Swap Measure¶

The $T$-forward measure simplifies the pricing of derivatives with a single payment date. Many interest rate products, however, involve streams of cashflows over multiple dates --- most notably, interest rate swaps and swaptions. The swap measure (also called the annuity measure) uses the present value of the annuity stream as numéraire, rendering the forward swap rate a martingale and producing the standard Black formula for swaptions.

Motivation¶

From Single to Multiple Payment Dates¶

Under the $T$-forward measure, the zero-coupon bond $P(t,T)$ is the numéraire. This is natural for payoffs at a single date $T$. For a swaption, however, the payoff at exercise is

\[ \text{Payoff} = \sum_{j=1}^{n} \delta_j P(T_0, T_j)\bigl(S(T_0) - K\bigr)^+ = A(T_0)\bigl(S(T_0) - K\bigr)^+ \]

where $A(T_0) = \sum_{j=1}^{n} \delta_j P(T_0, T_j)$ is the annuity factor. The annuity $A(T_0)$ multiplies the swap rate payoff, suggesting that $A(t)$ is the natural numéraire.

The Key Insight¶

If we choose $A(t)$ as the numéraire, the forward swap rate $S(t)$ becomes a martingale. Under lognormal dynamics, this leads directly to Black's formula for swaptions, exactly paralleling how the $T$-forward measure produces Black's caplet formula.

Definition of the Annuity Factor¶

Swap Structure¶

Fix a tenor structure with payment dates $T_1, T_2, \ldots, T_n$ and accrual fractions $\delta_j = T_j - T_{j-1}$. The swap start date is $T_0$.

Annuity Factor¶

The annuity factor (or present value of a basis point, PVBP) is

\[ A(t) = \sum_{j=1}^{n} \delta_j \, P(t, T_j) \]

This represents the time-$t$ value of receiving one unit of currency at each payment date, weighted by the accrual fractions.

Properties of the Annuity Factor¶

The annuity factor $A(t)$ is:

Strictly positive for $t < T_1$ (sum of positive bond prices)
Tradable --- it is the price of a portfolio of zero-coupon bonds
Self-financing --- the portfolio is static (fixed bond positions), so no rebalancing is required

These properties qualify $A(t)$ as a valid numéraire.

The Forward Swap Rate¶

Definition¶

The forward swap rate at time $t$ for a swap over $[T_0, T_n]$ is the fixed rate $S(t)$ that makes the swap have zero value:

\[ S(t) = \frac{P(t, T_0) - P(t, T_n)}{A(t)} = \frac{P(t, T_0) - P(t, T_n)}{\sum_{j=1}^{n} \delta_j \, P(t, T_j)} \]

Derivation¶

The value of the floating leg at time $t$ is $P(t, T_0) - P(t, T_n)$ (this follows from the telescoping property of forward rates). The value of the fixed leg paying rate $K$ is $K \cdot A(t)$. Setting the two equal and solving for $K$ gives $S(t)$.

Relationship to Forward LIBOR Rates¶

The forward swap rate can be expressed as a weighted average of forward LIBOR rates:

\[ S(t) = \sum_{i=0}^{n-1} w_i(t) \, L_i(t) \]

where the weights are

\[ w_i(t) = \frac{\delta_i \, P(t, T_{i+1})}{A(t)} \]

and $L_i(t)$ is the forward LIBOR rate for period $[T_i, T_{i+1}]$. The weights sum to one: $\sum_{i=0}^{n-1} w_i(t) = 1$.

The Swap Measure¶

Definition¶

The swap measure $\mathbb{Q}^A$ is the probability measure associated with the numéraire $A(t)$. Under $\mathbb{Q}^A$, for any tradable asset $V_t$,

\[ \frac{V_t}{A(t)} \quad \text{is a } \mathbb{Q}^A\text{-martingale} \]

Radon--Nikodym Derivative¶

The change of measure from the risk-neutral measure $\mathbb{Q}$ to the swap measure $\mathbb{Q}^A$ is given by

\[ \frac{d\mathbb{Q}^A}{d\mathbb{Q}}\bigg|_{\mathcal{F}_t} = \frac{A(t) / A(0)}{B_t} \]

where $B_t = \exp\left(\int_0^t r_s \, ds\right)$ is the money-market account.

