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Mean square error for the Leland–Lott hedging strategy: convex pay-offs

Author

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  • Emmanuel Denis
  • Yuri Kabanov

Abstract

Leland's approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim VT using the classical Black Scholes formulae with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e. a sequence of models with the transaction costs coefficients kn and n is the number of the portfolio revision dates. The enlarged volatility, in general, depends on n except the case which was investigated in details by Lott to whom belongs the first rigorous result on convergence of the approximating portfolio value to the pay-off. In this paper we consider only the Lott case alpha= 1/2. We prove first, for an arbitrary pay-off VT = G(ST ) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n^1/2 in L2 and find the first order term of asymptotics. We are working in the setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are t_ni=g(i_n) where the strictly increasing scale function g : [0; 1] -> [0; 1] and its inverse f are continuous with their first and second derivatives on the whole interval. We show that the sequence of approximate error converges in law to a random variable which is the terminal value of a component of two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Emmanuel Denis & Yuri Kabanov, 2010. "Mean square error for the Leland–Lott hedging strategy: convex pay-offs," Finance and Stochastics, Springer, vol. 14(4), pages 625-667, December.
    • Handle: RePEc:spr:finsto:v:14:y:2010:i:4:p:625-667
    DOI: 10.1007/s00780-010-0130-z
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    References listed on IDEAS

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    1. E. R. Grannan & G. H. Swindle, 1996. "Minimizing Transaction Costs Of Option Hedging Strategies," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 341-364, October.
    2. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    3. Emmanuel Temam & Emmanuel Gobet, 2001. "Discrete time hedging errors for options with irregular payoffs," Finance and Stochastics, Springer, vol. 5(3), pages 357-367.
    4. Zhao, Yonggan & Ziemba, William T., 2007. "Hedging errors with Leland's option model in the presence of transaction costs," Finance Research Letters, Elsevier, vol. 4(1), pages 49-58, March.
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    Citations

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    Cited by:

    1. Emmanuel Lépinette & Duc Thinh Vu, 2023. "Dynamic programming principle and computable prices in financial market models with transaction costs," Post-Print hal-03284655, HAL.
    2. Thai Huu Nguyen & Serguei Pergamenshchikov, 2015. "Approximate hedging problem with transaction costs in stochastic volatility markets," Papers 1505.02546, arXiv.org.
    3. Wang, Xiao-Tian & Li, Zhe & Zhuang, Le, 2017. "Risk preference, option pricing and portfolio hedging with proportional transaction costs," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 111-130.
    4. Tommi Sottinen & Lauri Viitasaari, 2018. "Conditional-Mean Hedging Under Transaction Costs In Gaussian Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(02), pages 1-15, March.
    5. Shokrollahi, Foad & Sottinen, Tommi, 2017. "Hedging in fractional Black–Scholes model with transaction costs," Statistics & Probability Letters, Elsevier, vol. 130(C), pages 85-91.
    6. Masaaki Fukasawa, 2014. "Efficient discretization of stochastic integrals," Finance and Stochastics, Springer, vol. 18(1), pages 175-208, January.
    7. Foad Shokrollahi & Tommi Sottinen, 2017. "Hedging in fractional Black-Scholes model with transaction costs," Papers 1706.01534, arXiv.org, revised Jul 2017.
    8. repec:hal:wpaper:hal-03284655 is not listed on IDEAS
    9. Romuald Elie & Emmanuel Lépinette, 2015. "Approximate hedging for nonlinear transaction costs on the volume of traded assets," Finance and Stochastics, Springer, vol. 19(3), pages 541-581, July.
    10. Jiatu Cai & Masaaki Fukasawa, 2016. "Asymptotic replication with modified volatility under small transaction costs," Finance and Stochastics, Springer, vol. 20(2), pages 381-431, April.
    11. Tommi Sottinen & Lauri Viitasaari, 2017. "Conditional-Mean Hedging Under Transaction Costs in Gaussian Models," Papers 1708.03242, arXiv.org.
    12. Masaaki Fukasawa, 2012. "Efficient Discretization of Stochastic Integrals," Papers 1204.0637, arXiv.org.
    13. Huu Thai Nguyen & Serguei Pergamenchtchikov, 2012. "Approximate hedging problem with transaction costs in stochastic volatility markets," Working Papers hal-00747689, HAL.
    14. Jiatu Cai & Masaaki Fukasawa, 2014. "Asymptotic replication with modified volatility under small transaction costs," Papers 1408.5677, arXiv.org.
    15. Xavier Warin, 2017. "Variance optimal hedging with application to Electricity markets," Papers 1711.03733, arXiv.org, revised Aug 2018.
    16. Serguei Pergamenchtchikov & Alena Shishkova, 2020. "Hedging problems for Asian options with transactions costs," Papers 2001.01443, arXiv.org.

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    More about this item

    Keywords

    Black–Scholes formula; European option; Transaction costs; Leland–Lott strategy; Approximate hedging; Martingale limit theorem; Diffusion approximation; 60G44; G11; G13;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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