Swaptions

A mortgage rate lock lets you fix today's rate for a house you'll close on in sixty days, without forcing you to actually take that rate if a better one shows up before closing — you have the right, not the obligation, to use it. A swaption is the institutional cousin of exactly that idea. It gives the holder the right, but not the obligation, to enter into an interest rate swap on a fixed future date at a rate agreed today. If market rates move in your favor by then, you walk away and deal in the open market instead; if they move against you, you exercise the swaption and lock in the rate you were promised. It costs money up front, like any option, precisely because that right is valuable.

Payer, receiver, and what's actually being optioned

A payer swaption gives the holder the right to enter a swap paying the fixed rate and receiving floating — useful if you're worried rates will rise, because it caps how high your fixed borrowing cost can go. A receiver swaption is the mirror image: the right to receive fixed and pay floating, useful if you're worried rates will fall. What's actually random here isn't a stock price — it's the forward swap rate, the fixed rate the market expects the underlying swap to carry when it starts. Price a swaption the same way you'd price a stock option, substituting the forward swap rate for the stock price, using a version of Black's formula:

$$\text{Payer value} = A \times \big[ F\, N(d_1) - K\, N(d_2) \big], \qquad d_1 = \frac{\ln(F/K) + \tfrac{1}{2}\sigma^2 T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}.$$

In plain English: $F$ is the forward swap rate — the fixed rate the market currently expects the underlying swap to be struck at when it starts. $K$ is the strike rate written into the swaption contract. $\sigma$ is the swap rate's volatility and $T$ the time until the swaption's expiry. $A$, the annuity (or PVBP, present value of a basis point), is the present value today of receiving $1 per year for every remaining payment date on the underlying swap — it plays the same role vega and the stock price play together in an equity option, converting a rate difference into an actual dollar value. The whole expression says: the payer swaption is worth the annuity times the expected benefit of paying the (probably lower) strike rate instead of the (probably higher) market rate, exactly the way a call option is worth the stock's expected excess over the strike.

Worked example 1 — pricing an at-the-money payer swaption

A corporation wants the right to enter a 5-year swap in one year, paying a fixed rate of 3.50% (the strike, set equal to today's 1-year-forward 5-year swap rate — at-the-money) on $10,000,000 notional. Swaption implied volatility is 20%, time to expiry $T = 1$. The annuity, from summing discount factors across the five annual payment dates of the underlying swap, works out to $A = 4.50$ (per $1 of notional, per year — this number comes straight out of the OIS discounting curve from OIS discounting).

Since $F = K = 3.50\%$, $\ln(F/K) = 0$, so $d_1 = \tfrac12 (0.20)^2 (1) / (0.20\sqrt{1}) = 0.02 / 0.20 = 0.10$, and $d_2 = 0.10 - 0.20 = -0.10$. Look up $N(0.10) \approx 0.5398$ and $N(-0.10) \approx 0.4602$. The bracket: $F \cdot N(d_1) - K \cdot N(d_2) = 0.035 \times 0.5398 - 0.035 \times 0.4602 = 0.035 \times (0.5398 - 0.4602) = 0.035 \times 0.0796 = 0.002786$. Multiply by the annuity, 4.50: $0.002786 \times 4.50 = 0.012537$, or 1.2537% of notional. On $10,000,000: $10{,}000{,}000 \times 0.012537 =$ $125{,}370 — the up-front premium for the right to lock in 3.50% fixed a year from now.

Worked example 2 — exercise at expiry, both directions

Fast-forward to the swaption's expiry, one year later. Two scenarios, same swaption, same 3.50% strike, same $10,000,000 notional and annuity of 4.50.

Rates rose: the actual forward swap rate at expiry is 4.20%. The payer swaption is in the money — you get to pay 3.50% fixed when the market rate is 4.20%, saving 0.70% per year for five years. Intrinsic value: $A \times (F - K) = 4.50 \times 0.0070 = 0.0315$, or 3.15% of notional: $10{,}000{,}000 \times 0.0315 =$ $315{,}000. You exercise, and immediately have a swap worth $315,000 more than a market-rate swap.

Rates fell: the actual forward swap rate at expiry is 2.80%. Now the payer swaption is worthless — nobody would pay 3.50% fixed when the market only demands 2.80% — and it expires unexercised, the $125,370 premium a sunk cost. But a receiver swaption with the same 3.50% strike would now be in the money by the same math flipped: $A \times (K - F) = 4.50 \times 0.0070 = 0.0315$, again $315,000. The payer protects against rates rising; the receiver protects against rates falling, and only one of the pair pays off on any given path.

Think of the payer swaption's payoff the same shape as the call payoff above, but with the swap rate on the horizontal axis instead of a stock price, and the strike at 3.50% instead of a dollar level. Below the strike it's worthless; above it, the payoff rises with how far rates have moved past 3.50%, scaled by the annuity.

What this means in practice

Corporate treasurers use payer swaptions to cap borrowing costs without giving up the chance of a better rate if rates fall; pension funds use receiver swaptions to protect the value of their liabilities if rates drop. Banks use both to hedge the optionality embedded in products like mortgages, which effectively contain swaption-like prepayment options. The swaption market is one of the largest and most liquid option markets in the world by notional, precisely because so much of the fixed income world has this kind of embedded, rate-contingent optionality to hedge.

Practice

1. Using worked example 1's numbers but a 15% implied vol instead of 20%, recompute $d_1$, $d_2$, and the swaption premium. (You'll need $N(0.075) \approx 0.5299$ and $N(-0.075) \approx 0.4701$.)
2. A receiver swaption, strike 4.00%, annuity 3.80, notional $5,000,000, expires with the forward swap rate at 3.20%. Is it in the money, and what's its intrinsic value at expiry?