Caplet and Swaption Formulas Under CIR

Interest rate caps and swaptions are the two most actively traded interest rate options, and any short-rate model used in practice must be able to price them. The CIR model achieves this through two key observations: first, a caplet is equivalent to a put option on a zero-coupon bond; second, Jamshidian's decomposition reduces a swaption to a portfolio of bond options. Since the CIR model provides closed-form bond option formulas via the non-central chi-squared distribution, both caps and swaptions can be priced analytically. This section develops these connections in detail.

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Caps and caplets

Cap structure

An interest rate cap protects the holder against rising rates. It consists of a series of caplets, each covering one reset period. A cap with notional \(N\), strike rate \(K\), and reset dates \(T_0 < T_1 < \cdots < T_n\) pays at each date \(T_{i+1}\):

\[ N\,\delta_i\,(L(T_i, T_{i+1}) - K)^+ \]

where \(\delta_i = T_{i+1} - T_i\) is the day count fraction and \(L(T_i, T_{i+1})\) is the simply compounded spot rate over \([T_i, T_{i+1}]\).

The cap price is the sum of individual caplet prices:

\[ \text{Cap}(t) = \sum_{i=0}^{n-1} \text{Caplet}_i(t) \]

Caplet as a bond put

The simply compounded rate \(L(T_i, T_{i+1})\) is related to bond prices by

\[ L(T_i, T_{i+1}) = \frac{1}{\delta_i}\left(\frac{1}{P(T_i, T_{i+1})} - 1\right) \]

Substituting into the caplet payoff at \(T_{i+1}\):

\[ N\,\delta_i\,(L(T_i, T_{i+1}) - K)^+ = N\left(1 - (1 + K\delta_i)P(T_i, T_{i+1})\right)^+ \]

\[ = N(1 + K\delta_i)\left(\frac{1}{1 + K\delta_i} - P(T_i, T_{i+1})\right)^+ \]

This is \((1 + K\delta_i)\) units of a put option on a zero-coupon bond maturing at \(T_{i+1}\), with exercise date \(T_i\) and strike \(\tilde{K}_i = \frac{1}{1 + K\delta_i}\).

Caplet-put equivalence

The caplet paid at \(T_{i+1}\) with rate strike \(K\) is equivalent to \(N(1 + K\delta_i)\) put options on the \(T_{i+1}\)-bond with exercise at \(T_i\) and bond strike \(\tilde{K}_i = 1/(1 + K\delta_i)\). This equivalence holds model-independently --- it is a payoff identity, not an approximation.

Caplet pricing under CIR

Discounting the caplet payoff to time \(t\):

\[ \text{Caplet}_i(t) = N(1 + K\delta_i)\,\text{Put}_{\text{ZCB}}(t;\,T_i,\,T_{i+1},\,\tilde{K}_i) \]

where \(\text{Put}_{\text{ZCB}}\) is the CIR zero-coupon bond put formula from the bond options section:

\[ \text{Put}_{\text{ZCB}}(t;\,T_i,\,T_{i+1},\,\tilde{K}_i) = \tilde{K}_i\,P(t,T_i)\left[1 - \chi^2(x_2;\,d,\,\lambda_2)\right] - P(t,T_{i+1})\left[1 - \chi^2(x_1;\,d,\,\lambda_1)\right] \]

with parameters computed using option expiry \(T_i\), bond maturity \(T_{i+1}\), and strike \(\tilde{K}_i\).

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Floors and floorlets

An interest rate floor protects against falling rates. A floorlet pays \(N\delta_i(K - L(T_i, T_{i+1}))^+\) at \(T_{i+1}\). By the same argument as above, this is equivalent to \(N(1 + K\delta_i)\) call options on the \(T_{i+1}\)-bond:

\[ \text{Floorlet}_i(t) = N(1 + K\delta_i)\,\text{Call}_{\text{ZCB}}(t;\,T_i,\,T_{i+1},\,\tilde{K}_i) \]

Cap-floor parity follows from call-put parity for bond options:

\[ \text{Cap}(t) - \text{Floor}(t) = \text{Swap}(t) \]

where \(\text{Swap}(t)\) is the value of the corresponding payer interest rate swap.

