Heath Jarrow Morton and the drift restriction

Definition Suppose that at time $t$ the zero-coupon bond prices $P(t,T)$ for maturities $T \ge t$ are positive, satisfy $P(t,t) = 1$ and are differentiable in $T$. The instantaneous forward rate for maturity $T$ is

and integrating it over maturity recovers the bond price,

The forward rate is annualized and continuously compounded. It is the limit, as $\delta \downarrow 0$, of the simple forward rate $\bigl(P(t,T)/P(t,T+\delta) - 1\bigr)/\delta$ for lending over $[T, T+\delta]$ agreed at time $t$. The short rate is its value on the diagonal, $r_t = f(t,t)$, and the continuously compounded zero yield $y(t,T) = -\ln P(t,T)/(T-t)$ is the average of the forward rates over $[t,T]$. Nothing forces a forward rate to be positive. It is negative at the maturities where the bond price increases with maturity.

Definition Fix a horizon $T^*$ and a filtered probability space carrying a $d$-dimensional Brownian motion $W^{\mathbb{Q}}$ under a measure $\mathbb{Q}$ equivalent to $\mathbb{P}$. In the Heath-Jarrow-Morton framework the forward rate of each maturity $T \le T^*$ evolves as

for $0 \le t \le T$. The initial curve $f(0,\cdot)$ is today’s forward curve, a given deterministic function that is integrable on $[0,T^*]$, so the model reproduces today’s bond prices by construction. The drift $\alpha(t,T)$ is real valued and the forward-rate volatility $\sigma(t,T)$ takes values in $\mathbb{R}^d$. Both are jointly measurable in the outcome, the time and the maturity, adapted in $t$ for each fixed $T$, and integrable enough for the ordinary and the stochastic Fubini theorems to apply to integrals over maturity, which holds for example when $\alpha$ and $\sigma$ are bounded. The short rate $r_t = f(t,t)$ is assumed well defined with $\int_0^{T^*} \lvert r_t \rvert\,dt < \infty$, and the money-market account is $B_t = \exp\bigl(\int_0^t r_s\,ds\bigr)$. The $k$th component of $\sigma(t,T)$ is the response of the forward rate to the $k$th Brownian driver, and the length $\lVert \sigma(t,T) \rVert$ is its absolute volatility in rate units per square root year. Two maturities move with the instantaneous correlation

when both volatilities are nonzero. With one factor, $d = 1$, this correlation is $1$ or $-1$, so the increments of all forward rates are perfectly correlated up to sign, and several factors are needed for the curve to twist or bend. The framework is written under $\mathbb{Q}$ with the intention that $\mathbb{Q}$ be a pricing measure for the numeraire $B$, and the next two cards find the drifts that make it one.

Definition Under the hypotheses of the framework, define for $t \le T$ the integrated drift and the integrated volatility

For every maturity $T \le T^*$ and $0 \le t \le T$,

and Itô’s lemma for the exponential gives

To prove the first identity, insert the integral form of $f(t,u)$ into $\ln P(t,T) = -\int_t^T f(t,u)\,du$, exchange the order of the time and maturity integrals, by Fubini’s theorem for the drift and by the stochastic Fubini theorem for the Brownian term, and collect the pieces with maturities below $t$ into $\int_0^t f(s,s)\,ds = \int_0^t r_s\,ds$. The term with $r_s$ is the contribution of the moving lower limit. As calendar time advances, the front of the interval $[t,T]$ drops out of the integral, which in differential form reads, formally, $f(t,t)\,dt - \int_t^T df(t,u)\,du$ for the increment of $-\int_t^T f(t,u)\,du$. The logarithm has Brownian loading $-\Sigma(t,T)$, so its quadratic variation grows at the rate $\lVert \Sigma(t,T) \rVert^2$, and the exponential adds half of it to the drift. Dividing by the money-market account removes $r_t$, so the discounted bond price satisfies

Definition Under the hypotheses of the framework, the discounted bond prices $P(t,T)/B_t$ are local martingales under $\mathbb{Q}$ for every maturity $T \le T^*$ if and only if

for every $T$, for almost every $t \le T$, almost surely. This follows from the previous card, because a continuous semimartingale is a local martingale exactly when its drift vanishes. When $\alpha$ and $\sigma$ are continuous in maturity, differentiating in $T$ with $\partial_T A(t,T) = \alpha(t,T)$ and $\partial_T \tfrac12 \lVert \Sigma(t,T) \rVert^2 = \sigma(t,T) \cdot \Sigma(t,T)$ gives the equivalent pointwise form, the HJM drift restriction

and integrating it back from $T = t$, where $A$ and $\Sigma$ vanish, recovers the first form. Once the initial curve and the volatility are chosen, the drift under $\mathbb{Q}$ is determined, so it is not a parameter to fit. Under the restriction, with $\Sigma = \Sigma(s,T)$ inside the integrals,

A positive local martingale is a supermartingale, and this one is a true martingale under an additional condition such as Novikov’s,

which holds whenever $\sigma$ is bounded. With true martingales, $P(t,T) = \mathbb{E}^{\mathbb{Q}}\bigl[e^{-\int_t^T r_s\,ds} \mid \mathcal{F}_t\bigr]$, and together with admissibility conditions on trading strategies such a measure rules out arbitrage among the bonds. The restriction does not make the market complete, and it says nothing about the drift under $\mathbb{P}$, which contains a market price of risk. Forward rates are in general not martingales under $\mathbb{Q}$, since their drift $\alpha(t,T)$ is exactly what cancels the convexity term of the bond.

Definition The exponential volatility kernel is the deterministic one-factor volatility

with constants $\eta > 0$, in rate units per square root year, and $a > 0$, per year. It is bounded and deterministic, so the hypotheses of the framework and Novikov’s condition hold. With $\tau = T - t$,

The last line comes from integrating $\alpha$ directly, using

and it equals $\tfrac12 \Sigma(t,T)^2$, as the restriction requires. As $a \to 0$ the kernel tends to the constant $\eta$, with $\Sigma \to \eta\tau$ and $\alpha \to \eta^2\tau$, the Ho-Lee values. With this kernel the short rate $r_t = f(t,t)$ follows the Hull-White model with mean reversion $a$ and volatility $\eta$, fitted to the initial curve.