Interest Rate options & FRA

I’m getting confused with the below professors note from Schweser

" For the exam you need to know that a long interest rate call combined with a short interest rate put can have the same payoff as a long position in an FRA"

I can’t put together how a risk reversal on an interest option equals an FRA. Thought the FRA was just rate at exp.less the forward rate.

Thanks in advance

A long interest rate call with a strike of X% has a payoff of:

  • 0%, when the market rate is ≤ X%
  • Market rate – X%, when the market rate > X%

A short interest rate put with a strike of X% has a payoff of:

  • Market rate – X%, when the market rate ≤ X%
  • 0%, when the market rate is > X%

A combination of a long call and a short put, therefore, has a payoff of:

  • 0% + (market rate – X%) = market rate – X%, when the market rate is ≤ X%
  • (Market rate – X%) + 0% = market rate – X%, when the market rate is > X%

Thus, a combination of a long call and a short put has a payoff of:

market rate – X%

The long position on an FRA with a fixed rate of X% has a payoff (at the expiration of the FRA: the beginning of the loan period) of:

PV(market rate – X%)

So, a long position in an FRA has the same payoff as the present value (as of the expiration of the FRA) of the payoff on a long interest rate call + a short interest rate put, both expiring at the end of the FRA loan period, both with a strike equal to the fixed rate in the FRA.

So just to confirm, an FRA has the same payoff as a long Interest rate call and a short interest rate put with the same strike %.

An interest rate collar however, is the combination of an long interest rate call (cap) and a short interest rate put (floor) at different strikes…correct ?

Correct.

(Note: a collar could also be a short put and a long call, again with different strikes.)

What do you mean “short interest rate put”? Do you mean that you are writing a put option, or do you mean you are actually sellig short a put option?

Writing a put option.