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Swaptions

Swaption structure

A payer swaption with expiry \(T_0\) gives the holder the right to enter a payer swap (paying fixed, receiving floating) with fixed rate \(K\) on dates \(T_1, \ldots, T_n\). The payoff at \(T_0\) is

\[ \left(\sum_{i=1}^{n} \delta_{i-1}\,(L(T_{i-1}, T_i) - K)\,P(T_0, T_i)\right)^+ \]

which simplifies to

\[ \left(1 - P(T_0, T_n) - K\sum_{i=1}^{n}\delta_{i-1}\,P(T_0, T_i)\right)^+ \]

Writing \(c_i = K\delta_{i-1}\) for \(i = 1, \ldots, n-1\) and \(c_n = 1 + K\delta_{n-1}\), this is

\[ \left(1 - \sum_{i=1}^{n} c_i\,P(T_0, T_i)\right)^+ \]

The swaption is exercised when the swap has positive value, which occurs when the coupon bond \(\sum c_i P(T_0, T_i)\) is worth less than par.

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Jamshidian's decomposition

Jamshidian (1989) observed that in any one-factor model where bond prices are monotone decreasing functions of the short rate, a European option on a coupon bond can be decomposed into a portfolio of options on zero-coupon bonds.

The key insight

Since each \(P(T_0, T_i) = A_i\,e^{-B_i\,r_{T_0}}\) is a decreasing function of \(r_{T_0}\) (where \(A_i = A(T_i - T_0)\) and \(B_i = B(T_i - T_0)\)), the coupon bond value

\[ V(r) = \sum_{i=1}^{n} c_i\,A_i\,e^{-B_i\,r} \]

is also a decreasing function of \(r\). Therefore, there exists a unique critical rate \(r^*\) satisfying

\[ \sum_{i=1}^{n} c_i\,P(T_0, T_i)\big|_{r_{T_0} = r^*} = 1 \]

equivalently \(V(r^*) = 1\).

Decomposition of the swaption

At the critical rate \(r^*\), define the individual bond strikes:

\[ K_i = P(T_0, T_i)\big|_{r_{T_0} = r^*} = A_i\,e^{-B_i\,r^*} \]

By construction, \(\sum_{i=1}^n c_i K_i = 1\). Since each \(P(T_0, T_i)\) is monotone decreasing in \(r_{T_0}\):

• When \(r_{T_0} < r^*\): all \(P(T_0, T_i) > K_i\) and the swaption is in the money
• When \(r_{T_0} > r^*\): all \(P(T_0, T_i) < K_i\) and the swaption is out of the money

The swaption payoff therefore decomposes as

\[ \left(1 - \sum_{i=1}^{n} c_i\,P(T_0, T_i)\right)^+ = \sum_{i=1}^{n} c_i\left(K_i - P(T_0, T_i)\right)^+ \]

This is a portfolio of \(n\) put options on zero-coupon bonds.

Proof of the decomposition

For any \(r_{T_0}\), define \(g(r) = 1 - \sum c_i P_i(r)\) and \(h(r) = \sum c_i (K_i - P_i(r))^+\). When \(r < r^*\): \(g(r) < 0\) (so \(g(r)^+ = 0\)) and each \(K_i - P_i(r) < 0\) (so \(h(r) = 0\)). When \(r > r^*\): \(g(r) > 0\) and each \(K_i - P_i(r) > 0\), so

\[ h(r) = \sum c_i(K_i - P_i(r)) = \sum c_i K_i - \sum c_i P_i(r) = 1 - V(r) = g(r) \]

Therefore \(g(r)^+ = h(r)\) for all \(r\). \(\square\)

Finding the critical rate

The critical rate \(r^*\) is the solution of

\[ \sum_{i=1}^{n} c_i\,A_i\,e^{-B_i\,r^*} = 1 \]

This is a one-dimensional root-finding problem that can be solved efficiently by Newton's method or bisection. The function on the left is strictly decreasing and continuous, guaranteeing existence and uniqueness of \(r^*\).

