$\rho$-arbitrage and $\rho$-consistent pricing for star-shaped risk measures

TL;DR

This paper explores mean-risk portfolio optimization using star-shaped risk measures, introducing new axioms and characterizations to ensure optimal portfolios and derive consistent pricing intervals.

Contribution

It introduces a sensitivity axiom for large losses, characterizes $ ho$-arbitrage, and derives $ ho$-consistent price intervals for financial contracts.

Findings

Sensitivity axiom ensures existence of optimal portfolios

Characterization of $ ho$-arbitrage conditions

Explicit derivation of $ ho$-consistent price intervals

Abstract

This paper revisits mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure $\rho$. We make three contributions. First, we introduce the new axiom of sensitivity to large expected losses and show that it is key to ensure the existence of optimal portfolios. Second, we give primal and dual characterisations of (strong) $\rho$-arbitrage. Finally, we use our conditions for the absence of (strong) $\rho$-arbitrage to explicitly derive the (strong) $\rho$-consistent price interval for an external financial contract.