Risk-Neutral Measure in the Hull-White Model

The Hull-White model is typically specified directly under the risk-neutral measure \(\mathbb{Q}\), where the short rate dynamics take the familiar mean-reverting form. However, understanding how this measure relates to the physical (real-world) measure \(\mathbb{P}\) through the market price of risk is essential for interpreting model parameters and connecting to observable data. This section presents the Hull-White dynamics under both measures and derives the Girsanov transformation that links them.

Physical Measure Dynamics

Under the real-world (physical) measure \(\mathbb{P}\), the short rate in the Hull-White framework satisfies

\[ dr(t) = \lambda^{\mathbb{P}}\!\left(\theta^{\mathbb{P}}(t) - r(t)\right)dt + \sigma\,dW^{\mathbb{P}}(t) \]

where \(\lambda^{\mathbb{P}}\) is the physical mean reversion speed, \(\theta^{\mathbb{P}}(t)\) is the physical mean reversion level, and \(W^{\mathbb{P}}(t)\) is a standard Brownian motion under \(\mathbb{P}\). These parameters govern the actual statistical behavior of the short rate observed in the market.

The volatility parameter \(\sigma\) is the same under both measures because Girsanov's theorem changes the drift but not the diffusion coefficient.

Market Price of Interest Rate Risk

To move from \(\mathbb{P}\) to the risk-neutral measure \(\mathbb{Q}\), we introduce the market price of interest rate risk \(\gamma(t)\), which quantifies the excess return per unit of volatility that the market demands for bearing interest rate risk.

Definition: Market Price of Risk

The market price of interest rate risk \(\gamma(t)\) is a (possibly time-dependent) adapted process such that

\[ dW^{\mathbb{Q}}(t) = dW^{\mathbb{P}}(t) + \gamma(t)\,dt \]

defines a standard Brownian motion \(W^{\mathbb{Q}}(t)\) under the risk-neutral measure \(\mathbb{Q}\).

The Radon-Nikodym derivative that defines the change of measure is

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\Bigg|_{\mathcal{F}(t)} = \exp\!\left(-\int_0^t \gamma(s)\,dW^{\mathbb{P}}(s) - \frac{1}{2}\int_0^t \gamma(s)^2\,ds\right) \]

By Girsanov's theorem, this is a valid change of measure provided the Novikov condition \(\mathbb{E}^{\mathbb{P}}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \gamma(s)^2\,ds\right)\right] < \infty\) is satisfied.

Hull-White Dynamics Under Q

Substituting \(dW^{\mathbb{P}}(t) = dW^{\mathbb{Q}}(t) - \gamma(t)\,dt\) into the physical dynamics:

\[\begin{array}{lllll} \displaystyle dr(t) &=&\displaystyle \lambda^{\mathbb{P}}\!\left(\theta^{\mathbb{P}}(t) - r(t)\right)dt + \sigma\left(dW^{\mathbb{Q}}(t) - \gamma(t)\,dt\right) \\[6pt] &=&\displaystyle \left[\lambda^{\mathbb{P}}\!\left(\theta^{\mathbb{P}}(t) - r(t)\right) - \sigma\gamma(t)\right]dt + \sigma\,dW^{\mathbb{Q}}(t) \end{array}\]

Theorem: Risk-Neutral Hull-White SDE

Under the risk-neutral measure \(\mathbb{Q}\), the Hull-White short rate satisfies

\[ dr(t) = \lambda\!\left(\theta^{\mathbb{Q}}(t) - r(t)\right)dt + \sigma\,dW^{\mathbb{Q}}(t) \]

where the risk-neutral parameters relate to the physical parameters and the market price of risk through

\[\begin{array}{lllll} \displaystyle \lambda &=& \lambda^{\mathbb{P}} \\[4pt] \displaystyle \lambda\,\theta^{\mathbb{Q}}(t) &=&\displaystyle \lambda^{\mathbb{P}}\theta^{\mathbb{P}}(t) - \sigma\gamma(t) \end{array}\]

