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Higher-Order Error Estimates of the Discrete-Time Clark–Ocone Formula

Author

Listed:
  • Tsubasa Nishimura

    (Kojimachi-odori Building 12F)

  • Kenji Yasutomi

    (Ritsumeikan University)

  • Tomooki Yuasa

    (Ritsumeikan University)

Abstract

In this article, we investigate the convergence rate of the discrete-time Clark–Ocone formula provided by Akahori–Amaba–Okuma (J Theor Probab 30: 932–960, 2017). In that paper, they mainly focus on the $$L_{2}$$ L 2 -convergence rate of the first-order error estimate related to the tracking error of the delta hedge in mathematical finance. Here, as two extensions, we estimate “the higher order error” for Wiener functionals with an integrability index 2 and “an arbitrary differentiability index.”

Suggested Citation

  • Tsubasa Nishimura & Kenji Yasutomi & Tomooki Yuasa, 2022. "Higher-Order Error Estimates of the Discrete-Time Clark–Ocone Formula," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2518-2539, December.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01134-0
    DOI: 10.1007/s10959-021-01134-0
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    References listed on IDEAS

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    1. Jirô Akahori & Takafumi Amaba & Kaori Okuma, 2017. "A Discrete-Time Clark–Ocone Formula and its Application to an Error Analysis," Journal of Theoretical Probability, Springer, vol. 30(3), pages 932-960, September.
    2. Akahori, Jirô & Kinuya, Masahiro & Sawai, Takashi & Yuasa, Tomooki, 2021. "An efficient weak Euler–Maruyama type approximation scheme of very high dimensional SDEs by orthogonal random variables," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 540-565.
    3. E. Temam, 2003. "Analysis of Error with Malliavin Calculus: Application to Hedging," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 201-214, January.
    4. Takaki Hayashi & Per A. Mykland, 2005. "Evaluating Hedging Errors: An Asymptotic Approach," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 309-343, April.
    5. Takafumi Amaba, 2014. "A Discrete-Time Clark-Ocone Formula for Poisson Functionals," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 21(2), pages 97-120, May.
    6. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107039124, January.
    7. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107611986, January.
    8. Emmanuel Temam & Emmanuel Gobet, 2001. "Discrete time hedging errors for options with irregular payoffs," Finance and Stochastics, Springer, vol. 5(3), pages 357-367.
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