Computing the implied volatility in local and stochastic volatility models

Henri Berestycki* (Ecole des Hautes Etudes en Sciences Sociales)

Abstract.
Determining the implied Black-Scholes volatility for call or put options in
local and stochastic volatility models is a challenging theoretical problem
as well as one of practical relevance, for instance in calibration issues. I
present here a joint work with Jerome Busca and Igor Florent, in which we
develop a PDE approach to directly compute this volatility. A nonlinear
parabolic equation is derived for the implied volatility. It reveals some
singular limits elucidating in particular the structure of implied
volatility near expiry. I will also present a recent joint work with Jerome
Busca where this approach is used to determine the implied volatility of
long options.