[Paper Review] On representing claims for coherent risk measures

Saul Jacka, Abdelkarem Berkaoui|ArXiv.org|Aug 3, 2007

Risk and Portfolio Optimization16 references4 citations

TL;DR

This paper establishes the equivalence between time-consistency and m-stability for coherent risk measures in markets with proportional transaction costs, proving that any coherent risk measure can be represented by a portfolio of assets under separability, and vice versa—every arbitrage-free market with transaction costs corresponds to a coherent risk measure. The results generalize Delbaen's m-stability and provide a unified framework for risk representation and hedging in frictional markets.

ABSTRACT

We consider the problem of representing claims for coherent risk measures. For this purpose we introduce the concept of (weak and strong) time-consistency with respect to a portfolio of assets, generalizing the one defined by Delbaen. In a similar way we extend the notion of m-stability, by introducing weak and strong versions. We then prove that the two concepts of m-stability and time-consistency are still equivalent, thus giving necessary and sufficient conditions for a coherent risk measure to be represented by a market with proportional transaction costs. We go on to deduce that, under a separability assumption, any coherent risk measure is strongly time-consistent with respect to a suitably chosen countable portfolio, and show the converse: that any market with proportional transaction costs is equivalent to a market priced by a coherent risk measure, essentially establishing the equivalence of the two concepts.

Motivation & Objective

• To generalize Delbaen's concept of m-stability and time-consistency to portfolios of assets in markets with proportional transaction costs.
• To establish necessary and sufficient conditions under which a coherent risk measure can be represented by a market with transaction costs.
• To show that under separability, any coherent risk measure is strongly time-consistent with respect to a countable portfolio of assets.
• To demonstrate that every arbitrage-free market with proportional transaction costs corresponds to a coherent risk measure, thus establishing a duality between the two concepts.

Proposed method

• Introduces weak and strong versions of time-consistency and m-stability with respect to a portfolio of assets, generalizing Delbaen's original definitions.
• Defines v-denominated risk measures for different numéraires v, enabling multi-currency or multi-commodity risk representation.
• Uses the dual cone of consistent price processes (as in Schachermayer and Kabanov) to characterize attainable claims in multi-asset markets.
• Applies closure and null strategy arguments in multi-period models to prove that the claim set is closed under arbitrage-free conditions.
• Employs conditional risk measures and essential infimum representations to define time-consistent risk assessments across periods.
• Leverages Theorem 4.14 and 4.16 from [11] to establish duality between the cone of attainable claims and the set of consistent price processes.

Experimental results

Research questions

• RQ1 Under what conditions is a coherent risk measure time-consistent with respect to a portfolio of assets in a market with proportional transaction costs?
• RQ2 How does the generalized notion of m-stability relate to time-consistency in multi-asset, frictional markets?
• RQ3 Can every coherent risk measure be represented by a market with proportional transaction costs, and vice versa?
• RQ4 What role does separability play in ensuring that all acceptable claims can be attained via a fixed countable portfolio?
• RQ5 To what extent can multi-currency or multi-commodity numéraires be used to represent and hedge claims under a coherent risk measure?

Key findings

• The paper proves that weak time-consistency and m-stability are equivalent under the same conditions as in Delbaen’s original result, but generalized to portfolios of assets.
• Under a separability assumption, every coherent risk measure is strongly time-consistent with respect to a suitably chosen countable portfolio of assets.
• The paper establishes that any market with proportional transaction costs is equivalent to a market priced by a coherent risk measure, thus proving a duality between the two frameworks.
• The set of attainable claims in a multi-period market with transaction costs is characterized as the set of claims X for which ρ(Y·X) ≤ 0, where ρ is a coherent risk measure and Y is a claim in the numéraire.
• The dual cone of consistent price processes is shown to correspond exactly to the set of probability measures under which the risk measure is represented as a supremum of expectations.
• The claim set 𝒞ₜ^∞ is σ(ℒ∞,ℒ¹)-closed, ensuring robustness and consistency in the representation of acceptable positions across time.