Caplet and Swaption Formulas¶
Caps, floors, and swaptions are the primary volatility instruments in interest rate markets. A cap is a strip of caplets, each protecting against a floating rate exceeding a strike, while a swaption is an option to enter an interest rate swap. In the Vasicek model, both reduce to portfolios of zero-coupon bond options, which have closed-form solutions. This section derives the caplet formula by showing it is equivalent to a put on a ZCB, extends the result to caps and floors, and applies Jamshidian's trick to obtain swaption prices.
Caplet as a bond option¶
Caplet payoff¶
A caplet on the simply compounded rate \(L(T_{k-1}, T_k)\) over the accrual period \([T_{k-1}, T_k]\) with day count fraction \(\delta_k = T_k - T_{k-1}\) and strike rate \(\kappa_{\text{cap}}\) pays at time \(T_k\):
Reduction to a ZCB put¶
The simply compounded rate satisfies \(L(T_{k-1}, T_k) = (1/P(T_{k-1}, T_k) - 1)/\delta_k\), so
Multiplying and dividing:
Define the adjusted strike \(K_k = \frac{1}{1 + \delta_k\,\kappa_{\text{cap}}}\). Then the caplet payoff at \(T_k\) is
This is \((1 + \delta_k\,\kappa_{\text{cap}})\) units of a European put on the ZCB \(P(T_{k-1}, T_k)\) with strike \(K_k\) and expiry \(T_{k-1}\), but with the payoff received at \(T_k\).
Discounting the \(T_k\)-payoff to \(T_{k-1}\), the value at time \(t\) is
where \(\text{Put}_{\text{ZCB}}(t; K, T, S) = K\,P(t,T)\,\Phi(-d_2) - P(t,S)\,\Phi(-d_1)\) is the Vasicek ZCB put formula.
Cap and floor formulas¶
Cap¶
A cap with strike \(\kappa_{\text{cap}}\) on reset dates \(T_0, T_1, \ldots, T_{n-1}\) and payment dates \(T_1, \ldots, T_n\) is a strip of caplets:
Floor¶
By analogous reasoning, a floorlet (paying \(\delta_k(\kappa_{\text{floor}} - L(T_{k-1}, T_k))^+\) at \(T_k\)) is equivalent to a ZCB call:
A floor is the sum of floorlets:
Cap-floor parity¶
Cap minus floor equals a payer swap when both use the same strike:
This identity serves as a consistency check for implementations.
Forward rate volatility¶
The volatility parameter entering the caplet formula is the bond option volatility
For short accrual periods \(\delta_k \ll 1\), \(B(\delta_k) \approx \delta_k\), and the bond option volatility becomes
The implied forward rate volatility (the volatility of \(L(T_{k-1}, T_k)\) under the \(T_k\)-forward measure) can be extracted by matching the Vasicek caplet price to the Black formula:
where \(F_k(t) = (P(t, T_{k-1})/P(t, T_k) - 1)/\delta_k\) is the forward rate. In the Vasicek model, this implied volatility is approximately
Vasicek produces a flat volatility smile
Since the Vasicek short rate is Gaussian, forward rates are approximately Gaussian (not log-normal), and the implied Black volatility from the Vasicek formula is approximately strike-independent. The model therefore produces a flat implied volatility smile for caplets, which is inconsistent with the observed cap volatility skew. Stochastic volatility or local volatility extensions are needed to capture smile effects.
Swaption pricing via Jamshidian¶
Swaption payoff¶
A payer swaption with expiry \(T_0\) gives the holder the right to enter a payer swap with fixed rate \(K\) on a swap with payment dates \(T_1, \ldots, T_n\). The payoff at \(T_0\) is
An equivalent representation uses the coupon bond decomposition. The payer swaption payoff is
which can be rewritten as
where \(c_k = K\,\delta_k\) for \(k = 1, \ldots, n-1\) and \(c_n = 1 + K\,\delta_n\). This is a put on a coupon bond with cash flows \(c_k\) at times \(T_k\) and strike \(1\).
Application of Jamshidian's trick¶
Since the swaption payoff is a put on a coupon bond, Jamshidian's trick decomposes it into a portfolio of ZCB puts.
Step 1: Find the critical rate \(r^*\) solving
Step 2: Compute individual strikes \(K_k = P(T_0, T_k)\big|_{r_{T_0} = r^*}\).
