Options and volatility · O10 · 9 min
Options across asset classes
Name a currency option, value a swaption or an interest rate cap at expiry, and explain why barrier options are cheaper.
In the news: “Borrowers rushed to buy caps before the rate decision”: options on interest rates work like options on shares.
The essentials
- Currency options
- Every currency option is a call on one currency and a put on the other, so traders name both: a EUR call / USD put is the right to buy euros with dollars. Struck at 1.05, it is in the money with EUR/USD at 1.08.
- Caps and floors
- A pays a borrower whenever a floating rate such as fixes above its strike; a floor pays when the rate fixes below. Selling a floor to pay for a cap makes an interest rate collar.
- Swaptions
- A is the right to enter an later at a rate agreed now. A payer (pay fixed) gains when rise, a receiver when they fall; a payer on gains as spreads widen.
- What a swaption pays
- The rate gap for every year of the swap, in today’s money: a payer struck at 4%, with the swap rate at 4.6%, an of 4.38 and £100m, pays about £2.63m.
- Barrier options
- A switches on or off at a set level: a knock-out dies if the price touches it; a knock-in comes alive only if it does. Knock-outs cost less, but the protection can vanish in the spike it was bought for.
Rates options at expiry
Payer swaption = max(S − K, 0) × annuity × N; caplet = max(L − K, 0) × N × days ÷ 365
- S
- the swap rate at expiry
- K
- the strike rate
- annuity
- present value of 1 a year over the swap’s life, in years
- N
- notional
- L
- the floating rate for the period, such as compounded SONIA (ACT/365)
How it works
1. Currency options
The right to buy euros with dollars is also the right to sell dollars for euros. So every currency option is a call on one currency and a put on the other, and traders name both: a EUR call / USD put. That avoids any doubt about which way the trade goes.
2. Moneyness in currencies
works as it does for shares: a EUR call / USD put struck at 1.05 is in the money with EUR/USD at 1.08, because buying euros at 1.05 beats the market rate. A call’s strike can sit below today’s rate. Premiums are quoted in or as a % of the amount.
3. Caps on interest rates
A borrower paying a floating rate such as fears rates rising. A pays them whenever the rate for a period comes in above a strike. With a 4% cap on £20m and SONIA at 4.6% for 91 days, it pays 0.6% × £20m × 91 ÷ 365 = £29,918.
4. Floors
A floor is the mirror image: it pays when the rate comes in below the strike, so lenders and investors use it to protect their income. A borrower can sell a floor to pay for a cap, giving up savings below the floor: a collar on interest rates.
5. Swaptions
A is the right to enter an later, at a fixed rate agreed now. A payer swaption gives the right to pay fixed, so it gains when swap rates rise. A receiver gives the right to receive fixed, and gains when they fall.
6. What a swaption pays
At expiry a payer swaption is worth the rate gap for every year of the swap, in today’s money: (swap rate − strike) × annuity × notional. The is the value today of 1 a year over the swap. Struck at 4%, with the swap rate at 4.6%, an annuity of 4.38 and £100m: 0.6% × 4.38 × £100m = £2.628m.
7. Options on credit
Options on work the same way, with a in place of a rate. A payer is the right to buy protection at a set spread and gains if spreads widen; a receiver gains if they tighten.
8. Barrier options
A switches on or off if the price touches a set level. A knock-out dies if the barrier trades; a knock-in comes alive only if it does. A call struck at 100 that knocks out at 120 is worthless after a rally to 121, just when a plain call would pay most.
9. Why barriers are cheaper
Because a knock-out can die, it pays out in fewer cases than a plain option, so it costs less. A knock-in plus the matching knock-out is worth exactly the plain option. The catch is that the protection can vanish in exactly the spike it was bought for.
Worked example
A UK company will borrow £100m for five years in six months’ time and fears rates rising before then. It buys a six-month payer swaption on a five-year SONIA swap, struck at 4.00%. At expiry the five-year swap rate is 4.60% and the annuity is 4.38.
- Swap rate minus strike: 4.60% − 4.00% = 0.60% a year.
- As a decimal: 0.0060.
- Times the annuity: 0.0060 × 4.38 = 0.02628.
- Times £100m: about £2.63m.
- Had rates fallen instead, the swaption would expire worthless and the company would borrow at the lower rate, having paid only the premium.
About £2.63m: the swaption pays the rate rise for every year of the swap, in today’s money.
Check yourself
9 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
A borrower fears rising rates but wants to benefit if they fall. What do you sell them?
What’s a payer swaption?
Why are knock-out options cheaper, and what’s the catch?
Remember
Payer = max(S − K, 0) × annuity × N
Rates, FX, credit and commodity options follow the same rules as equity options; only the underlying and the naming change.