Year Fractions in Year-on-Year Inflation Caplet Pricing

Summary

The document clarifies the role of the year fraction in a pricing formula for an inflation-indexed caplet or floorlet. In the usual market context, the formula is for a year-on-year option: the dates index successive years, so the day-count fraction is typically close to one. Monthly observations in the inflation index do not, by themselves, mean the option period is one month.

The same formula could describe a month-on-month option if its dates were set to successive months, in which case the fraction would reflect that shorter accrual period. The answer cautions that such options are probably not what is meant by commonly traded inter-dealer inflation options. It also frames each caplet as a forward-starting option over one period, with a cap represented as a sum of caplets. The discussion is conceptual and gives no derivation, numerical example, or treatment of alternative conventions, so practitioners would need to check the relevant contract and day-count conventions.

Key ideas

• The year fraction measures the option period specified by the dates in the pricing formula.
• In the usual year-on-year inflation caplet, successive dates represent years, so the fraction is near one.
• Monthly inflation index observations do not automatically make the caplet period monthly.
• A cap can be viewed as a sum of forward-starting caplets over successive periods.

Tags

# Inflation Indexed Caplet/Floorlet


# 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,

## Answer by BrownianBread (score 2)

https://quant.stackexchange.com/a/51436

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.