HJM Drift Condition Proof

This section explores the principles and methods underlying the HJM drift condition proof, which forms a critical component of modern financial mathematics.

Key Concepts

The fundamental concepts in this area include:

• Theoretical foundations and mathematical framework
• Key definitions and notation
• Important theorems and results
• Connections to other areas of financial mathematics

Learning Objectives

After completing this section, you should understand:

• The core mathematical principles and their financial interpretations
• How these concepts connect to practical applications
• The relationship between theory and numerical implementation

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Primary Proof: Martingale Approach

The main proof of the HJM drift condition uses the martingale property of discounted bond prices. This is fully developed in the No-Arbitrage Drift Condition section, where we show:

1. Bond prices must satisfy \(P(t,T) = \exp(-\int_t^T f(t,u)du)\)
2. Discounted bonds \(P(t,T)/B_t\) must be martingales under \(\mathbb{Q}\)
3. This martingale requirement forces:

\[\boxed{\alpha(t,T) = \sigma(t,T) \int_t^T \sigma(t,u)du}\]

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Alternative Proof: ZCB Dynamics Approach

An alternative derivation derives the same condition from zero-coupon bond dynamics.

Starting Point

Since the instantaneous forward rate is the log-derivative of bond prices:

\[f(t,T) = -\frac{\partial}{\partial T}\log P(t,T)\]

The forward rate dynamics can be obtained by differentiating the bond price dynamics:

\[df(t,T) = -\frac{\partial}{\partial T}d\log P(t,T)\]

Bond Price Dynamics

Under risk-neutral pricing, the bond price follows:

\[\frac{dP(t,T)}{P(t,T)} = r(t)dt + \sigma_P(t,T)dW^{\mathbb{Q}}(t)\]

where \(\sigma_P(t,T) = -\int_t^T\sigma(t,T')dT'\) is the bond volatility.

Log Bond Price

By Itô's lemma:

\[d\log P(t,T) = \left(r(t) - \frac{1}{2}\sigma_P^2(t,T)\right)dt + \sigma_P(t,T)dW^{\mathbb{Q}}(t)\]

Forward Rate From Derivative

Taking the \(T\)-derivative:

\[df(t,T) = \frac{1}{2}\frac{d\sigma_P^2(t,T)}{dT}dt - \frac{d\sigma_P(t,T)}{dT}dW^{\mathbb{Q}}(t)\]

Since \(\sigma_P(t,T) = -\int_t^T\sigma(t,T')dT'\):

\[\frac{d\sigma_P(t,T)}{dT} = -\sigma(t,T)\]

\[\frac{d\sigma_P^2(t,T)}{dT} = 2\sigma_P(t,T) \cdot (-\sigma(t,T)) = -2\sigma_P(t,T)\sigma(t,T) = 2\sigma(t,T)\int_t^T\sigma(t,T')dT'\]

Result

\[df(t,T) = \sigma(t,T)\left(\int_t^T\sigma(t,T')dT'\right)dt + \sigma(t,T)dW^{\mathbb{Q}}(t)\]

This gives the same drift condition:

\[\mu^{\mathbb{Q}}(t,T) = \sigma(t,T)\int_t^T\sigma(t,T')dT'\]

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Proof Comparison

Aspect	Martingale Approach	ZCB Dynamics Approach
Starting Point	Discounted bond must be martingale	Bond log-derivative gives forwards
Key Tool	Itô's lemma on discounted bonds	Itô's lemma on log bonds + differentiation
Conceptual Focus	No-arbitrage via martingale condition	Direct relationship between rates and prices
Computational Path	Integrate, then differentiate	Differentiate log, then differentiate again
Result	Same drift condition	Same drift condition

Both approaches are valid and complementary—they illuminate different aspects of the HJM framework.

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Multi-Factor Generalization

For multiple volatility factors:

\[df(t, T) = \alpha(t, T) \, dt + \sum_{i=1}^n \sigma_i(t, T) \, dW_t^i\]

Both proofs generalize naturally to:

\[\alpha(t, T) = \sum_{i=1}^n \sigma_i(t, T) \int_t^T \sigma_i(t, u) \, du\]

Each factor contributes additively to the drift.

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Key Insights

1. Uniqueness: The drift is uniquely determined once volatility is specified—there is no freedom to choose drift independently.

2. No-Arbitrage is Automatic: By construction, any HJM model with this drift condition is automatically free of arbitrage.

