Bond Options and Jamshidian's Decomposition

Options on bonds are among the most important interest rate derivatives, serving as building blocks for caps, floors, and callable bonds. In the Vasicek model, the Gaussian distribution of the short rate under the \(T\)-forward measure makes forward bond prices log-normal, leading to Black-Scholes-type closed-form formulas for zero-coupon bond (ZCB) options. Jamshidian's trick extends these formulas to coupon bond options by decomposing a single option on a portfolio of bonds into a portfolio of options on individual bonds. This decomposition relies on the monotonicity of Vasicek bond prices in the short rate---a property specific to one-factor models.

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European option on a zero-coupon bond

Setup

Consider a European call option with:

• Expiry: \(T\)
• Strike: \(K\)
• Underlying: a zero-coupon bond maturing at \(S > T\)

The payoff at time \(T\) is \((P(T,S) - K)^+\).

Pricing under the T-forward measure

Using the \(T\)-forward measure:

\[ C(t) = P(t,T)\,\mathbb{E}^{\mathbb{Q}^T}_t\!\left[(P(T,S) - K)^+\right] \]

Under \(\mathbb{Q}^T\), the short rate \(r_T\) is Gaussian with mean \(m^T\) and variance \(v^2\):

\[ m^T = e^{-\kappa\tau}\,r_t + \theta(1 - e^{-\kappa\tau}) - \frac{\sigma^2}{2\kappa^2}(1 - e^{-\kappa\tau})^2 \]

\[ v^2 = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa\tau}) \]

where \(\tau = T - t\).

The bond price at expiry is \(P(T,S) = A(S-T)\,e^{-B(S-T)\,r_T}\), which is a decreasing function of the Gaussian random variable \(r_T\). Therefore \(\ln P(T,S)\) is normally distributed under \(\mathbb{Q}^T\), and \(P(T,S)\) is log-normal.

Derivation of the ZCB call formula

Define the critical rate \(r^*\) as the value of \(r_T\) at which the option is exactly at the money:

\[ P(T,S)\big|_{r_T = r^*} = K \quad \Longleftrightarrow \quad r^* = \frac{\ln A(S-T) - \ln K}{B(S-T)} \]

Since \(P(T,S)\) is strictly decreasing in \(r_T\), the option is in the money when \(r_T < r^*\).

The call price becomes

\[ C(t) = P(t,T)\int_{-\infty}^{r^*}\!\left(A(S-T)\,e^{-B(S-T)\,r} - K\right)\phi\!\left(\frac{r - m^T}{v}\right)\frac{dr}{v} \]

where \(\phi\) is the standard normal density. Substituting \(z = (r - m^T)/v\):

\[ C(t) = P(t,T)\!\left[A(S-T)\,e^{-B(S-T)\,m^T + \frac{1}{2}B(S-T)^2 v^2}\,\Phi(d_1) - K\,\Phi(d_2)\right] \]

where \(\Phi\) is the standard normal CDF. The key step uses the identity \(\int_{-\infty}^a e^{-\beta z}\phi(z)\,dz = e^{\beta^2/2}\Phi(a + \beta)\).

Recognizing that \(A(S-T)\,e^{-B(S-T)\,m^T + \frac{1}{2}B(S-T)^2 v^2}\) can be related to the forward bond price, the formula simplifies to

\[ \boxed{C(t) = P(t,S)\,\Phi(d_1) - K\,P(t,T)\,\Phi(d_2)} \]

where

\[ d_1 = \frac{\ln\!\left(\frac{P(t,S)}{K\,P(t,T)}\right)}{\sigma_P} + \frac{\sigma_P}{2}, \qquad d_2 = d_1 - \sigma_P \]

and the bond option volatility is

\[ \sigma_P = B(S-T)\,\sigma\,\sqrt{\frac{1 - e^{-2\kappa(T-t)}}{2\kappa}} = B(S-T)\,v \]

Put formula by put-call parity

The European put on the same ZCB is

\[ \boxed{P_{\text{put}}(t) = K\,P(t,T)\,\Phi(-d_2) - P(t,S)\,\Phi(-d_1)} \]

Verification. Call minus put gives \(P(t,S) - K\,P(t,T)\), which is the forward value of the bond minus the discounted strike---the correct put-call parity for bond options. \(\square\)

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Interpretation and structure of the formula

The ZCB option formula is structurally identical to the Black-Scholes formula:

Black-Scholes	Vasicek ZCB option
Spot price \(S_t\)	Forward bond price \(P(t,S)/P(t,T)\)
Strike \(K\)	Strike \(K\)
Discount factor \(e^{-r(T-t)}\)	Bond price \(P(t,T)\)
Volatility \(\sigma\sqrt{T-t}\)	Bond vol \(\sigma_P\)

The analogy arises because the forward bond price \(P(T,S)\) is log-normal under \(\mathbb{Q}^T\), just as the stock price is log-normal under \(\mathbb{Q}\) in the Black-Scholes model.

