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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 "European Journal of Operational Research")

Author

Listed:
  • Kenichiro Shiraya

    (Graduate School of Economics, the University of Tokyo.)

  • Akihiko Takahashi

    (Graduate School of Economics, the University of Tokyo.)

Abstract

This paper presents a new control variate method for general multi-dimensional stochastic differential equations (SDEs) including jumps in order to reduce the variance of Monte Carlo method. Our control variate method is based on an asymptotic expansion technique, and does not require an explicit characteristic function nor a closed form probability density function of SDEs. This is the first one which derives the control variate method for such general models. Moreover, in our control variate method, the regression estimators can be chosen for each number of jump times, and improve the efficiency of the variance reduction. This paper also provides a variance estimate of our method in terms of its terminal time and a small noise parameter used in an asymptotic expansion method. For an application of our method, we evaluate multi-asset options under general local stochastic volatility with jumps models in finance, and show calculation scheme of control variates for Greeks. In numerical experiments, we apply the new control variate method to pricing basket options for ZABR type local stochastic volatility model with jumps, and confirm that our method works very well.

Suggested Citation

  • 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.
    • Handle: RePEc:cfi:fseres:cf382
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    References listed on IDEAS

    as
    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 Approximation Formula for Basket Option Prices under Local Stochastic Volatility with Jumps: an Application to Commodity Markets," CIRJE F-Series CIRJE-F-973, CIRJE, Faculty of Economics, University of Tokyo.
    3. 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.
    4. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    5. Dingeç, Kemal Dinçer & Hörmann, Wolfgang, 2013. "Control variates and conditional Monte Carlo for basket and Asian options," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 421-434.
    6. Caldana, Ruggero & Fusai, Gianluca, 2013. "A general closed-form spread option pricing formula," Journal of Banking & Finance, Elsevier, vol. 37(12), pages 4893-4906.
    7. 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.
    8. Griselda Deelstra & Jan Liinev & Michèle Vanmaele, 2004. "Pricing of arithmetic basket options by conditioning," ULB Institutional Repository 2013/7600, ULB -- Universite Libre de Bruxelles.
    9. 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.
    10. 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.
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