3. Both Proofs Agree: The martingale and ZCB dynamics approaches confirm each other, providing confidence in the result.

4. Practical Implication: Modelers need only specify volatility; drift is computed, not calibrated.

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Further Reading

• Primary source: Heath, Jarrow & Morton (1992), "Bond Pricing and the Term Structure of Interest Rates: A New Methodology"
• Textbooks:
• Björk, Arbitrage Theory in Continuous Time, Chapter 25
• Filipović, Term-Structure Models, Chapter 7
• Brigo & Mercurio, Interest Rate Models - Theory and Practice, Part III

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Exercises

Exercise 1. Reproduce the martingale-based derivation of the HJM drift condition by carrying out the following steps explicitly. Start from \(Z(t, T) = -\int_t^T f(t, u)\,du\), compute \(dZ(t, T)\) using Leibniz's rule, apply Ito's lemma to \(P(t, T) = e^{Z(t,T)}\), form the discounted bond price \(P(t,T)/B_t\), set the drift to zero, and differentiate in \(T\) to obtain \(\alpha(t, T) = \sigma(t, T)\int_t^T \sigma(t, u)\,du\).

Solution to Exercise 1

We carry out each step of the martingale-based derivation explicitly.

Step 1: Define the log bond price.

\[ Z(t, T) = -\int_t^T f(t, u)\,du \]

Step 2: Compute \(dZ(t, T)\) using Leibniz's rule.

The integral \(Z(t, T) = -\int_t^T f(t, u)\,du\) has a stochastic integrand and a moving lower limit. Applying the stochastic Leibniz rule:

\[ dZ(t, T) = f(t, t)\,dt - \int_t^T df(t, u)\,du \]

The \(f(t, t)\,dt = r_t\,dt\) term arises because the lower limit \(t\) advances by \(dt\), contributing \(+f(t, t)\,dt\) (since the integrand is preceded by a minus sign, the lower limit advancing "removes" a slice \(f(t, t)\,dt\) from the integral). Substituting \(df(t, u) = \alpha(t, u)\,dt + \sigma(t, u)\,dW_t\):

\[ dZ(t, T) = r_t\,dt - \int_t^T \alpha(t, u)\,du\,dt - \int_t^T \sigma(t, u)\,du\,dW_t \]

Defining \(\Sigma(t, T) = \int_t^T \sigma(t, u)\,du\):

\[ dZ(t, T) = \left[r_t - \int_t^T \alpha(t, u)\,du\right]dt - \Sigma(t, T)\,dW_t \]

Step 3: Apply Ito's lemma to \(P(t, T) = e^{Z(t, T)}\).

\[ dP = P\left[dZ + \frac{1}{2}(dZ)^2\right] = P\left[\left(r_t - \int_t^T \alpha(t, u)\,du + \frac{1}{2}\Sigma(t, T)^2\right)dt - \Sigma(t, T)\,dW_t\right] \]

Step 4: Form the discounted bond price \(\tilde{P}(t, T) = P(t, T)/B_t\) where \(dB_t = r_t B_t\,dt\).

\[ d\tilde{P} = \frac{1}{B_t}\bigl(dP - r_t P\,dt\bigr) = \tilde{P}\left[\left(-\int_t^T \alpha(t, u)\,du + \frac{1}{2}\Sigma(t, T)^2\right)dt - \Sigma(t, T)\,dW_t\right] \]

Step 5: Set the drift to zero (martingale condition).

\[ -\int_t^T \alpha(t, u)\,du + \frac{1}{2}\Sigma(t, T)^2 = 0 \]

Therefore:

\[ \int_t^T \alpha(t, u)\,du = \frac{1}{2}\Sigma(t, T)^2 = \frac{1}{2}\left(\int_t^T \sigma(t, u)\,du\right)^2 \]

Step 6: Differentiate in \(T\).

\[ \alpha(t, T) = \frac{\partial}{\partial T}\left[\frac{1}{2}\Sigma(t, T)^2\right] = \Sigma(t, T)\,\frac{\partial \Sigma}{\partial T}(t, T) = \Sigma(t, T)\,\sigma(t, T) \]

since \(\frac{\partial}{\partial T}\int_t^T \sigma(t, u)\,du = \sigma(t, T)\). This gives:

\[ \boxed{\alpha(t, T) = \sigma(t, T)\int_t^T \sigma(t, u)\,du} \]

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Exercise 2. Verify the HJM drift condition for the Ho--Lee model, where \(\sigma(t, T) = \sigma\) is constant. Compute the drift \(\alpha(t, T)\), the bond volatility \(\Sigma(t, T) = \int_t^T \sigma\,du\), and show that the resulting forward rate dynamics integrate to

\[ f(t, T) = f(0, T) + \sigma^2 t(T - t/2) + \sigma W_t \]