Swap Rate Is a Martingale¶

Proposition. Under $\mathbb{Q}^A$, the forward swap rate $S(t)$ is a martingale.

Proof. The numerator $P(t, T_0) - P(t, T_n)$ is the value of a tradable portfolio (long the $T_0$-bond, short the $T_n$-bond). Under $\mathbb{Q}^A$, any tradable asset divided by $A(t)$ is a martingale. Therefore

\[ \frac{P(t, T_0) - P(t, T_n)}{A(t)} = S(t) \]

is a $\mathbb{Q}^A$-martingale. $\square$

Swaption Pricing Under the Swap Measure¶

Payer Swaption¶

A payer swaption with exercise date $T_0$ and strike $K$ gives the right to enter a payer swap. Its payoff at $T_0$ is

\[ V_{T_0} = A(T_0) \max(S(T_0) - K, 0) \]

Pricing Formula¶

Under the risk-neutral measure:

\[ V_0 = \mathbb{E}^{\mathbb{Q}}\!\left[e^{-\int_0^{T_0} r_s \, ds} \, A(T_0) \max(S(T_0) - K, 0)\right] \]

Switching to the swap measure with numéraire $A(t)$:

\[ V_0 = A(0) \, \mathbb{E}^{\mathbb{Q}^A}\!\left[\max(S(T_0) - K, 0)\right] \]

The stochastic discount factor and the annuity factor have been absorbed into the numéraire.

Lognormal Swap Rate Dynamics¶

Assuming the swap rate follows a geometric Brownian motion under $\mathbb{Q}^A$:

\[ \frac{dS(t)}{S(t)} = \sigma_S(t) \, dW_t^A \]

where $\sigma_S(t)$ is the instantaneous volatility and $W_t^A$ is a Brownian motion under $\mathbb{Q}^A$. Since $S(t)$ is a martingale, the drift is zero.

Black's Swaption Formula¶

The terminal swap rate $S(T_0)$ is lognormally distributed:

\[ \ln S(T_0) \sim N\!\left(\ln S(0) - \tfrac{1}{2}v_S^2, \; v_S^2\right) \]

where $v_S^2 = \int_0^{T_0} \sigma_S(t)^2 \, dt$. Black's formula gives

\[ \boxed{V_0^{\text{payer}} = A(0)\bigl[S(0) \, N(d_1) - K \, N(d_2)\bigr]} \]

\[ d_1 = \frac{\ln(S(0)/K) + \tfrac{1}{2} v_S^2}{v_S}, \qquad d_2 = d_1 - v_S \]

Receiver Swaption¶

By put--call parity for swaptions:

\[ \boxed{V_0^{\text{receiver}} = A(0)\bigl[K \, N(-d_2) - S(0) \, N(-d_1)\bigr]} \]

Girsanov Transformation Details¶

From Risk-Neutral to Swap Measure¶

Under $\mathbb{Q}$, the dynamics of $A(t)$ involve the bond price volatilities. Since

\[ A(t) = \sum_{j=1}^{n} \delta_j P(t, T_j) \]

the volatility of $A(t)$ is a weighted combination:

\[ \frac{dA(t)}{A(t)} = r_t \, dt + \sigma_A(t) \, dW_t^{\mathbb{Q}} \]

where

\[ \sigma_A(t) = -\frac{\sum_{j=1}^{n} \delta_j P(t, T_j) \, \Sigma(t, T_j)}{A(t)} \]

and $\Sigma(t, T_j) = \int_t^{T_j} \sigma(t, u) \, du$ is the bond volatility.

