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Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing

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  • Leif Andersen
  • Jesper Andreasen
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    Abstract

    This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly in the dynamics of the implied volatility surface. The paper derives a forward PIDE (PartialIntegro-Differential Equation) and demonstrates how this equationcan be used to fit the model to European option prices. For numerical pricing of general contingent claims, we develop an ADI finite difference method that is shown to be unconditionally stable and, if combined with Fast Fourier Transform methods, computationally efficient. The paper contains several detailed examples fromthe S&P500 market. Copyright Kluwer Academic Publishers 2000

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    Bibliographic Info

    Article provided by Springer in its journal Review of Derivatives Research.

    Volume (Year): 4 (2000)
    Issue (Month): 3 (October)
    Pages: 231-262

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    Handle: RePEc:kap:revdev:v:4:y:2000:i:3:p:231-262

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    Web page: http://www.springerlink.com/link.asp?id=102989

    Related research

    Keywords: jump-diffusion process; local time; forward equation; volatility smile; ADI finite difference method; fast Fourier transform;

    References

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    1. Charles Quanwei Cao & Gurdip S. Bakshi & Zhiwu Chen, 1997. "Empirical Performance of Alternative Option Pricing Models," Yale School of Management Working Papers, Yale School of Management ysm65, Yale School of Management.
    2. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
    3. Charles Quanwei Cao & Gurdip S. Bakshi & Zhiwu Chen, 1997. "Empirical Performance of Alternative Option Pricing Models," Yale School of Management Working Papers, Yale School of Management ysm54, Yale School of Management.
    4. Boyle, Phelim & Broadie, Mark & Glasserman, Paul, 1997. "Monte Carlo methods for security pricing," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 21(8-9), pages 1267-1321, June.
    5. 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.
    6. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. " Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, American Finance Association, vol. 52(5), pages 2003-49, December.
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    Cited by:
    1. Andrey Itkin, 2014. "High-Order Splitting Methods for Forward PDEs and PIDEs," Papers 1403.1804, arXiv.org.
    2. Stefano Galluccio & Yann Le Cam, 2005. "Implied Calibration of Stochastic Volatility Jump Diffusion Models," Finance, EconWPA 0510028, EconWPA.
    3. Carol Alexander & Leonardo M. Nogueira, 2004. "Hedging with Stochastic and Local Volatility," ICMA Centre Discussion Papers in Finance, Henley Business School, Reading University icma-dp2004-10, Henley Business School, Reading University, revised Dec 2004.
    4. C. He & J. Kennedy & T. Coleman & P. Forsyth & Y. Li & K. Vetzal, 2006. "Calibration and hedging under jump diffusion," Review of Derivatives Research, Springer, Springer, vol. 9(1), pages 1-35, January.
    5. Ron Chan & Simon Hubbert, 2014. "Options pricing under the one-dimensional jump-diffusion model using the radial basis function interpolation scheme," Review of Derivatives Research, Springer, Springer, vol. 17(2), pages 161-189, July.
    6. 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.
    7. Nomikos, Nikos K. & Kyriakou, Ioannis & Papapostolou, Nikos C. & Pouliasis, Panos K., 2013. "Freight options: Price modelling and empirical analysis," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 51(C), pages 82-94.
    8. Farshid Jamshidian, 2004. "Numeraire-invariant option pricing and american, bermudan, trigger stream rollover (v1.6)," Finance, EconWPA 0407015, EconWPA.
    9. Chen, Z. & Vetzal, K. & Forsyth, P.A., 2008. "The effect of modelling parameters on the value of GMWB guarantees," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 165-173, August.
    10. H. A. Windcliff & P. A. Forsyth & K. R. Vetzal, 2006. "Numerical Methods and Volatility Models for Valuing Cliquet Options," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 13(4), pages 353-386.
    11. Erhan Bayraktar & Hao Xing, 2009. "Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions," Computational Statistics, Springer, Springer, vol. 70(3), pages 505-525, December.
    12. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
    13. Carl Chiarella & Andrew Ziogas, 2006. "American Call Options on Jump-Diffusion Processes: A Fourier Transform Approach," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 174, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Windcliff, H. & Forsyth, P.A. & Vetzal, K.R., 2006. "Pricing methods and hedging strategies for volatility derivatives," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 409-431, February.

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