Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy

We introduce a new coherent risk measure called the minimal entropy risk measure. This measure is based on the minimal entropy σ-martingale measure, which itself is inspired by the minimal entropy martingale measure well-known in option pricing. While the minimal entropy martingale measure is commonly used for pricing and hedging, the minimal entropy σ-martingale measure has not previously been studied, nor has it been analyzed as a traditional risk measure. We address this gap by clearly defining this new risk measure and examining its fundamental properties. In addition, we revisit the entropic risk measure, typically expressed through an exponential formula. We provide an alternative definition using a supremum over Kullback–Leibler divergences, making its connection to entropy clearer. We verify important properties of both risk measures, such as convexity and coherence, and extend these concepts to dynamic situations. We also illustrate their behavior in scenarios involving optimal risk transfer. Our results link entropic concepts with incomplete-market pricing and demonstrate how both risk measures share a unified entropy-based foundation once market constraints are considered.