Payer Swaption Pricing and Vega Under Black's Model

Worked Solution

How to Think About It: A payer swaption is a call option on the forward swap rate, but the natural numeraire is not the money market account -- it is the swap annuity. Once you switch to the annuity measure, the forward swap rate becomes a martingale and you land in Black's formula territory. Before writing anything down, think about what the answer should look like qualitatively. The swaption price must be proportional to $A(0)$, because the annuity is what converts rate-space payoffs into dollars. Vega should also scale with $A(0)$ for the same reason. In the limiting cases: deep ITM ($F \gg K$), the swaption approaches $A(0)(F - K)$, just the intrinsic value of the swap. Deep OTM ($F \ll K$), the price goes to zero. At expiry ($t \to 0$), the price collapses to $A(0)\max(F - K, 0)$ with zero time value and zero vega. These checks should be immediate for anyone on a rates desk.

Quick Sanity Checks:

• Price must be proportional to $A(0)$ -- a bigger annuity means a bigger dollar payoff for the same rate move.
• Deep ITM: price $\to A(0)(F - K)$, delta $\to A(0)$, vega $\to 0$.
• Deep OTM: price $\to 0$, delta $\to 0$, vega $\to 0$.
• At-the-money ($F = K$): vega is maximized, since $\phi(d_1)$ peaks when $d_1$ is near zero.
• Vega must be positive -- a payer swaption, like any vanilla option, benefits from higher volatility.
• Two swaptions with the same rate dynamics but different annuities should have vegas in the same ratio as their annuities.

Derivation:

*Part (a): Payer Swaption Price*

Under the annuity measure $\mathbb{Q}^{A}$, the forward swap rate $F(t)$ is a martingale:

$dF(t) = \sigma F(t) \, dW_t^{A}$

The time-0 value of the payer swaption is the expected payoff under $\mathbb{Q}^{A}$, discounted by the numeraire $A(0)$:

$V(0) = A(0) \, \mathbb{E}^{\mathbb{Q}^{A}}\!\left[\max(F(t) - K, \, 0)\right]$

Since $F(t)$ is lognormal under $\mathbb{Q}^{A}$, this evaluates via Black's formula. Define:

$d_1 = \frac{\ln(F/K) + \tfrac{1}{2}\sigma^2 t}{\sigma\sqrt{t}}, \qquad d_2 = d_1 - \sigma\sqrt{t}$

Then:

$V(0) = A(0)\bigl[F\,\Phi(d_1) - K\,\Phi(d_2)\bigr]$

where $\Phi(\cdot)$ is the standard normal CDF. This is Black's formula for a payer swaption, with $A(0)$ playing the role of the discounted notional.

*Part (b): Vega*

Differentiate $V(0)$ with respect to $\sigma$:

$\frac{\partial V}{\partial \sigma} = A(0)\left[F\,\phi(d_1)\frac{\partial d_1}{\partial \sigma} - K\,\phi(d_2)\frac{\partial d_2}{\partial \sigma}\right]$

where $\phi(\cdot)$ is the standard normal PDF. The key simplification uses a standard identity.

*Identity:* $F\,\phi(d_1) = K\,\phi(d_2)$.

*Proof:* Since $d_1 - d_2 = \sigma\sqrt{t}$, write:

$\frac{\phi(d_1)}{\phi(d_2)} = e^{-(d_1^2 - d_2^2)/2} = e^{-(d_1 + d_2)(d_1 - d_2)/2}$

Now $(d_1 + d_2) = 2d_1 - \sigma\sqrt{t}$ and $d_1 \cdot \sigma\sqrt{t} = \ln(F/K) + \tfrac{1}{2}\sigma^2 t$, so:

$(d_1 + d_2)\sigma\sqrt{t}/2 = d_1 \cdot \sigma\sqrt{t} - \sigma^2 t/2 = \ln(F/K)$

Thus $\phi(d_1)/\phi(d_2) = e^{-\ln(F/K)} = K/F$, confirming $F\,\phi(d_1) = K\,\phi(d_2)$.

Using this identity and the fact that $\partial d_2 / \partial \sigma = \partial d_1 / \partial \sigma - \sqrt{t}$, the terms involving $\partial d_1 / \partial \sigma$ cancel, leaving:

$\mathcal{V} = \frac{\partial V}{\partial \sigma} = A(0) \, F \, \phi(d_1) \, \sqrt{t}$

This is the swaption analogue of the Black-Scholes vega for a vanilla call. Quick check: vega is proportional to $\sqrt{t}$ (longer-dated options have more vega), proportional to $F$ (higher rates mean larger absolute moves for the same lognormal vol), and proportional to $A(0)$ (bigger annuity means bigger dollar impact).

*Part (c): Comparing Vegas Across Different Annuities*

Let swaptions 1 and 2 share the same $F$, $K$, $\sigma$, $t$ but have annuities $A_1(0)$ and $A_2(0)$. From part (b):

$\mathcal{V}_i = A_i(0) \cdot F \, \phi(d_1) \, \sqrt{t}$

Since $d_1$ depends only on $F$, $K$, $\sigma$, $t$ -- all identical -- $\phi(d_1)$ is the same for both. Therefore:

$\frac{\mathcal{V}_1}{\mathcal{V}_2} = \frac{A_1(0)}{A_2(0)}$

Vegas are proportional to the annuities. The swaption on the swap with more payment dates or longer tenor has a larger annuity and hence a larger dollar vega.

Practical Interpretation: The annuity $A(0)$ converts a swap-rate move into dollar P&L. It is essentially the DV01 of the underlying swap -- the dollar value of a one-basis-point move in the swap rate. A 10-year semi-annual swap has a much bigger annuity than a 2-year annual swap, so its swaption's dollar vega is proportionally larger even if the rate dynamics are identical. This means you cannot simply match notionals when constructing vega-neutral hedges across swaptions of different tenors -- you must account for the annuity ratio. In practice, rates vol desks often normalize vega by the annuity and quote in basis-point-vol (normal vol) units precisely to make cross-tenor comparisons apples-to-apples. When a trader says "I am long 500k vega in 5y10y swaptions," the annuity is already baked into that dollar figure.

Answer:

• (a) $V(0) = A(0)\bigl[F\,\Phi(d_1) - K\,\Phi(d_2)\bigr]$ with $d_1 = \frac{\ln(F/K) + \frac{1}{2}\sigma^2 t}{\sigma\sqrt{t}}$ and $d_2 = d_1 - \sigma\sqrt{t}$.
• (b) $\mathcal{V} = A(0) \, F \, \phi(d_1) \, \sqrt{t}$.
• (c) Vega scales linearly with $A(0)$. The swaption on the swap with the larger annuity has proportionally higher dollar vega, because the annuity acts as the rate-to-dollar conversion factor.