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Swaption pricing under CIR

Combining Jamshidian's decomposition with the CIR bond put formula:

\[ \text{Payer Swaption}(t) = \sum_{i=1}^{n} c_i\,\text{Put}_{\text{ZCB}}(t;\,T_0,\,T_i,\,K_i) \]

Each put \(\text{Put}_{\text{ZCB}}(t;\,T_0,\,T_i,\,K_i)\) is evaluated using the CIR closed-form formula with:

• Option expiry: \(T_0\)
• Bond maturity: \(T_i\)
• Bond strike: \(K_i = A(T_i - T_0)\,e^{-B(T_i - T_0)\,r^*}\)

The receiver swaption (right to enter a receiver swap) decomposes into a portfolio of call options:

\[ \text{Receiver Swaption}(t) = \sum_{i=1}^{n} c_i\,\text{Call}_{\text{ZCB}}(t;\,T_0,\,T_i,\,K_i) \]

Swaption parity:

\[ \text{Payer}(t) - \text{Receiver}(t) = \text{Swap}(t) \]

Jamshidian works only for one-factor models

The decomposition relies on the monotonicity of all bond prices in a single state variable \(r_{T_0}\). In multi-factor models, different bonds can move in different directions for the same state change, and the decomposition fails. Multi-factor swaption pricing requires different techniques (e.g., Monte Carlo or Bermudan swaption methods).

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Summary of pricing chain

The following diagram summarizes how CIR prices caps and swaptions.

Derivative	Decomposition	Building Block
Caplet	\(\longrightarrow\) ZCB put	CIR bond option formula (\(\chi^2\))
Cap	\(= \sum\) caplets	CIR bond option formula (\(\chi^2\))
Floorlet	\(\longrightarrow\) ZCB call	CIR bond option formula (\(\chi^2\))
Floor	\(= \sum\) floorlets	CIR bond option formula (\(\chi^2\))
Payer swaption	Jamshidian \(\longrightarrow \sum\) ZCB puts	CIR bond option formula (\(\chi^2\))
Receiver swaption	Jamshidian \(\longrightarrow \sum\) ZCB calls	CIR bond option formula (\(\chi^2\))

Every derivative in this table ultimately reduces to the CIR zero-coupon bond option formula involving the non-central chi-squared distribution.

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Summary

Caplets are equivalent to put options on zero-coupon bonds through a model-independent payoff identity, with bond strike \(\tilde{K} = 1/(1+K\delta)\). Swaptions decompose into portfolios of zero-coupon bond options via Jamshidian's trick, which exploits the monotonicity of bond prices in the short rate. Under the CIR model, all building-block bond options have closed-form prices in terms of the non-central chi-squared distribution, making caps and swaptions analytically tractable. This analytical pricing chain --- from swaption to bond option portfolio to chi-squared CDF evaluation --- is a major advantage of the CIR model over non-affine alternatives that require numerical methods at every stage.

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Exercises

Exercise 1. A caplet has strike \(K = 4\%\), reset date \(T_i = 2\) years, payment date \(T_{i+1} = 2.5\) years, and notional \(N = \$1{,}000{,}000\). Compute the day count fraction \(\delta_i\) and the equivalent bond put strike \(\tilde{K}_i = 1/(1 + K\delta_i)\). How many units of the bond put does the caplet correspond to?

Solution to Exercise 1

The day count fraction is:

\[ \delta_i = T_{i+1} - T_i = 2.5 - 2 = 0.5 \text{ years} \]

The equivalent bond put strike is:

\[ \tilde{K}_i = \frac{1}{1 + K\delta_i} = \frac{1}{1 + 0.04 \times 0.5} = \frac{1}{1.02} \approx 0.98039 \]

The number of units of the bond put is:

\[ N(1 + K\delta_i) = 1{,}000{,}000 \times 1.02 = 1{,}020{,}000 \]

So the caplet is equivalent to 1,020,000 put options on the zero-coupon bond maturing at \(T_{i+1} = 2.5\) years, with exercise date \(T_i = 2\) years and bond strike \(\tilde{K}_i \approx 0.98039\).

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Exercise 2. Derive the caplet-put equivalence step by step. Starting from the caplet payoff \(N\delta_i(L(T_i, T_{i+1}) - K)^+\), substitute \(L(T_i, T_{i+1}) = \frac{1}{\delta_i}(1/P(T_i, T_{i+1}) - 1)\) and algebraically manipulate to arrive at \(N(1 + K\delta_i)(\tilde{K}_i - P(T_i, T_{i+1}))^+\). Identify which step uses the fact that \((ax - b)^+ = a(x - b/a)^+\) for \(a > 0\).

