Derivatives

CHAPTER 49Equilibrium Models: Term Structure

Aims

• To show how bond prices and option prices can be derived from a model of the short-rate, using risk-neutral valuation.
• To investigate the properties of different time-series models of the short-rate.
• To demonstrate how continuous time models of the term structure can be used to price bonds and options on bonds.

Black's model for pricing fixed income derivatives such as European options on bonds, on bond futures, as well as caps, floors, and swaptions, assumes the underlying variable is lognormal at expiration of the option. Black's model cannot price derivatives which depend on the evolution of interest rates through time – such as American style options, callable bonds and other path-dependent options.

The equilibrium yield curve approach assumes a specific continuous time stochastic process for the one period (short) rate. The parameters of this process are then estimated from historical data. Bond prices and fixed income derivatives prices can then be derived mathematically and these prices depend directly on the estimated parameters of the stochastic process for the short-rate. Hence, in the equilibrium approach the current term structure is an output from the short-rate model. (In contrast, in the no-arbitrage BOPM applied to fixed-income assets the current term structure is used as an input to calibrate the lattice for the short-rate.) The sequence of steps in the equilibrium model approach are:

• Choose a continuous time ...