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Solved

As pointed by @dm63 in the comments, the implied swap rate can be derived by solving the caplet (floorlet) formula for the the interest rate, where you set the formula equal to the Swap NPV (due to the put-call parity).

Update (Original post below): After a bit of reasearch I have found that the put call parity on caps and floors works as it follows:

cap - floor = swap

Where cap, floor and swap are expressed in their NPV values.

So, using the below model of a fixed rate bond, I could back out the implied YoY swap rate by setting the NPV of the bond equal to the difference cap - floor and solve for the yield to maturity. My doubt at this point is: should the coupon on the bond be equal to zero, so that the yield to maturity would effectively be the average rate expected over the life of the bond?

Original Post: I am trying to get the implied swap rate of cap on a YoYIIS. As per any other option, I wanted to use the put-call parity relationship of the B&S model.

In this case tho the underlying of the option is a swap that pays a the difference between the Year-on-Year inflation and the strike rate. My process then was to:

  1. model a fixed rate bond with coupon rate equal to the strike rate of the YOYIIS swap and maturity equal to the tenor of the swap,
  2. get the NPV of the bond.
  3. get the Yield To Maturity of the bond.

As per any swap, the swap rate should be the rate that set to zero the NPV of the swap, so I assumed that the yield to maturity of a bond equal to the fixed rate leg of the swap would be the implied swap rate (i.e. the implied YoY inflation rate over the tenor of the swap).

Am I wrong in my assumption? Also, should I use the Yield to Maturity calculated as above in the put-call parity expression

                       `S = K(discounted) - p +c`

in place of K(discounted) to get the final implied swap rate (I have the put(floor) and call(cap) prices)? Or the Yield To Maturity itself is the implied swap rate?

Thanks

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  • $\begingroup$ Please mark the answer by @dm63 as correct to indicate that this is solved. $\endgroup$
    – Bob Jansen
    Commented Jul 1, 2022 at 12:40
  • $\begingroup$ @dm63 answer below solved my query. $\endgroup$ Commented Jul 1, 2022 at 14:20
  • $\begingroup$ There is a little check mark you should click then $\endgroup$
    – Bob Jansen
    Commented Jul 1, 2022 at 14:20

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You may be over complicating this by introducing bonds. As you say, in PV terms you have the identity: cap - floor = swap , where swap is the value of a swap where you pay a fixed rate K (equal to the strike rate of the cap and floor) vs yoy inflation. The implied market rate for the underlying inflation swap is then $$ K+ Swap PV/DV01$$, where DV01= the value of a 1bp annuity paid annually over the underlying period. This needs to be calculated separately using nominal discounting.

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    $\begingroup$ dm63, this was helpful. I am a bit rusty with reading throught the pricing formula. I have applied your formula and it works. I went to recover the pricing formula for the cap and the implied swap rate is in fact derived by solving for the value of the interest rate. $\endgroup$ Commented Jul 1, 2022 at 10:46

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