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Rates & curves · advanced

Hull–White one-factor model

Fitting today’s term structure while modelling mean-reverting Gaussian short-rate dynamics

BY THE END, YOU CAN

01Relate Ho–Lee, Vasicek and Hull–White short-rate dynamics.

02Derive the conditional Gaussian distribution of the mean-reverting factor.

03Explain how the time-dependent drift fits the initial curve.

04Price or calibrate caps/swaptions while diagnosing negative-rate and one-factor limitations.

01
INTUITION

Identify the state variables and the behavior they add.

Hull–White adds a deterministic time-dependent drift to an Ornstein–Uhlenbeck short-rate process. That shift fits today’s discount curve exactly, while mean reversion and volatility govern future rate distributions.

01

Curve fit and dynamic calibration are separate tasks.

02

Gaussian rates permit analytic bond formulas and negative outcomes.

03

One factor moves the entire curve through a single source of randomness, limiting decorrelation across tenors.

02
WHY MARKETS CARE

Ask which instruments can identify the dynamics.

Hull–White remains a transparent benchmark for callable structures, Bermudan exercise, exposure simulation and rate-option calibration.

INSTRUMENTS

caps and floors

European swaptions

callable bonds

Bermudan swaptions

QUOTE CONVENTION

Rate-option calibration must state normal/lognormal/shifted volatility convention, forward, annuity, expiry, tenor, strike, settlement and collateral curves.

03
MATHEMATICS

Write the dynamics before interpreting parameters.

Formula · Full derivation

Hull–White dynamics

drt=[θ(t)−art]dt+σdWtdr_t=[\theta(t)-a r_t]dt+\sigma dW_t

A time-inhomogeneous Gaussian short rate mean reverts at speed a with volatility σ.

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Formula · Full derivation

Conditional variance

Var⁡t(rT)=σ22a(1−e−2a(T−t))\operatorname{Var}_t(r_T)=\frac{\sigma^2}{2a}\left(1-e^{-2a(T-t)}\right)

Mean reversion bounds long-horizon short-rate variance when a>0.

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Full derivation
Full derivation

Solving the mean-reverting Gaussian factor

Apply an integrating factor to the linear short-rate SDE and separate deterministic drift from stochastic innovation.

  1. 01

    Multiply by the integrating factor

    Apply e^{at} so the mean-reversion term becomes an exact differential.

    d(eatrt)=eatθ(t)dt+σeatdWtd(e^{at}r_t)=e^{at}\theta(t)dt+\sigma e^{at}dW_t
  2. 02

    Integrate from t to T

    Solve for r_T as decayed current state plus deterministic drift integral and a Gaussian stochastic integral.

    rT=e−aτrt+∫tTe−a(T−u)θ(u)du+σ∫tTe−a(T−u)dWur_T=e^{-a\tau}r_t+\int_t^T e^{-a(T-u)}\theta(u)du+\sigma\int_t^T e^{-a(T-u)}dW_u
  3. 03

    Compute the variance

    Itô isometry converts the stochastic-integral variance into an ordinary integral.

    σ2∫tTe−2a(T−u)du=σ2(1−e−2aτ)/(2a)\sigma^2\int_t^T e^{-2a(T-u)}du=\sigma^2(1-e^{-2a\tau})/(2a)
  4. 04

    Fit the initial term structure

    Choose θ(t), or an equivalent deterministic shift φ(t), so model P(0,T) equals the observed discount curve at every maturity.

  5. 05

    Calibrate option dynamics

    Fit a and σ—or piecewise volatility—to a governed cap/swaption basket after the curve fit.

Hull–White separates exact initial-curve fit from a parsimonious one-factor Gaussian dynamics used for optionality and scenarios.

Inputs
  • r_t: short rate
  • a: mean-reversion speed
  • \sigma: short-rate volatility
  • \theta(t): curve-fitting drift
Assumptions and limits
  • Gaussian short rates permit arbitrarily negative values.
  • One factor cannot reproduce independent curve twists.
  • Parameters can be weakly identified and regime dependent.
Formula · Full derivation

Affine bond price

P(t,T)=A(t,T)e−B(t,T)rt,B(t,T)=1−e−a(T−t)aP(t,T)=A(t,T)e^{-B(t,T)r_t},\quad B(t,T)=\frac{1-e^{-a(T-t)}}{a}

Zero-coupon bond prices are exponential-affine in the short rate.

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05
MODEL / PRICING

Calibrate, compute, and challenge the dynamics.

METHOD

Fit the deterministic shift to the discount curve, calibrate dynamic parameters to a stated option-volatility convention and use analytic, tree or Monte Carlo engines appropriate to exercise style.

CALIBRATION

Calibrate with bid/ask-aware weights, multiple starts and stability diagnostics. Compare caps and swaption tenors not used in the fit.

Implementation with current QuantLib

Construct HullWhite from the accepted yield-curve handle, then use model-consistent engines and CalibrationHelpers for caps/swaptions. Recheck calibration after relinking the curve and inspect helper errors individually.

