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Numerical Analysis On Local Risk-Minimization For Exponential Lévy Models



    () (Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan)


    () (Department of Mathematics, Waseda University, 3-4-1 Okubo, Shinjyuku-ku, Tokyo 169-8555, Japan)


    () (Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan)


We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. Here, LRM is a popular hedging method through a quadratic criterion for contingent claims in incomplete markets. Arai & Suzuki (2015) have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform (FFT) method suggested by by Carr & Madan (1999). Considering Merton jump-diffusion models and variance gamma models as typical examples of exponential Lévy models, we provide the forms for the FFT explicitly; and compute the values of LRM numerically for given parameter sets. Furthermore, we illustrate numerical results for a variance gamma model with estimated parameters from the Nikkei 225 index.

Suggested Citation

  • Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2016. "Numerical Analysis On Local Risk-Minimization For Exponential Lévy Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(02), pages 1-27, March.
  • Handle: RePEc:wsi:ijtafx:v:19:y:2016:i:02:n:s0219024916500084
    DOI: 10.1142/S0219024916500084

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    References listed on IDEAS

    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.
    2. Takuji Arai & Ryoichi Suzuki, 2015. "Local risk-minimization for Lévy markets," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-28.
    3. Ewald, Christian-Oliver & Nawar, Roy & Siu, Tak Kuen, 2013. "Minimal variance hedging of natural gas derivatives in exponential Lévy models: Theory and empirical performance," Energy Economics, Elsevier, vol. 36(C), pages 97-107.
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    5. Abdou Kélani & François Quittard-Pinon, 2014. "Pricing, Hedging and Assessing Risk in a General Lévy Context," Bankers, Markets & Investors, ESKA Publishing, issue 131, pages 30-42, July-Augu.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Kiseop Lee & Seongjoo Song, 2007. "Insiders' hedging in a jump diffusion model," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 537-545.
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    Cited by:

    1. Takuji Arai & Yuto Imai & Ryo Nakashima, 2018. "Numerical analysis on quadratic hedging strategies for normal inverse Gaussian models," Papers 1801.05597,
    2. Takuji Arai, 2019. "Pricing And Hedging Of Vix Options For Barndorff-Nielsen And Shephard Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-26, December.
    3. Takuji Arai & Ryoichi Suzuki, 2019. "A Clark-Ocone type formula via Ito calculus and its application to finance," Papers 1906.06648,
    4. Takuji Arai, 2019. "Pricing and hedging of VIX options for Barndorff-Nielsen and Shephard models," Papers 1904.12260,

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