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Semi-closed form cubature and applications to financial diffusion models

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  • Christian Bayer
  • Peter Friz
  • Ronnie Loeffen

Abstract

Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [ Adv. Math. Econ ., 2004, 6 , 69--83] and Lyons--Victoir [ Proc. R. Soc. Lond. Ser. A , 2004, 460 , 169--198], involve the solution to numerous auxiliary ordinary differential equations (ODEs). With focus on the Ninomiya--Victoir algorithm [ Appl. Math. Finance , 2008, 15 , 107--121], which corresponds to a concrete level 5 cubature method, we study some parametric diffusion models motivated from financial applications, and show the structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies by introducing a (model-dependent) variation of the Ninomiya--Victoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time.

Suggested Citation

  • Christian Bayer & Peter Friz & Ronnie Loeffen, 2012. "Semi-closed form cubature and applications to financial diffusion models," Quantitative Finance, Taylor & Francis Journals, vol. 13(5), pages 769-782, October.
    • Handle: RePEc:taf:quantf:v:13:y:2012:i:5:p:769-782
    DOI: 10.1080/14697688.2012.752102
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    References listed on IDEAS

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    1. Bin Chen & Cornelis W. Oosterlee & Hans Van Der Weide, 2012. "A Low-Bias Simulation Scheme For The Sabr Stochastic Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(02), pages 1-37.
    2. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    3. Mariko Ninomiya & Syoiti Ninomiya, 2009. "A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method," Finance and Stochastics, Springer, vol. 13(3), pages 415-443, September.
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