Hull-White and HJM: Time-Dependent Rates and the No-Arbitrage Condition

One-factor Gaussian models like Vasicek and CIR are elegant, but they cannot fit an arbitrary initial term structure. If today’s discount curve is humped, inverted, or otherwise non-trivial, a constant-parameter model will give the wrong prices. The fix is to make the parameters time-dependent. Hull and White (1990, 1993) did this for the short rate, and Heath, Jarrow, and Morton (1992) did it for the entire forward rate curve. Together, they form the workhorse framework for practitioner interest-rate modeling.

This article derives both, shows the no-arbitrage condition that links the drift of forward rates to the volatility, and explains why HJM is the natural general framework of which Hull-White is a special case.

The Hull-White Extension

Hull-White modifies Vasicek by adding time dependence to the parameters:

Note the difference from Vasicek: the long-run mean is now a time-dependent function $\theta(t) / \kappa$, not a constant. The volatility is still constant. This gives the model enough flexibility to fit any initial discount curve.

The mean reversion $\kappa$ can also be made time-dependent, but the standard formulation keeps it constant and varies $\theta$. The bond price formula remains exponential-affine:

with

The same $B$ as Vasicek. But $A(t, T)$ now depends on $\theta(\cdot)$:

where $P^M(0, T)$ is the market discount bond price at time 0 for maturity $T$, and $f^M(0, t)$ is the market instantaneous forward rate. So $A(t, T)$ is chosen to make the model fit the initial market curve exactly.

This is the key innovation: $\theta(t)$ is not a free parameter but is determined by the requirement that $P(0, T) = P^M(0, T)$ for all $T$. With this choice, the model fits the current curve perfectly, while preserving the closed-form bond price and tractable dynamics.

Why Hull-White Fits the Curve

At time 0, the bond price formula gives

For this to equal $P^M(0, T)$ for all $T$, we need

Substituting into the expression for $A(t, T)$ in terms of $A(0, T)$ and $B(t, T)$, we get the formula above. The result is that the model fits the curve exactly at $t = 0$.

For $t > 0$, the model evolves. The bond price depends on the realized $r_t$, and the dynamics are governed by $\theta(t)$, which encodes the path of $r$ that the market is pricing in.

This is a clever trick: rather than estimating the parameters from historical data (which would give a wrong curve at the calibration date), Hull-White uses the current market curve as input. The only free parameters are $\kappa$ and $\sigma$, which control the dynamics of rates, not the level.

The Hull-White Tree

For pricing interest-rate derivatives that depend on the future path of rates (mortgage-backed securities, callable bonds, Bermudan swaptions), Hull-White is often implemented as a trinomial tree. The tree discretizes the short-rate dynamics, and at each node, the bond prices are computed using the closed-form formula.

The tree handles path-dependent payoffs by storing path information at each node. For American-style or Bermudan-style exercise features, the tree can compute the optimal exercise strategy by backward induction.

The volatility $\sigma$ controls how the tree branches. The mean reversion $\kappa$ controls the tendency to revert toward $\theta(t)$. The tree can be calibrated to fit both the initial curve (via $\theta$) and the swaption volatility surface (via $\sigma$ as a function of strike and expiry, in extensions).

The Hull-White tree is fast, accurate, and widely used in industry. Many banks have proprietary variants with more factors or time-dependent vol.

The Heath-Jarrow-Morton Framework

Heath, Jarrow, and Morton (1992) took a different approach. Instead of modeling the short rate $r_t$ and inferring the forward curve, they model the instantaneous forward rate $f(t, T)$ directly. The forward rate $f(t, T)$ is the rate at which one can lock in a contract at time $t$ for delivery at time $T > t$.

The HJM model specifies that for each maturity $T$, the forward rate satisfies

Here $\alpha(t, T)$ is the drift and $\sigma(t, T)$ is the volatility, both of which can depend on $t$ and $T$. The HJM framework lets you specify the volatilities freely and infers the drift from no-arbitrage.

The forward rates satisfy a key relationship:

i.e., the short rate equals the instantaneous forward rate at the same time. This connects the HJM framework back to the short-rate models we have seen.

The HJM No-Arbitrage Drift Condition

The important result of HJM is the no-arbitrage drift condition. For the model to be arbitrage-free, the drift of each forward rate must satisfy

for the case of one Brownian motion. In the multi-factor case with a vector of Brownian motions $\mathbf{W}_t$ and volatility vector $\boldsymbol{\sigma}(t, T)$,

This says: the drift of the forward rate is the inner product of its instantaneous volatility with the cumulative volatility integral. This is not a free choice — it is forced by no-arbitrage.

