Varadhan's formula, conditioned diffusions, and local volatilities

2013/11/06 by De Marco, Stefano, Friz, Peter
#60H30 #65C30 #91G20 #91G80 #Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Pricing of Securities (q-fin.PR) #Probability (math.PR)

paper · doi:10.48550/arxiv.1311.1545

Abstract

Motivated by marginals-mimicking results for Itô processes via SDEs and by their applications to volatility modeling in finance, we discuss the weak convergence of the law of a hypoelliptic diffusions conditioned to belong to a target affine subspace at final time, namely L(Z_t|Y_t = y) if X_⋅=(Y_⋅,Z_⋅). To do so, we revisit Varadhan-type estimates in a small-noise regime (as opposed to small-time), studying the density of the lower-dimensional component Y. The application to stochastic volatility models include the small-time and, for certain models, the large-strike asymptotics of the Gyongy-Dupire's local volatility function. The final product are asymptotic formulae that can (i) motivate parameterizations of the local volatility surface and (ii) be used to extrapolate local volatilities in a given model.

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