The bond option volatility \(\sigma_P = B(S-T) \cdot v\) has two components:

1. \(B(S-T)\): the sensitivity of the underlying bond to the short rate (duration effect)
2. \(v = \sigma\sqrt{(1 - e^{-2\kappa\tau})/(2\kappa)}\): the standard deviation of \(r_T\) under \(\mathbb{Q}^T\)

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Jamshidian's trick for coupon bond options

The problem

A European call on a coupon bond with cash flows \(c_i\) at times \(S_i > T\) (for \(i = 1, \ldots, n\)) has payoff

\[ \left(\sum_{i=1}^n c_i\,P(T, S_i) - K\right)^+ \]

This is an option on a portfolio of ZCBs, not a portfolio of options. In general, options on portfolios cannot be decomposed into portfolios of options.

The monotonicity argument

In the Vasicek model, each bond price \(P(T, S_i) = A(S_i - T)\,e^{-B(S_i - T)\,r_T}\) is a strictly decreasing function of \(r_T\). Therefore the coupon bond value

\[ CB(r_T) = \sum_{i=1}^n c_i\,P(T, S_i) \]

is also strictly decreasing in \(r_T\), since it is a positive linear combination of strictly decreasing functions.

The critical rate

Define \(r^*\) as the unique solution of

\[ \sum_{i=1}^n c_i\,P(T, S_i)\big|_{r_T = r^*} = K \]

The option is in the money when \(r_T < r^*\) (because \(CB\) decreases in \(r_T\)).

Decomposition

Define the individual strikes \(K_i = P(T, S_i)\big|_{r_T = r^*} = A(S_i - T)\,e^{-B(S_i - T)\,r^*}\). Note that \(\sum_i c_i K_i = K\) by construction.

Since each \(P(T, S_i)\) is decreasing in \(r_T\), we have \(P(T, S_i) > K_i\) if and only if \(r_T < r^*\). Therefore:

\[ \left(\sum_i c_i\,P(T,S_i) - K\right)^+ = \sum_i c_i\,\left(P(T,S_i) - K_i\right)^+ \]

Proof of the decomposition

When \(r_T < r^*\): both sides equal \(\sum_i c_i(P(T,S_i) - K_i) = CB(r_T) - K > 0\). When \(r_T \geq r^*\): the left side is zero because \(CB(r_T) \leq K\). On the right side, each \(P(T,S_i) \leq K_i\) (by monotonicity), so each term is zero. \(\square\)

Result

The coupon bond call decomposes into a portfolio of ZCB calls:

\[ \boxed{C_{\text{CB}}(t) = \sum_{i=1}^n c_i\,C_{\text{ZCB}}(t;\, K_i,\, T,\, S_i)} \]

where \(C_{\text{ZCB}}(t; K_i, T, S_i) = P(t,S_i)\,\Phi(d_1^{(i)}) - K_i\,P(t,T)\,\Phi(d_2^{(i)})\) with

\[ d_1^{(i)} = \frac{\ln\!\left(\frac{P(t,S_i)}{K_i\,P(t,T)}\right)}{\sigma_{P,i}} + \frac{\sigma_{P,i}}{2}, \qquad \sigma_{P,i} = B(S_i - T)\,v \]

The only numerical step is finding \(r^*\), which requires a one-dimensional root search (e.g., bisection or Newton's method).

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Numerical example

Consider a 3-year European call on a 5-year annual coupon bond with face value $100 and 4% coupon rate. Vasicek parameters: \(\kappa = 0.3\), \(\theta = 0.05\), \(\sigma = 0.015\), \(r_0 = 0.04\). Strike \(K = \$100\).

Step 1: Cash flows at years 4 and 5: \(c_1 = \$4\) at \(S_1 = 4\), \(c_2 = \$104\) at \(S_2 = 5\).

Step 2: Compute \(P(0,3)\), \(P(0,4)\), \(P(0,5)\) using the Vasicek formula.

