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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,

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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.

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