Chapter 9
The Heath, Jarrow, Morton Model
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
with
where 
is the market price of risk (see Vasicek, 1977, Chapter 6 of this volume). This can be written as
where
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.
References
- Heath, D., R. Jarrow, and A. Morton. (1992). “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation.” Econometrica 60, pp. 77–105.
- Vasicek, Oldrich A. (1977). “An Equilibrium Characterization of the Term Structure.” Journal of Financial Economics, 5, 177–188.
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