There are a number of volatilities defined differently under the LIBOR Market Model:
1. Forward volatility (of a cap): v(T_j)_cap
a single volatility for each cap that makes the cap value (which is a sum of caplet values) agree with the market price. Aka "flat volatility."
2. Forward forward volatility (of a caplet): v(T_j)_caplet
a set of volatilites, different for each caplet, that makes the cap value agree with the market price. Can be bootstrapped from the forward volatility.
3. Instantaneous volatility (of a forward LIBOR dynamic): sigma
the 'sigma' that appears in the SDE of a certain LIBOR. Can be used to parametrize the forward forward volatility.
4. Average volatility (between two points in time): V(T_j,T_k)
a volatility of which forward forward volatility is a special case. V(0,tau)=v(tau)_caplet
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