Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

6.1 Risk Sharing
◮ Suppose that there are two economic agents A and B measuring risk using
a general entropy convex measure of risk ρ A and ρ B with γ A , γ B ∈ R + .
◮ Let ¯ρ = −ρ, ēγ,Q = −eγ,Q and ¯c = −c.
◮ Suppose that A owns a financial payoff X A and B owns a financial payoff
X B .
◮ We solve explicitly the problem of optimal risk sharing given by
R A,B (X A ,X B ) = sup
F ∈L∞ {¯ρ A (X A − F + Π F ) + ¯ρ B (X B + F − Π F )}
= sup
¯F ∈L∞ {¯ρ A (X A + X B − ¯ F) + ¯ρ B ( ¯ F)} =: ¯ρ A ¯ρ B (X A + X B ),
where Π F is the agreed price of the financial derivative (risk transfer) F
and where we have set ¯F := F + X B .
◮ In particular, under technical conditions (see paper), the optimal risk
sharing is attained in the derivative F ∗ = γB
γ A +γ B X A − γA
γ A +γ B X B .
Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 37/40