Derivation from HJM Framework

The Heath-Jarrow-Morton (HJM) framework provides the most fundamental approach to interest rate modeling by specifying the dynamics of the entire forward rate curve. The Hull-White model arises as a special case of HJM when the forward rate volatility takes the specific exponentially decaying form \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\). This derivation is important because it demonstrates that the Hull-White model is not merely an ad hoc specification but is the unique short-rate model consistent with the HJM framework under this volatility structure. Moreover, the HJM perspective automatically ensures no-arbitrage pricing and makes the role of the initial term structure explicit.

Prerequisites

• HJM framework: forward rate dynamics, drift condition (Chapter 19)
• Hull-White SDE and mean reversion (previous section)
• Stochastic calculus: Ito's formula, stochastic Fubini theorem
• Instantaneous forward rates and their relationship to bond prices

Learning Objectives

By the end of this section, you will be able to:

1. State the HJM forward rate dynamics and the no-arbitrage drift condition
2. Specify the Hull-White volatility structure and derive the resulting drift
3. Extract the short rate dynamics from the forward rate equation via \(r_t = f(t,t)\)
4. Identify \(\theta(t)\) in terms of the initial forward curve
5. Verify that the derived SDE matches the Hull-White model

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The HJM Framework

The HJM framework models the evolution of the instantaneous forward rate curve \(f(t,T)\) for all maturities \(T \geq t\) simultaneously.

Definition: HJM Forward Rate Dynamics

Under the risk-neutral measure \(\mathbb{Q}\), the instantaneous forward rate satisfies

\[ df(t,T) = \alpha(t,T)\, dt + \sigma_f(t,T)\, dW_t^{\mathbb{Q}} \]

where \(\sigma_f(t,T)\) is the forward rate volatility (which may depend on \(t\) and \(T\) but is assumed deterministic for the Hull-White case) and \(\alpha(t,T)\) is the drift.

The fundamental result of the HJM theory is that the no-arbitrage condition uniquely determines the drift in terms of the volatility.

Theorem: HJM Drift Condition

Under the risk-neutral measure \(\mathbb{Q}\), absence of arbitrage requires

\[ \alpha(t,T) = \sigma_f(t,T) \int_t^T \sigma_f(t,u)\, du \]

That is, the drift of the forward rate at maturity \(T\) is completely determined by the volatility structure.

This is a powerful constraint: choosing the volatility function \(\sigma_f(t,T)\) determines the entire model, including the drift, leaving no additional freedom.

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Hull-White Volatility Specification

The Hull-White model corresponds to a specific choice of forward rate volatility.

Definition: Hull-White Volatility Structure

The Hull-White forward rate volatility is

\[ \sigma_f(t,T) = \sigma\, e^{-a(T-t)} \]

where \(\sigma > 0\) is a constant and \(a > 0\) is the mean-reversion speed.

This is a deterministic function of \(t\) and \(T\) that decays exponentially with the time to maturity \(T - t\). The economic interpretation is that near-term forward rates are more volatile than distant forward rates, with the rate of decay governed by \(a\). When \(a = 0\), the volatility is constant across all maturities (\(\sigma_f = \sigma\)), recovering the Ho-Lee model.

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Deriving the Forward Rate Drift

Applying the HJM drift condition to the Hull-White volatility structure yields the forward rate drift.

Proposition: Hull-White Forward Rate Drift

Under the Hull-White volatility \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\), the HJM no-arbitrage drift is

\[ \alpha(t,T) = \frac{\sigma^2}{a}\, e^{-a(T-t)}\bigl(1 - e^{-a(T-t)}\bigr) \]

Proof

Apply the HJM drift condition:

\[ \alpha(t,T) = \sigma_f(t,T) \int_t^T \sigma_f(t,u)\, du = \sigma\, e^{-a(T-t)} \int_t^T \sigma\, e^{-a(u-t)}\, du \]

Evaluate the integral with the substitution \(s = u - t\):

\[ \int_t^T \sigma\, e^{-a(u-t)}\, du = \sigma \int_0^{T-t} e^{-as}\, ds = \sigma \left[-\frac{1}{a}\, e^{-as}\right]_0^{T-t} = \frac{\sigma}{a}\bigl(1 - e^{-a(T-t)}\bigr) \]

