Multiscale Stochastic Volatility Models

Multiscale stochastic volatility models were first introduced by Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar around 1999. At that time, the model assumed that only one fast time-scale governed the evolution of the stochastic volatility of a financial asset. The most recent version of the model assumes two time-scales, one slow and one fast when compared to the usual maturities of derivatives on the given asset. 

 

Such models lead to a first-order approximation of derivatives prices. This approximation is composed by the leading-order term given by the Black-Scholes price with the averaged effective volatility and the first-order correction only involves Greeks of this leading-term.

 

In terms of implied volatility, this perturbation analysis translates into an affine approximation in the log-moneyness to maturity ratio (LMMR). Subsequently, this leads to a simple calibration procedure of the group market parameters, that are also used to compute the first-order  approximation of the price of exotic derivatives.

 

My research contributions in this topic are:

  • perturbation analysis to price of derivatives on futures using an inversion argument. Differently from previous methods, this leads to a simple calibration procedure and drops some unnecessary smoothness assumptions.

  • application of Functional Itô Calculus to perturbation analysis of path-dependent derivatives.