Convex risk measures for good deal bounds

TL;DR

This paper explores convex risk measures that define bounds for good deals in financial markets, characterizing their properties, existence conditions, and relation to no-arbitrage and no-free-lunch conditions.

Contribution

It provides a comprehensive characterization of good deal valuations, linking them to risk indifference prices and market conditions, and discusses their relevance and existence criteria.

Findings

Good deal valuations are characterized by risk indifference prices.

Existence of a relevant good deal valuation is equivalent to the NFL condition.

Conditions for all good deal valuations to be relevant are investigated.

Abstract

We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no-free-lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further we investigate conditions under which any good deal valuation is relevant.