# Payoff of Interest Rate Options

Can someone explain to me how interest rate options pay off. Take an example of a option on 90 Day LIBOR. Say the strike is 5%, LIBOR is 6% at expiration, and the notional is \$10MM Is it: a) At expiration, (6% - 5%)*(90/360)*\$10MM = \$25,000 or b) At expiration, the present value of \$25,000 paid in 90 days. In the reading on interest rate derivatives, it suggests that it is a), but in the reading on options, the Black model calculates it as b (although not exactly). Can someone clarify this for me because I’m stumped!

the simple interest rate call/put (sometimes called interest rate guarantee IRG) is settled as FRA, two days after fixing date (expiry date) discounted interest difference. interest rate cap and floor is settled as IRS, the interest difference is settled at the end of relevant interest rate period. it is common pratice the interest is supposed to be paid at the end of the interest rate period, that is how the interest rates are quoted. therefore if you want to move payment backwards you need to discount it, always. (for instance FRA) FRA and IRG are settled upfront because they have just one interest rate period, you (as buyer) dont need to wait for settled next couple of months. it decreases possible credit risk.

But it doesn’t read like that in the reading on interest rate derivatives. It specifically says, the difference between the reference rate and the strike rate on “settlement date” and not “expiration date”. They don’t calculate present values in any of the examples in this reading.

I would say that you will see payment at the end of interest period usually. As for Black model, there is discounting due to the fact that you want to know value of premium that would be paid now…

It’s always the PV value that’s accurate at expiration, if that’s what you are asked to compute. It’s also possible to talk about it as a FV of the amount due…it just depends on what the question is asing. Your question above didn’t ask at what point in time, although you should always assume the present time.

Hi Dreary, what your saying to me makes total sense. The problem is, it’s not worked out like that in the examples CFAI give on interest rate derivatives. It’s just plain: C = [Max 0, Strike - Reference*(t/360)]* notional. And this is on settlement date.

Check page 224 of derivatives, example 12. They specifically say that the formula assumes the payoff at expiration, which they say is false, and that it should be discounted to the present, which is what we expect.