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

$$ Z(t,T) = \exp \left( \int_{t}^{T} f(t,s) {d} s \right) \quad \textrm{for all } T \in [0,\tau],t \in [0,T], $$

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 $t$

$$ r(t) = f(t,t) \quad \textrm{for all } t \in [0,\tau]. $$

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

$$ B(t) = \exp \left( \int_{0}^{t} r(s) {d} s \right) \quad \textrm{for all } t \in [0,\tau] $$

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

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