Proof

Comparing the \(\mathbb{Q}\)-dynamics

\[ dr(t) = \left[\lambda^{\mathbb{P}}\theta^{\mathbb{P}}(t) - \lambda^{\mathbb{P}}r(t) - \sigma\gamma(t)\right]dt + \sigma\,dW^{\mathbb{Q}}(t) \]

with the standard form \(dr(t) = \lambda(\theta^{\mathbb{Q}}(t) - r(t))dt + \sigma\,dW^{\mathbb{Q}}(t)\), matching the \(r(t)\) coefficient gives \(\lambda = \lambda^{\mathbb{P}}\) (the mean reversion speed is unchanged). Matching the remaining drift terms gives \(\lambda\theta^{\mathbb{Q}}(t) = \lambda^{\mathbb{P}}\theta^{\mathbb{P}}(t) - \sigma\gamma(t)\). \(\square\)

Affine Market Price of Risk

A common specification assumes the market price of risk is affine in the short rate:

\[ \gamma(t) = \gamma_0 + \gamma_1\,r(t) \]

Under this assumption the risk-neutral drift becomes

\[\begin{array}{lllll} \displaystyle \lambda\theta^{\mathbb{Q}}(t) - \lambda r(t) &=&\displaystyle \lambda^{\mathbb{P}}\theta^{\mathbb{P}}(t) - \sigma\gamma_0 - (\lambda^{\mathbb{P}} + \sigma\gamma_1)r(t) \end{array}\]

This implies that the risk-neutral mean reversion speed is \(\lambda = \lambda^{\mathbb{P}} + \sigma\gamma_1\), which generally differs from \(\lambda^{\mathbb{P}}\) when \(\gamma_1 \neq 0\). In particular:

• Constant market price of risk (\(\gamma_1 = 0\)): The mean reversion speed is the same under \(\mathbb{P}\) and \(\mathbb{Q}\). Only \(\theta\) shifts.
• Rate-dependent market price of risk (\(\gamma_1 \neq 0\)): Both the mean reversion speed and level change under the measure change.

Common Convention

In practice, the Hull-White model is almost always specified directly under \(\mathbb{Q}\) with \(\lambda\) and \(\theta^{\mathbb{Q}}(t)\) as primary parameters. The physical measure parameters and market price of risk are then inferred from time series data if needed. The calibration to the initial yield curve determines \(\theta^{\mathbb{Q}}(t)\) without any reference to \(\mathbb{P}\).

Bond Pricing Under Q

Under \(\mathbb{Q}\), the discounted bond price process \(P(t,T)/M(t)\) is a martingale, where \(M(t) = \exp(\int_0^t r(s)\,ds)\) is the money market account. This martingale property is the fundamental reason the risk-neutral measure is the natural setting for derivative pricing:

\[ P(t,T) = \mathbb{E}^{\mathbb{Q}}\!\left[\frac{M(t)}{M(T)}\,\Big|\,\mathcal{F}(t)\right] = \mathbb{E}^{\mathbb{Q}}\!\left[e^{-\int_t^T r(s)\,ds}\,\Big|\,\mathcal{F}(t)\right] \]

Any derivative with payoff \(V(T)\) at time \(T\) is priced as

\[ V(t) = \mathbb{E}^{\mathbb{Q}}\!\left[e^{-\int_t^T r(s)\,ds}\,V(T)\,\Big|\,\mathcal{F}(t)\right] \]

Numerical Example

Suppose the physical dynamics have \(\lambda^{\mathbb{P}} = 0.03\), \(\theta^{\mathbb{P}}(t) = 0.05\), and \(\sigma = 0.01\). If the market price of risk is constant at \(\gamma = -0.15\) (negative, reflecting a positive risk premium for interest rate risk), then the risk-neutral parameters are

\[\begin{array}{lllll} \displaystyle \lambda &=& 0.03 \\[4pt] \displaystyle \theta^{\mathbb{Q}}(t) &=& 0.05 - \frac{0.01 \times (-0.15)}{0.03} = 0.05 + 0.05 = 0.10 \end{array}\]

The risk-neutral mean reversion level is higher than the physical level, reflecting the fact that the market demands compensation for bearing interest rate risk, which shifts the drift upward under \(\mathbb{Q}\).

