Hull-White and the LMM Framework

Setup

Why Interest Rate Models Differ from Equity Models

In equity models, the underlying (stock price) is directly observable and traded. In interest rate models, the "underlying" is an entire yield curve — an infinite-dimensional object. This creates structural differences:

1. Consistency with the initial curve. A model must, by construction, produce bond prices consistent with the observed term structure today. Black-Scholes for equities does not face this constraint (there is only one observable — the spot price). An interest rate model that misprices the initial curve is not miscalibrated: it is wrong.

2. Multi-asset nature. Interest rates at different tenors are correlated but not perfectly. A one-factor model cannot capture realistic correlation structure across the curve.

3. No-arbitrage drift conditions. Under the risk-neutral (or any equivalent) measure, the drift of interest rate processes is not free — it is constrained by the requirement that discounted bond prices are martingales (Heath-Jarrow-Morton framework).

Notation and Conventions

Throughout:

• $P(t, T)$: price at time $t$ of a zero-coupon bond paying 1 at $T$. $P(0, T)$ observed from the initial curve.
• $f(t, T)$: instantaneous forward rate at time $t$ for maturity $T$, related to the bond price by $P(t, T) = \exp\!\left(-\int_t^T f(t,s)\,ds\right)$.
• $r_t = f(t, t)$: short rate (instantaneous spot rate).
• $L(t; T, T+\tau)$: simply-compounded Libor rate for period $[T, T+\tau]$ set at time $t$.
• $\delta_i = T_{i+1} - T_i$: tenor of the $i$-th period.
• All rates are continuously compounded unless explicitly stated "simply compounded."

Hull-White One-Factor Model

SDE and Structure

The Hull-White (HW1F) model (Hull and White, 1990) specifies short-rate dynamics under the risk-neutral measure $\mathbb{Q}$:

$dr_t = (\theta(t) - a\, r_t)\, dt + \sigma\, dW_t.$

Parameters:

• $a > 0$: mean-reversion speed (constant).
• $\sigma > 0$: short-rate volatility (constant).
• $\theta(t)$: time-dependent drift, calibrated to exactly fit the initial term structure.

The model is affine: the bond price takes the form

$P(t, T) = \exp\!\left(A(t, T) - B(t, T)\, r_t\right),$

where $B(t,T)$ and $A(t,T)$ are deterministic functions.

Exact Calibration to the Initial Curve

For the bond price formula to be consistent with the observed $P(0, T)$, the drift $\theta(t)$ must satisfy:

$\theta(t) = \frac{\partial f^M(0, t)}{\partial t} + a\, f^M(0, t) + \frac{\sigma^2}{2a}(1 - e^{-2at}),$

where $f^M(0, t) = -\frac{\partial}{\partial t}\ln P^M(0, t)$ is the market instantaneous forward rate. This is derived by demanding $P_{\text{model}}(0, T) = P^M(0, T)$ for all $T$ and differentiating.

Key insight. The $\theta(t)$ term absorbs the entire shape of the initial forward curve. Once calibrated, the model prices all zero-coupon bonds exactly, by construction. This is the defining feature that distinguishes HW from Vasicek (which does not fit the initial curve exactly).

Bond Price Formula

$B(t, T) = \frac{1 - e^{-a(T-t)}}{a},$

$A(t, T) = \ln\frac{P^M(0, T)}{P^M(0, t)} + B(t, T)\, f^M(0, t) - \frac{\sigma^2}{4a}\left(1 - e^{-2at}\right) B(t,T)^2.$

This follows from the affine structure of the SDE combined with the exact calibration condition.

Caplet Pricing in Closed Form

A caplet is a call option on the Libor rate $L(T_0; T_0, T_1)$, paying $\delta \max(L(T_0; T_0, T_1) - K, 0)$ at $T_1$. Since Libor is related to bond prices by

$1 + \delta\, L(T_0; T_0, T_1) = \frac{1}{P(T_0, T_1)},$

a caplet is equivalently a put on a zero-coupon bond $P(T_0, T_1)$ with strike $X = 1/(1 + \delta K)$.

Under HW1F, the bond price $P(T_0, T_1) = e^{A(T_0,T_1) - B(T_0,T_1)r_{T_0}}$ is log-normal (since $r_{T_0}$ is Gaussian). The caplet has a Black's formula-type closed form:

$\text{Caplet}(0) = P(0, T_1)\, N(-d_2) \cdot \frac{1}{X} - P(0, T_0) \cdot N(-d_1) \cdot \frac{1}{X},$

Wait — more precisely, using the bond-put expression, the caplet price is:

$\text{Caplet}(0) = (1 + \delta K)\cdot P(0, T_1)\, N(-d_2) - P(0, T_0)\, N(-d_1),$

$d_1 = \frac{\ln\left(\frac{P(0,T_1)(1+\delta K)}{P(0,T_0)}\right)}{\sigma_P} + \frac{\sigma_P}{2}, \qquad d_2 = d_1 - \sigma_P,$

$\sigma_P = \sigma B(T_0, T_1) \sqrt{\frac{1 - e^{-2aT_0}}{2a}}.$

Here $\sigma_P$ is the standard deviation of $\ln P(T_0, T_1)$ under the $T_0$-forward measure.

Limitations of HW1F

• Gaussian rates. Because $r_t$ is Gaussian (driven by a Brownian motion with no lower bound), the model allows negative short rates with positive probability. This was historically considered a theoretical flaw but became empirically relevant post-2012 with negative ECB and BOJ rates.
• One factor. The entire curve moves as a single linear factor (multiplied by $B(t,T)$). Realistic co-movements across short and long tenors require multi-factor extensions (two-factor Hull-White, G2++ model).
• Constant $\sigma$ and $a$. A constant volatility structure cannot capture the volatility humps observed in caps and swaptions (vol peaked at 1–3 year tenors in most markets). Piecewise-constant $\sigma(t)$ is one extension.