# convexity adjustment in YOY inflation swap , compared with TRS, and considering autocorrelation

a YOY inflation swaplet payoff is S2/S1 - 1 , where Si is the CPI at time i and a TRS (total return swaplet) asset leg payoff is also the same except the underlying is an asset.

So it seems to me that the modelling for both should be identical, assuming you use the same model for each.

but i understand that typically , in TRS modelling S is GBM , with drift and vol dependent on t so E(S2/S1) = E(S2)/E(S1)=F2/F1 (ie no convexity adj) whereas in inflation , there is or is NOT a convexity adjustment , dependent on the model used!

i do not understand this for inflation - surely it cannot be that there can be 2 answers!

You get a convexity adjustment from forward correlations only if you model separately the forwards and they are not perfectly correlated on the time interval $[0, T_1]$, as is the case in inflation market models where each forward CPI index is modelled separately from the others, with a global instantaneous correlation structure, not set to identity, similar to the correlation structure in the Libor Market Model.
In the case of a TRS swaplet where $S_1$ and $S_2$ represent the prices of the same asset on times $T_1$ and $T_2$, and assuming deterministic rates and dividends and a diffusion process for the asset price $S_t$, the forwards are perfectly correlated on the time interval $[0, T_1]$ so there is no convexity adjustment that would come from decorrelation.
There is of course another convexity adjustment that arises when rates are stochastic because the expectation is computed under the $T_2$ terminal measure and the first forward is a martingale only under the $T_1$ measure.