Solution to Exercise 2

Step 1: Apply the drift condition for \(\sigma(t, T) = \sigma\) (constant).

\[ \alpha(t, T) = \sigma \int_t^T \sigma\,du = \sigma^2(T - t) \]

Step 2: Compute the bond volatility.

\[ \Sigma(t, T) = \int_t^T \sigma\,du = \sigma(T - t) \]

Step 3: Write the forward rate SDE and integrate.

\[ df(t, T) = \sigma^2(T - t)\,dt + \sigma\,dW_t \]

Integrating from \(0\) to \(t\):

\[ f(t, T) = f(0, T) + \sigma^2 \int_0^t (T - s)\,ds + \sigma W_t \]

The deterministic integral evaluates to:

\[ \int_0^t (T - s)\,ds = Tt - \frac{t^2}{2} = t\left(T - \frac{t}{2}\right) \]

Therefore:

\[ f(t, T) = f(0, T) + \sigma^2 t\!\left(T - \frac{t}{2}\right) + \sigma W_t \]

This confirms the stated result. The forward rate is Gaussian (normally distributed), and the model is Ho--Lee.

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Exercise 3. For the Hull--White (extended Vasicek) volatility \(\sigma(t, T) = \sigma e^{-\kappa(T-t)}\), compute the drift

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

and verify that setting \(T = t\) gives \(\alpha(t, t) = 0\). Explain why the drift at the short end of the curve vanishes.

Solution to Exercise 3

Step 1: Compute the drift.

With \(\sigma(t, T) = \sigma e^{-\kappa(T-t)}\):

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

Evaluate the integral:

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

Therefore:

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

Step 2: Verify \(\alpha(t, t) = 0\).

Setting \(T = t\):

\[ \alpha(t, t) = \frac{\sigma^2}{\kappa}e^{0}\bigl(1 - e^{0}\bigr) = \frac{\sigma^2}{\kappa}(1 - 1) = 0 \]

Step 3: Explain why the drift vanishes at the short end.

The HJM drift at maturity \(T\) is \(\alpha(t, T) = \sigma(t, T)\,\Sigma(t, T)\) where \(\Sigma(t, T) = \int_t^T \sigma(t, u)\,du\). At \(T = t\), the bond volatility \(\Sigma(t, t) = \int_t^t \sigma(t, u)\,du = 0\), because the "bond" with zero time-to-maturity is the money-market account, which has zero volatility. Since the drift is the product of the forward rate volatility and the bond volatility, and the bond volatility is zero at \(T = t\), the drift must vanish at the short end regardless of the volatility specification. This is a universal feature of the HJM drift condition.

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Exercise 4. Consider a two-factor HJM model with \(\sigma_1(t, T) = \sigma_1\) and \(\sigma_2(t, T) = \sigma_2\,e^{-\kappa(T-t)}\). Write down the multi-factor drift condition

\[ \alpha(t, T) = \sum_{i=1}^{2}\sigma_i(t, T)\int_t^T \sigma_i(t, u)\,du \]

and compute \(\alpha(t, T)\) explicitly. Show that it reduces to the single-factor cases when either \(\sigma_1 = 0\) or \(\sigma_2 = 0\).

Solution to Exercise 4

Step 1: Apply the multi-factor drift condition.

\[ \alpha(t, T) = \sigma_1(t, T)\int_t^T \sigma_1(t, u)\,du + \sigma_2(t, T)\int_t^T \sigma_2(t, u)\,du \]

Step 2: Compute each factor's contribution.

Factor 1 (\(\sigma_1(t, T) = \sigma_1\), constant):

\[ \sigma_1 \int_t^T \sigma_1\,du = \sigma_1^2(T - t) \]

Factor 2 (\(\sigma_2(t, T) = \sigma_2\,e^{-\kappa(T-t)}\)):

\[ \sigma_2 e^{-\kappa(T-t)} \int_t^T \sigma_2 e^{-\kappa(u-t)}\,du = \sigma_2^2 e^{-\kappa(T-t)} \cdot \frac{1 - e^{-\kappa(T-t)}}{\kappa} = \frac{\sigma_2^2}{\kappa}e^{-\kappa(T-t)}\bigl(1 - e^{-\kappa(T-t)}\bigr) \]

Step 3: Combine.

\[ \alpha(t, T) = \sigma_1^2(T - t) + \frac{\sigma_2^2}{\kappa}e^{-\kappa(T-t)}\bigl(1 - e^{-\kappa(T-t)}\bigr) \]

Step 4: Verify limiting cases.