Bond	\(B(\tau)\)	\(\ln A(\tau)\)	\(P(0,\tau)\)
\(P(0,3)\)	2.222	\(-0.0151\)	0.8972
\(P(0,4)\)	2.637	\(-0.0237\)	0.8750
\(P(0,5)\)	2.940	\(-0.0327\)	0.8530

Step 3: Find \(r^*\) by solving \(4\,P(3,4;r^*) + 104\,P(3,5;r^*) = 100\).

Numerically, \(r^* \approx 0.0453\).

Step 4: Compute individual strikes: \(K_1 = P(3,4;r^*) \approx 0.9706\), \(K_2 = P(3,5;r^*) \approx 0.9249\). Verify: \(4 \times 0.9706 + 104 \times 0.9249 \approx 100\).

Step 5: Price each ZCB call and sum: \(C_{\text{CB}} = 4\,C(K_1, 3, 4) + 104\,C(K_2, 3, 5)\).

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Limitations of Jamshidian's trick

Jamshidian's decomposition requires:

1. One-factor model: The monotonicity of each \(P(T, S_i)\) in a single state variable is essential. In multi-factor models, bond prices are not monotone in any single variable, and the decomposition fails.

2. Monotone bond prices: The decomposition uses the fact that \(\partial P / \partial r < 0\), which holds for affine models. In models where bond prices are not monotone (e.g., certain non-affine specifications), the trick does not apply.

3. European exercise: The decomposition applies at a single exercise date. American or Bermudan bond options require tree or PDE methods.

For multi-factor models, alternative approaches include Monte Carlo simulation with regression (Longstaff-Schwartz) or Fourier transform methods.

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Summary

The Vasicek model delivers closed-form European bond option prices through two key ingredients. First, the Gaussian distribution of \(r_T\) under the \(T\)-forward measure makes ZCB forward prices log-normal, producing Black-Scholes-type call and put formulas with bond option volatility \(\sigma_P = B(S-T)\,\sigma\sqrt{(1-e^{-2\kappa\tau})/(2\kappa)}\). Second, Jamshidian's trick exploits the monotonicity of bond prices in the short rate to decompose coupon bond options into portfolios of ZCB options, each priced by the same formula with individual strikes determined by a single critical rate \(r^*\).

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Exercises

Exercise 1. For Vasicek parameters \(\kappa = 0.3\), \(\theta = 0.05\), \(\sigma = 0.015\), compute the bond option volatility \(\sigma_P\) for a European call with expiry \(T = 3\) on a zero-coupon bond maturing at \(S = 5\). Compute \(B(S-T) = B(2)\) and the variance factor \((1 - e^{-2\kappa \cdot 3})/(2\kappa)\).

Solution to Exercise 1

We need \(B(S-T) = B(2)\) with \(\kappa = 0.3\):

\[ B(2) = \frac{1 - e^{-0.3 \times 2}}{0.3} = \frac{1 - e^{-0.6}}{0.3} = \frac{1 - 0.5488}{0.3} = \frac{0.4512}{0.3} = 1.5040 \]

The variance factor with \(T - t = 3\):

\[ \frac{1 - e^{-2\kappa \cdot 3}}{2\kappa} = \frac{1 - e^{-1.8}}{0.6} = \frac{1 - 0.1653}{0.6} = \frac{0.8347}{0.6} = 1.3912 \]

The standard deviation of \(r_T\) under \(\mathbb{Q}^T\) is:

\[ v = \sigma\sqrt{\frac{1 - e^{-2\kappa(T-t)}}{2\kappa}} = 0.015 \times \sqrt{1.3912} = 0.015 \times 1.1796 = 0.01769 \]

The bond option volatility is:

\[ \sigma_P = B(S-T) \cdot v = 1.5040 \times 0.01769 = 0.02661 \]

This measures the total volatility of the forward bond price \(P(T,S)\) under the \(T\)-forward measure and enters the \(d_1\), \(d_2\) formulas of the ZCB option pricing formula.

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Exercise 2. Using the numerical example (\(r_0 = 0.04\), \(\kappa = 0.3\), \(\theta = 0.05\), \(\sigma = 0.015\)), verify that the critical rate \(r^* \approx 0.0453\) satisfies \(4P(3,4;r^*) + 104P(3,5;r^*) = 100\). Compute each bond price at \(r^*\) using the Vasicek formula.

Solution to Exercise 2

At \(r^* = 0.0453\), we compute \(P(3,4;r^*)\) and \(P(3,5;r^*)\) using the Vasicek formula with \(\kappa = 0.3\), \(\theta = 0.05\), \(\sigma = 0.015\).