In practice, \(\theta^{\mathbb{Q}}(t)\) is not set this way but is instead calibrated to match the observed term structure \(P^M(0,T)\), yielding the time-dependent function derived in the consistency section.

From HJM to Risk-Neutral Hull-White

An alternative route to the risk-neutral Hull-White SDE starts from the HJM framework. Under \(\mathbb{Q}\), the HJM drift condition ensures no-arbitrage:

\[ \mu^{\mathbb{Q}}(t,T) = \sigma(t,T)\int_t^T \sigma(t,u)\,du \]

For the Hull-White volatility specification \(\sigma(t,T) = \sigma\,e^{-\lambda(T-t)}\), the HJM drift condition determines the forward rate dynamics uniquely under \(\mathbb{Q}\), and the short rate process \(r(t) = f(t,t)\) inherits the Hull-White SDE.

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Summary

The Hull-White model under \(\mathbb{Q}\) is obtained from the physical measure \(\mathbb{P}\) via Girsanov's theorem with market price of risk \(\gamma(t)\). The mean reversion speed is preserved (for constant \(\gamma\)), while the mean reversion level shifts according to \(\lambda\theta^{\mathbb{Q}}(t) = \lambda^{\mathbb{P}}\theta^{\mathbb{P}}(t) - \sigma\gamma(t)\). In practice, the model is calibrated directly under \(\mathbb{Q}\) by fitting \(\theta^{\mathbb{Q}}(t)\) to the market term structure, bypassing explicit specification of the physical measure.

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Exercises

Exercise 1. The Novikov condition requires \(\mathbb{E}^{\mathbb{P}}[\exp(\frac{1}{2}\int_0^T \gamma(s)^2\,ds)] < \infty\). For a constant market price of risk \(\gamma\), show that this condition is always satisfied and compute the Radon-Nikodym derivative explicitly.

Solution to Exercise 1

For a constant market price of risk \(\gamma\), the Novikov condition becomes

\[ \mathbb{E}^{\mathbb{P}}\!\left[\exp\!\left(\frac{1}{2}\int_0^T \gamma^2\,ds\right)\right] = \exp\!\left(\frac{1}{2}\gamma^2 T\right) < \infty \]

Since \(\gamma\) is a finite constant and \(T < \infty\), the exponential of a finite number is always finite. Therefore the Novikov condition is automatically satisfied for any constant \(\gamma\).

The Radon-Nikodym derivative simplifies to

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\Bigg|_{\mathcal{F}(t)} = \exp\!\left(-\gamma\,W^{\mathbb{P}}(t) - \frac{1}{2}\gamma^2 t\right) \]

This is a standard exponential martingale \(\mathcal{E}(-\gamma W^{\mathbb{P}})_t\). One can verify it is a martingale by checking that \(\mathbb{E}^{\mathbb{P}}[\frac{d\mathbb{Q}}{d\mathbb{P}}\big|_{\mathcal{F}(t)}] = 1\), which follows from the moment generating function of the normal distribution: \(W^{\mathbb{P}}(t) \sim \mathcal{N}(0, t)\), so

\[ \mathbb{E}^{\mathbb{P}}\!\left[e^{-\gamma W^{\mathbb{P}}(t) - \frac{1}{2}\gamma^2 t}\right] = e^{-\frac{1}{2}\gamma^2 t}\,e^{\frac{1}{2}\gamma^2 t} = 1 \]

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Exercise 2. Suppose \(\lambda^{\mathbb{P}} = 0.04\), \(\theta^{\mathbb{P}}(t) = 0.06\), \(\sigma = 0.012\), and \(\gamma = -0.20\). Compute \(\lambda\) and \(\theta^{\mathbb{Q}}(t)\) under the risk-neutral measure. Is \(\theta^{\mathbb{Q}}(t)\) higher or lower than \(\theta^{\mathbb{P}}(t)\), and why?

