The Term Structure of Volatility: Parametric and Piecewise-Constant Structures

Black's caplet formula shows that the price of a caplet on $F_k$ depends only on the root-mean-square vol $v_k = \sqrt{\frac{1}{T_{k-1}}\int_0^{T_{k-1}}\sigma_k^2(t)\,dt}$. But the instantaneous vol function $\sigma_k(t)$ has many degrees of freedom — many different $\sigma_k(t)$ functions give the same $v_k$ and therefore the same caplet price. The choice of $\sigma_k(t)$ matters because it determines: (a) the term structure of volatility — how the caplet vol $v_k$ evolves over time as some caplets expire; and (b) the terminal correlation between different forward rates — relevant for swaption pricing. This lesson characterises the piecewise-constant and parametric volatility structures of the LFM, identifies which ones generate realistic hump-shaped vol term structures that persist over time, and derives the conditions for time-homogeneity (stability of the vol structure as the curve "rolls down").