If \(\sigma_1 = 0\): \(\alpha(t, T) = \frac{\sigma_2^2}{\kappa}e^{-\kappa(T-t)}(1 - e^{-\kappa(T-t)})\), which is the Hull--White single-factor drift.

If \(\sigma_2 = 0\): \(\alpha(t, T) = \sigma_1^2(T - t)\), which is the Ho--Lee single-factor drift.

Both limiting cases are correctly recovered. \(\square\)

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Exercise 5. In the alternative ZCB dynamics proof, the key step is computing \(d\sigma_P^2(t, T)/dT\) where \(\sigma_P(t, T) = -\int_t^T \sigma(t, u)\,du\). Carry out this computation for a general volatility \(\sigma(t, T)\) and verify that

\[ \frac{d\sigma_P^2(t, T)}{dT} = 2\sigma(t, T)\int_t^T \sigma(t, u)\,du \]

Explain why the factor of 2 appears and how it leads to the drift condition after dividing by 2.

Solution to Exercise 5

Step 1: Compute \(d\sigma_P^2(t, T)/dT\).

We have \(\sigma_P(t, T) = -\int_t^T \sigma(t, u)\,du\). By the fundamental theorem of calculus:

\[ \frac{d\sigma_P(t, T)}{dT} = -\sigma(t, T) \]

Now apply the chain rule to \(\sigma_P^2(t, T) = [\sigma_P(t, T)]^2\):

\[ \frac{d\sigma_P^2(t, T)}{dT} = 2\,\sigma_P(t, T) \cdot \frac{d\sigma_P(t, T)}{dT} = 2\,\sigma_P(t, T) \cdot (-\sigma(t, T)) \]

Substituting \(\sigma_P(t, T) = -\int_t^T \sigma(t, u)\,du\):

\[ \frac{d\sigma_P^2(t, T)}{dT} = 2\left(-\int_t^T \sigma(t, u)\,du\right)(-\sigma(t, T)) = 2\,\sigma(t, T)\int_t^T \sigma(t, u)\,du \]

Step 2: Explain the factor of 2.

The factor of 2 arises from the chain rule: \(\frac{d}{dT}[g(T)]^2 = 2\,g(T)\,g'(T)\). This is the standard derivative of a squared function.

Step 3: Connection to the drift condition.

In the ZCB dynamics approach, the forward rate drift is

\[ \text{drift of } df(t, T) = \frac{1}{2}\frac{d\sigma_P^2(t, T)}{dT} = \frac{1}{2} \cdot 2\,\sigma(t, T)\int_t^T \sigma(t, u)\,du = \sigma(t, T)\int_t^T \sigma(t, u)\,du \]

The factor of \(\frac{1}{2}\) in the Ito correction (from \(d\log P\)) cancels with the factor of 2 from the chain rule, yielding the clean drift condition \(\alpha(t, T) = \sigma(t, T)\int_t^T \sigma(t, u)\,du\) without any numerical prefactor.

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Exercise 6. The HJM drift condition states that drift is uniquely determined by volatility. Does this mean that two different volatility specifications can never produce the same bond price dynamics? Construct a counterexample or prove that the mapping from \(\sigma(t, T)\) to bond price dynamics is injective.

Solution to Exercise 6

Claim: The mapping from \(\sigma(t, T)\) to bond price dynamics is injective (i.e., different volatility specifications cannot produce the same bond price dynamics).

Proof:

Under the risk-neutral measure, bond price dynamics are:

\[ \frac{dP(t, T)}{P(t, T)} = r_t\,dt - \Sigma(t, T)\,dW_t \]

where \(\Sigma(t, T) = \int_t^T \sigma(t, u)\,du\).

The drift is always \(r_t\) (risk-neutral pricing), so the dynamics are completely characterized by the bond volatility \(\Sigma(t, T)\).

Suppose two volatility functions \(\sigma\) and \(\tilde{\sigma}\) produce the same bond price dynamics. Then they must produce the same bond volatility:

\[ \int_t^T \sigma(t, u)\,du = \int_t^T \tilde{\sigma}(t, u)\,du \quad \text{for all } T \geq t \]

Differentiating both sides with respect to \(T\):

\[ \sigma(t, T) = \tilde{\sigma}(t, T) \quad \text{for all } T \geq t \]

(assuming \(\sigma\) and \(\tilde{\sigma}\) are continuous in \(T\)). Therefore the volatility functions must be identical.

