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1. The Hull-White Model

The Hull-White model, also known as the extended Vasicek model, is one of the most widely used interest rate models in practice. Introduced by John Hull and Alan White in 1990, it extends the Vasicek model by incorporating time-dependent parameters, enabling perfect calibration to the current term structure of interest rates.

1.1 Model Specification

The Hull-White model describes the short rate dynamics under the risk-neutral measure as:

\[ \begin{equation} dr(t) = \left[\theta(t) - a(t)r(t)\right]dt + \sigma(t)\,dW(t) \label{eq:hw_sde_general} \end{equation} \]

where:

  • \(r(t)\) is the instantaneous short rate at time \(t\)
  • \(a(t) > 0\) is the time-dependent mean reversion speed
  • \(\theta(t)\) is the time-dependent drift function
  • \(\sigma(t) > 0\) is the time-dependent volatility
  • \(W(t)\) is a standard Brownian motion under the risk-neutral measure \(\mathbb{Q}\)

In practice, the most common specification uses constant mean reversion and volatility:

\[ \begin{equation} dr(t) = \left[\theta(t) - ar(t)\right]dt + \sigma\,dW(t) \label{eq:hw_sde} \end{equation} \]

with constant \(a\) and \(\sigma\), but time-varying \(\theta(t)\)

1.2 The Key Innovation: Fitting the Yield Curve

The function \(\theta(t)\) is chosen to match the current observed market yield curve exactly. This is the defining innovation of the Hull-White model and what makes it such a powerful tool in practice. Unlike the Vasicek model, which uses a constant long-term mean \(b\) that is estimated from historical data, Hull-White allows the entire drift function to be time-varying. This means the model can reproduce the initial term structure of interest rates perfectly, by construction.

The economic intuition is compelling. The market's current yield curve encodes the collective expectations of all market participants about where interest rates will go in the future. The Vasicek model ignores this information and instead fits to just a long-term average. Hull-White respects the market's forward rate expectations by building them directly into the model. When \(\theta(t)\) is calibrated to the market's forward curve, the model prices every zero-coupon bond at time \(t=0\) exactly as the market does. This eliminates one major source of pricing error.

The drift function \(\theta(t)\) is determined from the initial forward curve \(f^M(0,t)\) observed in the market through the formula:

\[ \begin{equation} \theta(t) = \frac{\partial f^M(0,t)}{\partial t} + af^M(0,t) + \frac{\sigma^2}{2a}\left(1 - e^{-2at}\right) \label{eq:theta_calibration} \end{equation} \]

This formula arises from ensuring that bond prices, when computed under the Hull-White dynamics, match market prices at the initial time. The first term captures how the market's forward curve is changing with maturity—the market's explicit expectation about rates. The remaining terms are adjustments that account for the mean reversion and volatility already specified in the model 1.

Once \(\theta(t)\) is calibrated this way, only two parameters remain to be estimated: the mean reversion speed \(a\) and the volatility \(\sigma\). These are calibrated to market prices of interest rate derivatives such as caps, floors, or swaptions. This two-stage calibration process is pragmatic and powerful. The first stage ensures consistency with the current term structure, and the second stage ensures consistency with the market's view of interest rate dynamics as revealed through derivatives prices.

It is important to recognize that this approach to fitting the yield curve is intimately connected to the broader Heath-Jarrow-Morton (HJM) framework. In the HJM framework, the entire forward curve evolves as a stochastic process subject to a no-arbitrage drift restriction. When the HJM framework is specialized to have exponential volatility structure (as Hull and White chose), the resulting short-rate dynamics are exactly those of the Hull-White model, and the function \(\theta(t)\) that ensures no-arbitrage in the HJM context is precisely the drift function we calibrate to the market yield curve here. In this sense, Hull-White can be viewed as the most practical implementation of the HJM framework—it retains the theoretical rigor of HJM while achieving analytical tractability and implementability. For a detailed exploration of the HJM framework and its connection to Hull-White, see The Heath-Jarrow-Morton Framework.

1.3 Solution of the SDE

Despite the time-varying drift, we can still solve the Hull-White SDE explicitly. Using an integrating factor approach (similar to Vasicek):

\[ \begin{equation} r(t) = r(s)e^{-a(t-s)} + \int_s^t \theta(u)e^{-a(t-u)}du + \sigma\int_s^t e^{-a(t-u)}dW(u) \label{eq:hw_solution} \end{equation} \]
Where does \(\ref{eq:hw_solution}\) come from?