The Girsanov change of measure from $\mathbb{Q}$ to $\mathbb{Q}^A$ is

\[ dW_t^A = dW_t^{\mathbb{Q}} - \sigma_A(t) \, dt \]

From T-Forward to Swap Measure¶

The measure change from $\mathbb{Q}^{T_j}$ to $\mathbb{Q}^A$ involves the ratio $A(t) / P(t, T_j)$:

\[ dW_t^A = dW_t^{T_j} - \bigl[\sigma_A(t) - \Sigma(t, T_j)\bigr] dt \]

Swap Rate Volatility in the LMM¶

Rebonato's Approximation¶

In the LIBOR Market Model, the swap rate $S(t) = \sum_i w_i(t) L_i(t)$ is not exactly lognormal. The effective swap rate volatility is approximated by freezing the weights at their initial values:

\[ v_S^2 \, T_0 \approx \sum_{i,j=0}^{n-1} \frac{w_i(0) \, w_j(0) \, L_i(0) \, L_j(0)}{S(0)^2} \, \rho_{ij} \int_0^{T_0} \sigma_i(t) \, \sigma_j(t) \, dt \]

This is Rebonato's swaption volatility formula, which expresses the Black swaption implied volatility in terms of the LMM parameters (forward rate volatilities and correlations).

Interpretation¶

The diagonal terms ($i = j$) capture individual forward rate contributions
The off-diagonal terms ($i \neq j$) reflect the impact of correlation between forward rates
Lower correlation reduces the effective swap rate volatility (diversification effect)

Practical Applications¶

ATM Swaption Pricing¶

At-the-money swaptions have $K = S(0)$, so $d_1 = v_S/2$ and $d_2 = -v_S/2$:

\[ V_0^{\text{ATM}} = A(0) \cdot S(0) \bigl[N(v_S/2) - N(-v_S/2)\bigr] = A(0) \cdot S(0) \bigl[2N(v_S/2) - 1\bigr] \]

For small volatilities, $V_0^{\text{ATM}} \approx A(0) \cdot S(0) \cdot v_S / \sqrt{2\pi}$.

Swaption Straddle¶

A swaption straddle (long payer + long receiver at the same strike) is valued as

\[ V_0^{\text{straddle}} = V_0^{\text{payer}} + V_0^{\text{receiver}} \]

ATM straddles are pure volatility bets:

\[ V_0^{\text{ATM straddle}} = 2 \, V_0^{\text{ATM payer}} \approx 2 \, A(0) \cdot S(0) \cdot \frac{v_S}{\sqrt{2\pi}} \]

Market Conventions¶

Swaption markets quote in terms of Black implied volatility $\sigma_S^{\text{Black}}$ or normal (Bachelier) implied volatility $\sigma_S^{(n)}$. The swaption vol cube is organized by:

Expiry ($T_0$): e.g., 1M, 3M, 6M, 1Y, 2Y, 5Y, 10Y
Tenor ($T_n - T_0$): e.g., 1Y, 2Y, 5Y, 10Y, 30Y
Strike offset from ATM: e.g., -200bp, -100bp, ATM, +100bp, +200bp

Key Takeaways¶

The annuity factor $A(t) = \sum \delta_j P(t, T_j)$ serves as the natural numéraire for swap-related products
The swap measure $\mathbb{Q}^A$ is the probability measure under which the forward swap rate $S(t)$ is a martingale
Swaption pricing reduces to a Black-type formula: $V_0 = A(0)[S(0)N(d_1) - KN(d_2)]$
The Radon--Nikodym derivative involves the ratio of the annuity to the money-market account
In the LMM, the swap rate volatility is approximated via Rebonato's formula, linking forward rate volatilities and correlations to swaption prices
The swap measure framework extends naturally to receiver swaptions via put--call parity

Exercises¶

Exercise 1. Consider a swap with semiannual payments over 3 years ($T_0 = 0$, payment dates $T_1 = 0.5, T_2 = 1.0, \ldots, T_6 = 3.0$). The zero-coupon bond prices are $P(0, 0.5) = 0.985$, $P(0, 1.0) = 0.968$, $P(0, 1.5) = 0.950$, $P(0, 2.0) = 0.930$, $P(0, 2.5) = 0.910$, $P(0, 3.0) = 0.889$. Compute the annuity factor $A(0)$ and the forward swap rate $S(0)$.

