Extension theorems for linear operators on $L_\infty$ and application to price systems

TL;DR

This paper develops extension theorems for linear operators in $L_$ spaces and applies them to analyze various price systems in financial mathematics, including bounds, bid-ask dynamics, and no-good-deal measures.

Contribution

It introduces new extension theorems for linear operators and applies them to characterize and analyze different classes of price systems in finance.

Findings

Extension theorems for majorant and sandwich preserving operators.

Characterization of price bounds via linear minorants and majorants.

Introduction of dynamic no-good-deal pricing measures.

Abstract

In an $L_\infty$-framework, we present a few extension theorems for linear operators. We focus the attention on majorant preserving and sandwich preserving types of extensions. These results are then applied to the study of price systems derived by a reasonable restriction of the class of equivalent martingale measures applicable. First we consider equivalent martingale measures with bounds on densities and the corresponding prices bounded by linear minorant and majorant. Then we consider prices bounded by bid-ask dynamics. Finally we study price systems consistent with no-good-deal pricing measures for given bounds on the Sharpe ratio. Within this study we introduce the definition of dynamic no-good-deal pricing measure.