LIBOR Market Model Dynamics
Consider a zero coupon bond numeraire whose maturity coincides with the maturity of the forward rate. The measure associated with is called forward measure. Terminal measure is a forward measure where the maturity of the bond numeraire matches the terminal date .
For brevity, we discuss the one-factor LMM only. The one-factor LMM (Brace et al. [1997]) under forward measure can be expressed as
If , (3a)
If , (3b)
If , (3c)
where is a Brownian motion.
There is no requirement for what kind of instantaneous volatility structure should be chosen during the life of the caplet. All that is required is (see Hull-White [2000]):
(4)
where denotes the market Black caplet volatility and denotes the strike. Given this equation, it is obviously not possible to uniquely pin down the instantaneous volatility function. In fact, this specification allows an infinite number of choices. People often assume that a forward rate has a piecewise constant instantaneous volatility. Here we choose the forward rate has constant instantaneous volatility regardless oft (see Brigo-Mercurio [2006]).