Problems for ’Mathematical methods in finance’

(non strict second order dominance) iff Eu(A) ≥ Eu(B) for all utility functions u on [a, b] such that u′ ≥ 0 and u′′ ≤ 0 . 3. Second order dominance and stop loss. Show that (1) holds (i) iff E(B−d)− ≥ E(A− d)− for all d ∈ [a, b] or (ii) iff Emin(d,B) ≤ Emin(d,A) for all d ∈ [a, b]. 4. Utility and ARA. (a) Show that there exists a unique utility function u on R with a given ARA ρ(x) and such that u(0) = 0, u′(0) = 1. Express u in terms of ρ. (b) Let ui, i = 1, 2, be two utility functions with ARA ρi(x) and such that ui(0) = 0, ui(0) = 1. Show that if ρ1(x) ≥ ρ2(x) for |x| ≤ δ, then u1(x) ≤ u2(x) for all these x. Similarly, if ρ1(x) > ρ2(x) for |x| ≤ δ, then u1(x) < u2(x) for x ∈ (−δ, 0) ∪ (0, δ). ∗Department of Statistics, University of Warwick, Coventry CV4 7AL UK, Email: [email protected]