to find out the spot vol from known flat volsuppose we have one 1-yr cap, 3M resetassuming vol for the 4 caplets are v1,v2,v3,v4, and flat vol is Vwe have to meetpremium(v1)+premium(v2)+premium(v3)+premium(v4) = premium(V)however, i am confused when calculating premium(V)i know the same V will be used for the 4 caplets, but how about maturity here?is the same maturity as well? i.e. 0.25yr or i have to use 0.25, 0.5,0.75, 1 respectively?

Quoteis the same maturity as well? Since your 1y cap consists of the caplets with different maturities you will have to use your maturities of 0.25,0.5,0.75. Also (unless the yield curve is flat) you will need to use a different forward rate for each caplet.BTW on most currencies the 1yr cap is not composed of 4 caplets but by three: the rate is already known for the first period so convention for most markets is to ignore the first caplet, so the 1y covers the periods:3mx6m (expiry in around 0.25 years)6mx9m (expiry in around 0.50 years)9mx12m (expiryin around 0.75 years).Unless you are on one of those notable currencies where the first caplet is used in which case you will compute its intrinsic value (I think KRW is one of those)

thx gcif spot vol are used to price caplets, i understand that the maturity used should be 0.25, 0.5, 0.75but if flat vol are used to price caplets, should the maturity for all caplets to be 0.25????

Quotebut if flat vol are used to price caplets, should the maturity for all caplets to be 0.25????No: they are still the same caplets Flat vols is simply a market convention by which the same volatility is used for all caplets belonging to the cap in consideration.