Solution to Exercise 1

Annuity factor. With semiannual payments ($\delta_j = 0.5$ for all $j$) and 6 payment dates:

\[ A(0) = \sum_{j=1}^{6} \delta_j P(0, T_j) = 0.5 \times (0.985 + 0.968 + 0.950 + 0.930 + 0.910 + 0.889) \]

\[ A(0) = 0.5 \times 5.632 = 2.816 \]

Forward swap rate. With $T_0 = 0$, we have $P(0, T_0) = P(0, 0) = 1$ and $P(0, T_n) = P(0, 3.0) = 0.889$:

\[ S(0) = \frac{P(0, T_0) - P(0, T_n)}{A(0)} = \frac{1 - 0.889}{2.816} = \frac{0.111}{2.816} = 0.03942 \]

The forward swap rate is approximately 3.94%.

Exercise 2. Using the data from Exercise 1, price a 3-year ATM payer swaption (exercise at $T_0 = 0$, but assume the swaption expires in 1 year with $T_0 = 1$, payment dates $T_1 = 1.5, \ldots, T_4 = 3.0$) with Black implied volatility $\sigma_S = 25\%$. Recompute $A(0)$ and $S(0)$ for this sub-swap and apply Black's swaption formula.

Solution to Exercise 2

Recompute for the sub-swap. The swaption expires at $T_0 = 1$ year, and the underlying swap has payment dates $T_1 = 1.5, T_2 = 2.0, T_3 = 2.5, T_4 = 3.0$ with $\delta_j = 0.5$.

Annuity factor for the sub-swap:

\[ A(0) = 0.5 \times (P(0, 1.5) + P(0, 2.0) + P(0, 2.5) + P(0, 3.0)) \]

\[ = 0.5 \times (0.950 + 0.930 + 0.910 + 0.889) = 0.5 \times 3.679 = 1.8395 \]

Forward swap rate for the sub-swap:

\[ S(0) = \frac{P(0, T_0) - P(0, T_4)}{A(0)} = \frac{P(0, 1.0) - P(0, 3.0)}{1.8395} = \frac{0.968 - 0.889}{1.8395} = \frac{0.079}{1.8395} = 0.04294 \]

So $S(0) \approx 4.294\%$.

ATM swaption: $K = S(0) = 0.04294$. The integrated volatility is

\[ v_S = \sigma_S \sqrt{T_0} = 0.25 \times \sqrt{1} = 0.25 \]

For ATM, $d_1 = v_S / 2 = 0.125$ and $d_2 = -v_S / 2 = -0.125$.

Using the standard normal CDF: $N(0.125) \approx 0.5498$ and $N(-0.125) \approx 0.4502$.

Payer swaption price:

\[ V_0 = A(0)\left[S(0)\,N(d_1) - K\,N(d_2)\right] \]

Since $K = S(0)$:

\[ V_0 = 1.8395 \times 0.04294 \times \left[N(0.125) - N(-0.125)\right] \]

\[ = 1.8395 \times 0.04294 \times (0.5498 - 0.4502) = 1.8395 \times 0.04294 \times 0.0996 \]

\[ = 1.8395 \times 0.004274 = 0.007863 \]

The ATM payer swaption is worth approximately 0.79% of notional, or about $\$78.63$ per $\$10{,}000$ notional.

Exercise 3. Prove that the forward swap rate $S(t) = (P(t, T_0) - P(t, T_n))/A(t)$ can be written as a weighted average of forward LIBOR rates

\[ S(t) = \sum_{i=0}^{n-1} w_i(t)\,L_i(t), \qquad w_i(t) = \frac{\delta_i\,P(t, T_{i+1})}{A(t)} \]

and verify that $\sum_{i=0}^{n-1} w_i(t) = 1$. State what additional assumption is needed for this decomposition to hold exactly.

Solution to Exercise 3

Express the floating leg using forward LIBOR rates. The simply-compounded forward LIBOR rate for the period $[T_i, T_{i+1}]$ is

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

Rearranging: $P(t, T_i) = P(t, T_{i+1})(1 + \delta_i L_i(t))$.

