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A family of density expansions for L\'evy-type processes

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  • Matthew Lorig
  • Stefano Pagliarani
  • Andrea Pascucci

Abstract

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Levy measure. Generalizing and extending the novel adjoint expansion technique of Pagliarani, Pascucci, and Riga (2013), we derive a family of asymptotic expansions for the transition density of the underlying as well as for European-style option prices and defaultable bond prices. For the density expansion, we also provide error bounds for the truncated asymptotic series. Our method is numerically efficient; approximate transition densities and European option prices are computed via Fourier transforms; approximate bond prices are computed as finite series. Additionally, as in Pagliarani et al. (2013), for models with Gaussian-type jumps, approximate option prices can be computed in closed form. Sample Mathematica code is provided.

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File URL: http://arxiv.org/pdf/1312.7328
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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1312.7328.

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Date of creation: Dec 2013
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Handle: RePEc:arx:papers:1312.7328

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Web page: http://arxiv.org/

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  1. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
  2. Peter Christoffersen & Kris Jacobs & Chayawat Ornthanalai, 2009. "Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options," CIRANO Working Papers, CIRANO 2009s-34, CIRANO.
  3. Bjørn Eraker, 2004. "Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices," Journal of Finance, American Finance Association, American Finance Association, vol. 59(3), pages 1367-1404, 06.
  4. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Pricing approximations and error estimates for local L{\'e}vy-type models with default," Papers 1304.1849, arXiv.org, revised May 2014.
  5. Peter Carr & Vadim Linetsky, 2006. "A jump to default extended CEV model: an application of Bessel processes," Finance and Stochastics, Springer, Springer, vol. 10(3), pages 303-330, September.
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Cited by:
  1. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2014. "Asymptotics for $d$-dimensional L\'evy-type processes," Papers 1404.3153, arXiv.org.
  2. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A Taylor series approach to pricing and implied vol for LSV models," Papers 1308.5019, arXiv.org.
  3. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org.
  4. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org.

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