Swaption

A swaption is an option to enter an interest-rate swap on stated terms at a stated future date. It is among the most heavily traded interest-rate options, and it is what a treasurer reaches for when an issuance has to be hedged before it prints.

A payer swaption gives the holder the right to enter as the fixed payer at strike K. It gains value when swap rates rise, so it is the instrument for anyone short of rate protection: a future borrower, a bond issuer, a portfolio that will need to shorten duration.

A receiver swaption gives the holder the right to enter as the fixed receiver at K. It gains value when rates fall, so it is the instrument for a future lender, and for anyone who has sold a call on a bond they hold.

Two conventions have to be read off the term sheet before any price means anything. 1y × 5y means the option expires in one year and the swap that follows runs five years from that expiry. The first number is the option; the second is the underlying. And exercise style in this market is usually European, exercisable only on the expiry date, though Bermudan swaptions exercisable on a schedule of dates exist and are what a callable bond's embedded option really is.

How is a swaption priced?

The pricing follows the Black model once you identify the two right objects.

The underlying is the forward swap rate: the fixed rate that makes a swap starting at the option's expiry worth zero today. Off a discount curve with payment dates T1 … Tn starting after expiry T0,

S = [ D(T0) - D(Tn) ] / A        where   A = sum of tau_i · D(Ti)

The numeraire is the annuity A, the present value of one unit of fixed rate paid on the swap's schedule. It is the natural discount factor here because the swap's own value is N·(S - K)·A, so measuring everything in annuity units makes the payoff linear in the rate.

Put those together and the payer swaption is

payer    = N · A · [ S·N(d1) - K·N(d2) ]
receiver = N · A · [ K·N(-d2) - S·N(-d1) ]

with d1 and d2 built from S, K, the option's expiry T and the swap-rate volatility sigma. The annuity has already discounted to today, so no separate discount factor appears.

Payer-receiver parity falls out the same way cap-floor parity did:

payer - receiver = N · A · (S - K) = the value of a forward-starting payer swap at K

which is again model-free in the volatility, and again the fastest check on an implementation. When K = S, the two options cost exactly the same, which is what "at the money" means for a swaption and is not the same as "at today's spot swap rate".

How are swaptions settled and quoted?

Settlement comes in two styles and the choice is economic, not administrative. Physical settlement delivers the actual swap on exercise, which is what a hedger of a real exposure wants. Cash settlement pays the value of that swap in a single amount, computed off an agreed formula, which is what someone taking a rate view wants because it avoids carrying a swap they never wanted. The values are the same at exercise; the residual position is not.

One quoting convention has to be converted before it can be argued with. Swaption premiums are quoted in basis points of notional, up front. The payer swaption in Worked example 2 costs 180.04 basis points of notional, which sounds enormous until it is spread over the five-year annuity of the underlying swap, where it is 43.63 basis points a year. Converting a premium into a running spread is the only way to compare an option hedge against a swap hedge honestly, and it is the calculation most treasury memos skip.

What swaptions are actually used for

Three uses cover most of the market, and each has a distinct shape.

The first is pre-hedging an issuance. A company that will issue fixed-rate debt in six or twelve months is short the rate: if yields rise before it prints, its coupon rises with them. Two hedges are available. A forward-starting payer swap locks the rate now at the forward, costs nothing, and gives away every basis point of a fall. A payer swaption costs a premium and protects only above the strike, leaving the fall intact.

The comparison is not close in the way people assume. Take the module's 1y × 5y swaption struck at 5.25%, premium $900,201.39, against a forward-starting swap struck at the forward of 5.126616% for no premium, both on $50 million.

The swaption's loss is capped at the premium, future-valued to expiry as $942,577.84, and it wins in every large fall. The swap wins whenever rates rise at all and in the whole middle of the distribution. The two are worth the same at a rate of 4.694859%. A treasurer who says "the option is safer" has to mean safer against a specific outcome, and the outcome the option protects against is the one where rates fall and the company is glad it did not lock.

How do swaptions hedge callable debt?

The second use is hedging the short call inside callable debt. An issuer of a callable bond has bought the right to redeem early, which means the holder sold it, and the issuer calls when rates fall. An investor holding that bond has sold an option and is short exactly the payoff a receiver swaption has. Buying a receiver swaption of matching expiry and tenor reconstructs the option the investor gave away, and its premium is the honest market price of the option cost that a tree-based option-adjusted spread computes as Z-spread minus OAS. Practice item P15 prices both sides of that comparison and converts the premium into an annual running spread so the two numbers are commensurable. When the swaption market's price of the option and the model's option cost disagree materially, one of the two is wrong, and the model is the one with an assumed volatility in it.

How do swaptions trade rate volatility?

The third use is a pure position in rate volatility. Buy a payer and a receiver at the same strike, set at the forward, same expiry, same underlying swap. The two deltas very nearly cancel, leaving 9.5517% of a forward swap's rate sensitivity at inception, and what remains is a bet that the swap rate will move more than the premium implies. The breakeven has a pleasant closed form:

breakeven absolute move in the swap rate = straddle premium / (N × A)

The forward annuity cancels between premium and payoff, so the answer is just the premium expressed in annuity units. On the module's numbers the straddle costs $2,020,678.01 and the breakeven is 97.9356 basis points, meaning the five-year swap rate has to finish outside 4.147259% to 6.105972% for the buyer to profit. The standard rule of thumb, 0.7979 × sigma × sqrt(T) × S, gives 98.1707 basis points, within 0.24% of the exact answer, which is why the rule of thumb survives.