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Marginal density expansions for diffusions and stochastic volatility, part I: Theoretical Foundations

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  • J. D. Deuschel
  • P. K. Friz
  • A. Jacquier
  • S. Violante

Abstract

Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, at 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. Our small noise expansion allows for a "second order" exponential factor. As application, new light is shed on the Takanobu--Watanabe expansion of Brownian motion and Levy's stochastic area. Further applications include tail and implied volatility asymptotics in some stochastic volatility models, discussed in a compagnion paper.

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File URL: http://arxiv.org/pdf/1111.2462
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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1111.2462.

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Date of creation: Nov 2011
Date of revision: May 2013
Handle: RePEc:arx:papers:1111.2462

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Web page: http://arxiv.org/

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  1. Jim Gatheral & Antoine Jacquier, 2011. "Convergence of Heston to SVI," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1129-1132.
  2. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480.
  3. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12.
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Cited by:
  1. John Armstrong & Martin Forde & Matthew Lorig & Hongzhong Zhang, 2013. "Small-time asymptotics for a general local-stochastic volatility model with a jump-to-default: curvature and the heat kernel expansion," Papers 1312.2281, arXiv.org, revised Jun 2014.
  2. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2014. "Asymptotics for $d$-dimensional L\'evy-type processes," Papers 1404.3153, arXiv.org.
  3. Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
  4. Stefano De Marco & Peter Friz, 2013. "Varadhan's formula, conditioned diffusions, and local volatilities," Papers 1311.1545, arXiv.org.

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