Change of Measure for the CIR Model¶
Pricing interest rate derivatives often requires computing expectations of the form \(\mathbb{E}^{\mathbb{Q}}[D(t,T)\,h(r_T)]\), where \(D(t,T) = \exp(-\int_t^T r_s\,ds)\) is the stochastic discount factor and \(h\) is a payoff function. Direct evaluation of this expectation is difficult because both the discount factor and the payoff depend on the entire path of \(r_s\). The \(T\)-forward measure \(\mathbb{Q}^T\) eliminates this coupling by absorbing the discount factor into the change of measure, reducing the problem to a simple expectation of the payoff. This section defines the \(T\)-forward measure, derives the CIR dynamics under this new measure, and shows that the process remains in the CIR family with modified parameters --- a key property that enables closed-form bond option pricing.
Motivation: simplifying derivative pricing¶
Under the risk-neutral measure \(\mathbb{Q}\), the time-\(t\) price of a European claim paying \(h(r_T)\) at time \(T\) is
The stochastic discount factor and the payoff are correlated (both depend on the rate path), making this expectation analytically intractable in general. The \(T\)-forward measure removes the discount factor from inside the expectation by rewriting
where \(P(t,T)\) is the known zero-coupon bond price. The pricing problem thus reduces to computing an expectation of \(h(r_T)\) alone, without the path-dependent discount factor.
Definition of the T-forward measure¶
The \(T\)-forward measure \(\mathbb{Q}^T\) is defined by its Radon-Nikodym derivative with respect to \(\mathbb{Q}\) on \(\mathcal{F}_T\):
The numeraire associated with \(\mathbb{Q}^T\) is the zero-coupon bond \(P(t,T)\). Under \(\mathbb{Q}^T\), any asset price divided by \(P(t,T)\) is a martingale. The Radon-Nikodym density process (restricted to \(\mathcal{F}_t\)) is
Girsanov kernel for the CIR model¶
To apply Girsanov's theorem, we need the volatility of \(L_t\). Since \(P(t,T) = A(\tau)e^{-B(\tau)r_t}\) with \(\tau = T - t\), applying Ito's lemma to \(\ln P(t,T)\) and extracting the diffusion coefficient:
The martingale \(L_t\) satisfies
By Girsanov's theorem, the process
is a standard Brownian motion under \(\mathbb{Q}^T\).
CIR dynamics under the T-forward measure¶
The CIR SDE under \(\mathbb{Q}\) is
Substituting \(dW_t^{\mathbb{Q}} = dW_t^T - B(\tau)\sigma\sqrt{r_t}\,dt\):
Defining the time-dependent adjusted parameters:
the dynamics become
CIR stays CIR
The crucial observation is that the short rate under \(\mathbb{Q}^T\) still follows a CIR-type process --- the diffusion coefficient \(\sigma\sqrt{r_t}\) is unchanged, and only the drift parameters are modified. However, these modified parameters are now time-dependent through \(B(\tau) = B(T-t)\), so the process is a time-inhomogeneous CIR process. This preservation of the square-root structure is essential for obtaining closed-form bond option formulas.
Properties of the adjusted parameters¶
The adjusted mean-reversion speed \(\kappa^T(\tau) = \kappa + \sigma^2 B(\tau)\) satisfies:
- \(\kappa^T(0) = \kappa\) (at maturity, no adjustment)
- \(\kappa^T(\tau) > \kappa\) for all \(\tau > 0\) (the forward measure increases mean reversion)
- \(\kappa^T(\tau) \to \kappa + \sigma^2 B_\infty\) as \(\tau \to \infty\)
The adjusted long-run mean \(\theta^T(\tau) = \kappa\theta/\kappa^T(\tau)\) satisfies:
- \(\theta^T(0) = \theta\) (no adjustment at maturity)
- \(\theta^T(\tau) < \theta\) for all \(\tau > 0\) (the forward measure lowers the long-run mean)
The economic interpretation is that the \(T\)-forward measure tilts the rate distribution downward: since bond prices decrease in rates, conditioning on the bond numeraire (which favors states with low rates) pulls the expected rate path lower.
Feller condition under the forward measure¶
The Feller condition under \(\mathbb{Q}^T\) requires \(2\kappa^T(\tau)\theta^T(\tau) \geq \sigma^2\), which simplifies to \(2\kappa\theta \geq \sigma^2\) --- the same condition as under \(\mathbb{Q}\). The change of measure preserves the Feller condition.
Application to derivative pricing¶
The forward-measure pricing formula
reduces derivative pricing to computing expectations under the \(\mathbb{Q}^T\) dynamics. For European-style payoffs \(h(r_T)\):
-
Bond options: \(h(r_T) = (P(T,S) - K)^+\) for a call on a zero-coupon bond maturing at \(S > T\). Since \(P(T,S)\) is a known function of \(r_T\), this reduces to computing the distribution of \(r_T\) under \(\mathbb{Q}^T\).
