IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v19y2016i02ns0219024916500084.html
   My bibliography  Save this article

Numerical Analysis On Local Risk-Minimization For Exponential Lévy Models

Author

Listed:
  • TAKUJI ARAI

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

  • YUTO IMAI

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

  • RYOICHI SUZUKI

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

Abstract

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
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024916500084
    Download Restriction: Access to full text is restricted to subscribers

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    4. P. Leoni & N. Vandaele & M. Vanmaele, 2014. "Hedging strategies for energy derivatives," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1725-1737, October.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Takuji Arai & Yuto Imai & Ryo Nakashima, 2018. "Numerical analysis on quadratic hedging strategies for normal inverse Gaussian models," Papers 1801.05597, arXiv.org.
    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, arXiv.org.
    4. Takuji Arai, 2019. "Pricing and hedging of VIX options for Barndorff-Nielsen and Shephard models," Papers 1904.12260, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2015. "Numerical analysis on local risk-minimization forexponential L\'evy models," Papers 1506.03898, arXiv.org.
    2. Yeap, Claudia & Kwok, Simon S. & Choy, S. T. Boris, 2016. "A Flexible Generalised Hyperbolic Option Pricing Model and its Special Cases," Working Papers 2016-14, University of Sydney, School of Economics.
    3. Jose Cruz & Daniel Sevcovic, 2020. "On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models," Papers 2003.03851, arXiv.org.
    4. Yongxin Yang & Yu Zheng & Timothy M. Hospedales, 2016. "Gated Neural Networks for Option Pricing: Rationality by Design," Papers 1609.07472, arXiv.org, revised Mar 2020.
    5. Zura Kakushadze, 2016. "Volatility Smile as Relativistic Effect," Papers 1610.02456, arXiv.org, revised Feb 2017.
    6. Björn Lutz, 2010. "Pricing of Derivatives on Mean-Reverting Assets," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-02909-7, December.
    7. Geman, Helyette, 2002. "Pure jump Levy processes for asset price modelling," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1297-1316, July.
    8. Yishen Li & Jin Zhang, 2004. "Option pricing with Weyl-Titchmarsh theory," Quantitative Finance, Taylor & Francis Journals, vol. 4(4), pages 457-464.
    9. Cosma, Antonio & Galluccio, Stefano & Pederzoli, Paola & Scaillet, Olivier, 2020. "Early Exercise Decision in American Options with Dividends, Stochastic Volatility, and Jumps," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 55(1), pages 331-356, February.
    10. Yacine Aït-Sahalia & Jean Jacod, 2012. "Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data," Journal of Economic Literature, American Economic Association, vol. 50(4), pages 1007-1050, December.
    11. Jin Zhang & Yi Xiang, 2008. "The implied volatility smirk," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 263-284.
    12. Oliver X. Li & Weiping Li, 2015. "Hedging jump risk, expected returns and risk premia in jump-diffusion economies," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 873-888, May.
    13. Li, Minqiang, 2008. "Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern," MPRA Paper 11530, University Library of Munich, Germany.
    14. Buckley, Winston & Long, Hongwei & Marshall, Mario, 2016. "Numerical approximations of optimal portfolios in mispriced asymmetric Lévy markets," European Journal of Operational Research, Elsevier, vol. 252(2), pages 676-686.
    15. Antonio Cosma & Stefano Galluccio & Paola Pederzoli & O. Scaillet, 2012. "Valuing American Options Using Fast Recursive Projections," Swiss Finance Institute Research Paper Series 12-26, Swiss Finance Institute.
    16. Shuang Li & Yanli Zhou & Yonghong Wu & Xiangyu Ge, 2017. "Equilibrium approach of asset and option pricing under Lévy process and stochastic volatility," Australian Journal of Management, Australian School of Business, vol. 42(2), pages 276-295, May.
    17. Laura Ballotta, 2009. "Pricing and capital requirements for with profit contracts: modelling considerations," Quantitative Finance, Taylor & Francis Journals, vol. 9(7), pages 803-817.
    18. Wang, Xiaohu & Yu, Jun, 2016. "Double asymptotics for explosive continuous time models," Journal of Econometrics, Elsevier, vol. 193(1), pages 35-53.
    19. Madan, Dilip B. & Wang, King, 2016. "Nonrandom price movements," Finance Research Letters, Elsevier, vol. 17(C), pages 103-109.
    20. Göncü, Ahmet & Karahan, Mehmet Oğuz & Kuzubaş, Tolga Umut, 2016. "A comparative goodness-of-fit analysis of distributions of some Lévy processes and Heston model to stock index returns," The North American Journal of Economics and Finance, Elsevier, vol. 36(C), pages 69-83.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wsi:ijtafx:v:19:y:2016:i:02:n:s0219024916500084. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Tai Tone Lim). General contact details of provider: http://www.worldscinet.com/ijtaf/ijtaf.shtml .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.