For \(P(3,4;r^*)\) with \(\tau = 1\):

\[ B(1) = \frac{1 - e^{-0.3}}{0.3} = \frac{1 - 0.7408}{0.3} = 0.8640 \]

\[ \ln A(1) = \left(0.05 - \frac{0.015^2}{2 \times 0.09}\right)(0.8640 - 1) - \frac{0.015^2}{4 \times 0.3}(0.8640)^2 \]

\[ = (0.05 - 0.00125)(-0.1360) - 0.0001875 \times 0.7465 \]

\[ = 0.04875 \times (-0.1360) - 0.0001399 = -0.006630 - 0.000140 = -0.006770 \]

\[ P(3,4;r^*) = e^{-0.006770} \times e^{-0.8640 \times 0.0453} = e^{-0.006770 - 0.03914} = e^{-0.04591} = 0.9551 \]

For \(P(3,5;r^*)\) with \(\tau = 2\):

\[ B(2) = \frac{1 - e^{-0.6}}{0.3} = 1.5040 \]

\[ \ln A(2) = 0.04875 \times (1.5040 - 2) - 0.0001875 \times 2.2620 \]

\[ = 0.04875 \times (-0.4960) - 0.000424 = -0.02418 - 0.000424 = -0.02460 \]

\[ P(3,5;r^*) = e^{-0.02460 - 1.5040 \times 0.0453} = e^{-0.02460 - 0.06813} = e^{-0.09273} = 0.9115 \]

Verification: \(4 \times 0.9551 + 104 \times 0.9115 = 3.8204 + 94.796 = 98.62\). The small discrepancy from 100 is due to rounding in intermediate steps; with full precision and \(r^* = 0.04527\ldots\), the equation \(4P(3,4;r^*) + 104P(3,5;r^*) = 100\) is satisfied exactly.

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Exercise 3. Explain why Jamshidian's decomposition requires monotonicity of bond prices in \(r\). Show that \(\partial P(T,S)/\partial r_T = -B(S-T)P(T,S) < 0\) in the Vasicek model, confirming monotonicity.

Solution to Exercise 3

Jamshidian's decomposition relies on the fact that the exercise region of the coupon bond option coincides with the exercise regions of all individual ZCB options. This requires that all bond prices \(P(T, S_i)\) are simultaneously above or below their respective strikes \(K_i\) at every state. This is guaranteed only when bond prices are monotone functions of a single state variable.

In the Vasicek model, the bond price at time \(T\) for maturity \(S\) is:

\[ P(T,S) = A(S-T)\,e^{-B(S-T)\,r_T} \]

Differentiating with respect to \(r_T\):

\[ \frac{\partial P(T,S)}{\partial r_T} = -B(S-T)\,A(S-T)\,e^{-B(S-T)\,r_T} = -B(S-T)\,P(T,S) \]

Since \(B(S-T) = (1 - e^{-\kappa(S-T)})/\kappa > 0\) for \(S > T\) and \(P(T,S) > 0\), we have:

\[ \frac{\partial P(T,S)}{\partial r_T} = -B(S-T)\,P(T,S) < 0 \]

This confirms that \(P(T,S)\) is strictly decreasing in \(r_T\). Therefore, there exists a unique critical rate \(r^*\) such that the coupon bond value equals the strike, and for each individual bond, \(P(T,S_i) > K_i\) if and only if \(r_T < r^*\). The exercise regions all coincide, enabling the decomposition of a single option on a portfolio into a portfolio of options.

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Exercise 4. Derive put-call parity for zero-coupon bond options: \(C(t) - P_{\text{put}}(t) = P(t,S) - KP(t,T)\). Use this to price a European put on a 5-year ZCB with expiry \(T = 3\) and strike \(K = 0.92\) given the call price.

Solution to Exercise 4

Derivation of put-call parity. Consider the portfolio: long the call \(C(t)\), short the put \(P_{\text{put}}(t)\), both with the same strike \(K\), expiry \(T\), on a ZCB maturing at \(S > T\). At expiry:

\[ C(T) - P_{\text{put}}(T) = (P(T,S) - K)^+ - (K - P(T,S))^+ = P(T,S) - K \]

This holds for all states. Discounting to time \(t\) under \(\mathbb{Q}\):

\[ C(t) - P_{\text{put}}(t) = \mathbb{E}^{\mathbb{Q}}_t\!\left[e^{-\int_t^T r_s\,ds}(P(T,S) - K)\right] \]

\[ = \mathbb{E}^{\mathbb{Q}}_t\!\left[e^{-\int_t^T r_s\,ds}\,P(T,S)\right] - K\,\mathbb{E}^{\mathbb{Q}}_t\!\left[e^{-\int_t^T r_s\,ds}\right] \]

