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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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  • Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A family of density expansions for L\'evy-type processes," Papers 1312.7328, arXiv.org.
    • Handle: RePEc:arx:papers:1312.7328
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    File URL: http://arxiv.org/pdf/1312.7328
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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
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    3. Antoine Jacquier & Matthew Lorig, 2012. "The Smile of certain L\'evy-type Models," Papers 1207.1630, arXiv.org, revised Apr 2013.
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    10. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    11. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Pricing approximations and error estimates for local L\'evy-type models with default," Papers 1304.1849, arXiv.org, revised Nov 2014.
    12. Agostino Capponi & Stefano Pagliarani & Tiziano Vargiolu, 2014. "Pricing vulnerable claims in a Lévy-driven model," Finance and Stochastics, Springer, vol. 18(4), pages 755-789, October.
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    Cited by:

    1. Stefano Pagliarani & Andrea Pascucci, 2017. "The exact Taylor formula of the implied volatility," Finance and Stochastics, Springer, vol. 21(3), pages 661-718, July.
    2. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2014. "Asymptotics for $d$-dimensional L\'evy-type processes," Papers 1404.3153, arXiv.org, revised Nov 2014.
    3. Jos'e E. Figueroa-L'opez & Yankeng Luo, 2015. "Small-time expansions for state-dependent local jump-diffusion models with infinite jump activity," Papers 1505.04459, arXiv.org, revised Dec 2015.
    4. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A Taylor series approach to pricing and implied vol for LSV models," Papers 1308.5019, arXiv.org.
    5. Matthew Lorig & Ronnie Sircar, 2015. "Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations & Implied Sharpe Ratio," Papers 1506.06180, arXiv.org.
    6. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2017. "Explicit Implied Volatilities For Multifactor Local-Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 926-960, July.
    7. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org, revised Nov 2014.
    8. Figueroa-López, José E. & Luo, Yankeng, 2018. "Small-time expansions for state-dependent local jump–diffusion models with infinite jump activity," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4207-4245.
    9. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org, revised Apr 2015.
    10. Agostino Capponi & Stefano Pagliarani & Tiziano Vargiolu, 2014. "Pricing vulnerable claims in a Lévy-driven model," Finance and Stochastics, Springer, vol. 18(4), pages 755-789, October.

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