# 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

\[\frac{dZ(t,T)}{Z(t)} = r(t)dt + \sigma_Z(t,T)'dW(t),\]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 \(\mu(t,T)\)

\[df(t,T) = \mu(t,T)dt + \sigma_f(t,T)'dW(t)\]Let \(X(t) = -\int_t^T f(t,s) ds\), then

\[\begin{align} dX(t) &= f(t,t)dt -\int_t^T df(t,s) ds \\ &= r(t)dt - \int^T_t \left[ \mu(t,s)dt + \sigma_f(t,s)'dW(t) \right] ds \end{align}\]Next, by applying standard and stochastic Fubini theorem we get

\[\begin{align} dX(t) &= r(t) dt - \int^T_t \mu(t,s) ds\ dt - \int^T_t \sigma_f(t,s)' ds\ dW(t) \\ \\ &= r(t)dt - \alpha(t,T)dt - \sigma_B(t,T)' dW(t) \end{align}\]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

\[\begin{align} d Z(t,T) &= \exp ( X(t) ) dX(t) + \frac{1}{2} \exp ( X(t) ) (dX(t))^2 \\ \\ \frac{dZ(t,T)}{Z(t,T)}&= \left[ r(t) - \alpha(t,T) + \frac{1}{2} \sigma_B(t,T)' \sigma_B(t,T) \right] dt - \sigma_B(t,T)' dW(t) \end{align}\]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 \(T\)

\[\alpha(t,T) = \frac{1}{2} \sigma_B(t,T)' \sigma_B(t,T)\]By differentiating both sides with respect to \(T\) we obtain

\[\begin{align} \mu(t,T) &= \sigma_f(t,T)'\sigma_B(t,T) \\ \\ \mu(t,T) &= \sigma_f(t,T)' \int^T_t \sigma_f(t,s) ds \end{align}\]So the final stochastic differential equation for the instantaneous forward rate under the risk-free measure is of the following form

\[df(t,T) = \left[ \sigma_f(t,T)' \int^T_t \sigma_f(t,s)ds \right] dt + \sigma_f(t,T)'dW(t)\]