# Inflation Indexed Caplet/Floorlet

Can someone explain what is it with $$\psi_{i}$$ (year fraction in $$[T_{i-1},T_{i}]$$). The formula in Mercurio (2006) as is follows:

$$N\psi_{i}P_{n}(t,T_{i})\mathbb{E}_{n}^{T_{i}}\left[\left(\omega\left(\frac{I(T_{i})}{I(T_{i-1})}-K\right)\right)^{+}|\mathcal{F}\right]$$

If inflation index series are monthly, let's say $$T_{i-1}$$ is month 1 and $$T_{i}$$ is month 2, then should I multiply by 1/12 to get caplet value? This just doesn't make sense to me. In the end, this is an option, not a swap or something else to get rates as annual etc.

Best,

## 1 Answer

The formula above is usually the price for a year-on-year inflation indexed caplet, so the $$\psi_i$$ will be the day count fraction over periods $$[T_{i-1},T_i]$$ where these $$i$$'s index the year not the inflation index month. Therefore the $$\psi_i$$ should be close to $$1.0$$ since the day count will always be for successive years. You could use this formula for successive months, a month-on-month inflation index caplet/floorlet but these are probably not what you mean if you are talking about the commonly traded options in the inter-dealer market.

Think of this caplet as a forward starting option at time $$T_{i-1}$$ until $$T_i$$, the cap then becomes the sum of a series of caplets.