The intuition: if you want the forward rate at $T$ to have volatility $\sigma(t, T)$, the drift must compensate for the risk premium. The risk premium is the covariance of the forward rate with the discount bond, which by no-arbitrage equals the integral of vol-squared. So the drift is the inner product of vol with the integrated vol.

This is a beautiful structural result. It says: pick any volatility structure you like, and the model is arbitrage-free with the right drift. You can then calibrate the volatility to fit swaption prices or other instruments, and the model will be consistent.

Why HJM Generalizes Hull-White

Hull-White is a special case of HJM. In Hull-White, the short rate is Gaussian with constant volatility $\sigma$. The forward-rate volatility is related to the short-rate volatility by

This is the HJM volatility that produces the Hull-White short-rate dynamics. Plugging into the no-arbitrage drift condition:

The forward rate dynamics are then

The short rate $r_t = f(t, t)$ satisfies $df(t, t) = \alpha(t, t) \, dt + \sigma(t, t) \, dW_t$, which gives the Hull-White SDE with $\theta(t)$ chosen to match the curve.

So Hull-White is the HJM model with the specific volatility structure $\sigma(t, T) = \sigma e^{-\kappa(T-t)}$. Other choices of $\sigma(t, T)$ give other one-factor Gaussian short-rate models. The HJM framework is the most general Gaussian term structure model.

Multi-Factor HJM

In practice, you need multiple factors to fit the volatility surface. The multi-factor HJM model uses $d$ Brownian motions and volatility vectors $\boldsymbol{\sigma}_i(t, T)$. The drift condition becomes

A popular choice is to use principal components of the yield curve as factors. The first three PCs typically explain over 95% of yield-curve variation: level, slope, curvature. A three-factor HJM model with these factors captures most of the empirical dynamics.

Calibration: fit the factor loadings to historical yield changes, and the factor volatilities to historical factor changes. The model then has explicit dynamics and can be used to simulate yield curves.

For pricing interest-rate derivatives, the simulation approach is often easier than tree methods. Generate many paths of the yield curve under the risk-neutral measure, compute the derivative payoff on each path, and average. This is the standard Monte Carlo approach for HJM.

The Musiela Parameterization

A common reformulation of HJM uses Musiela’s parameterization, which keeps the state variables at a fixed time-to-maturity rather than at a fixed maturity. Define

for time-to-maturity $\tau \geq 0$. The dynamics of $x_t(\tau)$ are

The advantage: $x_t(\tau)$ is a process indexed by a fixed variable $\tau$, like the state of a functional. This is convenient for infinite-dimensional analysis and for some numerical methods (especially PDE-based pricing on a function space).

The Musiela parameterization is also the basis of the Brace-Gatarek-Musiela (BGM) model for LIBOR rates, which is the industry standard for pricing exotic interest-rate derivatives like swaptions, caps, and floors.

Pricing Caps and Floors under Hull-White

A cap on a floating rate pays the difference between the floating rate and a strike, floored at zero, at each reset date. The price of a cap is a portfolio of caplets, each of which is a call option on the future short rate.

Under Hull-White, the short rate is Gaussian, so the distribution of $r_{T_i}$ at the $i$-th reset date is Gaussian. The caplet price is therefore a Black-like formula with the Hull-White volatility parameter:

where $F_i$ is the forward rate for the period and $\sigma_i$ is related to the Hull-White volatility. The exact relationship involves an adjustment factor.

This is the same form as the Black formula, with the Hull-White vol playing the role of the Black vol. In practice, you back out the implied vol from market cap quotes, then fit Hull-White parameters to the vol surface.

The Ho-Lee Special Case

Ho and Lee (1986) proposed the simplest HJM model: $\sigma(t, T) = \sigma$ (constant). This corresponds to a short rate with no mean reversion:

The drift $\theta(t)$ is chosen to fit the initial curve. The model is Gaussian, has no mean reversion, and is rarely used in practice because it produces unrealistic rate dynamics (rates can wander without bound).

Ho-Lee is mostly of historical interest. It was the first no-arbitrage model to fit an arbitrary initial curve. It is the father of Hull-White, which added mean reversion.

A Calibration Recipe

Here is a working recipe for calibrating Hull-White to a discount curve and a swaption volatility surface.