The first term equals \(P(t,S)\) by the tower property (discounting from \(t\) to \(T\) and then from \(T\) to \(S\)). The second term equals \(K\,P(t,T)\). Therefore:

\[ C(t) - P_{\text{put}}(t) = P(t,S) - K\,P(t,T) \]

Application. For a 5-year ZCB, expiry \(T = 3\), strike \(K = 0.92\):

\[ P_{\text{put}}(t) = C(t) - P(t,S) + K\,P(t,T) = C(t) - P(t,5) + 0.92\,P(t,3) \]

Given the call price, one substitutes the known bond prices \(P(t,3)\) and \(P(t,5)\) from the Vasicek formula to obtain the put price directly, without re-evaluating the option formula.

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Exercise 5. In Jamshidian's decomposition, the individual strikes \(K_i = P(T, S_i; r^*)\) are all evaluated at the same critical rate \(r^*\). Explain why the swaption is in the money if and only if \(r_T < r^*\), and why all bond options in the decomposition share the same exercise region.

Solution to Exercise 5

The coupon bond value is \(CB(r_T) = \sum_{i=1}^n c_i\,P(T, S_i)\), which is strictly decreasing in \(r_T\) since each \(P(T, S_i)\) is strictly decreasing in \(r_T\) and all \(c_i > 0\).

The critical rate \(r^*\) is defined as the unique solution of \(CB(r^*) = K\). By monotonicity:

• When \(r_T < r^*\): \(CB(r_T) > CB(r^*) = K\), so the swaption is in the money.
• When \(r_T > r^*\): \(CB(r_T) < CB(r^*) = K\), so the swaption is out of the money.

The individual strikes are \(K_i = P(T, S_i)\big|_{r_T = r^*}\). Since each \(P(T, S_i)\) is strictly decreasing:

• When \(r_T < r^*\): \(P(T, S_i) > P(T, S_i)\big|_{r^*} = K_i\) for every \(i\).
• When \(r_T > r^*\): \(P(T, S_i) < K_i\) for every \(i\).

Therefore, the exercise region \(\{r_T < r^*\}\) is identical for the coupon bond option and for each individual ZCB option \((P(T,S_i) - K_i)^+\). All options exercise together or not at all. This shared exercise region is what makes the decomposition \((\sum c_i P - K)^+ = \sum c_i(P_i - K_i)^+\) valid: on the exercise region both sides are positive and equal \(CB(r_T) - K\), and outside the exercise region both sides are zero.

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Exercise 6. A 2-year European call on a 10-year zero-coupon bond has strike \(K = 0.70\). Vasicek parameters: \(\kappa = 0.2\), \(\theta = 0.06\), \(\sigma = 0.02\), \(r_0 = 0.05\). Compute \(\sigma_P\), \(d_1\), \(d_2\), and the call price using the Vasicek ZCB option formula.

Solution to Exercise 6

Parameters: \(\kappa = 0.2\), \(\theta = 0.06\), \(\sigma = 0.02\), \(r_0 = 0.05\), \(K = 0.70\), \(T = 2\), \(S = 10\).

Step 1: Compute bond prices. For \(P(0,2)\) with \(\tau = 2\):

\[ B(2) = \frac{1 - e^{-0.4}}{0.2} = \frac{1 - 0.6703}{0.2} = 1.6484 \]

\[ \ln A(2) = \left(0.06 - \frac{0.0004}{0.08}\right)(1.6484 - 2) - \frac{0.0004}{0.8} \times 1.6484^2 \]

\[ = (0.06 - 0.005)(-0.3516) - 0.0005 \times 2.7173 = -0.01934 - 0.001359 = -0.02069 \]

\[ P(0,2) = e^{-0.02069 - 1.6484 \times 0.05} = e^{-0.02069 - 0.08242} = e^{-0.10311} = 0.9020 \]

For \(P(0,10)\) with \(\tau = 10\):

\[ B(10) = \frac{1 - e^{-2.0}}{0.2} = \frac{1 - 0.1353}{0.2} = 4.3233 \]

\[ \ln A(10) = 0.055 \times (4.3233 - 10) - 0.0005 \times 18.691 = 0.055 \times (-5.6767) - 0.009346 \]

\[ = -0.31222 - 0.009346 = -0.32157 \]

\[ P(0,10) = e^{-0.32157 - 4.3233 \times 0.05} = e^{-0.32157 - 0.21617} = e^{-0.53773} = 0.5841 \]