Solution to Exercise 2

With \(\lambda^{\mathbb{P}} = 0.04\), \(\theta^{\mathbb{P}}(t) = 0.06\), \(\sigma = 0.012\), and constant \(\gamma = -0.20\):

Mean reversion speed: Since \(\gamma\) is constant (i.e., \(\gamma_1 = 0\) in the affine specification), the mean reversion speed is preserved:

\[ \lambda = \lambda^{\mathbb{P}} = 0.04 \]

Mean reversion level: From \(\lambda\theta^{\mathbb{Q}}(t) = \lambda^{\mathbb{P}}\theta^{\mathbb{P}}(t) - \sigma\gamma\):

\[ \theta^{\mathbb{Q}}(t) = \theta^{\mathbb{P}}(t) - \frac{\sigma\gamma}{\lambda} = 0.06 - \frac{0.012 \times (-0.20)}{0.04} = 0.06 + 0.06 = 0.12 \]

So \(\theta^{\mathbb{Q}}(t) = 0.12 > 0.06 = \theta^{\mathbb{P}}(t)\).

The risk-neutral mean reversion level is higher because \(\gamma < 0\), which means the market demands a positive risk premium for bearing interest rate risk. Under \(\mathbb{Q}\), the drift is shifted upward to compensate investors for this risk. The negative market price of risk implies that bond investors require extra compensation (higher expected rates under \(\mathbb{Q}\)) relative to the physical dynamics.

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Exercise 3. For the affine market price of risk \(\gamma(t) = \gamma_0 + \gamma_1 r(t)\), show that the risk-neutral mean reversion speed is \(\lambda = \lambda^{\mathbb{P}} + \sigma\gamma_1\), which differs from \(\lambda^{\mathbb{P}}\). Give a financial interpretation of why the mean reversion speed can change under a measure change when \(\gamma_1 \neq 0\).

Solution to Exercise 3

With \(\gamma(t) = \gamma_0 + \gamma_1 r(t)\), substituting into the \(\mathbb{Q}\)-drift:

\[\begin{array}{lllll} \displaystyle dr(t) &=&\displaystyle \left[\lambda^{\mathbb{P}}(\theta^{\mathbb{P}}(t) - r(t)) - \sigma(\gamma_0 + \gamma_1 r(t))\right]dt + \sigma\,dW^{\mathbb{Q}}(t) \\[6pt] &=&\displaystyle \left[\lambda^{\mathbb{P}}\theta^{\mathbb{P}}(t) - \sigma\gamma_0 - (\lambda^{\mathbb{P}} + \sigma\gamma_1)r(t)\right]dt + \sigma\,dW^{\mathbb{Q}}(t) \end{array}\]

Matching to the standard form \(dr(t) = \lambda(\theta^{\mathbb{Q}}(t) - r(t))dt + \sigma\,dW^{\mathbb{Q}}(t)\), the coefficient of \(r(t)\) gives

\[ \lambda = \lambda^{\mathbb{P}} + \sigma\gamma_1 \]

which differs from \(\lambda^{\mathbb{P}}\) whenever \(\gamma_1 \neq 0\).

Financial interpretation: When \(\gamma_1 \neq 0\), the market price of risk depends on the level of the short rate. This means the risk premium that investors demand varies with the interest rate environment. A positive \(\gamma_1\) increases the risk-neutral mean reversion speed relative to the physical speed: when rates are high, the larger risk premium creates stronger downward pressure under \(\mathbb{Q}\), and when rates are low, the smaller risk premium creates less upward pressure. This state-dependent adjustment to the drift effectively modifies how quickly the process reverts, changing the mean reversion speed under the new measure. By contrast, a constant market price of risk (\(\gamma_1 = 0\)) shifts the drift uniformly, affecting only the level to which the process reverts, not the speed.

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Exercise 4. Explain why the volatility parameter \(\sigma\) is the same under both \(\mathbb{P}\) and \(\mathbb{Q}\). What theorem guarantees this invariance, and what property of the diffusion coefficient is required?

