Hull-White Interest Rate Model Explained (2026)

The curve-fitting short-rate model

John Hull and Alan White's 1990 paper, Pricing Interest-Rate-Derivative Securities, fixed a practical defect in the Vasicek model. A constant-parameter short-rate model rarely reproduces today's discount curve. A pricing model that misses tradable bond prices before pricing the option is starting from the wrong place.

The one-factor Hull-White model under the money-market risk-neutral measure $\mathbb Q$ is

$$dr_t=[\theta(t)-a r_t]dt+\sigma dW_t^{\mathbb Q}, \qquad a>0,\quad\sigma>0.$$

The mean-reversion speed $a$ has units of inverse years. With rates expressed as decimals and time in years, $\sigma$ is an absolute rate volatility per square-root year. The deterministic drift $\theta(t)$ is chosen from the initial discount curve. It is not an extra free calibration parameter at every date.

Hull-White is useful when an exact initial-curve fit, Gaussian tractability and early-exercise machinery matter. It is less useful when positivity is essential or a one-factor volatility shape is plainly inadequate. The CIR model makes the opposite trade: positivity, but no exact fit without a shift.

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The exact initial-curve fit

Write $r_t=x_t+\alpha(t)$, where

$$dx_t=-a x_tdt+\sigma dW_t^{\mathbb Q}.$$

Then $\theta(t)=\alpha'(t)+a\alpha(t)$. If $P^M(0,T)$ is the market discount factor and

$$f^M(0,t)=-\frac{\partial}{\partial t}\log P^M(0,t),$$

the constant-volatility Hull-White fit is

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

and hence

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

The formula is exact for a smooth instantaneous forward curve. In production, differentiating a noisy bootstrap can make $\theta(t)$ jump about. Desks usually interpolate discount factors or forwards with a controlled scheme and keep the interpolation convention inside calibration and valuation.

The decomposition also exposes the distribution. Conditional on $x_s$,

$$x_t\mid x_s\sim \mathcal N\left( x_se^{-a(t-s)}, \frac{\sigma^2}{2a}[1-e^{-2a(t-s)}] \right).$$

Gaussian rates can be negative. That is a feature when the calibrated curve and scenarios require negative values, and a defect when the state represents something that cannot cross zero. The Ornstein-Uhlenbeck process is the mathematical core.

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Bond prices, PDE and options

A claim $V(t,r)$ with no intermediate cash flows solves

$$V_t+[\theta(t)-ar]V_r +\frac12\sigma^2V_{rr}-rV=0$$

under $\mathbb Q$. For a zero-coupon bond,

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

with

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

At $t=0$, the exponential correction cancels and $P(0,T)=P^M(0,T)$. This equality is a calibration identity, not a forecast that the future curve will stay near today's curve.

Suppose the initial curve is flat at 3%, $a=0.10$, $\sigma=0.01$, $t=1$, $T=5$ and the realised short rate at year one is 3.5%. Then $B=3.296800$, $A=0.978643$ and $P(1,5)=0.871991$. The remaining four-year continuously compounded yield is 3.4244%.

European options on zero-coupon bonds are also closed form. For a call expiring at $T_0$ on a bond maturing at $T>T_0$, with strike $K$ paid at $T_0$,

$$C(t)=P(t,T)N(d_1)-KP(t,T_0)N(d_2),$$ $$\sigma_P= \sigma B(T_0,T) \sqrt{\frac{1-e^{-2a(T_0-t)}}{2a}}, \quad d_1=\frac{\log[P(t,T)/(KP(t,T_0))]}{\sigma_P} +\frac12\sigma_P, \quad d_2=d_1-\sigma_P.$$

The time origin in $\sigma_P$ matters. Replacing $T_0-t$ by $T_0$ is only valid when pricing at $t=0$.

Farshid Jamshidian's 1989 exact bond-option formula extends this to coupon bonds. For a standard swaption, the receiver is a call on the fixed-leg coupon bond. The payer is a put. Because every component bond is monotone in the same one-factor state, one critical $r^*$ decomposes that coupon-bond option into zero-coupon options. Calling a payer swaption a call on the coupon bond reverses the payoff.

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Exact simulation and calibration

The Gaussian factor can be advanced exactly over each time step. There is no reason to use Euler for constant $a$ and $\sigma$.

import numpy as np


def simulate_hull_white(alpha, a, sigma, times, n_paths, seed=7):
    rng = np.random.default_rng(seed)
    x = np.zeros(n_paths)
    rates = np.empty((n_paths, len(times)))
    rates[:, 0] = alpha(times[0])

    for k, dt in enumerate(np.diff(times), start=1):
        decay = np.exp(-a * dt)
        stdev = sigma * np.sqrt((1.0 - decay**2) / (2.0 * a))
        x = decay * x + stdev * rng.standard_normal(n_paths)
        rates[:, k] = x + alpha(times[k])
    return rates

This assumes $x_0=0$, so $r_0=\alpha(0)=f^M(0,0)$. A different initial state must be carried explicitly.

