Chapter 14: Calibrating the Black, Derman and Toy discrete time model

Overview

In Chapter 8 we examined the formulation of the BDT model within a (discrete time) binomial lattice as well as its continuous time equivalent. The short- term interest rate process takes the form ^[1] :

where r ( t ), u ( t ), ƒ ( t ) and z ( t ) are the time t values of the short-term interest rate, the median of the (lognormal) short-term interest rate distribution, the short-term interest rate volatility and a standard Brownian motion. The continuous time equivalent of the BDT model is determined to be:

However, this formulation of the model is not analytically tractable (this is a characteristic of all lognormal models) it must be implemented by means of a binomial tree. The tree is constructed so as to approximate the above stochastic differential equation of the short-term interest rate. The short-term interest rate at each node of the tree is determined so as to be consistent with observed market prices of bonds .

^[1] This relationship is derived in §8.4 of Chapter 8.