Interest Rate Dynamics, Derivatives Pricing, and Risk Management

There are two types of tenn structure models in the literature: the equilibrium models and the no-arbitrage models. And there are, correspondingly, two types of interest rate derivatives pricing fonnulas based on each type of model of the tenn structure. The no-arbitrage models are characterized by the work of Ho and Lee (1986), Heath, Jarrow, and Morton (1992), Hull and White (1990 and 1993), and Black, Dennan and Toy (1990). Ho and Lee (1986) invent the no-arbitrage approach to the tenn structure modeling in the sense that the model tenn structure can fit the initial (observed) tenn structure of interest rates. There are a number of disadvantages with their model. First, the model describes the whole volatility structure by a sin­ gle parameter, implying a number of unrealistic features. Furthennore, the model does not incorporate mean reversion. Black-Dennan-Toy (1990) develop a model along tbe lines of Ho and Lee. They eliminate some of the problems of Ho and Lee (1986) but create a new one: for a certain specification of the volatility function, the short rate can be mean-fteeting rather than mean-reverting. Heath, Jarrow and Morton (1992) (HJM) construct a family of continuous models of the term struc­ ture consistent with the initial tenn structure data.

1 A Three-Factor Model of the Term Structure of Interest Rates.- 1.1 Introduction.- 1.2 The Model.- 1.3 Benchmark Case.- 1.4 Green’s Function.- 1.5 Derivatives Pricing.- 1.6 The Term Structure of Interest Rates.- 1.7 Expected Future Short Rate.- 1.8 Forward Rates.- 2 Pricing Interest Rate Derivatives.- 2.1 Introduction.- 2.2 Bond Options.- 2.3 Caps, Floors, and Collars.- 2.4 Futures Price and Forward Price.- 2.5 Swaps.- 2.6 Quality Delivery Options.- 2.7 Futures Options.- 2.8 American Options.- 3 Pricing Exotic Options.- 3.1 Introduction.- 3.2 Green’s Function in the Presence of Boundaries.- 3.3 Derivatives with Payoffs at Random Times.- 3.4 Barrier Options.- 3.5 Lookback Options.- 3.6 Yield Options.- 4 Fitting to a Given Term Structure.- 4.1 Introduction.- 4.2 Merging to the Heath-Jarrow-Morton Framework.- 4.3 Whole-Yield Model.- 5 A Discrete-Time Version of the Model.- 5.1 Introduction.- 5.2 Construction of the Four-Dimensional Lattice.- 5.3 Applications.- 6 Estimation of the Model.- 6.1 Introduction.- 6.2 Kaiman Filter.- 6.3 Maximum Likelihood.- 6.4 Method of Moments.- 6.5 Simulated Moments.- 7 Managing Interest Rate Risk.- 7.1 Introduction.- 7.2 Generalized Duration and Convexity.- 7.3 Hedging Ratios.- 7.4 Hedging: General Approach.- 7.5 Hedging Yield Curve Risk.- 8 Extensions of the Model.- 8.1 Introduction.- 8.2 Extension I: Jumping Mean and Diffusing Volatility.- 8.3 Extension II: Jumping Mean and Jumping Volatility.- 9 Concluding Remarks.- A Proof of Lemma 1.- B Proof of Proposition 2.- C Proof of Lemma 2.- D Proof of Proposition 8.- E Integral Equation for Derivative Prices.