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

David M. Kreps () and Walter Schachermayer ()
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David M. Kreps: Graduate School of Business, Stanford University
Walter Schachermayer: Faculty of Mathematics, University of Vienna

Theoretical Economics, 2021, vol. 16, issue 1

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.

Keywords: Market completeness; Black-Scholes-Merton model; synthesis of contingent claims (search for similar items in EconPapers)
JEL-codes: D0 G0 (search for similar items in EconPapers)
Date: 2021-01-15
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