Solution to Exercise 2

Starting from the caplet payoff at \(T_{i+1}\):

\[ N\delta_i(L(T_i, T_{i+1}) - K)^+ \]

Step 1: Substitute \(L(T_i, T_{i+1}) = \frac{1}{\delta_i}\left(\frac{1}{P(T_i, T_{i+1})} - 1\right)\):

\[ N\delta_i\left(\frac{1}{\delta_i}\left(\frac{1}{P(T_i, T_{i+1})} - 1\right) - K\right)^+ \]

\[ = N\left(\frac{1}{P(T_i, T_{i+1})} - 1 - K\delta_i\right)^+ \]

Step 2: Combine the constant terms:

\[ = N\left(\frac{1}{P(T_i, T_{i+1})} - (1 + K\delta_i)\right)^+ \]

Step 3: Factor out \(\frac{1}{P(T_i, T_{i+1})}\) from inside the positive part. Write:

\[ = N\left(\frac{1 - (1+K\delta_i)P(T_i, T_{i+1})}{P(T_i, T_{i+1})}\right)^+ \]

Since \(P(T_i, T_{i+1}) > 0\), this equals:

\[ = \frac{N}{P(T_i, T_{i+1})}\left(1 - (1 + K\delta_i)P(T_i, T_{i+1})\right)^+ \]

Step 4: This is the payoff at \(T_{i+1}\). To express as a bond put payoff at \(T_i\), note that the payoff \(\frac{N}{P(T_i, T_{i+1})}(\cdots)^+\) paid at \(T_{i+1}\) has the same present value at \(T_i\) as \(N(\cdots)^+\) paid at \(T_i\) (by discounting). Alternatively, work directly:

\[ N\left(1 - (1 + K\delta_i)P(T_i, T_{i+1})\right)^+ \]

Step 5: This is where we use the identity \((ax - b)^+ = a(x - b/a)^+\) for \(a > 0\). Set \(a = (1+K\delta_i)\):

\[ = N(1 + K\delta_i)\left(\frac{1}{1 + K\delta_i} - P(T_i, T_{i+1})\right)^+ \]

\[ = N(1 + K\delta_i)\left(\tilde{K}_i - P(T_i, T_{i+1})\right)^+ \]

This is \(N(1+K\delta_i)\) units of a put option on the \(T_{i+1}\)-bond with strike \(\tilde{K}_i\), payable at \(T_{i+1}\).

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Exercise 3. Explain why cap-floor parity \(\text{Cap}(t) - \text{Floor}(t) = \text{Swap}(t)\) holds model-independently. Start from the identity \((x - K)^+ - (K - x)^+ = x - K\) applied to \(x = L(T_i, T_{i+1})\) for each caplet-floorlet pair, and show that the resulting sum equals the swap value.

Solution to Exercise 3

For each caplet-floorlet pair on the interval \([T_i, T_{i+1}]\), the caplet pays \(N\delta_i(L_i - K)^+\) and the floorlet pays \(N\delta_i(K - L_i)^+\), where \(L_i = L(T_i, T_{i+1})\).

Using the identity \((x - K)^+ - (K - x)^+ = x - K\) for any \(x, K\):

\[ N\delta_i(L_i - K)^+ - N\delta_i(K - L_i)^+ = N\delta_i(L_i - K) \]

This holds for every realization of \(L_i\), so it is a model-independent payoff identity.

Summing over all periods and discounting to time \(t\):

\[ \text{Cap}(t) - \text{Floor}(t) = \sum_{i=0}^{n-1} \mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^{T_{i+1}} r_s\,ds}\,N\delta_i(L_i - K)\,\bigg|\,\mathcal{F}_t\right] \]

The right-hand side is exactly the value of a payer swap (receiving floating \(L_i\), paying fixed \(K\)), since each floating-minus-fixed cash flow \(N\delta_i(L_i - K)\) is paid at \(T_{i+1}\):

\[ \text{Cap}(t) - \text{Floor}(t) = \text{Swap}(t) \]

This identity is model-independent because it relies only on the payoff decomposition \((x - K)^+ - (K - x)^+ = x - K\), not on any distributional assumption about \(L_i\).

