Caps floors and swaptions

Definition Let $T_f$ be a fixing date and $T_p > T_f$ a payment date, and fix the accrual fraction $\delta > 0$ of the interest period, a notional $N$ and a strike $K$ quoted as a simple annual rate. A caplet pays at $T_p$ the amount

where $L(T_f)$ is the reference rate fixed at $T_f$, and a floorlet pays $N\delta\,\bigl(K - L(T_f)\bigr)^{+}$ at $T_p$. A cap is the sum of the caplets on a schedule of consecutive periods, and a floor is the sum of the floorlets, usually with one strike for all periods. Suppose the bond that pays one unit at $T_p$ has a discounted price that is a true martingale under a pricing measure $\mathbb{Q}$, so that it defines the forward measure $\mathbb{Q}^{T_p}$, and that the payoff is integrable. Then for $0 \le t \le T_f$ the caplet is worth

and the floorlet value $\operatorname{Fll}_t$ is the same with $(K - L(T_f))^{+}$. The payment date fixes the numeraire and the discounting, while the fixing date ends the uncertainty, so the horizon of the option is the time to fixing. In the classical single-curve model the fixing is the simple rate $L(T_f) = \bigl(1/P(T_f,T_p) - 1\bigr)/\delta$ for the period $[T_f, T_p]$. After $T_f$ the payment is a known amount, discounted with $P(t,T_p)$.

Definition Let a caplet and a floorlet have the same notional, accrual, strike, fixing date, payment date and settlement. Since $(x - K)^{+} - (K - x)^{+} = x - K$ for every real $x$, for $0 \le t \le T_f$

where $F_t = \mathbb{E}^{\mathbb{Q}^{T_p}}[L(T_f) \mid \mathcal{F}_t]$ is the forward rate of the fixing under the payment measure. The identity needs only the pricing formula of the previous card and an integrable fixing, not a lognormal or a normal law. In the classical single-curve model, where the fixing is the simple rate of its own period and is paid at the end of it, $F_t = \bigl(P(t,T_f)/P(t,T_p) - 1\bigr)/\delta$, and the right side is the value of a forward rate agreement at the fixed rate $K$ to the party that receives floating. Summed over a schedule, a long cap and a short floor with the same strike have the cash flows of a payer swap at the fixed rate $K$ over the same periods. A payment made at the fixing date, or a fixing whose period does not match the payment, is a different payoff, and its parity relation has to be derived from its own payoff and payment date.

Definition For $t \le s \le T_f$ let $F_s = \mathbb{E}^{\mathbb{Q}^{T_p}}[L(T_f) \mid \mathcal{F}_s]$ be the forward rate of the fixing, so that $F_{T_f} = L(T_f)$. Black’s model assumes $F_t > 0$ and

with a constant proportional volatility $\sigma_B > 0$ per square root year and a Brownian motion $W^{T_p}$ under $\mathbb{Q}^{T_p}$. With the time to fixing $\tau = T_f - t > 0$, the fixing $L(T_f) = F_t \exp\bigl(-\tfrac12 \sigma_B^2 \tau + \sigma_B \sqrt{\tau}\,Z\bigr)$ is lognormal, with $Z$ standard normal given $\mathcal{F}_t$, and for $K > 0$

The floorlet is $\operatorname{Fll}^{B}_t = N\delta\,P(t,T_p)\,\bigl[K\,\Phi(-d_2) - F_t\,\Phi(-d_1)\bigr]$. The positive-part integral is the one of the Black-Scholes formula, with the forward in place of the spot and no rates in $d_1$ and $d_2$, because the forward has no drift under the payment measure. As $\tau \to 0$ or $\sigma_B \to 0$ the price tends to the intrinsic value $N\delta\,P(t,T_p)(F_t - K)^{+}$, and for $K \le 0$ the positive fixing always exceeds the strike, so the caplet is worth $N\delta\,P(t,T_p)(F_t - K)$. The caplet price is strictly increasing in $\sigma_B$, with derivative $N\delta\,P(t,T_p)\,F_t\,\varphi(d_1)\sqrt{\tau} > 0$, and it runs from $N\delta\,P(t,T_p)(F_t - K)^{+}$ to $N\delta\,P(t,T_p)\,F_t$, so every price strictly between these bounds has exactly one Black implied volatility. The model does not allow $F_t \le 0$. A shifted lognormal model applies the formula to $F_t + s$ and $K + s$ for a chosen shift $s$ with $F_t + s > 0$ and $K + s > 0$, and its volatility is then a different number. In the market model of Forward rate market models with a constant volatility vector, $\sigma_B$ is the length of that vector.