Solution to Exercise 4

The volatility parameter \(\sigma\) is the same under \(\mathbb{P}\) and \(\mathbb{Q}\) because of Girsanov's theorem, which guarantees that a change of measure modifies only the drift of an Ito process, not the diffusion coefficient.

Specifically, Girsanov's theorem constructs the new Brownian motion \(W^{\mathbb{Q}}(t) = W^{\mathbb{P}}(t) + \int_0^t \gamma(s)\,ds\), so

\[ dW^{\mathbb{P}}(t) = dW^{\mathbb{Q}}(t) - \gamma(t)\,dt \]

Substituting into \(dr(t) = \mu^{\mathbb{P}}(t)\,dt + \sigma\,dW^{\mathbb{P}}(t)\):

\[ dr(t) = \left[\mu^{\mathbb{P}}(t) - \sigma\gamma(t)\right]dt + \sigma\,dW^{\mathbb{Q}}(t) \]

The \(dt\) coefficient (drift) changes, but the \(dW\) coefficient (diffusion) remains exactly \(\sigma\). This invariance holds because Girsanov's theorem operates by an absolutely continuous change of measure, which preserves the quadratic variation \(\langle r \rangle_t = \int_0^t \sigma^2\,ds\). The quadratic variation is a path-by-path property that does not depend on the probability measure, so the diffusion coefficient \(\sigma\) is measure-invariant.

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Exercise 5. Under \(\mathbb{Q}\), the discounted bond price \(P(t,T)/M(t)\) is a martingale. Verify this for the Hull-White model by showing that the drift of \(d(P(t,T)/M(t))\) vanishes. (Hint: use the bond dynamics \(dP/P = r\,dt + \sigma_P dW^{\mathbb{Q}}\) and the money market dynamics \(dM = rM\,dt\).)

Solution to Exercise 5

Define \(Z(t) = P(t,T)/M(t)\), the discounted bond price. We need to show \(Z(t)\) is a \(\mathbb{Q}\)-martingale, i.e., its drift under \(\mathbb{Q}\) vanishes.

The bond dynamics under \(\mathbb{Q}\) are \(dP = r P\,dt + \sigma_P P\,dW^{\mathbb{Q}}\) and the money market dynamics are \(dM = rM\,dt\).

Using the quotient rule (Ito's formula for \(Z = P/M\)):

\[\begin{array}{lllll} \displaystyle dZ &=&\displaystyle \frac{1}{M}\,dP - \frac{P}{M^2}\,dM + \frac{P}{M^3}(dM)^2 - \frac{1}{M^2}\,dP\,dM \end{array}\]

Since \(dM = rM\,dt\) has no Brownian component, \((dM)^2 = 0\) and \(dP\,dM = 0\). Therefore:

\[\begin{array}{lllll} \displaystyle dZ &=&\displaystyle \frac{1}{M}\!\left(rP\,dt + \sigma_P P\,dW^{\mathbb{Q}}\right) - \frac{P}{M^2}(rM\,dt) \\[6pt] &=&\displaystyle \frac{P}{M}\!\left(r\,dt + \sigma_P\,dW^{\mathbb{Q}} - r\,dt\right) \\[6pt] &=&\displaystyle Z\,\sigma_P\,dW^{\mathbb{Q}} \end{array}\]

The drift vanishes, so \(Z(t) = P(t,T)/M(t)\) is indeed a \(\mathbb{Q}\)-martingale.

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Exercise 6. Describe the alternative route from HJM to the risk-neutral Hull-White SDE. Starting from the HJM volatility \(\sigma(t,T) = \sigma e^{-\lambda(T-t)}\), derive the HJM drift condition and show that the short rate \(r(t) = f(t,t)\) satisfies the Hull-White SDE.

