Let be the instantaneous forward rate. Suppose the dynamic is
The idea of Heath-Jarrow-Morton is that the no-arbitrage drifts of the forward rates are uniquely specified once their volatilities and correlations are assigned. In particular, under risk-neutral measure we must have
Example. For constant and , consider the short rate dynamic
Zero bonds price and forward rate dynamic, respectively, are
1. Zero Bonds in HJM
We now show that in a HJM mode (1) the SDE for the zero-coupon bond price is
One might do the following calcucaltion to get the above equation.
Note that (1) implies
which is nothing but an identity
For simplicity we denote the integral by in the rest of this section. Then
Prop 1 In a HJM model, we have
For each iterated integral above we integrate once and obtain the formula.
Let be the bank account, i.e.,
where is the the instantaneous short rate at time . Note that
Then we do the same calculation as the proposition above and obtain
Cor 2 In a HJM model, we have
Rem 1 As a result, we now write down explicitly the formula for change of numeraires from the bank account to the zero bond for a fixed a maturity .
This is an exponential martingale with . It follows that
where is a Brownian motion under -forward measure.