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Annals of Finance

, Volume 15, Issue 1, pp 59–101 | Cite as

Extreme-strike asymptotics for general Gaussian stochastic volatility models

  • Archil Gulisashvili
  • Frederi ViensEmail author
  • Xin Zhang
Research Article
  • 35 Downloads

Abstract

We consider a stochastic volatility asset price model in which the volatility is the absolute value of a continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen–Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the asset price density for large or small values of the variable, and study the wing behavior of the implied volatility in these models. Our main result provides explicit expressions for the first three terms in the expansion of the implied volatility, based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the \(L^{2}\) norm of the projection of the mean function on the top eigenspace. Numerical illustrations using the Stein–Stein and fractional Stein–Stein models are presented, including strategies for parameter calibration.

Keywords

Stochastic volatility Implied volatility Large strike Karhunen–Loève expansion Chi-squared variates 

Mathematics Subject Classification

60G15 91G20 40E05 

JEL Classification

C6 G13 

Notes

Acknowledgements

We are grateful to the associate editor and the anonymous referee for their valuable suggestions and remarks, which significantly contributed to improving the paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsOhio UniversityAthensUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA

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