Step 3: The payer swaption price is
A receiver swaption (the right to receive fixed) is a call on the same coupon bond:
Swaption parity¶
Numerical example¶
Caplet pricing. Consider a caplet on the 6-month rate, resetting at \(T_0 = 1\) year, paying at \(T_1 = 1.5\) years, with strike \(\kappa_{\text{cap}} = 5\%\) and \(\delta = 0.5\). Vasicek parameters: \(\kappa = 0.3\), \(\theta = 0.05\), \(\sigma = 0.015\), \(r_0 = 0.04\).
- Adjusted strike: \(K = 1/(1 + 0.5 \times 0.05) = 0.97561\)
- Bond prices: \(P(0, 1) = 0.9617\), \(P(0, 1.5) = 0.9411\)
- Bond option volatility: \(\sigma_P = B(0.5) \cdot 0.015 \cdot \sqrt{(1 - e^{-0.6})/(0.6)} = 0.4758 \times 0.015 \times 0.8511 = 0.00607\)
- Caplet = \(1.025 \times \text{Put}(K = 0.97561, T = 1, S = 1.5)\)
Swaption pricing. A 2-year expiry payer swaption on a 3-year annual swap at strike 5%. Cash flows: \(c_1 = 0.05\), \(c_2 = 0.05\), \(c_3 = 1.05\) at years 3, 4, 5.
- Find \(r^*\) solving \(0.05\,P(2,3;r^*) + 0.05\,P(2,4;r^*) + 1.05\,P(2,5;r^*) = 1\)
- Compute \(K_1, K_2, K_3\) at \(r^*\)
- Sum three ZCB put prices
Summary¶
In the Vasicek framework, caplets reduce to puts on zero-coupon bonds via the identity \(\delta(L - \kappa_{\text{cap}})^+ = (1 + \delta\kappa_{\text{cap}})(K - P)^+\), and swaptions reduce to portfolios of ZCB puts (or calls) via Jamshidian's trick. All pricing is therefore reduced to the single ZCB option formula derived from the log-normality of forward bond prices under the \(T\)-forward measure. The model produces a flat implied volatility smile, which is its main limitation for calibrating to market cap and swaption volatilities.
Exercises¶
Exercise 1. Derive the caplet-put equivalence step by step. Starting from the caplet payoff \(N\delta(L(T_0, T_1) - K_{\text{cap}})^+\), substitute \(L = (1/P - 1)/\delta\) and show the result is \(N(1 + K_{\text{cap}}\delta)(\tilde{K} - P(T_0, T_1))^+\) with \(\tilde{K} = 1/(1 + K_{\text{cap}}\delta)\).
Solution to Exercise 1
Start with the caplet payoff at \(T_1\) (setting notional \(N = 1\) for simplicity):
Substitute the definition of the simply compounded rate \(L(T_0, T_1) = \frac{1}{\delta}\left(\frac{1}{P(T_0, T_1)} - 1\right)\):
Factor out \(1/P(T_0, T_1)\) inside the max:
Since \(P(T_0, T_1) > 0\), we can write this as:
However, it is more useful to multiply and divide by \((1 + \delta K_{\text{cap}})\):
Wait---let us redo this more carefully. From the expression \(\frac{1}{P} - 1 - \delta K_{\text{cap}}\), multiply through by \(P > 0\):
Now factor out \((1 + \delta K_{\text{cap}})\) from the numerator:
Define \(\tilde{K} = \frac{1}{1 + \delta K_{\text{cap}}}\). The payoff at \(T_1\) is:
Discounting from \(T_1\) to \(T_0\) by multiplying by \(P(T_0, T_1)\), the value at \(T_0\) of the \(T_1\)-payoff is:
This is \((1 + \delta K_{\text{cap}})\) units of a European put on \(P(T_0, T_1)\) with strike \(\tilde{K}\) and expiry \(T_0\), as claimed.
Exercise 2. Using the numerical example parameters (\(\kappa = 0.3\), \(\theta = 0.05\), \(\sigma = 0.015\), \(r_0 = 0.04\)), compute the caplet price for a 6-month caplet with reset \(T_0 = 1\), payment \(T_1 = 1.5\), and strike 5%. Follow all intermediate steps: adjusted strike, bond prices, bond option volatility, and put price.
