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On an approximation problem for stochastic integrals where random time nets do not help

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  • Geiss, Christel
  • Geiss, Stefan

Abstract

Given a geometric Brownian motion S=(St)t[set membership, variant][0,T] and a Borel measurable function such that g(ST)[set membership, variant]L2, we approximate bywhere 0=[tau]0[less-than-or-equals, slant]...[less-than-or-equals, slant][tau]n=T is an increasing sequence of stopping times and the vi-1 are -measurable random variables such that ( is the augmentation of the natural filtration of the underlying Brownian motion). In case that g is not almost surely linear, we show that one gets a lower bound for the L2-approximation rate of if one optimizes over all nets consisting of n+1 stopping times. This lower bound coincides with the upper bound for all reasonable functions g in case deterministic time-nets are used. Hence random time nets do not improve the rate of convergence in this case. The same result holds true for the Brownian motion instead of the geometric Brownian motion.

Suggested Citation

  • Geiss, Christel & Geiss, Stefan, 2006. "On an approximation problem for stochastic integrals where random time nets do not help," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 407-422, March.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:3:p:407-422
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    References listed on IDEAS

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    1. 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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    Cited by:

    1. Mats Brod'en & Peter Tankov, 2010. "Tracking errors from discrete hedging in exponential L\'evy models," Papers 1003.0709, arXiv.org.
    2. Alev{s} v{C}ern'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy Models and the Time Step Equivalent of Jumps," Papers 1309.7833, arXiv.org, revised Jul 2017.
    3. Tankov, Peter & Voltchkova, Ekaterina, 2009. "Asymptotic analysis of hedging errors in models with jumps," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 2004-2027, June.
    4. Masaaki Fukasawa, 2012. "Efficient Discretization of Stochastic Integrals," Papers 1204.0637, arXiv.org.
    5. Masaaki Fukasawa, 2014. "Efficient discretization of stochastic integrals," Finance and Stochastics, Springer, vol. 18(1), pages 175-208, January.
    6. Mats Brod'en & Magnus Wiktorsson, 2010. "Hedging Errors Induced by Discrete Trading Under an Adaptive Trading Strategy," Papers 1004.4526, arXiv.org.
    7. Thorsten Rheinländer & Jenny Sexton, 2011. "Hedging Derivatives," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8062, August.
    8. Stefan Geiss & Emmanuel Gobet, 2010. "Fractional smoothness and applications in finance," Papers 1004.3577, arXiv.org.
    9. Wang, Wensheng, 2019. "Asymptotics for discrete time hedging errors under fractional Black–Scholes models," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 160-170.
    10. Stefan Geiss & Emmanuel Gobet, 2011. "Fractional smoothness and applications in Finance," Post-Print hal-00474803, HAL.
    11. Rossella Agliardi & Ramazan Gençay, 2012. "Hedging through a Limit Order Book with Varying Liquidity," Working Paper series 12_12, Rimini Centre for Economic Analysis.

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