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Exercise 4. For a payer swaption with expiry \(T_0 = 1\) year into a 3-year annual swap with fixed rate \(K = 5\%\), write out the coupon bond that appears in the swaption payoff. Identify the cash flows \(c_1\), \(c_2\), \(c_3\) and the corresponding maturities \(T_1\), \(T_2\), \(T_3\). Explain why the swaption is exercised when this coupon bond is worth less than par.

Solution to Exercise 4

The payer swaption expiry is \(T_0 = 1\) year. The swap is 3-year annual, so payment dates are \(T_1 = 2\), \(T_2 = 3\), \(T_3 = 4\) years.

The cash flows of the fixed leg are:

• \(c_1 = K\delta_0 = 0.05 \times 1 = 0.05\) at \(T_1 = 2\)
• \(c_2 = K\delta_1 = 0.05 \times 1 = 0.05\) at \(T_2 = 3\)
• \(c_3 = 1 + K\delta_2 = 1 + 0.05 \times 1 = 1.05\) at \(T_3 = 4\)

The coupon bond appearing in the swaption payoff is:

\[ V(r_{T_0}) = c_1 P(T_0, T_1) + c_2 P(T_0, T_2) + c_3 P(T_0, T_3) = 0.05\,P(1,2) + 0.05\,P(1,3) + 1.05\,P(1,4) \]

The swaption payoff at \(T_0\) is \((1 - V(r_{T_0}))^+\).

The swaption is exercised when \(V(r_{T_0}) < 1\), meaning the coupon bond is worth less than par. This occurs when the swap has positive value to the payer: paying fixed at \(K = 5\%\) is advantageous when market rates are high enough that the present value of the fixed payments is less than par. Higher rates at \(T_0\) reduce bond prices \(P(T_0, T_i)\), reducing \(V\), making exercise more likely.

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Exercise 5. In Jamshidian's decomposition, the critical rate \(r^*\) solves \(\sum_{i=1}^n c_i A_i e^{-B_i r^*} = 1\). For a 2-year annual swap with \(K = 5\%\) (so \(c_1 = 0.05\) and \(c_2 = 1.05\)), and CIR functions \(A_1 = 0.99\), \(B_1 = 0.95\), \(A_2 = 0.97\), \(B_2 = 1.85\), set up the equation for \(r^*\) and describe how Newton's method would be applied. What is the derivative of the left-hand side with respect to \(r^*\)?

Solution to Exercise 5

The equation for \(r^*\) is:

\[ c_1 A_1 e^{-B_1 r^*} + c_2 A_2 e^{-B_2 r^*} = 1 \]

\[ 0.05 \times 0.99 \times e^{-0.95\,r^*} + 1.05 \times 0.97 \times e^{-1.85\,r^*} = 1 \]

\[ 0.0495\,e^{-0.95\,r^*} + 1.0185\,e^{-1.85\,r^*} = 1 \]

Define \(g(r) = 0.0495\,e^{-0.95 r} + 1.0185\,e^{-1.85 r}\).

Newton's method: Starting from an initial guess \(r^{(0)}\), iterate:

\[ r^{(n+1)} = r^{(n)} - \frac{g(r^{(n)}) - 1}{g'(r^{(n)})} \]

Derivative of the left-hand side:

\[ g'(r^*) = -c_1 B_1 A_1 e^{-B_1 r^*} - c_2 B_2 A_2 e^{-B_2 r^*} \]

\[ = -0.05 \times 0.95 \times 0.99 \times e^{-0.95\,r^*} - 1.05 \times 1.85 \times 0.97 \times e^{-1.85\,r^*} \]

\[ = -0.047025\,e^{-0.95\,r^*} - 1.88423\,e^{-1.85\,r^*} \]

The derivative is strictly negative (the function is strictly decreasing in \(r^*\)), which guarantees existence and uniqueness of the root and ensures Newton's method converges rapidly.

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Exercise 6. After finding \(r^*\) in Jamshidian's decomposition, the individual bond strikes are \(K_i = A_i e^{-B_i r^*}\). Verify that \(\sum c_i K_i = 1\) by construction. Then prove the key identity: when \(r_{T_0} > r^*\), we have \(P(T_0, T_i) < K_i\) for all \(i\) simultaneously, and when \(r_{T_0} < r^*\), we have \(P(T_0, T_i) > K_i\) for all \(i\). Why does this simultaneous crossing property depend on the model being one-factor?

