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