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Asymptotic synthesis of contingent claims with controlled risk in a sequence of discrete-time markets

Author

Listed:
  • Kreps, David M.

    (Graduate School of Business, Stanford University)

  • Schachermayer, Walter

    (Faculty of Mathematics, University of Vienna)

Abstract

We examine the connection between discrete-time models of financial markets and the celebrated Black--Scholes--Merton (BSM) continuous-time model in which ``markets are complete." Suppose that (a) the probability law of a sequence of discrete-time models converges to the law of the BSM model and (b) the largest possible one-period step in the discrete-time models converges to zero. We prove that, under these assumptions, every bounded and continuous contingent claim can be asymptotically synthesized, controlling for the risks taken in a manner that implies, for instance, that an expected-utility-maximizing consumer can asymptotically obtain as much utility in the (possibly incomplete) discrete-time economies as she can at the continuous-time limit. Hence, in economically significant ways, many discrete-time models with frequent trading resemble the complete-markets model of BSM.

Suggested Citation

  • Kreps, David M. & Schachermayer, Walter, 2021. "Asymptotic synthesis of contingent claims with controlled risk in a sequence of discrete-time markets," Theoretical Economics, Econometric Society, vol. 16(1), January.
    • Handle: RePEc:the:publsh:4034
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    References listed on IDEAS

    as
    1. B. Acciaio & M. Beiglbock & F. Penkner & W. Schachermayer & J. Temme, 2012. "A trajectorial interpretation of Doob's martingale inequalities," Papers 1202.0447, arXiv.org, revised Jul 2013.
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    More about this item

    Keywords

    Market completeness; Black-Scholes-Merton model; synthesis of contingent claims;
    All these keywords.

    JEL classification:

    • D0 - Microeconomics - - General
    • G0 - Financial Economics - - General

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