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When pricing an interest rate cap using Black model, price for each caplet is computed and summed to get cap price. Question is, should we use the same volatility for each caplet or use different volatility for each caplet?

No, each caplet has its own vol

Infoproviders publish flat volatilities; caplets volatilities must be bootstrapped.Cheers,Jordy

Each caplet has its own implied forward vol. You can extrapolate or bootstrap .. Though I have seen people using a scaled up flat vol too..

I echo the bootstrap idea... That's what I've always done in the past.

For quoting purposes (or to translate a quoted vol into a price) you generally use a flat vol, but for doing meaningful analytics you need to do some sort of bootstrapping to get at vols for the individual caplets. The main practical problem with this is dealing with vol skews, in particular if all you have to work with is a set of ATM quotes, since ATM forwards for different maturities will not line up. I honestly haven't done this in real life since the 90's, but I doubt the problem has gotten any simpler...

at least for the major currencies there are quotes for fixed strikes like 1%, 2%, ... (or nowadays -1%, -0.5%, 0%, 0.5% ...), which can be used for bootstrapping; you can then place the atm caplets in between those fixed strikes and interpolate in strike direction for bootstrapping from the atm flat volatilityif you price a series of spot caplets similar to the quoted market instrument it won't matter too much, if you use the flat vol for each caplet or the bootstrapped vols (by construction), but think about a situation where you price a single caplet with fixing in 5y, its volatility may have little to do the the average (flat) vol of the caplets with fixings 3m, 6m, 9m, ... 4y, 4y3m, 4y6m, 4y9m, 5y (if we are talking about a 3m index for example)

I don't quite understand the vol skew issue you mentioned. When bootstrapping caplet vol surface, the underlying assumption is that vol will be different for different combination of (maturity, strike) right?QuoteOriginally posted by: bearishFor quoting purposes (or to translate a quoted vol into a price) you generally use a flat vol, but for doing meaningful analytics you need to do some sort of bootstrapping to get at vols for the individual caplets. The main practical problem with this is dealing with vol skews, in particular if all you have to work with is a set of ATM quotes, since ATM forwards for different maturities will not line up. I honestly haven't done this in real life since the 90's, but I doubt the problem has gotten any simpler...

you got it... remember , an ATM Cap will use the same strike for all caplets which is likely to be diverging from the ATM for each caplet; i.e. the ATM strike for a 5y Caplet on 6m ibor , for example, will be the 5Y swap (with 6m projection)