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.

Libor Market Model (BGM)

Motivation

Hull-White models the short rate — an infinitesimally short maturity instrument. In practice, the liquid instruments are Libor rates and swap rates at discrete tenors (1M, 3M, 6M, 1Y, ...). The Brace-Gatarek-Musiela (BGM) model, also called the Libor Market Model (LMM), models the discrete Libor forward rates directly.

The central advantage: LMM is calibrated to market caplet prices exactly (by construction), because the Black caplet formula is built into the model specification.

Setup

Fix a tenor structure $0 = T_0 < T_1 < \cdots < T_N$, with $\delta_i = T_{i+1} - T_i$. Define the forward Libor rate:

$L_i(t) = L(t; T_i, T_{i+1}) = \frac{1}{\delta_i}\left(\frac{P(t, T_i)}{P(t, T_{i+1})} - 1\right), \qquad t \leq T_i.$

Each forward Libor $L_i(t)$ is a martingale under the $T_{i+1}$-forward measure $\mathbb{Q}^{T_{i+1}}$ (with numeraire $P(t, T_{i+1})$). Under this measure:

$dL_i(t) = \sigma_i(t) L_i(t) \, d\widetilde{W}_t^{(i)}, \qquad t \leq T_i,$

where $\sigma_i(t)$ is a (possibly time-dependent) scalar or vector volatility and $\widetilde{W}^{(i)}$ is a $\mathbb{Q}^{T_{i+1}}$-Brownian motion. The instantaneous correlation is:

$d\langle \widetilde{W}^{(i)}, \widetilde{W}^{(j)} \rangle_t = \rho_{ij}\, dt.$

Drift Under the Spot Measure

To simulate the entire forward Libor system simultaneously, one needs a common probability measure. The spot Libor measure $\mathbb{Q}^B$ uses as numeraire the discretely compounded money market account:

$B(t) = P(t, T_{\beta(t)}) \prod_{j=0}^{\beta(t)-1} (1 + \delta_j L_j(T_j)), \qquad \beta(t) = \min\{j : T_j > t\}.$

Under $\mathbb{Q}^B$, the drift of each forward Libor is uniquely determined by the no-arbitrage condition. Deriving it from a change-of-numeraire calculation:

$dL_i(t) = L_i(t)\left[\mu_i(t)\, dt + \boldsymbol{\sigma}_i(t)^\top d\mathbf{W}_t^B\right],$

$\mu_i(t) = \sum_{j=\beta(t)}^{i} \frac{\delta_j \rho_{ij} \sigma_i(t) \sigma_j(t) L_j(t)}{1 + \delta_j L_j(t)},$

where $\mathbf{W}_t^B$ is a $\mathbb{Q}^B$-Brownian motion vector and the correlation matrix $(\rho_{ij})$ is fixed.

Derivation structure. The drift arises from the Radon-Nikodym derivative between the $T_{i+1}$-forward measure and the spot measure. Each Libor $L_j(t)$ for $j \leq i$ contributes a covariance term $\rho_{ij}\sigma_i \sigma_j L_j/(1 + \delta_j L_j)$ to the drift of $L_i$. The sum runs from the current index $\beta(t)$ to $i$ — only the "alive" forward rates contribute.

Caplet Pricing: Recovering Black's Formula

Under the $T_{i+1}$-forward measure, $L_i(t)$ has zero drift and log-normal dynamics:

$dL_i(t) = \sigma_i(t) L_i(t)\, d\widetilde{W}_t^{(i)}.$

This gives $L_i(T_i) \sim L_i(0) \exp\!\left(-\frac{1}{2}\Sigma_i^2 T_i + \Sigma_i \sqrt{T_i} Z\right)$ where $\Sigma_i^2 = \frac{1}{T_i}\int_0^{T_i}\sigma_i^2(t)\,dt$ and $Z \sim \mathcal{N}(0,1)$. The caplet price is:

$\text{Caplet}_i = \delta_i P(0, T_{i+1}) \,\mathbb{E}^{\mathbb{Q}^{T_{i+1}}}\!\left[(L_i(T_i) - K)^+\right] = \delta_i P(0, T_{i+1})\, \text{Black}(L_i(0), K, \Sigma_i, T_i),$

where $\text{Black}(F, K, \sigma, T) = F\, N(d_1) - K\, N(d_2)$ with $d_1 = (\ln(F/K) + \sigma^2 T/2)/(\sigma\sqrt{T})$. This is exactly Black's caplet formula — the LMM recovers the market-standard pricing formula by construction.

Simulation of the LMM

To price path-dependent products (Bermudan swaptions, CMS, spread options), simulate under the spot measure:

1. Discretise time. Use times $t_k = T_k$ (the tenor dates) to avoid interpolation issues; finer grids if needed.

2. Euler-log discretisation. For each $i \geq \beta(t_k)$:

$\ln L_i(t_{k+1}) = \ln L_i(t_k) + \left(\mu_i(t_k) - \frac{1}{2}\sigma_i^2(t_k)\right)\Delta t_k + \sigma_i(t_k)\sqrt{\Delta t_k}\, \boldsymbol{\rho}_i^\top \mathbf{Z}_k,$

where $\mathbf{Z}_k \sim \mathcal{N}(\mathbf{0}, I)$ and $\boldsymbol{\rho}_i^\top \mathbf{Z}_k$ produces a correlated increment via the Cholesky decomposition of the correlation matrix.

