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

TL;DR

This paper develops a conditional version of the second fundamental theorem of asset pricing in discrete time, providing new insights into arbitrage-free prices, market completeness, and attainability using dynamic nonlinear expectations.

Contribution

It introduces a novel conditional second fundamental theorem of asset pricing, extending existing results to more general settings with dynamic nonlinear expectations.

Findings

Characterizes arbitrage-free prices conditionally on available information.

Provides conditions for market completeness and attainability.

Extends results to infinite probability spaces using time consistency of nonlinear expectations.

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 tool we use are 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.