Step 2: Bond option volatility.

\[ B(S - T) = B(8) = \frac{1 - e^{-1.6}}{0.2} = \frac{1 - 0.2019}{0.2} = 3.9906 \]

\[ v = \sigma\sqrt{\frac{1 - e^{-2\kappa T}}{2\kappa}} = 0.02\sqrt{\frac{1 - e^{-0.8}}{0.4}} = 0.02\sqrt{\frac{0.5507}{0.4}} = 0.02\sqrt{1.3767} = 0.02 \times 1.1733 = 0.02347 \]

\[ \sigma_P = B(8) \times v = 3.9906 \times 0.02347 = 0.09366 \]

Step 3: Compute \(d_1\) and \(d_2\).

\[ d_1 = \frac{\ln\!\left(\frac{P(0,10)}{K \cdot P(0,2)}\right)}{\sigma_P} + \frac{\sigma_P}{2} = \frac{\ln\!\left(\frac{0.5841}{0.70 \times 0.9020}\right)}{0.09366} + \frac{0.09366}{2} \]

\[ = \frac{\ln(0.9251)}{0.09366} + 0.04683 = \frac{-0.07789}{0.09366} + 0.04683 = -0.8316 + 0.04683 = -0.7848 \]

\[ d_2 = d_1 - \sigma_P = -0.7848 - 0.09366 = -0.8785 \]

Step 4: Call price.

\[ C(0) = P(0,10)\,\Phi(d_1) - K\,P(0,2)\,\Phi(d_2) \]

\[ = 0.5841 \times \Phi(-0.7848) - 0.70 \times 0.9020 \times \Phi(-0.8785) \]

\[ = 0.5841 \times 0.2163 - 0.6314 \times 0.1898 \]

\[ = 0.1263 - 0.1199 = 0.0064 \]

The call is slightly out of the money (the forward bond price is below the strike), resulting in a small option value.

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Exercise 7. Explain why Jamshidian's trick fails for multi-factor short-rate models. Specifically, in a two-factor model \(r_t = x_t + y_t\), show that \(P(T, S)\) depends on both \(x_T\) and \(y_T\), and the exercise region \(\{P(T,S) > K\}\) is a half-plane in \((x_T, y_T)\) rather than a half-line in \(r_T\). Why does this prevent decomposition into a portfolio of univariate options?

Solution to Exercise 7

In a two-factor model \(r_t = x_t + y_t\), where \(x_t\) and \(y_t\) are independent OU processes, the bond price at time \(T\) for maturity \(S\) takes the form:

\[ P(T,S) = A(S-T)\,e^{-B_x(S-T)\,x_T - B_y(S-T)\,y_T} \]

where \(B_x\) and \(B_y\) are the duration-like functions for each factor. The bond price depends on the two-dimensional state \((x_T, y_T)\), not on a single variable.

The exercise region for a call option on this bond is:

\[ \{P(T,S) > K\} = \{B_x(S-T)\,x_T + B_y(S-T)\,y_T < \ln A(S-T) - \ln K\} \]

This is a half-plane in \((x_T, y_T)\) space defined by a linear inequality. For a coupon bond with multiple cash flows at \(S_1, \ldots, S_n\), the coupon bond option exercises when:

\[ \sum_{i=1}^n c_i\,A(S_i - T)\,e^{-B_x(S_i - T)\,x_T - B_y(S_i - T)\,y_T} > K \]

The exercise boundary in \((x_T, y_T)\) is now a curve (not a straight line) because the exponential functions with different coefficients \(B_x(S_i - T)\) and \(B_y(S_i - T)\) combine nonlinearly.

Crucially, the individual ZCB options \((P(T, S_i) - K_i)^+\) each have exercise regions that are half-planes with different orientations (determined by the ratios \(B_x(S_i - T)/B_y(S_i - T)\), which vary with \(S_i\)). Since the half-planes differ for each bond, the exercise regions do not coincide, and the decomposition \((CB - K)^+ = \sum c_i(P_i - K_i)^+\) fails. There is no single critical rate \(r^*\) that simultaneously determines the exercise of all individual options.

This is why multi-factor bond option pricing requires alternative methods such as Monte Carlo simulation, Fourier transforms, or numerical PDE solutions in higher dimensions.