A conditional Version of the second fundamental theorem of asset pricing in discrete time

Lars Niemann; Thorsten Schmidt

doi:10.3934/fmf.2024006

Article Contents

2024, Volume 3, Issue 2: 239-269. Doi: 10.3934/fmf.2024006 open access

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A conditional Version of the second fundamental theorem of asset pricing in discrete time

Lars Niemann^, and
Thorsten Schmidt

University of Freiburg, Germany

^*Corresponding author: Lars Niemann
^*Corresponding author: Lars Niemann

Received: September 2022

Revised: January 2024

Early access: April 16, 2024

Published online: April 16, 2024

Financial support from the DFG in project SCHM 2160/13-1 is gratefully acknowledged. We thank two anonymous referees for their helpful comments and suggestions.

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Abstract

We consider a financial market in discrete time and study pricing and hedging conditional on the information available up to an arbitrary point in time. In this conditional framework, we determine the structure of arbitrage-free prices. Moreover, we characterize attainability and market completeness. We derive a conditional version of the second fundamental theorem of asset pricing, which, surprisingly, is not available up to now.

The main tools we use are the time consistency properties of dynamic nonlinear expectations, which we apply to the super- and subhedging prices. The results obtained extend existing results in the literature, where the conditional setting is considered in most cases only on finite probability spaces.

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Mathematics Subject Classification: 60G42, 91G15, 91G30.

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