Therefore:

\[ \alpha(t,T) = \sigma\, e^{-a(T-t)} \cdot \frac{\sigma}{a}\bigl(1 - e^{-a(T-t)}\bigr) = \frac{\sigma^2}{a}\, e^{-a(T-t)}\bigl(1 - e^{-a(T-t)}\bigr) \]

\(\square\)

The complete forward rate dynamics under the Hull-White model are therefore

\[ df(t,T) = \frac{\sigma^2}{a}\, e^{-a(T-t)}\bigl(1 - e^{-a(T-t)}\bigr)\, dt + \sigma\, e^{-a(T-t)}\, dW_t^{\mathbb{Q}} \]

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Extracting the Short Rate

The short rate is the forward rate at the current time: \(r_t = f(t,t)\). To extract its dynamics, we integrate the forward rate equation and then evaluate at \(T = t\).

Theorem: Forward Rate Solution

The forward rate at time \(t\) for maturity \(T\) is

\[ f(t,T) = f(0,T) + \int_0^t \alpha(s,T)\, ds + \int_0^t \sigma_f(s,T)\, dW_s^{\mathbb{Q}} \]

where \(f(0,T)\) is the initial (market-observed) forward curve.

Setting \(T = t\) gives the short rate:

\[ r_t = f(t,t) = f(0,t) + \int_0^t \alpha(s,t)\, ds + \int_0^t \sigma_f(s,t)\, dW_s^{\mathbb{Q}} \]

We need to compute each integral for the Hull-White volatility.

Theorem: Derivation of Hull-White SDE from HJM

Starting from the HJM forward rate dynamics with \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\), the short rate \(r_t = f(t,t)\) satisfies

\[ dr_t = \bigl[\theta(t) - a\, r_t\bigr]\, dt + \sigma\, dW_t^{\mathbb{Q}} \]

where

\[ \theta(t) = \frac{\partial f(0,t)}{\partial t} + a\, f(0,t) + \frac{\sigma^2}{2a}\bigl(1 - e^{-2at}\bigr) \]

Proof

Step 1: Compute the integrated drift.

\[ \int_0^t \alpha(s,t)\, ds = \int_0^t \frac{\sigma^2}{a}\, e^{-a(t-s)}\bigl(1 - e^{-a(t-s)}\bigr)\, ds \]

Substituting \(v = t - s\):

\[ = \frac{\sigma^2}{a} \int_0^t e^{-av}(1 - e^{-av})\, dv = \frac{\sigma^2}{a} \int_0^t \bigl(e^{-av} - e^{-2av}\bigr)\, dv \]

\[ = \frac{\sigma^2}{a}\left[\frac{1 - e^{-at}}{a} - \frac{1 - e^{-2at}}{2a}\right] = \frac{\sigma^2}{2a^2}\bigl(1 - e^{-at}\bigr)^2 \]

Step 2: Write the short rate in closed form.

\[ r_t = f(0,t) + \frac{\sigma^2}{2a^2}(1 - e^{-at})^2 + \sigma \int_0^t e^{-a(t-s)}\, dW_s^{\mathbb{Q}} \]

Step 3: Differentiate to obtain the SDE.

Apply Ito's formula to the expression for \(r_t\). Define \(\psi(t) = f(0,t) + \frac{\sigma^2}{2a^2}(1 - e^{-at})^2\) (the deterministic part) and \(\tilde{r}_t = \sigma \int_0^t e^{-a(t-s)}\, dW_s^{\mathbb{Q}}\) (the stochastic part).

For the deterministic part:

\[ \psi'(t) = f'(0,t) + \frac{\sigma^2}{a}\, e^{-at}(1 - e^{-at}) \]

For the stochastic integral, apply Ito's formula. Writing \(\tilde{r}_t = \sigma \int_0^t e^{-a(t-s)}\, dW_s\), differentiation via the Leibniz-Ito rule gives

\[ d\tilde{r}_t = -a\, \tilde{r}_t\, dt + \sigma\, dW_t^{\mathbb{Q}} \]

Combining:

\[ dr_t = \psi'(t)\, dt + d\tilde{r}_t = \left[f'(0,t) + \frac{\sigma^2}{a}\, e^{-at}(1 - e^{-at}) - a\, \tilde{r}_t\right] dt + \sigma\, dW_t^{\mathbb{Q}} \]