The mapping is injective: different volatility specifications always produce different bond price dynamics. \(\square\)

Remark: This injectivity holds for the diffusion coefficient. If one allows different numbers of factors (e.g., a one-factor model versus a two-factor model), the distributional properties of the bond price could potentially agree if the total instantaneous variance \(\sum_i \Sigma_i(t, T)^2\) is the same, but the driving noise structure would differ (one vs. two Brownian motions), so the dynamics in the strong (pathwise) sense would still be different.

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Exercise 7. Under the physical measure \(\mathbb{P}\), forward rates have drift \(\alpha^{\mathbb{P}}(t, T)\) and the market price of risk is \(\lambda(t)\), so that \(\alpha(t, T) = \alpha^{\mathbb{P}}(t, T) - \lambda(t)\,\sigma(t, T)\). If \(\lambda(t) = \lambda_0 + \lambda_1 r_t\) (affine in the short rate), discuss what constraints the HJM no-arbitrage condition imposes on the relationship between \(\alpha^{\mathbb{P}}(t, T)\) and \(\sigma(t, T)\). Is \(\alpha^{\mathbb{P}}(t, T)\) still uniquely determined by \(\sigma(t, T)\) alone?

Solution to Exercise 7

Step 1: Relate physical and risk-neutral drifts.

Under \(\mathbb{P}\), the forward rate dynamics are:

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

The Girsanov theorem relates the two measures via \(dW_t^{\mathbb{Q}} = dW_t^{\mathbb{P}} - \lambda(t)\,dt\), so under \(\mathbb{Q}\):

\[ df(t, T) = \bigl[\alpha^{\mathbb{P}}(t, T) + \lambda(t)\,\sigma(t, T)\bigr]\,dt + \sigma(t, T)\,dW_t^{\mathbb{Q}} \]

Wait -- let us be careful with signs. Under \(\mathbb{P}\): \(dW_t^{\mathbb{P}} = dW_t^{\mathbb{Q}} + \lambda(t)\,dt\). Substituting:

\[ df(t, T) = \bigl[\alpha^{\mathbb{P}}(t, T) + \lambda(t)\,\sigma(t, T)\bigr]\,dt + \sigma(t, T)\,dW_t^{\mathbb{Q}} \]

The risk-neutral drift is \(\alpha(t, T) = \alpha^{\mathbb{P}}(t, T) + \lambda(t)\,\sigma(t, T)\), which rearranges to:

\[ \alpha^{\mathbb{P}}(t, T) = \alpha(t, T) - \lambda(t)\,\sigma(t, T) = \sigma(t, T)\,\Sigma(t, T) - \lambda(t)\,\sigma(t, T) \]

(using the sign convention in the problem, \(\alpha(t, T) = \alpha^{\mathbb{P}}(t, T) - \lambda(t)\,\sigma(t, T)\) gives the same result upon rearrangement).

Step 2: Apply the HJM no-arbitrage condition.

The HJM condition determines the \(\mathbb{Q}\)-drift:

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

Therefore the physical drift must satisfy:

\[ \alpha^{\mathbb{P}}(t, T) = \sigma(t, T)\left[\int_t^T \sigma(t, u)\,du + \lambda(t)\right] \]

Step 3: Substitute the affine market price of risk \(\lambda(t) = \lambda_0 + \lambda_1 r_t\).

\[ \alpha^{\mathbb{P}}(t, T) = \sigma(t, T)\left[\int_t^T \sigma(t, u)\,du + \lambda_0 + \lambda_1 r_t\right] \]

Step 4: Is \(\alpha^{\mathbb{P}}\) determined by \(\sigma\) alone?

No. The physical drift \(\alpha^{\mathbb{P}}(t, T)\) depends on:

1. The volatility function \(\sigma(t, T)\) (which determines \(\Sigma(t, T)\) and the \(\mathbb{Q}\)-drift),
2. The market price of risk parameters \(\lambda_0\) and \(\lambda_1\), and
3. The current short rate \(r_t\) (through the \(\lambda_1 r_t\) term).

The no-arbitrage condition constrains \(\alpha^{\mathbb{P}}\) to have the form above, but the parameters \(\lambda_0\) and \(\lambda_1\) are not determined by \(\sigma(t, T)\). They must be estimated from historical data or specified as part of the model. Therefore, \(\alpha^{\mathbb{P}}(t, T)\) is not uniquely determined by \(\sigma(t, T)\) alone --- it requires additional information about the market price of risk.

This is the fundamental distinction: under \(\mathbb{Q}\), the drift is fully pinned down by no-arbitrage; under \(\mathbb{P}\), there is additional freedom parameterized by \(\lambda(t)\).