Definition The Bachelier model, or normal model, assumes

with a constant absolute volatility $\sigma_N > 0$ in rate units per square root year. Then $L(T_f) = F_t + \sigma_N \sqrt{\tau}\,Z$ is normal under $\mathbb{Q}^{T_p}$ given $\mathcal{F}_t$, with $\tau = T_f - t > 0$, and for every real $F_t$ and $K$

and the floorlet is $\operatorname{Fll}^{N}_t = N\delta\,P(t,T_p)\,\bigl[(K - F_t)\,\Phi(-d) + \sigma_N \sqrt{\tau}\,\varphi(d)\bigr]$. With $s = \sigma_N \sqrt{\tau}$, the payoff is positive exactly when $Z > -d$, and

because $\int_{-d}^{\infty} z\,\varphi(z)\,dz = \varphi(-d) = \varphi(d)$. The normal law gives negative fixings a positive probability, so the formula applies to negative forward rates and strikes. As $\tau \to 0$ or $\sigma_N \to 0$ the price tends to the intrinsic value. The price is strictly increasing in $\sigma_N$, with derivative $N\delta\,P(t,T_p)\sqrt{\tau}\,\varphi(d) > 0$. A Black volatility and a normal volatility have different units, and the relation $\sigma_N \approx F_t\,\sigma_B$ near the money is a local correspondence, not an exact conversion.

Definition Fix an exercise date $T_0$ and an underlying swap that starts at $T_0$, with fixed payments at $T_1 < \dots < T_n$, fixed-leg accrual fractions $\alpha_j > 0$, notional $N$ and fixed rate $K$. In a single-curve market, define for $t \le T_0$ the annuity and the forward swap rate

Here $P(t,T_0) - P(t,T_n)$ is the value of the floating leg per unit of notional, and $S(t)$ is the par rate of the forward-starting swap, as in Interest rate swaps and forward rate agreements. If the floating leg’s value includes the notional, $S(t)$ is that value divided by $N A(t)$, so the notional is not counted twice. A European payer swaption gives the right to enter, at $T_0$, the swap that pays the fixed rate $K$ and receives floating, and a receiver swaption gives the right to enter the swap that receives $K$. At $T_0$ the payer swap is worth $N A(T_0)\bigl(S(T_0) - K\bigr)$, so with physical settlement and exercise exactly when that value is positive, the payer and the receiver swaptions pay at $T_0$

A cash-settled swaption replaces $A(T_0)$ by a contractual annuity computed from the realized swap rate, which is a different payoff with its own treatment.

Definition Assume that the discounted bond prices $P(t,T_j)/B_t$ for $j = 0, \dots, n$ are true martingales under a pricing measure $\mathbb{Q}$. On $[0, T_0]$ the annuity $A$ is the value of a fixed portfolio of bonds with no payments before $T_1$, so it is a positive traded numeraire there. The annuity measure, also called the swap measure, is the measure $\mathbb{Q}^A$ with

for $t \le T_0$. A payoff $H$ at $T_0$ with $H/A(T_0)$ integrable under $\mathbb{Q}^A$ has the value $V_t = A(t)\,\mathbb{E}^{\mathbb{Q}^A}[H/A(T_0) \mid \mathcal{F}_t]$. The floating leg is also a portfolio of bonds, so the forward swap rate $S(t)$ is a martingale under $\mathbb{Q}^A$. For the payer swaption the random annuity cancels,

and the receiver swaption is the same with $(K - S(T_0))^{+}$, so a payer minus a receiver with the same strike is worth $N A(t)\bigl(S(t) - K\bigr)$, the value of the forward-starting payer swap. If $dS = \sigma_B S\,dW^A$ under $\mathbb{Q}^A$, Black’s formula with $S(t)$ in place of $F_t$, $N A(t)$ in place of $N\delta\,P(t,T_p)$ and $\tau = T_0 - t$ gives the price, and if $dS = \sigma_N\,dW^A$, the Bachelier formula does. These are assumptions about the law of the swap rate under $\mathbb{Q}^A$. A model in which the forward rates are lognormal under their own forward measures does not in general make the swap rate lognormal under $\mathbb{Q}^A$, so caps and swaptions are consistent with each other only within one joint model of the curve.