna Institute of Finance, a research institution in the field of Mathematical Finance, funded by the Vienna Science and Technology Fund (WWTF), and founded and co-funded by the University of Vienna and the Vienna University of Economics and Business Administration. Prior to this position he held the Chair of Financial and Insurance Mathematics at the University of Munich, and he was Assistant Professor at Princeton University. Moreover, he worked for the Swiss Federal Office of Private Insurance, where he co-developed the Swiss Solvency Test (SST) - a risk based solvency assessment for insurance undertakings - which was enacted in 2006. He also held visiting positions at ETH Zurich, Columbia University, Stanford University, and the Vienna University of Technology.

Contenu

1 Introduction.- 2 Interest Rates and Related Contracts.- 2.1 Zero-Coupon Bonds.- 2.2 Interest Rates.- 2.2.1 Market Example: LIBOR.- 2.2.2 Simple vs. Continuous Compounding.- 2.2.3 Forward vs. Future Rates.- 2.3 Bank Account and Short Rates.- 2.4 Coupon Bonds, Swaps and Yields.- 2.4.1 Fixed Coupon Bonds.- 2.4.2 Floating Rate Notes.- 2.4.3 Interest Rate Swaps.- 2.4.4 Yield and Duration.- 2.5 Market Conventions.- 2.5.1 Day-count Conventions.- 2.5.2 Coupon Bonds.- 2.5.3 Accrued Interest, Clean Price and Dirty Price.- 2.5.4 Yield-to-Maturity.- 2.6 Caps and Floors.- 2.7 Swaptions.- 3 Statistics of the Yield Curve.- 3.1 Principal Component Analysis (PCA).- 3.2 PCA of the Yield Curve.- 3.3 Correlation.- 4 Estimating the Yield Curve.- 4.1 A Bootstrapping Example.- 4.2 General Case.- 4.2.1 Bond Markets.- 4.2.2 Money Markets.- 4.2.3 Problems.- 4.2.4 Parameterized Curve Families.- 5 Arbitrage Theory.- 5.1 Self-Financing Portfolios.- 5.1.1 Financial Market.- 5.1.2 Self-financing Portfolios.- 5.1.3 Numeraires.- 5.2 Arbitrage and Martingale Measures.- 5.2.1 Contingent Claims.- 5.2.2 Arbitrage.- 5.2.3 Martingale Measures.- 5.2.4 Market Price of Risk.- 5.2.5 Admissible Strategies.- 5.2.6 The Fundamental Theorem of Asset Pricing.- 5.3 Hedging and Pricing.- 5.3.1 Attainable Claims.- 5.3.2 Complete Markets.- 5.3.3 Pricing.- 5.3.4 State-price Density.- 6 Short Rate Models.- Generalities.- 6.2 Diffusion Short Rate Models.- 6.2.1 Examples.- 6.3 Inverting the Yield Curve.- 6.4 Affine Term Structures.- 6.5 Some Standard Models.- 6.5.1 Vasicek Model.- 6.5.2 Cox-Ingersoll-Ross Model.- 6.5.3 Dothan Model.- 6.5.4 Ho-Lee Model.- 6.5.5 Hull-White Model.- 7 HJM Methodology.- Forward Curve Movements.- 7.2 Absence of Arbitrage .- 7.3 Short Rate Dynamics.- 7.4 Fubini's Theorem.- 7.5 Explosion of Lognormal Forward Rates.- 8 Forward Measures.- 8.1 T-Bond as Numeraire.- 8.2 An Expectation Hypothesis.- 8.3 Option Pricing in Gaussian HJM Models.- 8.4 Black-Scholes Model with Stochastic Short Rates.- 9Forwards and Futures.- 9.1 Forward Contracts.- 9.2 Futures Contracts.- 9.3 Interest Rate Futures.- 9.4 Forward vs. Futures in a Gaussian Setup.- 10 Consistent Term Structure Parameterizations.- 10.1 No-Arbitrage Condition.- 10.2 Affine Term Structures.- 10.3 Polynomial Term Structures.- 10.4 Exponential-Polynomial Families.- 10.4.1 Nelson{Siegel Family.- 10.2 Svensson Family.- 11 Affine Processes.- 11.1 Option Pricing in Affine Models.- 11.1.1 Vasicek Model.- 11.1.2 Cox-Ingersoll-Ross Model.- 12 Market Models.- 12.1 Models of Forward LIBOR Rates.- 12.1.1 Discrete-tenor Case.- 12.1.2 Continuous-tenor Case.- 13 Default Risk.- 13.1 Transition and Default Probabilities.- 13.1.1 Historical Method.- 13.1.2 Structural Approach.- 13.2 Intensity Based Method.- 13.2.1 Construction of Intensity Based Models.- 13.2.2 Computation of Default Probabilities.- 13.2.3 Pricing Default Risk.- 13.2.4 Measure Change.