-
Caplets: A caplet with strike \(K\) on the rate over \([T, T+\delta]\) has payoff \(\delta(L(T,T+\delta) - K)^+\), which is equivalent to a put on a zero-coupon bond.
-
General European claims: Any payoff that depends on \(r_T\) alone (not the entire path) can be priced using the \(\mathbb{Q}^T\) transition density.
Transition density under the forward measure¶
Under \(\mathbb{Q}^T\), the short rate \(r_T\) given \(r_t\) follows a non-central chi-squared distribution with time-dependent parameters. The conditional distribution has the form
where the scaling, degrees of freedom, and non-centrality parameters are
The degrees-of-freedom parameter \(d^T = 4\kappa\theta/\sigma^2\) is the same as under \(\mathbb{Q}\) (because the Feller condition is preserved), but the scaling and non-centrality parameters differ due to the time-dependent drift. The explicit computation of these parameters involves integrating the time-dependent coefficients \(\kappa^T(\tau)\) and \(\theta^T(\tau)\), which can be carried out in terms of the CIR bond price functions \(A\) and \(B\).
Physical to risk-neutral to forward measure¶
The full chain of measure changes in the CIR framework is:
| Measure | Drift of \(r_t\) | Volatility | Numeraire |
|---|---|---|---|
| \(\mathbb{P}\) (physical) | \(\kappa^{\mathbb{P}}(\theta^{\mathbb{P}} - r_t)\) | \(\sigma\sqrt{r_t}\) | None |
| \(\mathbb{Q}\) (risk-neutral) | \(\kappa(\theta - r_t)\) | \(\sigma\sqrt{r_t}\) | Money market account |
| \(\mathbb{Q}^T\) (\(T\)-forward) | \(\kappa^T(\tau)(\theta^T(\tau) - r_t)\) | \(\sigma\sqrt{r_t}\) | \(P(t,T)\) |
In each case, the square-root diffusion structure is preserved. The physical-to-risk-neutral change adjusts \(\kappa^{\mathbb{P}}\) and \(\theta^{\mathbb{P}}\) (via the market price of risk), and the risk-neutral-to-forward change further adjusts these parameters through \(B(\tau)\).
Market price of risk in CIR
The standard choice for the CIR market price of risk is \(\lambda(r_t) = \lambda_0 \sqrt{r_t}\), which ensures the process remains CIR under \(\mathbb{Q}\). Under this specification, \(\kappa = \kappa^{\mathbb{P}} + \lambda_0\) and \(\theta = \kappa^{\mathbb{P}}\theta^{\mathbb{P}}/\kappa\). The state-dependent form \(\lambda_0\sqrt{r_t}\) is required (rather than a constant) to preserve the CIR structure.
Summary¶
The \(T\)-forward measure \(\mathbb{Q}^T\) simplifies CIR derivative pricing by removing the stochastic discount factor from the pricing expectation. Under this measure, the CIR process retains its square-root diffusion structure with time-dependent drift parameters \(\kappa^T(\tau) = \kappa + \sigma^2 B(\tau)\) and \(\theta^T(\tau) = \kappa\theta/\kappa^T(\tau)\). The forward measure increases the speed of mean reversion and lowers the long-run mean, reflecting the bias toward low-rate states introduced by the bond numeraire. The Feller condition is preserved, and the transition density remains non-central chi-squared with modified parameters. This framework provides the foundation for the closed-form CIR bond option formulas developed in the next section.
Exercises¶
Exercise 1. Starting from the CIR bond price \(P(t,T) = A(\tau)e^{-B(\tau)r_t}\) with \(\tau = T - t\), apply Ito's lemma to \(\ln P(t,T)\) and identify the drift and diffusion terms. Confirm that the diffusion coefficient of \(\ln P\) is \(-B(\tau)\sigma\sqrt{r_t}\), which gives rise to the Girsanov kernel.
Solution to Exercise 1
The CIR bond price is \(P(t,T) = A(\tau)e^{-B(\tau)r_t}\) with \(\tau = T - t\). Define \(F(t,r) = \ln P(t,T) = \ln A(\tau) - B(\tau)r\).
Since \(\tau = T - t\), we have \(d\tau = -dt\), so:
Applying Ito's lemma to \(F = \ln P(t,T)\):
The last term vanishes since \(\partial^2 F/\partial r^2 = 0\). Substituting \(dr = \kappa(\theta - r)\,dt + \sigma\sqrt{r}\,dW^{\mathbb{Q}}\):
The diffusion (stochastic) part is:
This confirms that the diffusion coefficient of \(\ln P\) is \(-B(\tau)\sigma\sqrt{r_t}\), which is the Girsanov kernel that drives the measure change from \(\mathbb{Q}\) to \(\mathbb{Q}^T\).