Since \(r_t = \psi(t) + \tilde{r}_t\), we have \(\tilde{r}_t = r_t - \psi(t)\), so \(-a\tilde{r}_t = -ar_t + a\psi(t)\). Substituting:

\[ dr_t = \left[f'(0,t) + \frac{\sigma^2}{a}\, e^{-at}(1 - e^{-at}) + af(0,t) + \frac{\sigma^2}{2a}(1-e^{-at})^2 - a\, r_t\right] dt + \sigma\, dW_t^{\mathbb{Q}} \]

Collecting the \(\sigma^2\) terms:

\[ \frac{\sigma^2}{a}\, e^{-at}(1-e^{-at}) + \frac{\sigma^2}{2a}(1-e^{-at})^2 = \frac{\sigma^2}{2a}(1-e^{-at})\bigl[2e^{-at} + (1-e^{-at})\bigr] = \frac{\sigma^2}{2a}(1-e^{-2at}) \]

Therefore:

\[ dr_t = \left[\underbrace{f'(0,t) + af(0,t) + \frac{\sigma^2}{2a}(1-e^{-2at})}_{\theta(t)} - a\, r_t\right] dt + \sigma\, dW_t^{\mathbb{Q}} \]

\(\square\)

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Uniqueness of the Hull-White Model Under HJM

The derivation establishes a one-to-one correspondence between the volatility specification and the short-rate model.

Proposition: Markov Property from HJM

The forward rate volatility \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\) is the unique deterministic volatility of the separable form \(\sigma_f(t,T) = g(t) \cdot h(T-t)\) that produces a Markov short rate process with time-homogeneous diffusion coefficient.

The key structural reason is that the exponential function satisfies \(e^{-a(T-t)} = e^{-aT} \cdot e^{at}\), which allows the stochastic integral \(\int_0^t \sigma e^{-a(t-s)} dW_s\) to be expressed as a function of a single state variable. This is the Markov property: the future evolution of \(r_t\) depends on the past only through the current value \(r_t\), not through the entire forward curve history. General HJM models are infinite-dimensional (the state is the entire forward curve), but the Hull-White volatility reduces the state space to one dimension.

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The Role of the Initial Forward Curve

The HJM derivation makes the role of the initial term structure particularly transparent. The initial forward curve \(f(0,T)\) enters the Hull-White model through \(\theta(t)\):

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

This formula has three contributions:

1. \(\frac{\partial f(0,t)}{\partial t}\): the slope of the initial forward curve, capturing the direction in which forward rates are moving at each maturity
2. \(a\, f(0,t)\): the mean-reversion pull toward the current forward rate level, scaled by the reversion speed
3. \(\frac{\sigma^2}{2a}(1 - e^{-2at})\): a convexity correction arising from Jensen's inequality, which accounts for the difference between the expected short rate and the forward rate

Computing \(\theta(t)\) from a Flat Curve

For a flat forward curve \(f(0,t) = r_0\) for all \(t\):

• \(\partial_t f(0,t) = 0\)
• \(af(0,t) = ar_0\)
• Convexity correction: \(\frac{\sigma^2}{2a}(1 - e^{-2at})\)

Therefore \(\theta(t) = ar_0 + \frac{\sigma^2}{2a}(1 - e^{-2at})\). In the long run (\(t \to \infty\)), \(\theta(t) \to ar_0 + \frac{\sigma^2}{2a}\), which slightly exceeds \(ar_0\) due to the convexity correction. Setting \(\sigma = 0\) gives \(\theta(t) = ar_0\) constantly, recovering the Vasicek model with \(\theta_{\infty} = r_0\).

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Connection to the Forward Rate Dynamics Section

The forward rate dynamics derived here,

\[ df(t,T) = \frac{\sigma^2}{a}\, e^{-a(T-t)}(1 - e^{-a(T-t)})\, dt + \sigma\, e^{-a(T-t)}\, dW_t^{\mathbb{Q}} \]

are developed in full detail in the section on HJM volatility and drift condition. The instantaneous forward rate section derives the explicit formula for \(f(t,T)\) in terms of \(r_t\) and the initial curve.

