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Local risk-minimization for Barndorff-Nielsen and Shephard models

Author

Listed:
  • Takuji Arai

    (Keio University)

  • Yuto Imai

    (Waseda University)

  • Ryoichi Suzuki

    (Keio University)

Abstract

We obtain explicit representations of locally risk-minimizing strategies for call and put options in Barndorff-Nielsen and Shephard models, which are Ornstein–Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for Lévy processes, Arai and Suzuki (Int. J. Financ. Eng. 2:1550015, 2015) obtained a formula for locally risk-minimizing strategies for Lévy markets under many additional conditions. Supposing mild conditions, we make sure that the Barndorff-Nielsen and Shephard models satisfy all the conditions imposed in (Arai and Suzuki in Int. J. Financ. Eng. 2:1550015, 2015). Among others, we investigate the Malliavin differentiability of the density of the minimal martingale measure. Moreover, we introduce some numerical experiments for locally risk-minimizing strategies.

Suggested Citation

  • Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2017. "Local risk-minimization for Barndorff-Nielsen and Shephard models," Finance and Stochastics, Springer, vol. 21(2), pages 551-592, April.
    • Handle: RePEc:spr:finsto:v:21:y:2017:i:2:d:10.1007_s00780-017-0324-8
    DOI: 10.1007/s00780-017-0324-8
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    References listed on IDEAS

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    1. 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.
    2. Fred Benth & Nils Detering, 2015. "Pricing and hedging Asian-style options on energy," Finance and Stochastics, Springer, vol. 19(4), pages 849-889, October.
    3. Jan Kallsen & Richard Vierthauer, 2009. "Quadratic hedging in affine stochastic volatility models," Review of Derivatives Research, Springer, vol. 12(1), pages 3-27, April.
    4. Solé, Josep Lluís & Utzet, Frederic & Vives, Josep, 2007. "Canonical Lévy process and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 165-187, February.
    5. Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2015. "Local risk-minimization for Barndorff-Nielsen and Shephard models," Papers 1503.08589, arXiv.org, revised Jan 2016.
    6. Jouini,E. & Cvitanic,J. & Musiela,Marek (ed.), 2001. "Handbooks in Mathematical Finance," Cambridge Books, Cambridge University Press, number 9780521792370, October.
    7. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    8. Delong, Lukasz & Imkeller, Peter, 2010. "On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1748-1775, August.
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    Cited by:

    1. Xianfei Hui & Baiqing Sun & Indranil SenGupta & Yan Zhou & Hui Jiang, 2022. "Stochastic volatility modeling of high-frequency CSI 300 index and dynamic jump prediction driven by machine learning," Papers 2204.02891, arXiv.org, revised Jan 2023.
    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, 2021. "Approximate option pricing formula for Barndorff-Nielsen and Shephard model," Papers 2104.10877, arXiv.org.
    4. Masahiro Handa & Noriyoshi Sakuma & Ryoichi Suzuki, 2024. "A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$ L 1 -pure jump additive processes and its application to portfolio optimization," Annals of Finance, Springer, vol. 20(3), pages 329-352, September.
    5. Arai, Takuji & Imai, Yuto, 2024. "Monte Carlo simulation for Barndorff–Nielsen and Shephard model under change of measure," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 223-234.
    6. Takuji Arai, 2020. "Al\`os type decomposition formula for Barndorff-Nielsen and Shephard model," Papers 2005.07393, arXiv.org, revised Sep 2020.
    7. Takuji Arai & Yuto Imai, 2024. "Option pricing for Barndorff-Nielsen and Shephard model by supervised deep learning," Papers 2402.00445, arXiv.org.
    8. Takuji Arai & Yuto Imai, 2023. "Monte Carlo simulation for Barndorff-Nielsen and Shephard model under change of measure," Papers 2306.05750, arXiv.org.
    9. Suzuki, Ryoichi, 2018. "Malliavin differentiability of indicator functions on canonical Lévy spaces," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 183-190.
    10. Takuji Arai, 2019. "Pricing and hedging of VIX options for Barndorff-Nielsen and Shephard models," Papers 1904.12260, arXiv.org.

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    More about this item

    Keywords

    Local risk-minimization; Barndorff-Nielsen and Shephard models; Stochastic volatility models; Malliavin calculus; Lévy processes;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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