Marginal density expansions for diffusions and stochastic volatility, part II: Applications [to the Stein--Stein model]

TL;DR

This paper applies advanced density expansion techniques for multidimensional diffusions to financial models, providing tail and implied volatility asymptotics, and resolving an open problem in stochastic volatility modeling.

Contribution

It extends the density expansion framework to financial applications, specifically stochastic volatility models, and addresses an open problem in implied volatility asymptotics.

Findings

Derived tail and implied volatility asymptotics for stochastic volatility models

Solved an open problem by Gulisashvili and Stein (2009)

Provided a framework for applying density expansions in finance

Abstract

In the compagnion paper [Marginal density expansions for diffusions and stochastic volatility, part I] we discussed density expansions for multidimensional diffusions $(X^1,...,X^d)$, at fixed time $T$ and projected to their first $l$ coordinates, in the small noise regime. Global conditions were found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptotics. In the present paper we discuss financial applications; these include tail and implied volatility asymptotics in some correlated stochastic volatility models. In particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).