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Summary

The Hull-White model emerges from the HJM framework by choosing the forward rate volatility \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\). The HJM no-arbitrage drift condition determines \(\alpha(t,T) = \frac{\sigma^2}{a} e^{-a(T-t)}(1 - e^{-a(T-t)})\) uniquely, and extracting the short rate via \(r_t = f(t,t)\) yields the Hull-White SDE \(dr_t = [\theta(t) - ar_t]\, dt + \sigma\, dW_t^{\mathbb{Q}}\) with \(\theta(t) = f'(0,t) + af(0,t) + \frac{\sigma^2}{2a}(1 - e^{-2at})\). This derivation guarantees no-arbitrage consistency, makes the initial term structure's role explicit through \(\theta(t)\), and reveals that the Hull-White model is the unique Markov short-rate model within the HJM class for this volatility structure.

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Exercises

Exercise 1. Verify the HJM drift condition \(\alpha(t,T) = \sigma_f(t,T)\int_t^T \sigma_f(t,u)\,du\) for the Hull-White volatility \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\). Compute the integral explicitly and show that \(\alpha(t,T) = \frac{\sigma^2}{a}e^{-a(T-t)}(1 - e^{-a(T-t)})\).

Solution to Exercise 1

The HJM drift condition states \(\alpha(t,T) = \sigma_f(t,T)\int_t^T \sigma_f(t,u)\,du\). With \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\):

Step 1: Compute the integral:

\[ \int_t^T \sigma_f(t,u)\,du = \sigma \int_t^T e^{-a(u-t)}\,du \]

Substituting \(s = u - t\):

\[ = \sigma \int_0^{T-t} e^{-as}\,ds = \sigma\left[-\frac{1}{a}e^{-as}\right]_0^{T-t} = \frac{\sigma}{a}(1 - e^{-a(T-t)}) \]

Step 2: Multiply by \(\sigma_f(t,T)\):

\[ \alpha(t,T) = \sigma e^{-a(T-t)} \cdot \frac{\sigma}{a}(1 - e^{-a(T-t)}) = \frac{\sigma^2}{a}e^{-a(T-t)}(1 - e^{-a(T-t)}) \]

This confirms the stated formula. The drift is the product of the volatility at maturity \(T\) and the cumulative volatility effect integrated over \([t, T]\).

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Exercise 2. Show that the integrated HJM drift \(\int_0^t \alpha(s,t)\,ds = \frac{\sigma^2}{2a^2}(1 - e^{-at})^2\). Identify this as the convexity correction in the short rate formula \(r_t = f(0,t) + \frac{\sigma^2}{2a^2}(1 - e^{-at})^2 + \sigma\int_0^t e^{-a(t-s)}dW_s\).

Solution to Exercise 2

We need to compute \(\int_0^t \alpha(s,t)\,ds\) where \(\alpha(s,t) = \frac{\sigma^2}{a}e^{-a(t-s)}(1 - e^{-a(t-s)})\).

Substituting \(v = t - s\) (so \(ds = -dv\), and the limits change from \(s \in [0,t]\) to \(v \in [t,0]\)):

\[ \int_0^t \alpha(s,t)\,ds = \frac{\sigma^2}{a}\int_0^t e^{-av}(1 - e^{-av})\,dv = \frac{\sigma^2}{a}\int_0^t (e^{-av} - e^{-2av})\,dv \]

Evaluating each integral:

\[ \int_0^t e^{-av}\,dv = \frac{1 - e^{-at}}{a}, \qquad \int_0^t e^{-2av}\,dv = \frac{1 - e^{-2at}}{2a} \]

Therefore:

\[ \int_0^t \alpha(s,t)\,ds = \frac{\sigma^2}{a}\left[\frac{1 - e^{-at}}{a} - \frac{1 - e^{-2at}}{2a}\right] \]

Simplify using \(1 - e^{-2at} = (1 - e^{-at})(1 + e^{-at})\):

\[ = \frac{\sigma^2}{a}\cdot\frac{1 - e^{-at}}{a}\left[1 - \frac{1 + e^{-at}}{2}\right] = \frac{\sigma^2}{a}\cdot\frac{1 - e^{-at}}{a}\cdot\frac{1 - e^{-at}}{2} = \frac{\sigma^2}{2a^2}(1 - e^{-at})^2 \]

This is the convexity correction in the short rate formula

\[ r_t = f(0,t) + \frac{\sigma^2}{2a^2}(1 - e^{-at})^2 + \sigma\int_0^t e^{-a(t-s)}dW_s \]

It represents the drift accumulation due to the HJM no-arbitrage condition. The convexity correction grows from \(0\) at \(t=0\) and saturates at \(\sigma^2/(2a^2)\) as \(t \to \infty\), reflecting the difference between forward rates and expected future short rates caused by Jensen's inequality in the bond pricing formula.

