# 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?

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.

• So if for example $T_i$ is one year from now, and $T_{i+1}-T_i$ is 3M and I have a standard cap volatility surface from BBG that has dimension expiry x strike, which point should I be looking at? Should this be the 1 year expiry and ATM? – TDC Mar 27 '18 at 13:48
• It's a 3m caplet with expiration 1yr. If bberg has only cap volatilities , such as 1 year cap and 2yr cap, you have to do some work to extract a caplet vol. that's because a 1yr cap is a portfolio of caplets expiring in 3m,6m and 9m, and the 2yr cap has all those plus the ones expiring in 1yr, 15m, 18m and 21m. So you have to come up with a term structure of caplet vols which recalibrates the cap surface. Not too difficult. Yes, you can use ATM vols, although theoretically it does depend on the whole skew structure at a second order level. – dm63 Mar 27 '18 at 14:35