Coherent Risk Measure on $L^0$: NA Condition, Pricing and Dual Representation

TL;DR

This paper extends the fundamental theorem of asset pricing to models with risk measures on $L^0$, providing dual characterizations of super-hedging prices and risk measures under no-arbitrage conditions.

Contribution

It introduces a new version of the fundamental theorem of asset pricing for $L^0$-based risk measures and offers a dual representation of these measures.

Findings

Dual characterization of super-hedging prices

Closure of risk-hedging prices under NA

Dual representation of risk-measure on $L^0$

Abstract

The NA condition is one of the pillars supporting the classical theory of financial mathematics. We revisit this condition for financial market models where a dynamic risk-measure defined on $L^0$ is fixed to characterize the family of acceptable wealths that play the role of non negative financial positions. We provide in this setting a new version of the fundamental theorem of asset pricing and we deduce a dual characterization of the super-hedging prices (called risk-hedging prices) of a European option. Moreover, we show that the set of all risk-hedging prices is closed under NA. At last, we provide a dual representation of the risk-measure on $L^0$ under some conditions.