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Exercise 3. For a flat forward curve \(f(0,t) = 0.04\) with \(a = 0.1\) and \(\sigma = 0.01\), compute \(\theta(t)\) at \(t = 0, 5, 10, 50\). Verify that \(\theta(0) = a \times 0.04 = 0.004\) and identify the long-run limit of \(\theta(t)\).

Solution to Exercise 3

With \(f(0,t) = 0.04\) (flat), \(a = 0.1\), \(\sigma = 0.01\):

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

Since \(f(0,t) = 0.04\) is constant, \(\frac{\partial f(0,t)}{\partial t} = 0\).

\[ \theta(t) = 0 + 0.1 \times 0.04 + \frac{0.0001}{0.2}(1 - e^{-0.2t}) = 0.004 + 0.0005(1 - e^{-0.2t}) \]

At \(t = 0\): \(\theta(0) = 0.004 + 0.0005 \times 0 = 0.004 = a \times 0.04\). Verified.

At \(t = 5\): \(\theta(5) = 0.004 + 0.0005(1 - e^{-1.0}) = 0.004 + 0.0005 \times 0.6321 = 0.004316\)

At \(t = 10\): \(\theta(10) = 0.004 + 0.0005(1 - e^{-2.0}) = 0.004 + 0.0005 \times 0.8647 = 0.004432\)

At \(t = 50\): \(\theta(50) = 0.004 + 0.0005(1 - e^{-10.0}) = 0.004 + 0.0005 \times 0.99995 \approx 0.004500\)

Long-run limit: As \(t \to \infty\), \(e^{-2at} \to 0\), so

\[ \theta(\infty) = a\,f(0,\infty) + \frac{\sigma^2}{2a} = 0.004 + 0.0005 = 0.0045 \]

\(t\)	\(\theta(t)\)
\(0\)	\(0.00400\)
\(5\)	\(0.00432\)
\(10\)	\(0.00443\)
\(50\)	\(0.00450\)
\(\infty\)	\(0.00450\)

The slight increase from \(0.004\) to \(0.0045\) is the convexity correction: even with a flat forward curve, \(\theta(t)\) must increase slightly to compensate for the Jensen's inequality effect in bond pricing.

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Exercise 4. Explain why the exponential volatility structure \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\) produces a Markov short rate, while a general deterministic volatility \(\sigma_f(t,T) = g(t,T)\) does not. What algebraic property of the exponential function is essential?

Solution to Exercise 4

The exponential volatility \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\) produces a Markov short rate because of the separability of the exponential function: \(e^{-a(T-t)} = e^{-aT} \cdot e^{at}\).

The stochastic integral in the short rate is

\[ \sigma\int_0^t e^{-a(t-s)}dW_s = \sigma e^{-at}\int_0^t e^{as}dW_s \]

The key point is that \(\int_0^t e^{as}dW_s\) is a single real-valued process. Given \(r_t\), the entire history of \(W_s\) for \(s \leq t\) is not needed to determine the future of \(r\); only the current value \(r_t\) matters, because the SDE \(dr_t = [\theta(t) - ar_t]\,dt + \sigma\,dW_t\) involves only \(r_t\) and \(t\) in the coefficients.

Why a general \(\sigma_f(t,T) = g(t,T)\) fails: For a general deterministic volatility, the short rate is

\[ r_t = f(0,t) + \text{(drift terms)} + \int_0^t g(s,t)\,dW_s \]

and the forward rate at maturity \(T > t\) is

\[ f(t,T) = f(0,T) + \text{(drift terms)} + \int_0^t g(s,T)\,dW_s \]

The forward rate depends on \(\int_0^t g(s,T)\,dW_s\), which in general cannot be expressed as a function of \(r_t = f(0,t) + \cdots + \int_0^t g(s,t)\,dW_s\) alone, because the "weighting function" \(g(s,T)\) differs from \(g(s,t)\).

For the exponential case, \(g(s,T) = \sigma e^{-a(T-s)} = e^{-a(T-t)} \cdot \sigma e^{-a(t-s)}\), so \(\int_0^t g(s,T)\,dW_s = e^{-a(T-t)}\int_0^t \sigma e^{-a(t-s)}dW_s\), which is a deterministic multiple of the stochastic part of \(r_t\). This factorization is the essential algebraic property that produces the Markov property.

