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]).