Exercise 2. Derive the CIR dynamics under \(\mathbb{Q}^T\) step by step. Substitute \(dW_t^{\mathbb{Q}} = dW_t^T - B(\tau)\sigma\sqrt{r_t}\,dt\) into the risk-neutral CIR SDE and collect terms to obtain \(dr_t = [\kappa\theta - (\kappa + \sigma^2 B(\tau))r_t]\,dt + \sigma\sqrt{r_t}\,dW_t^T\). Verify that \(\kappa^T(\tau) = \kappa + \sigma^2 B(\tau)\) and \(\theta^T(\tau) = \kappa\theta/\kappa^T(\tau)\).
Solution to Exercise 2
The risk-neutral CIR SDE is:
By Girsanov's theorem, \(dW_t^{\mathbb{Q}} = dW_t^T - B(\tau)\sigma\sqrt{r_t}\,dt\). Substituting:
Expanding:
Collecting drift terms:
Define \(\kappa^T(\tau) = \kappa + \sigma^2 B(\tau)\). Then the drift becomes:
Defining \(\theta^T(\tau) = \kappa\theta/\kappa^T(\tau)\):
This confirms both adjusted parameters: \(\kappa^T(\tau) = \kappa + \sigma^2 B(\tau)\) and \(\theta^T(\tau) = \kappa\theta/\kappa^T(\tau)\).
Exercise 3. For CIR parameters \(\kappa = 0.5\), \(\theta = 0.06\), \(\sigma = 0.10\), compute \(B(\tau)\) for \(\tau = 5\) years using the CIR formula. Then compute \(\kappa^T(5) = \kappa + \sigma^2 B(5)\) and \(\theta^T(5) = \kappa\theta/\kappa^T(5)\). Compare these with the risk-neutral values \(\kappa\) and \(\theta\). Interpret the economic meaning of the changes.
Solution to Exercise 3
Given \(\kappa = 0.5\), \(\theta = 0.06\), \(\sigma = 0.10\).
Step 1: Compute \(\gamma\) and \(B(5)\).
Step 2: Compute \(\kappa^T(5)\).
Step 3: Compute \(\theta^T(5)\).
Comparison with risk-neutral values:
- \(\kappa^T(5) = 0.518 > \kappa = 0.5\) (increased by \(\sim 3.6\%\))
- \(\theta^T(5) = 0.0579 < \theta = 0.06\) (decreased by \(\sim 3.5\%\))
Economic interpretation: The forward measure increases the speed of mean reversion (the process reverts faster to its long-run mean) and lowers the long-run mean level. This reflects the bias introduced by the bond numeraire \(P(t,T)\): since bond prices are high when rates are low, the forward measure overweights low-rate scenarios. This tilts the expected rate path downward (lower \(\theta^T\)) and pulls the process more strongly toward this lower target (higher \(\kappa^T\)).
Exercise 4. Show that the Feller condition is preserved under the \(T\)-forward measure. Specifically, verify that \(2\kappa^T(\tau)\theta^T(\tau) = 2\kappa\theta\) for all \(\tau\), so the condition \(2\kappa\theta \geq \sigma^2\) is equivalent under both measures.
Solution to Exercise 4
We need to show \(2\kappa^T(\tau)\theta^T(\tau) = 2\kappa\theta\).
The \(\kappa + \sigma^2 B(\tau)\) factors cancel exactly, leaving \(2\kappa\theta\) regardless of \(\tau\).
Therefore, the Feller condition under \(\mathbb{Q}^T\) is \(2\kappa^T(\tau)\theta^T(\tau) \geq \sigma^2\), which simplifies to \(2\kappa\theta \geq \sigma^2\), exactly the same condition as under \(\mathbb{Q}\).
This is a remarkable and useful result: the change of measure modifies \(\kappa\) and \(\theta\) in a way that preserves their product \(\kappa\theta\), and since the diffusion coefficient \(\sigma\) is unchanged by the measure change, the Feller condition is invariant. The process under the forward measure has the same boundary behavior as under the risk-neutral measure.
Exercise 5. The forward measure tilts the rate distribution downward. Explain this economically: the numeraire is the zero-coupon bond \(P(t,T)\), which is high when rates are low. How does conditioning on a high numeraire value bias the distribution of \(r_T\)? Relate this to the decrease in \(\theta^T(\tau)\) compared to \(\theta\).
Solution to Exercise 5
Under \(\mathbb{Q}^T\), the numeraire is the zero-coupon bond \(P(t,T)\). Since \(P(t,T) = A(\tau)e^{-B(\tau)r_t}\) is a decreasing function of \(r_t\) (because \(B(\tau) > 0\)), the bond price is high when rates are low.