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Exercise 5. The Ho-Lee model corresponds to \(a = 0\) in the HJM volatility. Derive \(\alpha(t,T)\) for the Ho-Lee case by taking the limit \(a \to 0\) in the Hull-White drift formula. Show that \(\theta(t)\) becomes \(f'(0,t) + \sigma^2 t\).

Solution to Exercise 5

For the Ho-Lee model, \(\sigma_f(t,T) = \sigma\) (constant), which corresponds to \(a = 0\) in the Hull-White volatility. We take the limit \(a \to 0\).

Drift: Starting from \(\alpha(t,T) = \frac{\sigma^2}{a}e^{-a(T-t)}(1 - e^{-a(T-t)})\), expand for small \(a\):

\[ e^{-a(T-t)} \approx 1 - a(T-t), \qquad 1 - e^{-a(T-t)} \approx a(T-t) \]

\[ \alpha(t,T) \approx \frac{\sigma^2}{a}\cdot 1 \cdot a(T-t) = \sigma^2(T-t) \]

This is the Ho-Lee drift: \(\alpha^{\text{HL}}(t,T) = \sigma^2(T-t)\).

\(\theta(t)\) for Ho-Lee: Taking \(a \to 0\) in \(\theta(t) = f'(0,t) + af(0,t) + \frac{\sigma^2}{2a}(1 - e^{-2at})\):

The last term: \(\frac{\sigma^2}{2a}(1 - e^{-2at}) \approx \frac{\sigma^2}{2a}\cdot 2at = \sigma^2 t\) as \(a \to 0\).

The middle term: \(af(0,t) \to 0\) as \(a \to 0\).

Therefore:

\[ \theta^{\text{HL}}(t) = f'(0,t) + \sigma^2 t \]

The Ho-Lee SDE is \(dr_t = \theta^{\text{HL}}(t)\,dt + \sigma\,dW_t\) (no mean reversion term \(-ar_t\), since \(a = 0\)). The drift function \(\theta^{\text{HL}}(t) = f'(0,t) + \sigma^2 t\) grows linearly in \(t\) due to the \(\sigma^2 t\) convexity correction, which is unbounded -- a well-known drawback of the Ho-Lee model.

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Exercise 6. Starting from the stochastic part \(\tilde{r}_t = \sigma\int_0^t e^{-a(t-s)}dW_s\), apply the Leibniz-Ito rule to derive \(d\tilde{r}_t = -a\tilde{r}_t\,dt + \sigma\,dW_t\). This verifies that the stochastic component is an Ornstein-Uhlenbeck process.

Solution to Exercise 6

Define \(\tilde{r}_t = \sigma\int_0^t e^{-a(t-s)}dW_s\). We can write this as

\[ \tilde{r}_t = \sigma e^{-at}\int_0^t e^{as}dW_s \]

Define \(Z_t = \int_0^t e^{as}dW_s\), so \(\tilde{r}_t = \sigma e^{-at}Z_t\).

By the product rule (Ito's formula):

\[ d\tilde{r}_t = \sigma\,d(e^{-at}Z_t) = \sigma\bigl[-a e^{-at}Z_t\,dt + e^{-at}\,dZ_t\bigr] \]

Since \(dZ_t = e^{at}dW_t\):

\[ d\tilde{r}_t = \sigma\bigl[-a e^{-at}Z_t\,dt + e^{-at}\cdot e^{at}\,dW_t\bigr] = -a\,\sigma e^{-at}Z_t\,dt + \sigma\,dW_t \]

Recognizing \(\sigma e^{-at}Z_t = \tilde{r}_t\):

\[ d\tilde{r}_t = -a\,\tilde{r}_t\,dt + \sigma\,dW_t \]

This is the standard Ornstein-Uhlenbeck SDE with mean-reversion speed \(a\), long-run mean \(0\), volatility \(\sigma\), and initial condition \(\tilde{r}_0 = 0\). The stochastic component of the Hull-White short rate is therefore a zero-mean OU process, confirming the decomposition \(r_t = \psi(t) + \tilde{r}_t\) where \(\psi(t)\) is deterministic and \(\tilde{r}_t\) is a mean-zero Gaussian process.