Calibration has two layers. First, bootstrap the discount curve and compute $\theta(t)$ for any proposed $a,\sigma$. Second, fit $a$ and $\sigma$, or piecewise volatilities, to caplet or swaption prices. Price or vega-weighted errors are preferable to treating every quoted volatility as equally informative.

An exact curve fit does not imply an exact option fit. Constant $a$ and $\sigma$ provide only two volatility parameters, so one-factor Hull-White cannot reproduce an arbitrary swaption cube or smile. Piecewise $\sigma(t)$ improves the term structure of volatility, but the model remains Gaussian and one-factor.

Building a calibration that survives production

The curve fit begins before the model optimiser. Instrument conventions, calendars, day counts, collateral currency and interpolation all enter the bootstrapped discount factors. In a modern multi-curve framework, the curve used for discounting need not be the curve used to project an index fixing. The short-rate dynamics and numeraire must belong to the chosen discounting setup, while forwarding curves need their own consistent treatment. Substituting a single curve into a multi-curve trade can create a larger error than changing $a$ or $\sigma$.

Do not estimate $\theta(t)$ by taking raw finite differences of market quotes. Build a smooth, arbitrage-controlled representation of $P^M(0,T)$ or $f^M(0,T)$, differentiate that representation, and test the reconstructed discount factors. A local interpolation change can alter forward derivatives substantially while barely moving bond prices. For simulation, storing $\alpha(t)$ on the event grid is often more stable than repeatedly evaluating a noisy $\theta(t)$.

The volatility calibration should reflect the payoff being valued. A callable bond desk might select co-terminal swaptions near likely exercise dates. A cap book should include caplets or caps across the relevant reset tenors. Convert Black, shifted-Black or normal quotes into instrument prices with the exact market annuity and forward conventions, then minimise price residuals or residuals scaled by vega or bid-offer width. Comparing quoted volatilities directly can give arbitrary weight to instruments simply because their quotation systems differ.

Mean reversion is identified mainly by how volatility changes across option expiry and underlying maturity. A narrow strip of instruments often leaves a long valley of nearly equivalent $(a,\sigma)$ pairs. Use several starting points, inspect the objective around the optimum and report parameter stability across dates. A tight optimiser tolerance is not evidence that $a$ is economically identified. If piecewise $\sigma(t)$ is used, regularisation or a deliberately coarse time partition can prevent the model from fitting quote noise with alternating volatility buckets.

Calibration also needs a policy for stale or illiquid quotes. Bid-offer filters and fixed instrument-selection rules should be decided before seeing the parameter result. Otherwise daily parameter jumps may reflect changing inputs rather than changing market dynamics. The original Hull and White framework and the implementation treatment in Brigo and Mercurio both make the separation clear: the deterministic term fits the initial curve, while the stochastic parameters govern option dispersion.

Numerical and model validation

For small $a\Delta$, direct evaluation of $1-e^{-a\Delta}$ loses precision. Use $-\operatorname{expm1}(-a\Delta)$ for $B$, and the corresponding stable form for the conditional variance. The $a\to0$ limits should also be tested explicitly: $B(t,T)\to T-t$, and the factor variance over $\Delta$ tends to $\sigma^2\Delta$. These checks catch divisions by a small mean-reversion speed.

A production implementation should pass several independent tests:

1. Reconstruct $P^M(0,T)$ at every bootstrap pillar from the model at $t=0$, to machine precision apart from interpolation tolerance.
2. Check $P(t,t)=1$, the bond maturity payoff and the bond-option limits as volatility or time to expiry tends to zero.
3. Simulate $x_t$ and compare empirical conditional means and variances with the stated Gaussian transition. Test several irregular step sizes, since cash-flow grids are rarely uniform.
4. Compare closed-form bond-option prices with Monte Carlo estimates and confidence intervals. Use the same numeraire and discounting convention in both calculations.
5. Reprice the exact calibration portfolio from stored market inputs and independently recompute the reported residuals.
6. Bump the initial curve, $a$, each volatility bucket and the underlying quotes. Greeks should converge as bump sizes fall and should not depend materially on an unrelated interpolation choice.

