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Expansion formulae for local Lévy models


  • Stefano, Pagliarani
  • Pascucci, Andrea
  • Candia, Riga


We propose a novel method for the analytical approximation in local volatility models with Lèvy jumps. In the case of Gaussian jumps, we provide an explicit approximation of the transition density of the underlying process by a heat kernel expansion: the approximation is derived in two ways, using PIDE techniques and working in the Fourier space. Our second and main result is an expansion of the characteristic function for a local volatility model with general Lévy jumps. Combined with standard Fourier methods, such an expansion allows to obtain efficient and accurate pricing formulae. Numerical tests confirm the effectiveness of the method.

Suggested Citation

  • Stefano, Pagliarani & Pascucci, Andrea & Candia, Riga, 2011. "Expansion formulae for local Lévy models," MPRA Paper 34571, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:34571

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    References listed on IDEAS

    1. Xu, Guoping & Zheng, Harry, 2010. "Basket options valuation for a local volatility jump-diffusion model with the asymptotic expansion method," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 415-422, December.
    2. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    3. Guoping Xu & Harry Zheng, 2010. "Basket Options Valuation for a Local Volatility Jump-Diffusion Model with the Asymptotic Expansion Method," Papers 1003.1848,
    4. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    5. Rama Cont & Nicolas Lantos & Olivier Pironneau, 2011. "A reduced basis for option pricing," Post-Print hal-00522410, HAL.
    6. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    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. Sam Howison, 2005. "Matched asymptotic expansions in financial engineering," OFRC Working Papers Series 2005mf01, Oxford Financial Research Centre.
    9. Paolo Foschi & Stefano Pagliarani & Andrea Pascucci, 2011. "Black-Scholes formulae for Asian options in local volatility models," Quaderni di Dipartimento 7, Department of Statistics, University of Bologna.
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    Cited by:

    1. Marjon Ruijter & Kees Oosterlee (CWI), 2012. "Two-dimensional Fourier cosine series expansion method for pricing financial options," CPB Discussion Paper 225, CPB Netherlands Bureau for Economic Policy Analysis.

    More about this item


    Lévy process; local volatility; asymptotic expansion; partial-integro differential equation; Fourier methods;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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