The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 644 Wednesday, February 4, 2004 1:33 PM

644 The Mathematics of Financial Modeling and Investment Management

interest rate which is a simple forward interest rate defined over a dis-

crete time period. The BGM model, and the HJM from which it derives,

form a wide class of models which has been extensively explored in the

literature. Here we will only give a brief account of the BGM model.

First define L(t,0) as the rate of simple interest over a discrete period

δ so that an amount of D(t,δ ) dollars invested at time t in a bond with

maturity (t + δ ) become 1 dollar at maturity:

Dt ( δ, )[ 1 + δ Lt(, 0 )] = 1

Then define the forward LIBOR as follows:

Dt( , τδ+ ) (

---------------------------[^1 + δ Lt τ, )] =^1

Dt ( τ, )

It is possible to demonstrate that

+

∫

(

τ

τδ)

ft u( , ) ud

(

δ

Lt τ, ) = ------------------------------------------e –^1

where f is the continuously compounding forward rate.

Define now σ *(t,τ ) recursively as follows:

σ *( t, τδ+ ) = σ *( t τ, ) + δ Lt ( τ, )γ ( t τ, )

------------------------------------

1 + δ Lt ( τ, )

(

1

Lt τ, )γ ( t τ, ) = ---[ 1 + δ Lt ( τ, )][σ *( t, τδ+ ) – σ *( t τ, )]

δ

DISCRETIZATION OF ITÔ PROCESSES

Itô processes are stochastic differential equations that admit a forward

discretization scheme similar to that of ordinary differential equations.

Consider an Itô process that obeys the following SDE:

dXt = μ( Xt, t) dt + σ( Xt, t) dBt