Put Option Premiums and Coherent Risk Measures

This note defines the premium of a put option on the firm as a measure of insolvency risk. The put premium is not a coherent risk measure as defined by Artzner et al. (1999). It satisfies all the axioms for a coherent risk measure except one, the translation invariance axiom. However, it satisfies a weakened version of the translation invariance axiom that we label translation monotonicity. The put premium risk measure generates an acceptance set that satisfies the regularity Axioms 2.1–2.4 of Artzner et al. (1999). In fact, this is a general result for any risk measure satisfying the same risk measure axioms as the put premium. Finally, the coherent risk measure generated by the put premium's acceptance set is the minimal capital required to protect the firm against insolvency uniformly across all states of nature.