Motivated by Gyongy’s (86) projection result for Ito SDEs and by its applications in the context of volatility modeling in finance, we discuss some large deviation results for densities of diffusions in a small-noise regime, focusing on marginals (i.e. the densities of components of the process) and on the law of the diffusion conditioned to be in an affine subspace at final time.
For stochastic volatility models enjoying certain scaling properties (such as the Stein-Stein model), this eventually leads to the large-strike asymptotic behavior of their ‘equivalent’ local volatility function.
The asymptotic value can be used to patch the typical numerical instabilities affecting the local volatility extracted from Dupire’s formula.
In models with known moment generating function (Heston), the asymptotic result can be obtained from a careful application of the saddle point method.