Telescoping the floating leg. The floating leg value is $P(t, T_0) - P(t, T_n)$. Using a telescoping decomposition:

\[ P(t, T_0) - P(t, T_n) = \sum_{i=0}^{n-1}\left[P(t, T_i) - P(t, T_{i+1})\right] \]

From the LIBOR rate definition: $P(t, T_i) - P(t, T_{i+1}) = \delta_i L_i(t) P(t, T_{i+1})$. Therefore:

\[ P(t, T_0) - P(t, T_n) = \sum_{i=0}^{n-1} \delta_i L_i(t) P(t, T_{i+1}) \]

Dividing by the annuity factor:

\[ S(t) = \frac{P(t, T_0) - P(t, T_n)}{A(t)} = \frac{\sum_{i=0}^{n-1} \delta_i L_i(t) P(t, T_{i+1})}{\sum_{j=1}^{n} \delta_j P(t, T_j)} \]

Since $T_{i+1} = T_{j}$ when $j = i + 1$ (i.e., the payment dates in the numerator and denominator align):

\[ S(t) = \sum_{i=0}^{n-1} \frac{\delta_i P(t, T_{i+1})}{A(t)} L_i(t) = \sum_{i=0}^{n-1} w_i(t)\,L_i(t) \]

where $w_i(t) = \frac{\delta_i P(t, T_{i+1})}{A(t)}$.

Verifying the weights sum to one. Note that $A(t) = \sum_{j=1}^{n} \delta_j P(t, T_j) = \sum_{i=0}^{n-1} \delta_i P(t, T_{i+1})$ (re-indexing $j = i+1$, and using $\delta_j = \delta_{j-1}$ only if $\delta_j = T_j - T_{j-1}$ and $\delta_i = T_{i+1} - T_i$, which is the same). Therefore:

\[ \sum_{i=0}^{n-1} w_i(t) = \sum_{i=0}^{n-1} \frac{\delta_i P(t, T_{i+1})}{A(t)} = \frac{A(t)}{A(t)} = 1 \]

Additional assumption. The decomposition holds exactly under the assumption that the accrual fractions in the swap match the LIBOR rate tenors, i.e., $\delta_i = T_{i+1} - T_i$ for all $i$, and that the LIBOR rates are simply-compounded rates over exactly these periods. In practice, day-count conventions may introduce small discrepancies.

Exercise 4. Using Rebonato's frozen-weight approximation, compute the Black swaption implied volatility for a 2-year into 3-year swaption in an LMM with three annual forward rates. The parameters are: $L_0(0) = 4.0\%$, $L_1(0) = 4.2\%$, $L_2(0) = 4.5\%$; flat volatilities $\sigma_0 = \sigma_1 = \sigma_2 = 20\%$; and exponential correlation $\rho_{ij} = e^{-0.1|i-j|}$. The forward swap rate is $S(0) = 4.23\%$ with weights $w_0 = 0.34$, $w_1 = 0.33$, $w_2 = 0.33$.

Solution to Exercise 4

Setup. We have a 2-year into 3-year swaption with three annual forward rates ($n = 3$). Expiry $T_0 = 2$, annual payments at $T_1 = 3, T_2 = 4, T_3 = 5$.

Given parameters:

$L_0(0) = 4.0\%$, $L_1(0) = 4.2\%$, $L_2(0) = 4.5\%$
$\sigma_0 = \sigma_1 = \sigma_2 = 20\%$ (flat volatilities)
$\rho_{ij} = e^{-0.1|i-j|}$: $\rho_{00} = 1$, $\rho_{11} = 1$, $\rho_{22} = 1$, $\rho_{01} = \rho_{10} = e^{-0.1} = 0.9048$, $\rho_{02} = \rho_{20} = e^{-0.2} = 0.8187$, $\rho_{12} = \rho_{21} = e^{-0.1} = 0.9048$
$S(0) = 4.23\%$, $w_0 = 0.34$, $w_1 = 0.33$, $w_2 = 0.33$

Rebonato's formula. With flat volatilities and expiry $T_0 = 2$:

\[ v_S^2 \cdot T_0 = \sum_{i,j=0}^{2} \frac{w_i(0)\,w_j(0)\,L_i(0)\,L_j(0)}{S(0)^2}\,\rho_{ij}\int_0^{T_0}\sigma_i(t)\,\sigma_j(t)\,dt \]

Since $\sigma_i(t) = 0.20$ for all $i$ and $t$:

\[ \int_0^{2} \sigma_i(t)\sigma_j(t)\,dt = 0.04 \times 2 = 0.08 \]