Solution to Exercise 2
Parameters: \(\kappa = 0.3\), \(\theta = 0.05\), \(\sigma = 0.015\), \(r_0 = 0.04\), \(T_0 = 1\), \(T_1 = 1.5\), \(\delta = 0.5\), \(K_{\text{cap}} = 0.05\).
Adjusted strike:
Bond prices. For \(P(0,1)\) with \(\tau = 1\):
For \(P(0,1.5)\) with \(\tau = 1.5\):
Bond option volatility. With \(\delta = T_1 - T_0 = 0.5\):
Put price. \(\text{Put}(K, T_0, T_1) = K\,P(0,T_0)\,\Phi(-d_2) - P(0,T_1)\,\Phi(-d_1)\) where:
Caplet price:
The caplet price is approximately 10.9 basis points of notional.
Exercise 3. Explain why the Vasicek model produces a flat implied volatility smile for caplets. What property of the Gaussian distribution causes this? How does this compare with the Black-Karasinski model?
Solution to Exercise 3
The Vasicek model produces a flat implied volatility smile for caplets because the underlying forward rate \(L(T_0, T_1)\) is approximately Gaussian (not log-normal) under the forward measure.
In the Vasicek model, \(r_T\) is normally distributed under \(\mathbb{Q}^T\). The forward LIBOR rate \(L = (1/P(T_0,T_1) - 1)/\delta\) is a function of \(r_{T_0}\), and since \(P(T_0,T_1) = A(\delta)\,e^{-B(\delta)\,r_{T_0}}\) with \(r_{T_0}\) Gaussian, the rate \(L\) is approximately linear in \(r_{T_0}\) for small \(\delta\), hence approximately Gaussian. When a Gaussian variable is priced using the Black (log-normal) formula, the resulting implied volatility is approximately constant across strikes---the smile is flat.
More precisely, the Black formula assumes log-normality of the forward rate, which generates a symmetric smile in log-moneyness. Since the Vasicek forward rate is (approximately) normal rather than log-normal, its distribution is symmetric in the rate itself, producing no skew or smile when mapped through the Black formula.
The Black-Karasinski model \(d\ln r_t = \kappa(\ln\theta - \ln r_t)\,dt + \sigma\,dW_t\) models the log of the short rate as an OU process, making \(r_t\) log-normally distributed. This generates a non-trivial implied volatility smile because the forward rate distribution under the forward measure is no longer Gaussian. The log-normal specification introduces positive skewness (rates cannot be negative and have a fat right tail), which translates into an upward-sloping volatility skew for out-of-the-money caplets.
Exercise 4. For a 2Y-into-3Y payer swaption at strike 5%, write out the cash flow structure (\(c_1\), \(c_2\), \(c_3\)) and the Jamshidian decomposition. How many ZCB put options appear? What equation determines \(r^*\)?
Solution to Exercise 4
Cash flow structure. A 2Y-into-3Y payer swaption gives the right at \(T_0 = 2\) to enter a 3-year annual payer swap at strike \(K = 5\%\). The swap has payment dates \(T_1 = 3\), \(T_2 = 4\), \(T_3 = 5\) with \(\delta_k = 1\) for all \(k\).
The swaption payoff at \(T_0 = 2\) can be written as a put on a coupon bond:
where the cash flows are:
- \(c_1 = K\delta_1 = 0.05\) at \(T_1 = 3\)
- \(c_2 = K\delta_2 = 0.05\) at \(T_2 = 4\)
- \(c_3 = 1 + K\delta_3 = 1.05\) at \(T_3 = 5\)
Jamshidian decomposition. Three ZCB put options appear, one for each payment date.
Step 1: Find \(r^*\) solving:
This is a one-dimensional root-finding problem (e.g., bisection or Brent's method).
Step 2: Compute strikes \(K_k = P(2, T_k;r^*)\) for \(k = 1, 2, 3\).
Step 3: The payer swaption price is:
where each put is priced using the Vasicek ZCB put formula with bond option volatility \(\sigma_{P,k} = B(T_k - 2)\,\sigma\sqrt{(1 - e^{-2\kappa \cdot 2})/(2\kappa)}\).
Exercise 5. Prove cap-floor parity: \(\text{Cap}(t) - \text{Floor}(t) = \text{Swap}(t)\). Use the identity \((x-K)^+ - (K-x)^+ = x - K\) applied to each caplet-floorlet pair.
