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Asymptotics for $d$-dimensional L\'evy-type processes

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  • Matthew Lorig
  • Stefano Pagliarani
  • Andrea Pascucci

Abstract

We consider a general d-dimensional Levy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator A(t) of the Levy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stochastic volatility models with jumps are provided in order to illustrate the numerical accuracy of our approach. The methods described in this paper extend the results from Corielli et al. (2010), Pagliarani and Pascucci (2013) and Lorig et al. (2013a) for Markov diffusions to Markov processes with jumps.

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File URL: http://arxiv.org/pdf/1404.3153
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Paper provided by arXiv.org in its series Papers with number 1404.3153.

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Date of creation: Apr 2014
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Handle: RePEc:arx:papers:1404.3153

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Web page: http://arxiv.org/

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  1. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A Taylor series approach to pricing and implied vol for LSV models," Papers 1308.5019, arXiv.org.
  2. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
  3. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A family of density expansions for L\'evy-type processes," Papers 1312.7328, arXiv.org.
  4. J. D. Deuschel & P. K. Friz & A. Jacquier & S. Violante, 2011. "Marginal density expansions for diffusions and stochastic volatility, part I: Theoretical Foundations," Papers 1111.2462, arXiv.org, revised May 2013.
  5. Stefano Pagliarani & Andrea Pascucci, 2013. "Local Stochastic Volatility With Jumps: Analytical Approximations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(08), pages 1350050-1-1.
  6. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org.
  7. E. Benhamou & E. Gobet & M. Miri, 2009. "Smart expansion and fast calibration for jump diffusions," Finance and Stochastics, Springer, vol. 13(4), pages 563-589, September.
  8. 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.
  9. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
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