3. Drift correction. The Euler-log scheme introduces a bias in the drift. The predictor-corrector method (using the average of drift at $t_k$ and the Euler-predicted drift at $t_{k+1}$) reduces this bias significantly.

Limitations

LMM dimension. For $N = 20$ forward rates, simulation requires generating $N \times N_{\text{paths}} \times N_{\text{steps}}$ random numbers and applying a Cholesky decomposition to a $20 \times 20$ correlation matrix at each step. Computational cost scales as $O(N^2)$ per path step. High-dimensional LMM is expensive; low-rank approximations of the correlation matrix (e.g., via PCA) are standard.

Correlation matrix specification. The $\binom{N}{2}$ pairwise correlations must be specified and positive semi-definite. A common parametrisation is $\rho_{ij} = \rho_\infty + (1-\rho_\infty)e^{-\beta|T_i - T_j|}$ (exponential decay). More complex shapes (e.g., with a hump) require additional parameters.

Calibration to swaptions. LMM calibrates to caplets naturally but not to swaption prices — swaption payoffs involve sums of Libor rates, not a single one. Rebonato's calibration or a Cascade Calibration algorithm is needed to jointly fit the full swaption matrix. This remains an active area of practice.

Negative rates. The log-normal LMM requires $L_i > 0$ always. Post-2012, Libor rates went negative in EUR and JPY. The displaced-diffusion LMM ($L_i + \alpha$ follows a log-normal) or the SABR-LMM (stochastic vol extension) address this.

Markov property. LMM is not Markovian in a finite-dimensional sense: the drift at time $t$ depends on the full vector of surviving forward rates, not a low-dimensional state. This makes PDE pricing methods inapplicable; Monte Carlo is the primary tool.

Interview Angle

L1. What is the Hull-White model? How does $\theta(t)$ relate to the initial forward curve? State Black's caplet formula and explain why the caplet is equivalent to a put on a bond.

HW models the short rate as an Ornstein-Uhlenbeck process with a time-dependent mean level $\theta(t)$ chosen to match the initial term structure exactly. The connection: $\theta(t)$ is found by differentiating the bond price formula $P(0,T) = P_{\text{model}}(0,T)$ with respect to $T$.

Caplet as bond put. Payoff at $T_1$: $\delta \max(L(T_0;T_0,T_1) - K, 0)$. Using $1 + \delta L = P(T_0,T_1)^{-1}$: $L > K \iff P(T_0,T_1) < 1/(1+\delta K) \equiv X$. So the caplet is a put on $P(T_0,T_1)$ with strike $X$, payoff measured at $T_1$ (discounted by the bond itself). Under HW1F, $P(T_0,T_1)$ is log-normal, giving Black's formula.

L2. Derive the LMM drift under the spot Libor measure. Why must the drift be non-zero under this measure if the $T_{i+1}$-forward measure gives zero drift?

Change of numeraire. Under $\mathbb{Q}^{T_{i+1}}$, $L_i$ is a martingale (zero drift). Under the spot measure $\mathbb{Q}^B$, the Radon-Nikodym derivative relating the two measures involves the ratio of numeraires $P(t,T_{i+1})/B(t)$. By Girsanov, changing measure introduces a drift proportional to the covariation between $\ln L_i$ and $\ln(P(t,T_{i+1})/B(t))$. The numeraire ratio depends on $L_j$ for $j \leq i$ through the compounding formula for $B(t)$. Each alive $L_j$ contributes a term $\rho_{ij}\sigma_i\sigma_j L_j \delta_j / (1 + \delta_j L_j)$ — the standard LMM drift.

L3. Discuss the HJM framework. How does LMM relate to the HJM no-arbitrage condition? What are the Markov conditions under which an HJM model admits a finite-dimensional Markovian state representation?

HJM models the entire forward rate curve $f(t,T)$ via:

$df(t,T) = \alpha(t,T)\, dt + \boldsymbol{\sigma}(t,T)^\top d\mathbf{W}_t.$

The HJM no-arbitrage condition (under $\mathbb{Q}$) fixes the drift:

$\alpha(t,T) = \boldsymbol{\sigma}(t,T)^\top \int_t^T \boldsymbol{\sigma}(t,u)\, du.$

The drift is not free — it is determined by the volatility structure. LMM is a discrete-tenor version of an HJM model: each forward Libor satisfies a log-normal SDE whose drift (under the spot measure) is the discrete analogue of the HJM drift condition.

Markov reduction. The HJM model is generically non-Markovian (the drift depends on the entire volatility history). Cheyette (1992) and Ritchken-Sankarasubramanian (1995) identify conditions under which HJM reduces to a finite-dimensional Markovian system: the volatility must have a separable form $\sigma(t,T) = g(t)h(T)$ for deterministic functions $g, h$. Hull-White with constant $\sigma$ satisfies this, which is why it admits the Markovian affine representation. General HJM does not.