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Efficient discretization of stochastic integrals

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  • Masaaki Fukasawa

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    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

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    File URL: http://hdl.handle.net/10.1007/s00780-013-0215-6
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    Bibliographic Info

    Article provided by Springer in its journal Finance and Stochastics.

    Volume (Year): 18 (2014)
    Issue (Month): 1 (January)
    Pages: 175-208

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    Handle: RePEc:spr:finsto:v:18:y:2014:i:1:p:175-208

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    Related research

    Keywords: Itô integral; Riemann sum; Kurtosis; Skewness; Asymptotic efficiency; Discrete hedging; 60H05; 60F05; G11;

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    1. A.E. Whalley & P. Wilmott, 1999. "Optimal Hedging of Options with Small but Arbitrary Transaction Cost Structure," OFRC Working Papers Series 1999mf09, Oxford Financial Research Centre.
    2. 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.
    3. A. E. Whalley & P. Wilmott, 1997. "An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 7(3), pages 307-324.
    4. Umut Çetin & Robert Jarrow & Philip Protter, 2004. "Liquidity risk and arbitrage pricing theory," Finance and Stochastics, Springer, vol. 8(3), pages 311-341, 08.
    5. Karandikar, Rajeeva L., 1995. "On pathwise stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 11-18, May.
    6. Leland, Hayne E, 1985. " Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    7. 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.
    8. Bertsimas, Dimitris & Kogan, Leonid & Lo, Andrew W., 2000. "When is time continuous?," Journal of Financial Economics, Elsevier, vol. 55(2), pages 173-204, February.
    9. Fukasawa, Masaaki, 2010. "Realized volatility with stochastic sampling," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 829-852, June.
    10. Lépinette-Denis, Emmanuel & Kabanov, Yuri, 2010. "Mean square error for the Leland-Lott hedging strategy: convex pay-offs," Economics Papers from University Paris Dauphine 123456789/4654, Paris Dauphine University.
    11. Nicole El Karoui & Monique Jeanblanc-Picquè & Steven E. Shreve, 1998. "Robustness of the Black and Scholes Formula," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 93-126.
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