The forward measure \(\mathbb{Q}^T\) is defined by \(d\mathbb{Q}^T/d\mathbb{Q} \propto e^{-\int_0^T r_s\,ds} \cdot P(t,T)\), which upweights scenarios where the discount factor \(e^{-\int r_s\,ds}\) is large (i.e., where rates are low) and where \(P(t,T)\) is large (also when rates are low). This double bias toward low-rate states systematically tilts the distribution of \(r_T\) downward.
Quantitatively:
- The adjusted long-run mean \(\theta^T(\tau) = \kappa\theta/(\kappa + \sigma^2 B(\tau)) < \theta\) because \(\sigma^2 B(\tau) > 0\). This means the forward-measure process mean-reverts to a lower level.
- The stronger mean reversion (\(\kappa^T > \kappa\)) means the process is pulled more forcefully toward this lower target.
Together, these effects shift the entire distribution of \(r_T\) to the left (lower rates), consistent with conditioning on a numeraire that is more valuable in low-rate environments.
Exercise 6. In the chain \(\mathbb{P} \to \mathbb{Q} \to \mathbb{Q}^T\), the diffusion \(\sigma\sqrt{r_t}\) is unchanged at each step. Explain why the diffusion coefficient is invariant under measure changes in general (Girsanov's theorem only modifies the drift). If the market price of risk is \(\lambda(r_t) = \lambda_0\sqrt{r_t}\), compute the physical-measure drift parameters \(\kappa^{\mathbb{P}}\) and \(\theta^{\mathbb{P}}\) in terms of \(\kappa\), \(\theta\), and \(\lambda_0\).
Solution to Exercise 6
Girsanov's theorem and diffusion invariance: Girsanov's theorem states that changing from measure \(\mathbb{P}\) to \(\mathbb{Q}\) modifies the drift of a process by adding a term \(\sigma(r_t) \cdot \lambda(t)\) (where \(\lambda\) is the Girsanov kernel), but the diffusion coefficient \(\sigma(r_t)\) remains unchanged. This is a fundamental property: the quadratic variation \(\langle r \rangle_t = \int_0^t \sigma^2(r_s)\,ds\) is the same under all equivalent measures. The intuition is that the diffusion coefficient determines the "roughness" of sample paths, which is a pathwise property unaffected by reweighting probabilities.
Physical-measure parameters: If the market price of risk is \(\lambda(r_t) = \lambda_0\sqrt{r_t}\), the Girsanov kernel is \(\lambda_0\sqrt{r_t}\) and:
Under \(\mathbb{P}\): \(dr_t = \kappa^{\mathbb{P}}(\theta^{\mathbb{P}} - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t^{\mathbb{P}}\).
Substituting \(dW_t^{\mathbb{P}} = dW_t^{\mathbb{Q}} - \lambda_0\sqrt{r_t}\,dt\):
Matching with the \(\mathbb{Q}\) dynamics \(dr_t = \kappa(\theta - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t^{\mathbb{Q}}\):
Solving:
Exercise 7. A European digital bond option pays $1 if \(P(T, S) > K\) and $0 otherwise, with expiry \(T\) and underlying bond maturity \(S\). Using the \(T\)-forward measure, write the price of this digital option as \(P(t,T) \cdot \mathbb{Q}^T(r_T < r^*)\). Express this probability in terms of the non-central chi-squared CDF with the appropriate parameters under \(\mathbb{Q}^T\).
Solution to Exercise 7
The digital bond option pays $1 if \(P(T,S) > K\) and $0 otherwise. Under \(\mathbb{Q}\):
Switching to the \(T\)-forward measure:
Since \(P(T,S) = A(S-T)e^{-B(S-T)r_T}\) is decreasing in \(r_T\), the condition \(P(T,S) > K\) is equivalent to \(r_T < r^*\) where \(r^* = \frac{1}{B(S-T)}\ln\frac{A(S-T)}{K}\).
Therefore:
Under \(\mathbb{Q}^T\), the scaled rate \(2c^T r_T\) follows a non-central chi-squared distribution \(\chi^2(d, \lambda^T)\) where \(d = 4\kappa\theta/\sigma^2\) and \(\lambda^T\) and \(c^T\) are the forward-measure parameters. The probability is:
where \(\chi^2(x; d, \lambda)\) denotes the non-central chi-squared CDF. Here \(c^T = c^T(t,T)\) and \(\lambda^T = 2c^T r_t e^{-\int_t^T \kappa^T(s)\,ds}\) involve the integrated forward-measure drift coefficients.
In terms of the bond option parameters \(\phi\) and \(\psi\):
where \(\lambda_2 = 2\phi^2 r_t e^{\gamma(T-t)}/(\phi + \psi)\) and \(\phi\), \(\psi\) are as defined in the bond option formula. This is precisely the second term of the CIR call option formula (without the \(P(t,S)\) first term and without the strike).