1. Fit the initial curve. Use the market discount bond prices $P^M(0, T_i)$ for a set of maturities. The model automatically fits these via the $\theta(t)$ function.

2. Choose $\kappa$. This controls mean reversion. Typical values are 0.01 to 0.5 per year. Higher $\kappa$ means rates revert faster. Calibrate to ATM swaption prices: ATM vols depend on $\kappa$.

3. Choose $\sigma$. This is the short-rate volatility. Calibrate to ATM swaption prices or to a time series of historical rate changes. Typical values are 0.005 to 0.02 per year.

4. Time-dependent vol (optional). For non-ATM swaptions, you may need $\sigma(t)$ varying with time. This breaks the closed-form bond price but allows fitting the vol surface.

For a vanilla implementation, $\kappa$ and $\sigma$ are enough.

import numpy as np
from scipy.optimize import brentq

def hw_bond_price(r0, kappa, sigma, t, T, theta):
    B = (1 - np.exp(-kappa * (T - t))) / kappa
    integral = theta * (T - t) - (theta/kappa) * (1 - np.exp(-kappa * (T - t)))
    var_term = (sigma**2 / (2 * kappa**2)) * (T - t + (1/(2*kappa))*(1 - np.exp(-2*kappa*(T-t))) - (2/kappa)*(1 - np.exp(-kappa*(T-t))))
    return np.exp(integral + 0.5 * var_term - B * r0)

This is a numerical implementation that incorporates the time-dependent $\theta$. For practical use, you would compute $\theta(t)$ from the initial curve via

where $f^M(0, t)$ is the market instantaneous forward rate. This formula ensures the model fits the initial curve.

HJM as a General Framework

The HJM framework is the most general Gaussian term structure model with Markovian short-rate dynamics. Every Gaussian short-rate model (Vasicek, CIR, Hull-White, Ho-Lee, extended CIR) is a special case of HJM with a specific volatility structure $\sigma(t, T)$.

For non-Gaussian models (CIR, jump-diffusion, regime-switching), HJM does not directly apply. But the no-arbitrage idea — the drift is forced by the volatility — extends. In general, for any model of forward rates $df(t, T) = \alpha \, dt + \sigma \, dW$, no-arbitrage implies

regardless of whether the volatility is Gaussian or jump-driven.

This is the Brace-Gatarek-Musiela generalization, which underlies the LIBOR market model. We will not go into detail here, but the principle is the same: pick the volatility structure, the drift is forced.

Why Mean Reversion Matters

Hull-White’s mean reversion $\kappa$ has a real effect on prices and risk. With higher $\kappa$:

• Rates revert to $\theta$ faster, so long-horizon rates have lower vol.
• Bond prices are less sensitive to the current rate (because the rate is more likely to revert).
• Option prices (caps, floors) have lower implied vol for long maturities.

Empirically, mean reversion in rates is moderate. Half-lives of 2-5 years are typical. Calibrations that ignore mean reversion (Ho-Lee, BGM with $\kappa = 0$) overestimate long-horizon vol.

Mean reversion also affects risk management. A long-horizon bond position is less risky in Hull-White than in Ho-Lee, because rates are more likely to revert to a sensible level. For asset-liability matching (pension funds, insurance), mean reversion is essential for keeping long-horizon forecasts reasonable.

What to Take Away

Hull-White extends Vasicek by adding a time-dependent drift that fits any initial discount curve. The model has closed-form bond prices and tractable dynamics. The HJM framework generalizes to arbitrary forward-rate volatility structures, with the drift determined by a no-arbitrage condition. Hull-White is the most popular practitioner model; HJM is the most general Gaussian framework.

Both rely on a beautiful structural result: the drift of forward rates is the inner product of their volatility with the cumulative volatility integral. Pick any volatility, get a consistent model. The calibration is the easy part. The no-arbitrage condition does the heavy lifting.

References: Hull and White (1990, 1993); Heath, Jarrow, and Morton (1992); Brigo and Mercurio (2006), Chapters 3-4; Hull (2018), Ch. 32 (Hull-White one-factor model) and Ch. 33 (HJM framework). The HJM paper was awarded the Journal of Financial Econometrics Distinguished Author Award. It deserved the prize.

For the bond pricing, the key fact is that $P(t, T)$ is exponential-affine in $r_t$, with time-dependent coefficients that absorb the initial curve. For the no-arbitrage, the key fact is the drift-volatility relationship. Everything else follows from these two facts. Master them, and you have the basis of all Gaussian term structure models.

