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Distributional Replication

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  • Beare, Brendan K.

Abstract

Suppose that X and Y are random variables. We define a replicating function to be a function f such that f(X) and Y have the same distribution. In general, the set of replicating functions for a given pair of random variables may be infinite. Suppose we have some objective function, or cost function, defined over the set of replicating functions, and we seek to estimate the replicating function with the lowest cost. We develop an approach to estimating the cheapest replicating function that involves minimizing the cost function over an estimate of the set of replicating functions. Our estimated set of replicating functions is obtained by considering the functions f in some sieve space for which the empirical distributions of f(X) and Y are close. Under suitable conditions, we show that our estimated function comes close to achieving distributional replication, and close to achieving the minimum cost among replicating functions. We discuss the relevance of our results to the financial literature on hedge fund replication; in this case, X is the market return, Y is the return from a hedge fund or other asset, and our estimation procedure amounts to choosing the cheapest portfolio of options on X such that the returns from our portfolio have the same distribution as the hedge fund returns.

Suggested Citation

  • Beare, Brendan K., 2009. "Distributional Replication," University of California at San Diego, Economics Working Paper Series qt65k3m6x9, Department of Economics, UC San Diego.
    • Handle: RePEc:cdl:ucsdec:qt65k3m6x9
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    References listed on IDEAS

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    1. Maxwell B. Stinchcombe & Halbert White, 1992. "Some Measurability Results for Extrema of Random Functions Over Random Sets," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 59(3), pages 495-514.
    2. Dybvig, Philip H, 1988. "Distributional Analysis of Portfolio Choice," The Journal of Business, University of Chicago Press, vol. 61(3), pages 369-393, July.
    3. Jackwerth, Jens Carsten, 2000. "Recovering Risk Aversion from Option Prices and Realized Returns," The Review of Financial Studies, Society for Financial Studies, vol. 13(2), pages 433-451.
    4. Philip H. Dybvig, 1988. "Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market," The Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 67-88.
    5. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-887, September.
    6. Chen, Xiaohong, 2007. "Large Sample Sieve Estimation of Semi-Nonparametric Models," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 76, Elsevier.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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    Cited by:

    1. Beare, Brendan K., 2010. "Optimal Measure Preserving Derivatives," University of California at San Diego, Economics Working Paper Series qt78k062ns, Department of Economics, UC San Diego.

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