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Exercise 7. Consider a two-factor HJM volatility \(\sigma_f(t,T) = \sigma_1 e^{-a_1(T-t)} + \sigma_2 e^{-a_2(T-t)}\). Is the resulting short rate process Markov in \(r_t\) alone? If not, what is the minimal state vector needed? Discuss how this relates to the two-factor Hull-White model.

Solution to Exercise 7

With the two-factor volatility \(\sigma_f(t,T) = \sigma_1 e^{-a_1(T-t)} + \sigma_2 e^{-a_2(T-t)}\), the short rate involves two independent stochastic integrals (assuming two independent Brownian motions \(W_1\) and \(W_2\)):

\[ r_t = f(0,t) + \text{(drift terms)} + \sigma_1\int_0^t e^{-a_1(t-s)}dW_{1,s} + \sigma_2\int_0^t e^{-a_2(t-s)}dW_{2,s} \]

Markov property: The short rate \(r_t\) alone is not Markov because knowing \(r_t\) does not determine the individual stochastic integrals \(X_{1,t} = \sigma_1\int_0^t e^{-a_1(t-s)}dW_{1,s}\) and \(X_{2,t} = \sigma_2\int_0^t e^{-a_2(t-s)}dW_{2,s}\) separately. Since \(X_{1}\) and \(X_{2}\) have different mean-reversion speeds, their future evolution differs, and knowing only their sum \(r_t - \psi(t) = X_{1,t} + X_{2,t}\) is insufficient to predict the future.

Minimal state vector: The process \((X_{1,t}, X_{2,t})\) is Markov in \(\mathbb{R}^2\), since each component satisfies an independent OU equation:

\[ dX_{1,t} = -a_1 X_{1,t}\,dt + \sigma_1\,dW_{1,t}, \qquad dX_{2,t} = -a_2 X_{2,t}\,dt + \sigma_2\,dW_{2,t} \]

The minimal state vector is \((X_{1,t}, X_{2,t})\), which is two-dimensional. Equivalently, the state can be taken as \((r_t, X_{1,t})\) or \((r_t, X_{2,t})\) since \(X_{2,t}\) can be recovered from \(r_t\) and \(X_{1,t}\).

Connection to two-factor Hull-White: This is precisely the two-factor Hull-White model (also known as the G2++ model). The bond price becomes \(P(t,T) = \exp(A(t,T) + B_1(t,T)X_{1,t} + B_2(t,T)X_{2,t})\), an affine function of the two-dimensional state. The model can generate imperfect correlations between forward rates at different maturities, resolving the perfect correlation limitation of the one-factor model.

Correlation computation: For the one-factor case with two components summed, using a single Brownian motion, the correlation between forward rate changes at \(T_1\) and \(T_2\) involves:

\[ \text{Var}[\Delta f(t,T_i)] = \int_0^t (\sigma_1 e^{-a_1(T_i-s)} + \sigma_2 e^{-a_2(T_i-s)})^2\,ds \]

With the given parameters \(\sigma_1 = 0.008\), \(a_1 = 0.02\), \(\sigma_2 = 0.005\), \(a_2 = 0.30\), \(T_1 = 1\), \(T_2 = 10\), and assuming two independent factors, the correlation is

\[ \rho(T_1,T_2) = \frac{\text{Cov}[\Delta f(t,T_1), \Delta f(t,T_2)]}{\sqrt{\text{Var}[\Delta f(t,T_1)]\cdot\text{Var}[\Delta f(t,T_2)]}} \]

where

\[ \text{Cov} = \int_0^t \bigl[\sigma_1^2 e^{-a_1(T_1+T_2-2s)} + \sigma_2^2 e^{-a_2(T_1+T_2-2s)}\bigr]\,ds \]

For large \(t\) (stationary regime), this evaluates to

\[ \text{Cov} = \frac{\sigma_1^2}{2a_1}e^{-a_1(T_1+T_2-2t)} + \frac{\sigma_2^2}{2a_2}e^{-a_2(T_1+T_2-2t)} \]

The first factor (\(a_1 = 0.02\), slow) contributes significantly to both maturities, while the second factor (\(a_2 = 0.30\), fast) decays rapidly and contributes mainly to short maturities. This imbalance produces a correlation strictly less than 1, with the exact value depending on the observation horizon \(t\). For typical \(t\), the correlation between the 1-year and 10-year forward rates is approximately 0.85--0.95, well below the perfect correlation of the one-factor model.