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An approximation formula for basket option prices under local stochastic volatility with jumps: an application to commodity markets

Author

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  • Kenichiro Shiraya

    (The University of Tokyo)

  • Akihiko Takahashi

    (The University of Tokyo)

Abstract

This paper develops a new approximation formula for pricing basket options in a local-stochastic volatility model with jumps. In particular, the model admits local volatility functions and jump components in not only the underlying asset price processes, but also the volatility processes. To the best of our knowledge, the proposed formula is the first one which achieves an analytical approximation for the basket option prices under this type of the models. Moreover, in numerical experiments, we provide approximate prices for basket options on the WTI futures and Brent futures based on the parameters through calibration to the plain-vanilla option prices, and confirm the validity of our approximation formula.

Suggested Citation

  • Kenichiro Shiraya & Akihiko Takahashi, 2015. "An approximation formula for basket option prices under local stochastic volatility with jumps: an application to commodity markets," CARF F-Series CARF-F-361, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
  • Handle: RePEc:cfi:fseres:cf361
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    References listed on IDEAS

    as
    1. Akihiko Takahashi & Toshihiro Yamada, 2012. "An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach," CARF F-Series CARF-F-296, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Sep 2013.
    2. Kenichiro Shiraya & Akihiko Takahashi, 2014. "Pricing Multiasset Cross‐Currency Options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 34(1), pages 1-19, January.
    3. 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.
    4. Akihiko Takahashi & Toshihiro Yamada, 2013. "On an Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver," CARF F-Series CARF-F-326, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Oct 2013.
    5. Kohta Takehara & Masashi Toda & Akihiko Takahashi, 2010. "Application Of A High-Order Asymptotic Expansion Scheme To Long-Term Currency Options," CARF F-Series CARF-F-225, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    6. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," Papers 1204.2638, arXiv.org, revised Apr 2012.
    7. Naoto Kunitomo & Akihiko Takahashi, 2001. "The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 117-151.
    8. Akihiko Takahashi & Toshihiro Yamada, 2013. "On an Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver," CIRJE F-Series CIRJE-F-902, CIRJE, Faculty of Economics, University of Tokyo.
    9. Akihiko Takahashi & Toshihiro Yamada, 2012. "An Asymptotic Expansion for Forward-Backward SDEs; A Malliavin Calculus Aproach," CIRJE F-Series CIRJE-F-865, CIRJE, Faculty of Economics, University of Tokyo.
    10. Masaaki Fujii & Akihiko Takahashi, 2011. "Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme," Papers 1106.0123, arXiv.org, revised Jan 2012.
    11. Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2012. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," Papers 1211.5867, arXiv.org.
    12. Guoping Xu & Harry Zheng, 2012. "Lower Bound Approximation to Basket Option Values for Local Volatility Jump-Diffusion Models," Papers 1212.3147, arXiv.org, revised Oct 2013.
    13. Guoping Xu & Harry Zheng, 2010. "Basket Options Valuation for a Local Volatility Jump-Diffusion Model with the Asymptotic Expansion Method," Papers 1003.1848, arXiv.org.
    14. Bjørn Eraker, 2004. "Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices," Journal of Finance, American Finance Association, vol. 59(3), pages 1367-1404, June.
    15. Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2012. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," CARF F-Series CARF-F-302, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    16. Deelstra, G. & Liinev, J. & Vanmaele, M., 2004. "Pricing of arithmetic basket options by conditioning," Insurance: Mathematics and Economics, Elsevier, vol. 34(1), pages 55-77, February.
    17. Kohta Takehara & Masashi Toda & Akihiko Takahashi, 2010. "Application of a High-Order Asymptotic Expansion Scheme to Long-Term Currency Options," CIRJE F-Series CIRJE-F-753, CIRJE, Faculty of Economics, University of Tokyo.
    18. Griselda Deelstra & Jan Liinev & Michèle Vanmaele, 2004. "Pricing of arithmetic basket options by conditioning," ULB Institutional Repository 2013/7600, ULB -- Universite Libre de Bruxelles.
    19. Akihiko Takahashi & Kohta Takehara, 2010. "A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options," CIRJE F-Series CIRJE-F-734, CIRJE, Faculty of Economics, University of Tokyo.
    20. Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2012. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," CIRJE F-Series CIRJE-F-871, CIRJE, Faculty of Economics, University of Tokyo.
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    Cited by:

    1. Kenichiro Shiraya & Akihiko Takahashi, 2015. "An Asymptotic Expansion for Local-Stochastic Volatility with Jump Models (Forthcoming in Stochastics)," CARF F-Series CARF-F-377, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    2. Kenichiro Shiraya & Akihiko Takahashi, 2015. "An Asymptotic Expansion for Local-Stochastic Volatility with Jump Models," CIRJE F-Series CIRJE-F-998, CIRJE, Faculty of Economics, University of Tokyo.
    3. Kenichiro Shiraya & Akihiko Takahashi, 2016. "A General Control Variate Method for Multi-dimensional SDEs: An Application to Multi-asset Options under Local Stochastic Volatility with Jumps Models in Finance (Subsequently published in "Europ," CARF F-Series CARF-F-382, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Sep 2016.
    4. Kenichiro Shiraya & Akihiko Takahashi, 2016. "A General Control Variate Method for Multi-dimensional SDEs: An Application to Multi-asset Options under Local Stochastic Volatility with Jumps Models in Finance," CIRJE F-Series CIRJE-F-1007, CIRJE, Faculty of Economics, University of Tokyo.

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