To solve the Hull-White SDE, we employ the same integrating factor technique used for the Vasicek model. We begin with the SDE in the form:

\[ dr(t) = \left[\theta(t) - ar(t)\right]dt + \sigma\,dW(t) \]

Rearranging to isolate the mean-reverting term:

\[ dr(t) + ar(t)dt = \theta(t)dt + \sigma\,dW(t) \]

We multiply both sides by the integrating factor \(\style{color: var(--math-highlight-4)}{e^{at}}\), which transforms the left-hand side into a perfect differential:

\[ \style{color: var(--math-highlight-4)}{e^{at}}dr(t) + a\style{color: var(--math-highlight-4)}{e^{at}}r(t)dt = d\left(\style{color: var(--math-highlight-4)}{e^{at}}r(t)\right) = \style{color: var(--math-highlight-4)}{e^{at}}\theta(t)dt + \style{color: var(--math-highlight-4)}{e^{at}}\sigma\,dW(t). \]

Now we integrate both sides from time \(s\) to time \(t\):

\[ \int_s^t d\left(e^{au}r(u)\right) = \int_s^t e^{au}\theta(u)du + \int_s^t e^{au}\sigma\,dW(u) \]

Evaluating the left-hand side:

\[ e^{at}r(t) - e^{as}r(s) = \int_s^t e^{au}\theta(u)du + \sigma\int_s^t e^{au}dW(u) \]

Finally, multiply both sides by \(\style{color: var(--math-highlight-1)}{e^{-at}}\) to isolate \(r(t)\), which is some how the tip to revert the integrating factor:

\[ r(t) = e^{-a(\style{color: var(--math-highlight-1)}{t}-s)}r(s) + \style{color: var(--math-highlight-1)}{e^{-at}}\int_s^t e^{au}\theta(u)du + \style{color: var(--math-highlight-1)}{e^{-at}}\sigma\int_s^t e^{au}dW(u) \]

Eventually, we get to the equation \(\ref{eq:hw_solution}\)

\[ r(t) = r(s)e^{-a(t-s)} + \int_s^t \theta(u)e^{-a(t-u)}du + \sigma\int_s^t e^{-a(t-u)}dW(u) \]

1.3.1 Decomposition of \(r(t)\)

The solution \(\ref{eq:hw_solution}\) reveals the structure of how the short rate evolves. It consists of three distinct contributions, each playing a different role in determining \(r(t)\) at a future time \(t\), given its current value at time \(s\).

Initial Condition Component:

\[ \begin{equation} r_{\text{initial}}(t) = r(s)e^{-a(t-s)} \label{eq:hw_initial_component} \end{equation} \]

The initial short rate \(r(s)\) decays exponentially as time progresses. The decay factor \(e^{-a(t-s)}\) represents the mean-reverting pull of the process: the further we look into the future, the less the current rate influences the future rate. This is the deterministic memory of the initial condition, fading at a speed determined by the mean reversion coefficient \(a\). If \(a\) is large, rates forget their initial value quickly. If \(a\) is small, the initial condition persists longer into the future.

Drift Component:

\[ \begin{equation} r_{\text{drift}}(t) = \int_s^t \theta(u)e^{-a(t-u)}du \label{eq:hw_drift_component} \end{equation} \]

The time-varying drift function \(\theta(t)\) encodes the market's expectations about where interest rates should go.

Caution!

This integral accumulates the effect of the drift over the time interval \([s, t]\), but with an important remark: each contribution \(\theta(u)\) is weighted by an exponential factor \(e^{-a(t-u)}\) that depends on when the drift acts. Shocks to the drift that occur closer to time \(t\) are weighted more heavily because there is less time for mean reversion to absorb them.

Conversely, drift shocks far in the past are heavily discounted because mean reversion has had ample time to pull the rate back toward the mean. This weighting scheme captures the pragmatic insight that recent information about the market's expectations matters more than ancient history.

Stochastic Component:

\[ \begin{equation} r_{\text{stochastic}}(t) = \sigma\int_s^t e^{-a(t-u)}dW(u) \label{eq:hw_stochastic_component} \end{equation} \]

The randomness in the model enters through the stochastic integral of Brownian increments. Each infinitesimal shock \(dW(u)\) scaled by the volatility \(\sigma\) contributes to the future short rate. Like the drift component, recent shocks have a larger impact because of the exponential weighting \(e^{-a(t-u)}\).

