In late 1980s David Heath, Robert Jarrow and Andrew Morton worked on interest rate framework which would unify the theory for valuing contingent claims under a stochastic term structure of interest rates. The paper, Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation, was published in 1992 in Econometrica.
In this post we will quickly describe the model and derive the no-arbitrage condition in the risk-neutral measure in a single economy.
Definitions and Relationships
Let $f(t,T)$ be a nominal instantaneous forward rate at time with expiry , be a nominal zero coupon bond price at time with maturity . The relationship between the two is
where is the ending point in the trading interval. The nominal spot rate at time , , is the nominal instantaneous forward rate at time for date
The risk-free nominal money-market account (accumulation factor), initialized at time 0 with a one unit of currency investment is
Under the nominal risk neutral measure the dynamics of the nominal risk-free zero coupon bond is
where is a -dimensional standard -Brownian motion. Assuming following dynamics for the , the question now is to identify the drift term
Let , then
Next, by applying standard and stochastic Fubini theorem we get
where and . We are almost there, now we can put all things together and using the very first equation and Ito’s lemma
Please note that under the risk-neutral measure the drift term for must be equal to . We conclude following no-arbitrage condition which must hold for all
By differentiating both sides with respect to we obtain
So the final stochastic differential equation for the instantaneous forward rate under the risk-free measure is of the following form