Hull White and initial curve fitting

Definition Under a pricing measure $\mathbb{Q}$ for the money-market account $B_t = \exp\bigl(\int_0^t r_s \, ds\bigr)$, the one-factor Hull-White model is

with constants $a > 0$ per year and $\sigma \ge 0$ per square-root year, a known $r_0$, and a deterministic, locally integrable function $\theta$ in rate units per year. For $s \le t$ the solution is

In this convention $\theta(t)$ is the whole constant term of the drift. Another common convention writes the drift as $a\bigl(\bar{\theta}(t) - r_t\bigr)$ with a mean level $\bar{\theta} = \theta/a$, so the same symbol has a different numerical meaning there, and formulas from the two conventions must not be mixed. A constant $\theta(t) = a\theta_V$ gives the Vasicek model with speed $a$ and level $\theta_V$. This lesson keeps $a$ and $\sigma$ constant, and time-varying $a$ or $\sigma$ is a further extension.

Definition Write the rate as

where $\phi$ is a continuous deterministic function and the factor $x$ solves $dx_t = -a x_t \, dt + \sigma \, dW^{\mathbb{Q}}_t$. Then $x_t = \sigma \int_0^t e^{-a(t - u)} \, dW^{\mathbb{Q}}_u$ is Gaussian with mean $0$ and variance $\sigma^2(1 - e^{-2at})/(2a)$, so $r_t$ is Gaussian with mean $\phi(t)$ and the same variance, and given $\mathcal{F}_s$ it is Gaussian with mean $\phi(t) + x_s e^{-a(t - s)}$ and variance $\sigma^2(1 - e^{-2a(t - s)})/(2a)$. With the kernel $B(u,t) = (1 - e^{-a(t - u)})/a$, a stochastic Fubini interchange, valid because the kernel is deterministic and square integrable on every finite horizon, gives

a centered Gaussian variable. Expanding the square of the kernel inside the Itô isometry gives its variance

For $\sigma > 0$ and $t > 0$ the rate $r_t$ is negative with positive $\mathbb{Q}$-probability. That is a property of the Gaussian model, not a failure of its solution.

Definition In the representation $r = x + \phi$, the deterministic shift $\phi$ moves the rate distribution at each date by a fixed amount and leaves its randomness to the factor $x$. Since $-\int_0^T x_s \, ds$ is centered Gaussian with variance $V(T)$, the model’s initial bond prices are

Because $x_0 = 0$, the initial short rate is $r_0 = \phi(0)$. When $\phi$ is continuously differentiable, adding $d\phi = \phi'(t)\,dt$ to the equation for $x$ gives

which is the Hull-White model with $\theta = \phi' + a\phi$. Conversely, every Hull-White model has this form with $\phi(t) = \mathbb{E}^{\mathbb{Q}}[r_t] = r_0 e^{-at} + \int_0^t e^{-a(t - u)}\theta(u)\,du$, because subtracting this mean from the solution leaves exactly the factor $x$.

Definition Let the initial market curve satisfy $P^M(0,T) > 0$ for $0 \le T \le T^{*}$ and $P^M(0,0) = 1$, with instantaneous forward rate $f^M(0,T) = -\partial_T \ln P^M(0,T)$ continuous on $[0, T^{*}]$, so that $P^M(0,T) = \exp\bigl(-\int_0^T f^M(0,s)\,ds\bigr)$. The model fits the initial curve if $P(0,T) = P^M(0,T)$ for every $T \in [0, T^{*}]$. With the shift representation this means

for every such $T$. Both sides vanish at $T = 0$, so for a continuous $\phi$ the condition is equivalent to equality of the derivatives. With $V'(T) = \sigma^2(1 - e^{-aT})^2/a^2$, the unique fitting shift is

At $T = 0$ it gives $r_0 = \phi(0) = f^M(0,0)$, so the model’s initial short rate must equal the curve’s instantaneous forward rate at zero. The fit holds for every $a > 0$ and $\sigma \ge 0$, so it does not identify these parameters. Choosing them is a separate calibration to further instruments, such as caps or swaptions, under a declared objective and constraints.

Definition Assume in addition that $f^M(0,\cdot)$ is continuously differentiable on $[0, T^{*}]$. In the convention $dr_t = (\theta(t) - a r_t)\,dt + \sigma\,dW^{\mathbb{Q}}_t$ with $r_0 = f^M(0,0)$, the drift function that fits the initial curve is $\theta = \phi' + a\phi$ for the fitting shift, which works out to

The derivative of the shift contributes $(\sigma^2/a)(e^{-at} - e^{-2at})$, and $a$ times the shift contributes $(\sigma^2/(2a))(1 - 2e^{-at} + e^{-2at})$, and together these give the last term. It is the unique continuous $\theta$ with this initial rate for which the model reproduces $P^M(0,T)$ on $[0, T^{*}]$, since the fit determines $\phi$ and $\phi$ determines $\theta$. The drift depends on the slope of the forward curve, which is the second derivative of the log discount curve, and differentiation amplifies small errors in a bootstrapped curve. Smooth interpolation of the initial curve matters for this reason.

Definition In the fitted model, let $0 \le t \le T \le T^{*}$ and $B(t,T) = (1 - e^{-a(T - t)})/a$. Given $\mathcal{F}_t$, the integral $\int_t^T x_u \, du$ is Gaussian with mean $B(t,T)\,x_t$ and variance $v_{t,T} = \sigma^2 \int_t^T B(u,T)^2 \, du$. With $I_{t,T} = \int_t^T r_u \, du$ and $c_{t,T} = \int_t^T \phi(u) \, du$, the Gaussian exponential moment gives

Substituting the fitting shift and $x_t = r_t - \phi(t)$ and evaluating the exponential integrals gives, with $B = B(t,T)$,

where

The price is exponential-affine in $r_t$ with slope $-B(t,T)$ in the log, as in Vasicek. At $t = 0$ both terms of $G$ vanish, because $r_0 = f^M(0,0)$ and $1 - e^{0} = 0$, so $P(0,T) = P^M(0,T)$ for every maturity. The ratio $P^M(0,T)/P^M(0,t)$ is the forward bond price fixed today, while $P(t,T)$ is random through $r_t$.