Why Inflation Swaplets Can Have a Convexity Adjustment
Summary
The document compares the payoff of a year-on-year inflation swaplet, based on the ratio of two CPI levels, with a total return swaplet based on an asset price ratio. Although the payoffs have a similar form, their models need not produce the same convexity adjustment. In an inflation market model, separate forward CPI indices can have imperfect correlations over the period before the first fixing. Those correlations can create an adjustment to the expected ratio.
For an asset total return swaplet under deterministic rates and dividends, the forwards for the same asset are perfectly correlated over that interval in the described diffusion setting, so this source of convexity adjustment is absent. Stochastic interest rates can introduce a separate adjustment through the change of pricing measure between forward maturities. These conclusions rely on the stated model assumptions. The explanation also notes why market models are used for inflation: CPI itself is not a tradable asset, and modeling quoted forwards can better fit derivative prices.
Key ideas
- Similar ratio payoffs do not imply identical convexity adjustments across asset classes.
- Imperfect correlations among separately modeled CPI forwards can create an inflation convexity adjustment.
- Under deterministic rates and dividends, forwards on the same diffusive asset are perfectly correlated in the described setting.
- Stochastic interest rates can produce an additional adjustment through changes in pricing measure.
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Full text
# convexity adjustment in YOY inflation swap , compared with TRS, and considering autocorrelation # 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! ## Answer by Antoine Conze (score 2) https://quant.stackexchange.com/a/34304 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. The reason for using a market model approach for inflation is that the CPI itself is not a tradable asset anyway so we do not need a model that would start from the CPI as a state variable, and the market model provides a better fit to quoted derivatives. 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.
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