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Efficient discretization of stochastic integrals

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  • Masaaki Fukasawa

Abstract

Sharp asymptotic lower bounds on the expected quadratic variation of the discretization error in stochastic integration are given when the integrator admits a predictable quadratic variation and the integrand is a continuous semimartingale with nondegenerate local martingale part. The theory relies on inequalities for the kurtosis and skewness of a general random variable which are themselves seemingly new. Asymptotically efficient schemes which attain the lower bounds are constructed explicitly. The result is directly applicable to a practical hedging problem in mathematical finance; for hedging a payoff which is replicated by a continuous-time trading strategy, it gives an asymptotically optimal way to choose discrete rebalancing dates and portfolios with respect to transaction costs. The asymptotically efficient strategies in fact reflect the structure of the transaction costs. In particular, a specific biased rebalancing scheme is shown to be superior to unbiased schemes if the transaction costs follow a convex model. The problem is discussed also in terms of exponential utility maximization. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Masaaki Fukasawa, 2014. "Efficient discretization of stochastic integrals," Finance and Stochastics, Springer, vol. 18(1), pages 175-208, January.
    • Handle: RePEc:spr:finsto:v:18:y:2014:i:1:p:175-208
    DOI: 10.1007/s00780-013-0215-6
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Bouchard, Bruno & Muhle-Karbe, Johannes, 2022. "Simple bounds for utility maximization with small transaction costs," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 98-113.
    2. Bruno Bouchard & Johannes Muhle-Karbe, 2022. "Simple Bounds for Transaction Costs," Post-Print hal-01711371, HAL.
    3. Bruno Bouchard & Johannes Muhle-Karbe, 2018. "Simple Bounds for Utility Maximization with Small Transaction Costs," Papers 1802.06120, arXiv.org, revised Mar 2021.
    4. Masaaki Fukasawa & Mitja Stadje, 2017. "Perfect hedging under endogenous permanent market impacts," Papers 1702.01385, arXiv.org.
    5. Asaf Cohen & Yan Dolinsky, 2022. "A scaling limit for utility indifference prices in the discretised Bachelier model," Finance and Stochastics, Springer, vol. 26(2), pages 335-358, April.
    6. Masaaki Fukasawa, 2014. "Efficient price dynamics in a limit order market: an utility indifference approach," Papers 1410.8224, arXiv.org.
    7. Albert Altarovici & Johannes Muhle-Karbe & Halil Soner, 2015. "Asymptotics for fixed transaction costs," Finance and Stochastics, Springer, vol. 19(2), pages 363-414, April.
    8. Asaf Cohen & Yan Dolinsky, 2021. "A Scaling Limit for Utility Indifference Prices in the Discretized Bachelier Model," Papers 2102.11968, arXiv.org, revised Mar 2022.
    9. Masaaki Fukasawa & Mitja Stadje, 2018. "Perfect hedging under endogenous permanent market impacts," Finance and Stochastics, Springer, vol. 22(2), pages 417-442, April.
    10. Bruno Bouchard & Johannes Muhle-Karbe, 2018. "Simple Bounds for Transaction Costs," Working Papers hal-01711371, HAL.

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    More about this item

    Keywords

    Itô integral; Riemann sum; Kurtosis; Skewness; Asymptotic efficiency; Discrete hedging; 60H05; 60F05; G11;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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