Term-Structure Models

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.