Monetary utility over coherent risk ratios

Abstract

SUMMARY For a monetary utility functional U and a coherent risk measure ρ, both with compact scenario sets in Lq, we optimize the ratio α(V): = U(V)/ρ(V) over an (arbitrage-free) linear sub-space V⊆Lp, 1 ≤ p ≤ ∞, of attainable returns in an incomplete market model such that ρ > 0 on V \ {0}. If a solution Vˆ ∈ V with α(Vˆ) = α¯ V: = sup V∈Vα(V)∈[0,∞) exists, then the first order optimality condition allows to construct an absolutely continuous martingale measure for V as a convex combination Q¯+α¯VQ/1+α¯V of two probability measures Q¯, Q from the respective scenario sets defining U and ρ. Conversely, if α¯V ∈ [0,∞), then α¯V equals the smallest a∈[0,∞) such that Q¯+aQ/1+a is an absolutely continuous martingale measure for V for some probability measures Q¯, Q from the scenario sets defining U, ρ, and α¯V = ∞ holds iff such a convex combination does not exist.