# HJM No-Arbitrage Drift Condition

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 $t$ with expiry $T$, $Z(t,T)$ be a nominal zero coupon bond price at time $t$ with maturity $T$. The relationship between the two is

where $\tau$ is the ending point in the trading interval. The nominal spot rate at time $t$, $r(t)$, is the nominal instantaneous forward rate at time $t$ for date $t$

The risk-free nominal money-market account (accumulation factor), $B(t)$ initialized at time 0 with a one unit of currency investment is

## No-Arbitrage Conditions

Under the nominal risk neutral measure the dynamics of the nominal risk-free zero coupon bond is

where $W(t)$ is a $d$-dimensional standard $Q$-Brownian motion. Assuming following dynamics for the $f(t,T)$, the question now is to identify the drift term $\mu(t,T)$

Let $X(t) = -\int_t^T f(t,s) ds$, then

Next, by applying standard and stochastic Fubini theorem we get

where $\alpha(t,T) = \int^T_t \mu(t,s) ds$ and $\sigma_B(t,T)' = \int^T_t \sigma_f(t,s)' ds$. 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 $Z(t,T)$ must be equal to $r(t) Z(t,T)$. We conclude following no-arbitrage condition which must hold for all $T$

By differentiating both sides with respect to $T$ we obtain

So the final stochastic differential equation for the instantaneous forward rate under the risk-free measure is of the following form