This factor is common to all terms. Define $c_i = w_i(0) L_i(0) / S(0)$:

$c_0 = 0.34 \times 0.040 / 0.0423 = 0.3216$
$c_1 = 0.33 \times 0.042 / 0.0423 = 0.3277$
$c_2 = 0.33 \times 0.045 / 0.0423 = 0.3511$

Then:

\[ v_S^2 \cdot T_0 = 0.08 \times \sum_{i,j} c_i\,c_j\,\rho_{ij} \]

Computing the double sum:

\[ \sum_{i,j} c_i\,c_j\,\rho_{ij} = c_0^2 + c_1^2 + c_2^2 + 2c_0 c_1 \rho_{01} + 2c_0 c_2 \rho_{02} + 2c_1 c_2 \rho_{12} \]

$c_0^2 = 0.1034$
$c_1^2 = 0.1074$
$c_2^2 = 0.1233$
$2c_0 c_1 \rho_{01} = 2 \times 0.3216 \times 0.3277 \times 0.9048 = 0.1906$
$2c_0 c_2 \rho_{02} = 2 \times 0.3216 \times 0.3511 \times 0.8187 = 0.1849$
$2c_1 c_2 \rho_{12} = 2 \times 0.3277 \times 0.3511 \times 0.9048 = 0.2082$

\[ \sum_{i,j} c_i\,c_j\,\rho_{ij} = 0.1034 + 0.1074 + 0.1233 + 0.1906 + 0.1849 + 0.2082 = 0.9178 \]

Therefore:

\[ v_S^2 \cdot T_0 = 0.08 \times 0.9178 = 0.07342 \]

\[ v_S^2 = \frac{0.07342}{2} = 0.03671 \]

The Black swaption implied volatility is:

\[ \sigma_S^{\text{Black}} = \sqrt{v_S^2 / T_0} = \sqrt{0.03671 / 2} = \sqrt{0.01836} \approx 0.1355 = 13.55\% \]

Alternatively: $v_S = \sqrt{0.07342} = 0.2710$ and $\sigma_S^{\text{Black}} = v_S / \sqrt{T_0} = 0.2710/\sqrt{2} \approx 19.16\%$.

Wait --- let me recompute more carefully. We have $v_S^2 = \int_0^{T_0}\sigma_S(t)^2\,dt$. Rebonato's formula gives $v_S^2 T_0$ on the left side. Actually, reviewing the formula:

\[ v_S^2\,T_0 \approx \sum_{i,j} \frac{w_i w_j L_i L_j}{S(0)^2}\rho_{ij}\int_0^{T_0}\sigma_i(t)\sigma_j(t)\,dt \]

With $\sigma_S^{\text{Black}}$ defined as $v_S = \sigma_S^{\text{Black}}\sqrt{T_0}$, we get $(\sigma_S^{\text{Black}})^2 T_0 = v_S^2$. So:

\[ (\sigma_S^{\text{Black}})^2 T_0 \cdot T_0 = v_S^2 \cdot T_0 = 0.07342 \]

This gives $(\sigma_S^{\text{Black}})^2 = 0.07342 / T_0^2 = 0.07342 / 4 = 0.01836$, so $\sigma_S^{\text{Black}} = 13.5\%$.

However, the standard convention is that $v_S^2 = (\sigma_S^{\text{Black}})^2 T_0$, so $(\sigma_S^{\text{Black}})^2 = v_S^2 / T_0 = 0.07342 / 2 = 0.03671$, giving $\sigma_S^{\text{Black}} = \sqrt{0.03671} \approx 19.2\%$.

The Black swaption implied volatility is approximately 19.2%.

Note that this is slightly below the individual forward rate volatility of 20%, reflecting the diversification effect from imperfect correlations ($\rho_{ij} < 1$ for $i \neq j$).

Exercise 5. Derive the put--call parity relationship for swaptions. Show that the difference between a payer and a receiver swaption with the same strike and expiry equals

\[ V_0^{\text{payer}} - V_0^{\text{receiver}} = A(0)(S(0) - K) \]

Interpret this result in terms of the value of a forward-starting swap.