Solution to Exercise 6

Step 1: HJM forward rate dynamics. Under \(\mathbb{Q}\), the HJM framework specifies:

\[ df(t,T) = \mu^{\mathbb{Q}}(t,T)\,dt + \sigma(t,T)\,dW^{\mathbb{Q}}(t) \]

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

\[ \mu^{\mathbb{Q}}(t,T) = \sigma(t,T)\int_t^T \sigma(t,u)\,du \]

Step 2: Hull-White volatility specification. For \(\sigma(t,T) = \sigma e^{-\lambda(T-t)}\):

\[ \int_t^T \sigma(t,u)\,du = \int_t^T \sigma e^{-\lambda(u-t)}\,du = \frac{\sigma}{\lambda}\left(1 - e^{-\lambda(T-t)}\right) \]

So the drift is:

\[ \mu^{\mathbb{Q}}(t,T) = \sigma e^{-\lambda(T-t)} \cdot \frac{\sigma}{\lambda}\left(1 - e^{-\lambda(T-t)}\right) = \frac{\sigma^2}{\lambda}e^{-\lambda(T-t)}\left(1 - e^{-\lambda(T-t)}\right) \]

Step 3: Extract the short rate. Set \(r(t) = f(t,t)\). Using the chain rule:

\[ dr(t) = \left[\frac{\partial f}{\partial t}(t,t) + \frac{\partial f}{\partial T}(t,t)\right]dt + \text{(evaluated at } T = t\text{)} \]

Integrating the forward rate dynamics and differentiating with respect to \(T\), one obtains:

\[ dr(t) = \left[\frac{\partial f^M}{\partial T}(0,t) + \frac{\sigma^2}{2\lambda^2}\frac{\partial}{\partial t}(1 - e^{-\lambda t})^2 + \lambda(f^M(0,t) - r(t)) + \frac{\sigma^2}{2\lambda}(1 - e^{-2\lambda t})\right]dt + \sigma\,dW^{\mathbb{Q}}(t) \]

Collecting terms and defining \(\theta^{\mathbb{Q}}(t)\) appropriately, this reduces to the Hull-White SDE \(dr(t) = \lambda(\theta^{\mathbb{Q}}(t) - r(t))dt + \sigma\,dW^{\mathbb{Q}}(t)\). The HJM drift condition uniquely determines the drift, ensuring no-arbitrage.

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Exercise 7. In practice, \(\theta^{\mathbb{Q}}(t)\) is calibrated to the market term structure rather than derived from \(\theta^{\mathbb{P}}(t)\) and \(\gamma(t)\). Discuss the advantages and disadvantages of this approach. Under what circumstances would you need to estimate \(\gamma(t)\) explicitly?

Solution to Exercise 7

Advantages of direct calibration under \(\mathbb{Q}\):

• The function \(\theta^{\mathbb{Q}}(t)\) is uniquely determined by the observed market term structure \(P^M(0,T)\) through the relation \(\theta^{\mathbb{Q}}(t) = \frac{1}{\lambda}\frac{\partial f^M}{\partial T}(0,t) + f^M(0,t) + \frac{\sigma^2}{2\lambda^2}(1 - e^{-\lambda t})^2\). This ensures the model exactly reproduces all observed bond prices.
• No estimation of the market price of risk \(\gamma(t)\) or physical dynamics is needed, avoiding the statistical challenges of estimating drift parameters from noisy time series data.
• The approach is model-consistent for pricing: all derivative prices are expectations under \(\mathbb{Q}\), so only \(\mathbb{Q}\)-parameters matter.

Disadvantages:

• Physical dynamics are unknown, so the model cannot generate realistic rate scenarios for risk management (VaR, stress testing) without additional assumptions.
• The model provides no information about expected returns on bonds or risk premia.
• Changes in the yield curve over time lead to recalibration of \(\theta^{\mathbb{Q}}(t)\), and the time-series behavior of recalibrated parameters may be unrealistic.

When explicit estimation of \(\gamma(t)\) is needed:

• Risk management and scenario generation: Simulating future rate paths for VaR or expected shortfall requires the physical measure \(\mathbb{P}\).
• Real-world forecasting: Predicting future interest rate levels or evaluating investment strategies.
• Relative value analysis: Comparing model-implied risk premia to historical norms to identify mispriced securities.
• Multi-period portfolio optimization: Utility-based optimization requires expected returns under \(\mathbb{P}\), not \(\mathbb{Q}\).