Asymptotic Synthesis of Contingent Claims with Controlled Risk in a Sequence of Discrete-Time Markets

31 Pages Posted: 12 Jun 2019 Last revised: 3 Oct 2019

See all articles by David M. Kreps

David M. Kreps

Stanford Graduate School of Business

W. Schachermayer

University of Vienna

Date Written: October 2, 2019

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." We prove that if (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, then every bounded and continuous contingent claim can be asymptotically synthesized in a manner that controls 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. This implies that, in economically significant ways, many discrete-time models with frequent trading "resemble" the complete-markets model of BSM.

Suggested Citation

Kreps, David M. and Schachermayer, W., Asymptotic Synthesis of Contingent Claims with Controlled Risk in a Sequence of Discrete-Time Markets (October 2, 2019). Stanford University Graduate School of Business Research Paper No. 3795, June 2019. Available at SSRN: https://ssrn.com/abstract=3402645 or http://dx.doi.org/10.2139/ssrn.3402645

David M. Kreps (Contact Author)

Stanford Graduate School of Business ( email )

655 Knight Way
Stanford, CA 94305-5015
United States

W. Schachermayer

University of Vienna

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