Written in 1994; printed in Economic Notes, 36 (3) (2008), 205–207.
The price at time τ of a discount bond maturing at T is subject to the dynamics
where is the market price of risk (see Vasicek, 1977, Chapter 6 of this volume). This can be written as
Integrate Eq. (1) with respect to τ from 0 to t,
and differentiate with respect to T. This produces
where σF is the volatility of the forward rates and
Eq. (2) is the Heath, Jarrow, Morton (1992) model.
If are vectors, their products are interpreted as inner products.