Solution to Exercise 5

Payer swaption payoff: $V_{T_0}^{\text{payer}} = A(T_0)\max(S(T_0) - K, 0)$

Receiver swaption payoff: $V_{T_0}^{\text{receiver}} = A(T_0)\max(K - S(T_0), 0)$

Difference of payoffs. Using the identity $\max(x, 0) - \max(-x, 0) = x$:

\[ V_{T_0}^{\text{payer}} - V_{T_0}^{\text{receiver}} = A(T_0)\left[\max(S(T_0) - K, 0) - \max(K - S(T_0), 0)\right] = A(T_0)(S(T_0) - K) \]

Pricing both sides. Under the swap measure $\mathbb{Q}^A$ with numéraire $A(t)$:

\[ V_0^{\text{payer}} - V_0^{\text{receiver}} = A(0)\,\mathbb{E}^{\mathbb{Q}^A}\!\left[S(T_0) - K\right] \]

Since $S(t)$ is a $\mathbb{Q}^A$-martingale, $\mathbb{E}^{\mathbb{Q}^A}[S(T_0)] = S(0)$. Therefore:

\[ V_0^{\text{payer}} - V_0^{\text{receiver}} = A(0)(S(0) - K) \]

Financial interpretation. The right-hand side $A(0)(S(0) - K)$ is the value of a forward-starting swap where one pays fixed rate $K$ and receives floating, starting at $T_0$. The forward swap rate $S(0)$ is the break-even fixed rate. If $K < S(0)$, the payer swaption is worth more than the receiver (it is more likely to be exercised), and the difference equals the value of a swap at the off-market rate $K$.

This is the swaption analogue of the standard put--call parity for European options on stocks: $C - P = S_0 - Ke^{-rT}$, where the forward replaces the stock and the annuity factor replaces the discount factor.

Exercise 6. An ATM swaption straddle on a 5-year swap has $A(0) = 4.20$, $S(0) = 3.8\%$, and Black implied volatility $\sigma_S = 22\%$ with expiry $T_0 = 2$ years. Using the approximation $V_0^{\text{ATM straddle}} \approx 2A(0)\,S(0)\,v_S/\sqrt{2\pi}$, compute the straddle value and express it as a percentage of the notional.

Solution to Exercise 6

Given data: $A(0) = 4.20$, $S(0) = 3.8\% = 0.038$, $\sigma_S = 22\%$, $T_0 = 2$ years.

Integrated volatility:

\[ v_S = \sigma_S \sqrt{T_0} = 0.22 \times \sqrt{2} = 0.22 \times 1.4142 = 0.31113 \]

ATM straddle approximation:

\[ V_0^{\text{ATM straddle}} \approx 2\,A(0)\,S(0)\,\frac{v_S}{\sqrt{2\pi}} \]

Computing $v_S / \sqrt{2\pi}$:

\[ \frac{v_S}{\sqrt{2\pi}} = \frac{0.31113}{2.5066} = 0.12413 \]

Therefore:

\[ V_0^{\text{ATM straddle}} \approx 2 \times 4.20 \times 0.038 \times 0.12413 = 2 \times 4.20 \times 0.004717 \]

\[ = 2 \times 0.019811 = 0.039622 \]

As a percentage of notional: The straddle value is approximately 3.96% of the notional, or equivalently $\$396$ per $\$10{,}000$ notional.

Exercise 7. The swap measure $\mathbb{Q}^A$ can be viewed as a weighted combination of individual forward measures. Starting from the Radon--Nikodym derivative

\[ \frac{d\mathbb{Q}^A}{d\mathbb{Q}^{T_j}}\bigg|_{\mathcal{F}_t} = \frac{A(t)\,P(0, T_j)}{P(t, T_j)\,A(0)}, \]

derive the Girsanov drift adjustment $dW_t^A = dW_t^{T_j} - [\sigma_A(t) - \Sigma(t, T_j)]\,dt$, identifying the volatility $\sigma_A(t)$ of the annuity factor in terms of the individual bond volatilities $\Sigma(t, T_k)$.