Effect of long-term shocks

A large shock in the recent past affects \(r(t)\) substantially, but the same shock in the distant past has been mean-reverted away and barely influences the current rate. This structure reflects the economic reality that interest rates are sticky—they do not instantaneously return to a mean but instead drift back gradually through mean reversion.

1.4 Expected Value

Given information at time \(s\), the expected value of \(r(t)\) is:

\[ \begin{equation} \mathbb{E}\left[r(t) \mid \mathcal{F}_s\right] = r(s)e^{-a(t-s)} + \int_s^t \theta(u)e^{-a(t-u)}du \label{eq:hw_expectation} \end{equation} \]
Derivation of equation \(\ref{eq:hw_expectation}\) here!

Starting from the solution of the Hull-White SDE:

\[ r(t) = r(s)e^{-a(t-s)} + \int_s^t \theta(u)e^{-a(t-u)}du + \sigma\int_s^t e^{-a(t-u)}dW(u) \]

we take the conditional expectation given the filtration \(\mathcal{F}_s\) at time \(s\). By the tower property of expectations:

\[ \mathbb{E}\left[r(t) \mid \mathcal{F}_s\right] = \mathbb{E}\left[r(s)e^{-a(t-s)} \mid \mathcal{F}_s\right] + \mathbb{E}\left[\int_s^t \theta(u)e^{-a(t-u)}du \mid \mathcal{F}_s\right] + \mathbb{E}\left[\sigma\int_s^t e^{-a(t-u)}dW(u) \mid \mathcal{F}_s\right] \]

The first term is deterministic given \(\mathcal{F}_s\):

\[ \mathbb{E}\left[r(s)e^{-a(t-s)} \mid \mathcal{F}_s\right] = r(s)e^{-a(t-s)} \]

The second term is also deterministic (since \(\theta(u)\) is a known function):

\[ \mathbb{E}\left[\int_s^t \theta(u)e^{-a(t-u)}du \mid \mathcal{F}_s\right] = \int_s^t \theta(u)e^{-a(t-u)}du \]

The third term—the stochastic integral of Brownian increments—has expectation zero under the risk-neutral measure:

\[ \mathbb{E}\left[\sigma\int_s^t e^{-a(t-u)}dW(u) \mid \mathcal{F}_s\right] = 0 \]

Combining these three contributions yields the result.

This generalizes the Vasicek expectation to include time-varying drift. When \(\theta(t) = ab\) (constant), we recover the Vasicek formula.

1.5 Variance

The variance of \(r(t)\) given information at time \(s\) is:

\[ \begin{equation} \text{Var}\left[r(t) \mid \mathcal{F}_s\right] = \frac{\sigma^2}{2a}\left(1 - e^{-2a(t-s)}\right) \label{eq:hw_variance} \end{equation} \]
Derivation of equation \(\ref{eq:hw_variance}\) here!

Starting from the solution of the Hull-White SDE:

\[ r(t) = r(s)e^{-a(t-s)} + \int_s^t \theta(u)e^{-a(t-u)}du + \style{color: var(--math-highlight-4)}{\sigma\int_s^t e^{-a(t-u)}dW(u)} \]

we compute the conditional variance given the filtration \(\mathcal{F}_s\) at time \(s\). The first two terms are deterministic given \(\mathcal{F}_s\), so they contribute zero variance. Only the stochastic component varies:

\[ \text{Var}\left[r(t) \mid \mathcal{F}_s\right] = \text{Var}\left[\style{color: var(--math-highlight-4)}{\sigma\int_s^t e^{-a(t-u)}dW(u)} \mid \mathcal{F}_s\right] \]

Using the Itô isometry, which states that \(\mathbb{E}\left[\left(\int_s^t f(u)dW(u)\right)^2\right] = \mathbb{E}\left[\int_s^t f(u)^2 du\right]\):

\[ \text{Var}\left[\sigma\int_s^t e^{-a(t-u)}dW(u)\right] = \sigma^2 \style{color: var(--math-highlight-3)}{\int_s^t e^{-2a(t-u)}du} \]

To evaluate the integral we can substitute \(v = t - u\), so \(du = -dv\). Furthermore:

  • When \(u = s\) then \(v = t - s\)
  • and, when \(u = t\) then \(v = 0\):
\[ \begin{aligned} \style{color: var(--math-highlight-3)}{\int_s^t e^{-2a(t-u)}du} &= \int_{t-s}^0 e^{-2av}(-dv) \\ &= \int_0^{t-s} e^{-2av}dv \\ &= \left[-\frac{1}{2a}e^{-2av}\right]_0^{t-s} \\ &= \frac{1}{2a}\left(1 - e^{-2a(t-s)}\right) \end{aligned} \]

