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The coherent risk framework is linked to martingale valuation by adding hedgeinvariance as a fifth axiom, motivated by the concept of consistent hedging. The resulting subclass, called coherent pre-hedge (CoPr) measures, is characterized by a martingale condition on the test set that underlies a coherent measure. It is also made explicit how consistent hedging, optimal as well as non-optimal, transforms the test set of a given coherent measure into a martingale test set. These results are put in perspective of the fundamental theorems of asset pricing and the concept of valuation bounds.

The coherent risk framework is linked to martingale valuation by adding hedgeinvariance as a fifth axiom, motivated by the concept of consistent hedging. The resulting subclass, called coherent pre-hedge (CoPr) measures, is characterized by a martingale condition on the test set that underlies a coherent measure. It is also made explicit how consistent hedging, optimal as well as non-optimal, transforms the test set of a given coherent measure into a martingale test set. These results are put in perspective of the fundamental theorems of asset pricing and the concept of valuation bounds.

In this paper coherent risk measures and other currently used risk measures, notably Value-at-Risk (VaR), are studied from the perspective of the theory of coherent imprecise previsions. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. We also show that Value-at-Risk does not necessarily satisfy a weaker notion of coherence called ‘avoiding sure loss’ (ASL), and discuss both sufficient conditions for VaR to avoid sure loss and ways of modifying VaR into a coherent risk measure.

We consider a general class of optimization problems in financial markets with incomplete information and transaction costs. Under a no-arbitrage condition strictly weaker than the existence of a martingale measure, and when asset prices are quasi-left-continuous processes, we show the existence of optimal strategies. Applications include maximization of expected utility, minimization of coherent risk measures and hedging of contingent claims.