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Probability Distribution in the SABR Model of Stochastic Volatility Read chapter Read first chapter

2015 | OriginalPaper | Chapter

The Gärtner-Ellis Theorem, Homogenization, and Affine Processes

Authors : Archil Gulisashvili, Josef Teichmann

Published in: Large Deviations and Asymptotic Methods in Finance

Publisher: Springer International Publishing

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Abstract

We obtain a first order extension of the large deviation estimates in the Gärtner-Ellis theorem. In addition, for a given family of measures, we find a special family of functions having a similar Laplace principle expansion up to order one to that of the original family of measures. The construction of the special family of functions mentioned above is based on heat kernel expansions. Some of the ideas employed in the paper come from the theory of affine stochastic processes. For instance, we provide an explicit expansion with respect to the homogenization parameter of the rescaled cumulant generating function in the case of a generic continuous affine process. We also compute the coefficients in the homogenization expansion for the Heston model that is one of the most popular stock price models with stochastic volatility.

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previous chapter Extrapolation Analytics for Dupire’s Local Volatility
next chapter Asymptotics for -Dimensional Lévy-Type Processes
Footnotes
1
Note that the topology of \({\widehat{\mathcal {D}}}\) enters our assumptions in a subtle way: We require later that X is càdlàg on \({\widehat{\mathcal {D}}}\), which is a property for which the topology matters.
 
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Metadata
Title
The Gärtner-Ellis Theorem, Homogenization, and Affine Processes
Authors
Archil Gulisashvili
Josef Teichmann
Copyright Year
2015
Book
Large Deviations and Asymptotic Methods in Finance

Print ISBN: 978-3-319-11604-4

Electronic ISBN: 978-3-319-11605-1

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-11605-1_11