The Ho-Lee Tree, Briefly

Ho-Lee was the first no-arbitrage model to fit an initial curve. The model uses $\sigma(t, T) = \sigma$ (constant volatility). The corresponding short-rate SDE is

with $\theta(t)$ chosen to fit the curve. The original paper used a binomial tree for pricing bond options.

The tree is built by discretizing time into steps of length $\Delta t$. At each node, the short rate can move up or down by $\sigma \sqrt{3 \Delta t}$. The terminal tree values are set to fit the bond prices at each maturity. The tree is recombining (up-then-down equals down-then-up).

Ho-Lee trees are simple to implement but have poor accuracy for long-maturity claims because the discretization error accumulates. Hull-White trees improve on this by including mean reversion, which keeps rates in a sensible range.

The Brace-Gatarek-Musiela (BGM) Model

The Ho-Lee / Hull-White framework models the instantaneous short rate. The BGM model (1997) takes a different approach: model the forward LIBOR rates directly. This is more useful for pricing derivatives on LIBOR, which is the standard reference rate for floating-rate notes and interest-rate swaps.

The BGM dynamics for the forward LIBOR $L(t, T_i, T_{i+1})$ over the period $[T_i, T_{i+1}]$ are

where $\mu_i(t)$ is a drift determined by no-arbitrage and $\sigma_i(t)$ is the volatility. The volatility is typically parameterized as $\sigma_i(t) = \phi_i L(t, T_i, T_{i+1})$ or some similar function. The model is rich enough to fit the swaption volatility surface.

The BGM model has become the industry standard for pricing interest-rate derivatives, especially after the financial crisis when the LIBOR market model became essential for valuing exotic products. It generalizes HJM to a discrete-tenor setting (specific reset dates rather than continuous time).

Forward Rates and the Bond Price

The forward rate $f(t, T)$ and the bond price $P(t, T)$ are related by

For an arbitrage-free model, this relationship must hold. The HJM framework specifies dynamics for $f$, and the bond price is derived by integration:

for some reference maturity $T_0$. The short rate is the limit as $T \to t$:

In practice, you do not have continuous forward rates. You have discrete forward rates at specific tenors (1m, 3m, 6m, 1y, etc.). The HJM model in discrete form is the BGM model.

Multi-Factor Models in Practice

For accurate pricing of path-dependent or volatility-sensitive claims, you need multi-factor HJM. The standard approach:

1. Estimate principal components of historical yield changes.
2. Use 3 factors (level, slope, curvature) for a tractable model.
3. Specify the factor volatilities from historical data.
4. Calibrate the factor correlations from data.

The resulting model has 3-6 free parameters (factor vols, correlations, mean-reversion rates). It fits the volatility surface better than one-factor models and gives reasonable prices for path-dependent claims.

Multi-factor HJM is the basis of the LIBOR market model used by most banks for pricing exotics. The implementation can be Monte Carlo (most common), PDE (for low-dimensional cases), or quadrature (for analytic approximation).

The Drift Condition in Discrete Time

For a discrete set of forward rates $L_i(t)$ for tenors $T_i$, the HJM drift condition in discrete time is

where $\delta_j = T_{j+1} - T_j$ is the day count fraction and the sum is over all tenors $T_j$ that are observable at time $t$. This is the discrete version of the HJM drift condition.

This formula says: the drift of the $i$-th forward rate depends on the volatility of the $i$-th rate and on the volatilities of all later rates, weighted by their durations. The intuition: a movement in a later forward rate affects the value of the bond at that tenor, which affects the pricing of the $i$-th forward rate through the discount factor.

Implementing this in Monte Carlo is straightforward: simulate each forward rate using its drift and volatility, and the no-arbitrage condition ensures the prices are consistent.

A Historical Aside

Heath, Jarrow, and Morton’s 1992 paper was a breakthrough because it unified all earlier term structure models in one framework. Before HJM, each model (Vasicek, CIR, Ho-Lee) was a separate construction with its own dynamics and limitations. After HJM, every model was a special case of the general framework, and the focus shifted to specifying volatility structures.

The HJM drift condition is one of the most general no-arbitrage results in finance. It applies to Gaussian models, jump-diffusion models, regime-switching models, and even some stochastic-vol models. As long as the underlying noise is a martingale, the drift is forced by no-arbitrage.

This generality is what made HJM the foundation of the LIBOR market model and the basis for most modern interest-rate derivatives pricing.