Solution to Exercise 6

Verification that \(\sum c_i K_i = 1\): By construction, \(K_i = A_i e^{-B_i r^*}\), and \(r^*\) is defined as the solution to \(\sum c_i A_i e^{-B_i r^*} = 1\). Therefore \(\sum c_i K_i = \sum c_i A_i e^{-B_i r^*} = 1\).

Simultaneous crossing property: Each bond price \(P(T_0, T_i) = A_i e^{-B_i r_{T_0}}\) and its strike \(K_i = A_i e^{-B_i r^*}\).

When \(r_{T_0} > r^*\): Since \(B_i > 0\) for all \(i\), \(-B_i r_{T_0} < -B_i r^*\), so \(e^{-B_i r_{T_0}} < e^{-B_i r^*}\), which gives \(P(T_0, T_i) = A_i e^{-B_i r_{T_0}} < A_i e^{-B_i r^*} = K_i\) for all \(i\) simultaneously.

When \(r_{T_0} < r^*\): By the same logic, \(e^{-B_i r_{T_0}} > e^{-B_i r^*}\), so \(P(T_0, T_i) > K_i\) for all \(i\) simultaneously.

Why this depends on the model being one-factor: The simultaneous crossing property requires that all bond prices are monotone functions of the same state variable \(r_{T_0}\). In a one-factor model, each \(P(T_0, T_i)\) is determined by the single short rate \(r_{T_0}\), and since all \(B_i > 0\), they all decrease in \(r_{T_0}\) simultaneously.

In a multi-factor model, bond prices depend on multiple state variables (e.g., \(r_{T_0}\) and a slope factor \(x_{T_0}\)). Different bonds have different sensitivities to each factor, so it is possible for \(P(T_0, T_1)\) to increase while \(P(T_0, T_2)\) decreases (if the factors move in offsetting directions). This breaks the simultaneous crossing property, and Jamshidian's decomposition fails.

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Exercise 7. Consider pricing a 1Y-into-5Y payer swaption under CIR with semi-annual payment dates. How many zero-coupon bond put options appear in the Jamshidian decomposition? If each CIR bond put evaluation requires two non-central chi-squared CDF evaluations, how many \(\chi^2\) CDF calls are needed in total? Compare this computational cost to a non-affine model (e.g., Black-Karasinski) where each bond option requires a full tree backward induction.

Solution to Exercise 7

A 1Y-into-5Y payer swaption with semi-annual payments has:

• Expiry: \(T_0 = 1\) year
• Swap payment dates: \(T_1 = 1.5\), \(T_2 = 2.0\), \(T_3 = 2.5\), \(T_4 = 3.0\), \(T_5 = 3.5\), \(T_6 = 4.0\), \(T_7 = 4.5\), \(T_8 = 5.0\), \(T_9 = 5.5\), \(T_{10} = 6.0\)

So there are \(n = 10\) payment dates, yielding 10 zero-coupon bond put options in the Jamshidian decomposition.

Each CIR bond put uses the put formula, which involves two non-central chi-squared CDF evaluations (one for each of the \(\chi^2(x_1; d, \lambda_1)\) and \(\chi^2(x_2; d, \lambda_2)\) terms). Therefore:

\[ \text{Total } \chi^2 \text{ CDF calls} = 10 \times 2 = 20 \]

Plus one root-finding step to determine \(r^*\) (typically 5--10 function evaluations, each requiring 10 exponential evaluations).

Comparison with non-affine models: In a Black-Karasinski model (log-normal short rate), there is no closed-form bond option formula. Each of the 10 bond put options would require a full backward induction on a trinomial tree. If the tree has \(N\) time steps and \(O(N)\) nodes at each step, each bond option pricing requires \(O(N^2)\) operations. For \(N = 500\) (typical for accurate pricing), each option costs \(O(250{,}000)\) operations, and the total is \(10 \times 250{,}000 = 2{,}500{,}000\) operations.

By contrast, the CIR approach requires only 20 chi-squared CDF evaluations (each costing \(O(100)\) operations for the series evaluation), totaling \(O(2{,}000)\) operations. The CIR model is thus roughly three orders of magnitude faster for swaption pricing, which is a major practical advantage when calibrating to a large panel of swaption quotes.