Solution to Exercise 5
For each caplet-floorlet pair at payment date \(T_k\), the caplet pays \(\delta(L_k - K)^+\) and the floorlet pays \(\delta(K - L_k)^+\) where \(L_k = L(T_{k-1}, T_k)\). Using the identity for any real number \(x\):
Applied with \(x = L_k\):
The left side is \(\text{Caplet}_k - \text{Floorlet}_k\). The right side \(\delta(L_k - K)\) is the net payment at \(T_k\) of a payer swap (receiving floating \(L_k\), paying fixed \(K\)).
Summing over all \(k = 1, \ldots, n\):
Taking present values:
This is cap-floor parity. It holds model-independently (it is a consequence of the algebraic identity \((x-K)^+ - (K-x)^+ = x - K\)) and serves as a consistency check for any implementation of cap and floor pricing.
Exercise 6. The bond option volatility for a caplet is \(\sigma_P = B(\delta)\sigma\sqrt{(1 - e^{-2\kappa T_0})/(2\kappa)}\). Analyze how this volatility depends on (i) the accrual period \(\delta\), (ii) the reset date \(T_0\), and (iii) the mean reversion \(\kappa\). For large \(T_0\), what does \(\sigma_P\) converge to?
Solution to Exercise 6
The bond option volatility for the \(k\)-th caplet is:
(i) Dependence on accrual period \(\delta\). Since \(B(\delta) = (1 - e^{-\kappa\delta})/\kappa \approx \delta - \frac{1}{2}\kappa\delta^2\) for small \(\delta\), we have \(\sigma_P \approx \delta\,\sigma\sqrt{(1-e^{-2\kappa T_0})/(2\kappa)}\). The bond option volatility is approximately proportional to \(\delta\): longer accrual periods produce larger bond option volatilities because the underlying ZCB has more rate sensitivity.
(ii) Dependence on reset date \(T_0\). The factor \(\sqrt{(1 - e^{-2\kappa T_0})/(2\kappa)}\) is an increasing function of \(T_0\) that saturates at \(1/\sqrt{2\kappa}\) as \(T_0 \to \infty\). For short reset dates, \(\sigma_P\) grows approximately as \(\sigma_P \propto \sqrt{T_0}\), reflecting the accumulation of rate uncertainty. For large \(T_0\), the variance of \(r_{T_0}\) under the forward measure approaches its stationary value, and \(\sigma_P\) flattens.
(iii) Dependence on mean reversion \(\kappa\). Both \(B(\delta)\) and \(\sqrt{(1-e^{-2\kappa T_0})/(2\kappa)}\) are decreasing in \(\kappa\). Stronger mean reversion reduces rate uncertainty and bond sensitivity, thereby reducing \(\sigma_P\).
Limit for large \(T_0\):
This finite limit reflects the bounded stationary variance of the OU process.
Exercise 7. A receiver swaption with expiry \(T_0\) into a swap with payment dates \(T_1, \ldots, T_n\) decomposes into a portfolio of ZCB calls via Jamshidian's trick. Write the pricing formula analogous to the payer swaption decomposition. Verify swaption parity: Payer \(-\) Receiver \(=\) Swap value.
Solution to Exercise 7
The receiver swaption gives the right to receive fixed rate \(K\) and pay floating. Its payoff at \(T_0\) is:
where \(c_k = K\delta_k\) for \(k < n\) and \(c_n = 1 + K\delta_n\). This is a call on the coupon bond with strike 1.
By Jamshidian's trick, using the same critical rate \(r^*\) (solving \(\sum c_k P(T_0, T_k; r^*) = 1\)) and individual strikes \(K_k = P(T_0, T_k; r^*)\):
where \(\text{Call}_{\text{ZCB}}(t; K_k, T_0, T_k) = P(t, T_k)\,\Phi(d_1^{(k)}) - K_k\,P(t, T_0)\,\Phi(d_2^{(k)})\).
Swaption parity verification. We need \(\text{PSwaption} - \text{RSwaption} = \text{Swap}_{\text{payer}}\).
By put-call parity for ZCB options: \(\text{Put} - \text{Call} = K_k\,P(t, T_0) - P(t, T_k)\). Therefore:
Since \(\sum c_k K_k = 1\) (by definition of \(r^*\)), this equals:
confirming swaption parity.