A continuous selection for optimal portfolios under convex risk measures does not always exist

TL;DR

This paper demonstrates that continuous selections for optimal portfolios under convex risk measures may not exist, even in finite-dimensional, arbitrage-free markets, highlighting limitations in risk mitigation strategies.

Contribution

It reveals that lower semicontinuity and continuous selections cannot be guaranteed in finite-dimensional settings with convex risk measures, challenging assumptions in financial risk management.

Findings

Continuous selections may not exist even in finite-dimensional markets.

Lower semicontinuity of set-valued maps can fail in arbitrage-free markets.

The failure persists under convex risk measures.

Abstract

One of the crucial problems in mathematical finance is to mitigate the risk of a financial position by setting up hedging positions of eligible financial securities. This leads to focusing on set-valued maps associating to any financial position the set of those eligible payoffs that reduce the risk of the position to a target acceptable level at the lowest possible cost. Among other properties of such maps, the ability to ensure lower semicontinuity and continuous selections is key from an operational perspective. It is known that lower semicontinuity generally fails in an infinite-dimensional setting. In this note we show that neither lower semicontinuity nor, more surprisingly, the existence of continuous selections can be a priori guaranteed even in a finite-dimensional setting. In particular, this failure is possible under arbitrage-free markets and convex risk measures.