First things first, a swap is just a set of FRAs, so a long swap (fixed payer) is identical to a long FRA; i.e. you long on the interest rate and expect rates to go up (unfortunately, a lot of real life practitioners believe this is incorrect, but we’re in CFA-verse at the moment so we stick to the CFA-verse rules and laws for now)
therefore, we can deduce that an IR cap is an IR call and IR floor is an IR put.
makes sense, if you buy an IR call, you effectively put a ceiling on the IRs. once IRs go above a certain point, you exercise the call and enjoy the strike IR (exercise IR)
if you buy an IR put, you effectively put a floor on the IRs. once IRs go down beyond a certain point, you exercise the put offsetting the IR decline.
for the first equivalence you wrote, this is a matter of put-call parity (you can test this). c - p = S + X (remember that pay fixed is long S) with interest rate being the underlying. 2nd equivalence is just the same thing with different jargons as we established. 4 and 5 is just reverse of 1 and 2. therefore, we can establish 1,2,4,5 with pcp (not that kind of pcp)
swaption is a bit tricky because the underlying is a swap and this is a derivative-ception (a derivative for a derivative). but again, this can be explained using the pcp.
payer swaption is an option to enter a long swap, this is a call option as you gain when interest rates increase (you pay the agreed fixed swap rates but receive the higher floating spot interest rates). receiver swaption is an option to enter a short swap (i.e. you pay floating receive fixed), this is a put option as you gain when interest rates decrease (you receive the agreed fixed swap rates but pay the lower floating spot interest rates). so swaptions is a fancy workaround for caps and floors.
so a long swap is equivalent to a long call with a short put. the same goes for number 6.
for number 7, we can’t use pcp because there are two different underlying here (a bond and a swap), but remember the properties of a callable bond, which is usually called when interest rates decline. note that the call option in the callable is for the bond and not the interest rates, so when interest rates decline, the option is exercised, this is equal to an IR floor owned by the issuer, so it’s a short IR put in your perspective (bondholder). so a callable bond is long straight bond and a short IR put.
now I don’t know how to convert IR put into receiver swaption but you can take the easy way and see that number 2 and 3 is the same thing isn’t it? you can substitute IR put with a receiver swaption by the laws of math??
