Choosing Volatility Inputs for Libor-in-Arrears Convexity Adjustments

Summary

The document addresses which implied volatility to use in a convexity adjustment when valuing a Libor-in-arrears swap cash flow. The relevant volatility comes from an option whose expiry matches the rate reset date and whose underlying rate covers the accrual period beginning at that date.

For a short underlying rate period, such as a typical three- or six-month tenor, the suggested input is cap volatility. For longer periods, around a year or more, swaption volatility is indicated. The guidance identifies the appropriate instrument type by the rate tenor, but does not provide a worked numerical example, interpolation method, or further detail on market conventions and calibration. Users therefore still need to select a market quote consistent with the precise reset and underlying rate period in their swap.

Key ideas

• Match the volatility expiry to the rate reset date used in the in-arrears cash flow.
• The volatility should correspond to the underlying forward rate period beginning at that reset date.
• Cap volatility is suggested for short underlying rate periods.
• Swaption volatility is suggested when the underlying period is longer.
• The guidance does not specify calibration or interpolation details.

Tags

# Which volatility input for in-arrear convexity correction?


# Which volatility input for in-arrear convexity correction?












When pricing a Libor-in-arrear swap, I am using the following formula (for the cashflow covering the period $[T_{i-1}, T_i]$, ie. paid at $T_i$ and resetting at $T_i$):

$V(t) = P(t,T_i)F(t;T_i,T_{i+1})\left(1 + \frac{\tau F(t;T_i,T_{i+1})}{1+\tau F(t; T_i, T_{i+1})}(\exp(\sigma T_{i}) - 1) \right) $

where $P(t,.)$ is the discount factor, $F(t;.,.)$ the forward rate, $\tau$ the year fraction. I'm just not sure where I should get the volatility $\sigma$ from?

I read it should be from capfloors surface, but can someone give a more concrete example?

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/39022

It is the implied volatility of an instrument which has expiration $T_i$ with an underlying rate from $T_i$ to $T_i+1$. If $T_i+1 - T_i$ is a short period such as 3m or 6m, this is a cap volatility. If it is longer (1yr or more) then it is a swaption volatility.

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.