Therefore:

\[ \text{Var}\left[r(t) \mid \mathcal{F}_s\right] = \sigma^2 \cdot \frac{1}{2a}\left(1 - e^{-2a(t-s)}\right) = \frac{\sigma^2}{2a}\left(1 - e^{-2a(t-s)}\right) \]

Key insight: The variance is identical to the Vasicek model! The time-varying drift \(\theta(t)\) affects the mean but not the variance structure. This is because \(\theta(t)\) is deterministic.

1.6 Distribution of the Short Rate

Combining our results, the conditional distribution is:

\[ \begin{equation} r(t) \mid \mathcal{F}_s \sim \mathcal{N}\left(r(s)e^{-a(t-s)} + \int_s^t \theta(u)e^{-a(t-u)}du, \frac{\sigma^2}{2a}\left(1 - e^{-2a(t-s)}\right)\right) \label{eq:hw_distribution} \end{equation} \]

Like Vasicek, the Hull-White model is Gaussian and therefore allows negative rates (though less problematic when calibrated to positive market curves).

1.7 Connection to Bond Pricing

The Hull-White model maintains the affine structure, so zero-coupon bond prices still have an exponential-affine form. However, the functions \(A(t,T)\) and \(B(t,T)\) now incorporate the time-varying \(\theta(t)\).

Remarkably, we can still derive closed-form bond pricing formulas, making Hull-White extremely practical for derivatives pricing and risk management.

For the derivation of bond prices under the Hull-White model using the affine ansatz approach, see Affine Bond Pricing: Hull-White Model.

1.8 Advantages of Hull-White

Strengths:

  • Perfect yield curve fit: Exactly matches the current market term structure
  • Analytical tractability: Closed-form bond prices and many derivative formulas
  • Practical calibration: Only need to fit \(a\) and \(\sigma\) to caps/swaptions
  • Industry standard: Widely used and understood in practice
  • Flexible extensions: Can incorporate jumps, multi-factors, or stochastic volatility

Limitations:

  • Negative rates: Gaussian structure allows negative interest rates
  • Constant volatility: \(\sigma\) doesn't depend on rate level (can be extended to \(\sigma(t)\))
  • Mean reversion: Single factor may not capture all curve dynamics
  • Recalibration: \(\theta(t)\) changes as market curves evolve

1.9 Practical Implementation

In practice, implementing Hull-White involves:

  1. Bootstrapping \(\theta(t)\): Extract from market forward curve using equation \(\ref{eq:theta_calibration}\)
  2. Calibrating \(a\) and \(\sigma\): Fit to liquid caps, floors, or swaptions
  3. Monte Carlo simulation: Generate rate paths for exotic derivatives
  4. Risk management: Compute Greeks and hedge ratios analytically

The model strikes an excellent balance between realism, tractability, and market consistency.

1.10 Relationship to Other Models

The Hull-White model is central to the interest rate modeling landscape:

  • Vasicek: Special case with \(\theta(t) = ab\) (constant)
  • Ho-Lee: Special case with \(a = 0\) (no mean reversion)
  • Black-Karasinski: Log-normal extension preventing negative rates
  • Multi-factor Hull-White: Add independent factors for richer dynamics
  • G2++: Two-factor Gaussian model with cross-correlation

Hull-White provides the foundation for understanding these extensions and remains the benchmark for one-factor Gaussian models.

1.11 References

3 - Chapter 3.4: The Hull-White Extended Vasicek Model.

  • Chapter 5: Time-dependent parameters and yield curve fitting.

  1. See 2, Chapter 5: Time-dependent parameters and yield curve fitting. ↩

  2. Leif Andersen and Vladimir Piterbarg. Interest Rate Modeling. Volume 1-3. Atlantic Financial Press, 2010. ↩

  3. Damiano Brigo and Fabio Mercurio. Interest Rate Models – Theory and Practice: With Smile, Inflation and Credit. Springer, 2nd edition, 2006. URL: https://link.springer.com/book/10.1007/978-3-540-34604-3. ↩

  4. John Hull and Alan White. Pricing interest-rate-derivative securities. The Review of Financial Studies, 3(4):573–592, 1990. doi:10.1093/rfs/3.4.573. ↩