# Derive the convexity adjustment for inflation YoY swap with unconventional payoff

I'm trying to solve for the convexity adjustment for an inflation YoY swap with unconventional payoff, where $$I_i$$ is CPI at time i:

$$Notional * ([I_i/I_{i-1}]^{Day Count Fraction} - 1)$$

In the normal case, the payoff is: $$Notional * Day Count Fraction * (I_i/I_{i-1} - 1)$$ And we can show using the Jarrow-Yildirim model that the PV is as follows:

with convexity adjustment (e^D_i (t) ), where

However I am struggling with the payoff raised to the day count fraction instead of multiplied by day count fraction. How can I deal with this raised power in the convexity adjustment derivation?

• This is unnatural. Are you sure this is the payoff ?
– dm63
Commented Nov 24, 2023 at 3:55
• yes the payoff is accurate. Commented Nov 26, 2023 at 3:01
• What is the date of the payoff ? If the yoy swap is annual pay , then day count fraction is 1, so the convention is the same.
– dm63
Commented Nov 26, 2023 at 3:33
• the coupons pay quarterly, so the day count fraction for each coupon payment T_i will be from [T_i, 3m] Commented Nov 28, 2023 at 1:09