Solution to Exercise 7

Starting point. The Radon--Nikodym derivative from $\mathbb{Q}^{T_j}$ to $\mathbb{Q}^A$ is

\[ \frac{d\mathbb{Q}^A}{d\mathbb{Q}^{T_j}}\bigg|_{\mathcal{F}_t} = \frac{A(t)\,P(0, T_j)}{P(t, T_j)\,A(0)} = \frac{A(t)/A(0)}{P(t, T_j)/P(0, T_j)} \]

Dynamics of the ratio. Define $R(t) = A(t)/P(t, T_j)$. Under $\mathbb{Q}^{T_j}$, the Radon--Nikodym derivative is $R(t)/R(0)$, which must be a $\mathbb{Q}^{T_j}$-martingale.

Under $\mathbb{Q}^{T_j}$, the bond $P(t, T_j)$ dynamics have volatility $\Sigma(t, T_j)$, and the annuity factor dynamics have volatility $\sigma_A(t)$. By Itô's quotient rule, the volatility of $R(t)$ is $\sigma_A(t) - \Sigma(t, T_j)$.

Identifying $\sigma_A(t)$. The annuity factor $A(t) = \sum_{k=1}^{n}\delta_k P(t, T_k)$ is a portfolio of bonds. Under $\mathbb{Q}$ (or any equivalent measure), each bond satisfies

\[ \frac{dP(t, T_k)}{P(t, T_k)} = (\cdots)\,dt + \Sigma(t, T_k)\,dW_t \]

where $\Sigma(t, T_k) = -\int_t^{T_k}\sigma(t, u)\,du$ is the bond volatility. By the linearity of the portfolio:

\[ \frac{dA(t)}{A(t)} = (\cdots)\,dt + \sigma_A(t)\,dW_t \]

where the annuity volatility is the weighted average of bond volatilities:

\[ \sigma_A(t) = \sum_{k=1}^{n} \frac{\delta_k P(t, T_k)}{A(t)}\,\Sigma(t, T_k) = -\sum_{k=1}^{n} \frac{\delta_k P(t, T_k)}{A(t)}\int_t^{T_k}\sigma(t, u)\,du \]

The weights $\delta_k P(t, T_k)/A(t)$ are the fractional contributions of each bond to the annuity.

Girsanov transformation. The Radon--Nikodym derivative $R(t)/R(0)$ is an exponential martingale under $\mathbb{Q}^{T_j}$ with volatility $\sigma_A(t) - \Sigma(t, T_j)$. By Girsanov's theorem:

\[ dW_t^A = dW_t^{T_j} - \left[\sigma_A(t) - \Sigma(t, T_j)\right]dt \]

This drift adjustment converts a $\mathbb{Q}^{T_j}$-Brownian motion into a $\mathbb{Q}^A$-Brownian motion. The adjustment depends on:

$\sigma_A(t)$: the volatility of the entire annuity portfolio (a weighted average over all payment date bond volatilities)
$\Sigma(t, T_j)$: the volatility of the specific bond $P(t, T_j)$

The difference $\sigma_A(t) - \Sigma(t, T_j)$ captures the extent to which the annuity's risk profile differs from that of the single bond $P(t, T_j)$.

Swap Measure¶

Motivation¶

From Single to Multiple Payment Dates¶

The Key Insight¶

Definition of the Annuity Factor¶

Swap Structure¶

Annuity Factor¶

Properties of the Annuity Factor¶

The Forward Swap Rate¶

Definition¶

Derivation¶

Relationship to Forward LIBOR Rates¶

The Swap Measure¶

Definition¶

Radon--Nikodym Derivative¶

Swap Rate Is a Martingale¶

Swaption Pricing Under the Swap Measure¶

Payer Swaption¶

Pricing Formula¶

Lognormal Swap Rate Dynamics¶

Black's Swaption Formula¶

Receiver Swaption¶

Girsanov Transformation Details¶

From Risk-Neutral to Swap Measure¶

From T-Forward to Swap Measure¶

Swap Rate Volatility in the LMM¶

Rebonato's Approximation¶

Interpretation¶

Practical Applications¶

ATM Swaption Pricing¶

Swaption Straddle¶

Market Conventions¶

Key Takeaways¶

Further Reading¶

Exercises¶