API authority: upstream QuantLib reference pinned in the source registry.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
ARCHITECTURE
  • Keep market conventions and quote lineage at the boundary.
  • Solve curves and dynamics in framework-free deterministic kernels.
  • Return residuals, state and sensitivities with every value.
  • Test analytical limits, reconstruction identities and failure domains.
PYTHON 3 · NUMPY / SCIPY

Exact Hull–White factor transition

Simulate the mean-zero Ornstein–Uhlenbeck factor with the exact Gaussian variance.

REUSABLE EXAMPLE
01from __future__ import annotations
02
03import math
04import numpy as np
05
06def hw_transition(x: np.ndarray, a: float, sigma: float, dt: float, z: np.ndarray) -> np.ndarray:
07 if a <= 0 or sigma < 0 or dt <= 0 or x.shape != z.shape:
08 raise ValueError("invalid Hull-White transition")
09 decay = math.exp(-a * dt)
10 variance = sigma*sigma * (1.0 - math.exp(-2.0*a*dt)) / (2.0*a)
11 return decay*x + math.sqrt(variance)*z
12
13rng = np.random.default_rng(7)
14z = rng.standard_normal(200_000)
15x = hw_transition(np.zeros_like(z), 0.08, 0.01, 1.0, z)
16theory = 0.01**2 * (1-math.exp(-0.16)) / 0.16
17assert abs(np.var(x) - theory) / theory < 0.015
18print(f"sample variance={np.var(x):.8f} | theory={theory:.8f}")
EXPECTED OUTPUTSample variance within 1.5% of the exact transition variance.
SANITY CHECKS

✓ Mean reversion and time step are positive.

✓ Deterministic random seed is fixed.

✓ Monte Carlo moment matches the analytical reference.

07
INTERACTIVE LAB

Shock one parameter and trace the full response.

MEAN REVERSION AND GAUSSIAN RATE DISPERSION

Hull–White dynamics laboratory

Move mean reversion, volatility and policy shock; inspect short-rate distribution, bond response and curve-factor decay.

SYNTHETIC · CONTROLLED SCENARIOS
Mean reversion0.030
Long-run factor std2.86%
10Y persistence74.08%
Factor response / volatility by Horizon (years)

Slow reversion: Shocks persist across the curve.

  • shock persistence
  • short-rate std dev
Horizon (years): 0.0Y. shock persistence: 1.00000. short-rate std dev: 0.00000.

Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.

ACTIVE STATE

Slow reversion — Shocks persist across the curve. Move the control and inspect every series with pointer or touch.

08
FRONT OFFICE

Where the model meets the book.

ON THE DESK
“Hull–White fits today’s curve by construction; calibration quality is about tomorrow’s distribution and option basket.”
VISIBLE INPUTS

discount curve

swaption/cap volatility basket

volatility convention

mean reversion

model/engine discretisation

CALIBRATION

Fit curve deterministically, then calibrate dynamics with residual, parameter-bound and out-of-sample diagnostics.

RISK

mean-reversion risk

volatility buckets

one-factor basis

exercise/model risk

DAILY WORKFLOW
  1. freeze curve/vol snapshot
  2. fit deterministic shift
  3. calibrate a/σ
  4. validate helpers
  5. run scenarios and hedges
Production failure modes
  • normal/lognormal mismatch
  • curve relink without recalibration
  • parameter boundary
  • tree time-grid misalignment
09MACRO CONNECTIONOpen the transmission channel.
MACRO CONNECTION

Mean reversion and policy regimes

Policy shocks move the short end immediately; the mean-reversion parameter determines how quickly model scenarios pull that disturbance back toward the fitted term structure.

01Policy shocktransmits

moves current short-rate state

02Mean reversiontransmits

controls persistence

03Bond/option valuestransmits

respond through affine dynamics

04Callable riskoutput

changes exercise and convexity

10COMMON PITFALLSOpen the failure checklist.
01

Calling exact initial-curve fit a successful option calibration.

02

Comparing normal and lognormal swaption vols directly.

03

Ignoring negative-rate tails of a Gaussian model.

04

Using one factor for curve-basis risk it cannot span.

11SOURCES / FURTHER READINGOpen sources and continue the track.
research

Curve construction, multi-curve, short-rate and HJM lectures

Research map for term-structure theory and numerical experiments; all platform explanations and code are original.

Source
Financial Engineering: Interest Rates & xVA
Author
L. A. Grzelak
Ref
main
OPEN ORIGINAL SOURCE ↗
research

Monte Carlo, stochastic calculus and calibration lectures

Mathematical and numerical cross-reference for model dynamics and diagnostics.

Source
Computational Finance Course
Author
L. A. Grzelak
Ref
main
OPEN ORIGINAL SOURCE ↗
implementation reference

Current bootstrapping, interpolation, curve, model, cap/floor and swaption tests

Implementation authority for production object boundaries and regression-test patterns.

Source
QuantLib upstream
Author
QuantLib contributors
Ref
v1.42.1
OPEN ORIGINAL SOURCE ↗
CONTINUE THE TRACKCurve risk, carry and roll-down

Moving from one parallel DV01 to bucketed scenarios, hedge instruments and time passage

→HJM and market-model dynamics

Modelling the whole forward curve under a no-arbitrage drift restriction

→