Stochastic-Volatility Models

Abstract

LIBOR models with stochastic volatility are extensions of the LFM where the instantaneous volatility of relevant rates evolves according to a diffusion process driven by a Brownian motion that is possibly instantaneously correlated with those governing the rates’ evolution. Formally, the general forward rate F_j is assumed to evolve under its canonical measure Q^j according to

$$ \begin{gathered} dF_j (t) = a_j (t)\varphi (F_j (t))[V(t)]^\gamma dZ_j (t), \hfill \\ dV(t) = a_V dt + b_V dW(t), \hfill \\ \end{gathered} $$

(11.1)

where a_j and ϕ are deterministic functions, γ ∈ {1/2, 1}, a_V and b_V are adapted processes, and Z_j and W are possibly correlated Brownian motions.