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Efficient discretization of stochastic integrals

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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.

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Acknowledgements

The author is grateful to the anonymous referees for their careful reading and helpful comments. This work is supported by Japan Society for the Promotion of Science, KAKENHI Grant Numbers 24684006, 24300107 and 22243021.

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Authors and Affiliations

  1. Department of Mathematics and Center for the Study of Finance and Insurance, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, Japan

    Masaaki Fukasawa

  2. CREST, Japan Science and Technology Agency, Tokyo, Japan

    Masaaki Fukasawa

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  1. Masaaki Fukasawa
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Correspondence to Masaaki Fukasawa.

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Fukasawa, M. Efficient discretization of stochastic integrals. Finance Stoch 18, 175–208 (2014). https://doi.org/10.1007/s00780-013-0215-6

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