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Generalized sub-Riemannian cut loci and implied/local volatility smiles

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Density expansions for hypoelliptic diffusions $(X1,...,Xd)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T1,...,X_Tl)$, at fixed time $T>0$, with $l \leq d$. Global conditions are found which replace the well-known “not-in-cutlocus” condition known from heat-kernel asymptotics (Molchanov, Bismut, Ben Arous, ...). Our small noise expansion allows for a “second order” exponential factor, not present in small time expansions.

Applications include tail and implied/local volatility asymptotics in some correlated stochastic volatility models. In particular, we are able to analyze the Stein—Stein model in case of negative correlation (the typical case in equity markets), thereby solving a problem left open by A. Gulisashvili and E.M. Stein.

Joint work with Deuschel, Jacquier and Violante.

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