Outline of the book

Outline of the book

Part examines the various models mentioned above. Each chapter begins with the assumptions underlying each model and examines its derivation and, where analytical solutions exist, the derivation of the pricing formulae for contingent claims. Comparisons are drawn between the various models with the aim of explaining the significance of the various approaches. Part 12.8 describes the calibration procedure for the HW-extended Vasicek, BDT and HJM models. These models represent distinct approaches to term structure modelling and their calibration methodologies are representative of the general class of models to which each belongs.

Part I: Interest Rate Models

Chapter List

Chapter 1: The Vasicek Model
Chapter 2: The Cox, Ingersoll and Ross Model
Chapter 3: The Brennan and Schwartz Model
Chapter 4: Longstaff and Schwartz-A Two-Factor Equilibrium Model
Chapter 5: Langetieg's Multi-Factor Equilibrium Framework
Chapter 6: The Ball and Torous Model
Chapter 7: The Hull and White Model
Chapter 8: The Black, Derman and Toy One-Factor Interest Rate Model
Chapter 9: The Black and Karasinski Model
Chapter 10: The Ho and Lee Model
Chapter 11: The Heath, Jarrow and Morton Model
Chapter 12: Brace, Gatarek and Musiela Model

Chapter 1: The Vasicek Model

The initial formulation of Vasicek's model is very general, with the short- term interest rate being described by a diffusion process. An arbitrage argument, similar to that used to derive the Black-Scholes option pricing formula [ 8 ], is applied within this broad framework to determine the partial differential equation satisfied by any contingent claim. A stochastic representation of the bond price results from the solution to this equation. Vasicek then allows more restrictive assumptions to formulate the specific model with which his name is associated.

The consistency of the model specifications with an underlying economic equilibrium is not proved. Rather, it is implicitly assumed. The special case of the general model formulation, which Vasicek uses for illustrative purposes, was suggested by Merton [ 40 ] in a study of price dynamics in a continuous time, equilibrium economy. Equilibrium conditions imply that interest rates are such that the demand and supply of capital are equally matched.

1.1 Preliminaries

First, define the following variables :

P ( t, T )


time t price of a discount bond maturing at time T , t T , with P ( T , T )=1.

R ( t , )


time t rate of interest applicable for period . Intermsof the return on a discount bond, this rate is defined as the internal rate of return, at time t , on a bond with maturity date T = t + .

r ( t )


instantaneous rate of interest (short rate) at time t .

f ( t , T )


instantaneous forward interest rate i.e. time t assessment of the instantaneous rate of interest applicable at time T .

The following relationships apply:

or explicitly for the forward rate:

The short rate is defined as the instantaneous rate of interest at time t :

Vasicek makes the following three assumptions:

Assumption 1

start example

The current short interest rate is known with certainty . However, subsequent values of the short rate are not known. The assumption is made that r ( t ) follows a stochastic process. Also assume that r ( t ):

  • is a continuous function of time,

  • follows a Markovian process. That is, given its current value, future developments of the short rate are independent of past movements.

This implies that the short rate process is fully characterised by a single state variable, i.e. its current value, and the probability distribution of r ( t *), t * t is fully determined by r ( t ). A continuous Markovian process is called a diffusion process, which is described by the stochastic differential equation:

where v ( r , t ) is the instantaneous drift and s 2 ( r , t ) the instantaneous variance of r ( t ). z ( t ) is a Wiener process under a given measure Q .

end example

Assumption 2

start example

The time t price of a discount bond with maturity T , P ( t , T ), is fully determined by the time t assessment of { r ( t *), t t * T }, the segment of the short rate over the remaining term of the bond. Moreover, the development of the short rate over [ t , T ] is fully determined by its current value r ( t ), so the bond price may be written as a function of the current short rate:

Hence the entire term structure is determined by the short rate.

end example

Assumption 3

start example

The market is assumed to be efficient. This implies:

  • there are no transaction costs;

  • information is simultaneously distributed to all investors;

  • investors are rational with homogeneous expectations;

  • profitable, riskless arbitrage is not possible.

end example