Large deviations  for some fast stochastic volatility models by viscosity methods

Martino Bardi; Annalisa Cesaroni; Daria Ghilli

doi:10.3934/dcds.2015.35.3965

Article Contents

2015, Volume 35, Issue 9: 3965-3988. Doi: 10.3934/dcds.2015.35.3965

This issue Previous Article Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets Next Article State constrained $L^\infty$ optimal control problems interpreted as differential games

Large deviations for some fast stochastic volatility models by viscosity methods

1.
Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35133 Padova

Received: April 30, 2014

Revised: August 31, 2014

Published: May 2017

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Abstract

We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenization and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity.

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Mathematics Subject Classification: Primary: 93C70, 60F10, 49L25, 91G20, 35B25.

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