Interest Rate options & FRA

[m_onesti3]

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

 

[S2000magician]

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.

 

[kos]

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 ?

 

[S2000magician]

kos wrote: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.)

 

[ajb1]

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?

 

[S2000magician]

ajb1 wrote: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.