For a trinomial tree, convergence requires more than increasing the number of time steps. Confirm that the fitted state-price sums reproduce discount factors at each tree date, inspect the effect of state-space truncation and compare European claims with closed forms before trusting early-exercise values. For Monte Carlo exposure, compare simulated zero-coupon bond means under the appropriate forward measure or discounted asset means under $\mathbb Q$. Merely checking that the average short rate looks reasonable does not test no-arbitrage implementation.

Reserve instruments not used in calibration for out-of-sample validation. Errors by expiry, tenor and strike reveal whether the failure is a term-structure problem, a one-factor correlation problem or a missing smile. That diagnosis matters because adding volatility buckets cannot repair every failure.

For Bermudan swaptions and callable bonds, the attraction is numerical. Hull and White's 1994 trinomial construction calibrates time-dependent lattice shifts to market discount factors before backward induction. Merely placing $r=x+\alpha(t)$ on a truncated tree is not the full no-arbitrage tree algorithm.

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Where Hull-White earns its keep

One-factor Hull-White remains a sensible model for callable bonds, Bermudan swaptions and exposure calculations where speed, an exact initial curve and stable Greeks matter more than smile dynamics. Trees, finite differences and regression Monte Carlo all work with its low-dimensional state. The risk-neutral pricing convention must remain consistent across the curve, calibration instruments and payoff.

Its weaknesses are structural. A single Brownian shock imposes perfect instantaneous correlation between all model bond returns up to loading signs. Gaussian tails allow very negative rates. Constant volatility cannot generate a strike smile. And $a$ is often weakly identified because different $(a,\sigma)$ pairs can price a narrow calibration set similarly.

A common two-factor extension is G2++:

$$r_t=x_t+y_t+\phi(t),\qquad dx_t=-ax_tdt+\sigma dW_{1,t},\qquad dy_t=-by_tdt+\eta dW_{2,t},$$

with $d\langle W_1,W_2\rangle_t=\rho dt$. It adds separate decay scales and correlation while retaining Gaussian bond prices. It is not a six-parameter process plus another free curve-fit parameter: $\phi(t)$ is determined by the initial curve.

Model choice should begin with the risk that must be represented. Use one-factor Hull-White when an exact discount-curve fit, fast European formulas and dependable lattice or finite-difference valuation outweigh the need for smile and rich curve dynamics. It is particularly effective when one exercise driver dominates and many valuations or exposure paths must be produced quickly.

Choose G2++ when relative movements of short and long maturities materially affect the payoff or hedge. Choose a market model when the joint dynamics of several forward rates and a broad swaption calibration are central. Choose a smile model or a stochastic-volatility extension when strike dependence drives value. Choose CIR or another positive process when negative states are economically impossible. Extra factors are not automatically better: they add weakly identified parameters, calibration cost and hedging instability, so the retained instruments must demonstrate a material improvement.

For risk-neutral pricing, calibrate to option prices and test hedges. For forecasting, stress testing or asset-liability scenarios, the risk-neutral drift fitted to today's curve is not a statistical forecast. A physical-measure model or an explicit real-world overlay is required. Confusing those objectives can make a technically exact curve fit produce misleading long-horizon scenarios.

Use Hull-White when its numerical convenience matches the payoff. Do not use it merely because the first curve fit looks perfect. That part was guaranteed.

The original papers remain the best next stop: Hull and White (1990) for the short-rate construction, Jamshidian (1989) for the coupon-bond decomposition, Hull and White (1994) for the trinomial tree, and Brigo and Mercurio (2006) for the textbook development.

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Frequently Asked Questions

Does Hull-White fit today's yield curve exactly?

Yes, provided $\theta(t)$ is constructed from the same market discount curve used for valuation. This says nothing about the quality of the option-volatility fit.

Can Hull-White rates be negative?

Yes. The stochastic factor is Gaussian, so every future short rate has support on the whole real line. A deterministic shift does not remove that tail.

Is $\theta(t)$ the long-run mean?

Not directly in this notation. The instantaneous mean-reversion level is $\theta(t)/a$, while $\theta(t)$ also carries the initial-curve fit and convexity correction.

Is a payer swaption a call on a coupon bond?

No. Under the standard fixed-leg representation, a receiver swaption is a call on the coupon bond and a payer swaption is a put. Jamshidian's decomposition handles either option direction.

How are $a$ and $\sigma$ calibrated?

Fit model caplet or swaption prices to liquid market instruments after bootstrapping the discount curve. The result is risk-neutral and depends on the calibration set, weights and volatility convention.

When is G2++ preferable?

Use G2++ when one factor cannot represent the observed correlation between short and long maturities. It adds flexibility, but it remains Gaussian and does not create a smile by itself.