Marginal density expansions for diffusions and stochastic volatility, part I: Theoretical Foundations
AbstractDensity 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.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1111.2462.
Date of creation: Nov 2011
Date of revision: May 2013
Contact details of provider:
Web page: http://arxiv.org/
This paper has been announced in the following NEP Reports:
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 19(1), pages 1-12.
- Jim Gatheral & Antoine Jacquier, 2010.
"Convergence of Heston to SVI,"
- Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 14(3), pages 469-480.
- Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2014. "Asymptotics for $d$-dimensional L\'evy-type processes," Papers, arXiv.org 1404.3153, arXiv.org.
- 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, arXiv.org 1312.2281, arXiv.org, revised Jun 2014.
- Stefano De Marco & Peter Friz, 2013. "Varadhan's formula, conditioned diffusions, and local volatilities," Papers, arXiv.org 1311.1545, arXiv.org.
- Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers, arXiv.org 1303.4268, arXiv.org, revised Aug 2013.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators).
If references are entirely missing, you can add them using this form.