Risk Aversion Asymptotics for Power Utility Maximization

Transcription

1 Risk Aversion Asympoics for Power Uiliy Maximizaion Marcel Nuz ETH Zurich AnSAp10 Conference Vienna, Marcel Nuz (ETH) Risk Aversion Asympoics 1 / 15

2 Basic Problem Power uiliy funcion U(x) = 1 p x p, p (, 0) (0, 1) Uiliy maximizaion: max E [ T 0 ] U(c ) d + U(X T (π, c)) over rading and consumpion sraegies (π, c) Aim: Describe opimal sraegies in he limis p 0, i.e. relaive risk aversion 1. p, i.e. relaive risk aversion +. Marcel Nuz (ETH) Risk Aversion Asympoics 2 / 15

3 Ouline 1 Inroducion 2 Asympoics p 0 3 Asympoics p (0h order) 4 Asympoics p (1s order) Marcel Nuz (ETH) Risk Aversion Asympoics 3 / 15

4 Ouline 1 Inroducion 2 Asympoics p 0 3 Asympoics p (0h order) 4 Asympoics p (1s order) Marcel Nuz (ETH) Risk Aversion Asympoics 3 / 15

5 Uiliy Maximizaion Problem d risky asses: semimaringale R of sock reurns, R 0 = 0 spo prices S = ( E(R 1 ),..., E(R d ) ) saisfying NFLVR (Random) uiliy funcion U (x) = U (p) (x) = D 1 p x p, p (, 0) (0, 1), wih D > 0 càdlàg adaped and D, D 1 bounded For xed p and given iniial capial x 0 > 0: [ T ] u p (x 0 ) := sup E U (c ) d + U T (c T ) (π,c) A 0 }{{} U (c ) μ (d), wih μ :=d+δ {T } π L(R) rading sraegy, c 0 consumpion Wealh: X (π, c) = x 0 + X 0 s (π, c)π s dr s c 0 s ds Admissibiliy: (π, c) A if X (π, c) > 0 and c T = X T (π, c) Exisence [Karazas and šikovi AoP'03]: If u p (x 0 ) <, here exiss an opimal sraegy (ˆπ, ĉ) A, i.e., u p (x 0 ) = E [ T 0 U (ĉ ) μ (d) ]. Marcel Nuz (ETH) Risk Aversion Asympoics 4 / 15

6 Uiliy Maximizaion Problem d risky asses: semimaringale R of sock reurns, R 0 = 0 spo prices S = ( E(R 1 ),..., E(R d ) ) saisfying NFLVR (Random) uiliy funcion U (x) = U (p) (x) = D 1 p x p, p (, 0) (0, 1), wih D > 0 càdlàg adaped and D, D 1 bounded For xed p and given iniial capial x 0 > 0: [ T ] u p (x 0 ) := sup E U (c ) d + U T (c T ) (π,c) A 0 }{{} U (c ) μ (d), wih μ :=d+δ {T } π L(R) rading sraegy, c 0 consumpion Wealh: X (π, c) = x 0 + X 0 s (π, c)π s dr s c 0 s ds Admissibiliy: (π, c) A if X (π, c) > 0 and c T = X T (π, c) Exisence [Karazas and šikovi AoP'03]: If u p (x 0 ) <, here exiss an opimal sraegy (ˆπ, ĉ) A, i.e., u p (x 0 ) = E [ T 0 U (ĉ ) μ (d) ]. Marcel Nuz (ETH) Risk Aversion Asympoics 4 / 15

7 Uiliy Maximizaion Problem d risky asses: semimaringale R of sock reurns, R 0 = 0 spo prices S = ( E(R 1 ),..., E(R d ) ) saisfying NFLVR (Random) uiliy funcion U (x) = U (p) (x) = D 1 p x p, p (, 0) (0, 1), wih D > 0 càdlàg adaped and D, D 1 bounded For xed p and given iniial capial x 0 > 0: [ T ] u p (x 0 ) := sup E U (c ) d + U T (c T ) (π,c) A 0 }{{} U (c ) μ (d), wih μ :=d+δ {T } π L(R) rading sraegy, c 0 consumpion Wealh: X (π, c) = x 0 + X 0 s (π, c)π s dr s c 0 s ds Admissibiliy: (π, c) A if X (π, c) > 0 and c T = X T (π, c) Exisence [Karazas and šikovi AoP'03]: If u p (x 0 ) <, here exiss an opimal sraegy (ˆπ, ĉ) A, i.e., u p (x 0 ) = E [ T 0 U (ĉ ) μ (d) ]. Marcel Nuz (ETH) Risk Aversion Asympoics 4 / 15

8 Ouline 1 Inroducion 2 Asympoics p 0 3 Asympoics p (0h order) 4 Asympoics p (1s order) Marcel Nuz (ETH) Risk Aversion Asympoics 4 / 15

9 Asympoics p 0 for D 1 Propensiy o consume κ := Theorem (p 0) c X (π,c). Assume u p 0 (x 0) < for some p 0 > 0. Then ˆκ (p) If S is coninuous, T uniformly in, P-a.s. ˆπ(p) λ in L 2 loc (M). S coninuous + NFLVR srucure condiion: R = M + d M λ, some λ L 2 (M). loc Limi is he log-opimal sraegy. Marcel Nuz (ETH) Risk Aversion Asympoics 5 / 15

10 Asympoics p 0 for D 1 Propensiy o consume κ := Theorem (p 0) c X (π,c). Assume u p 0 (x 0) < for some p 0 > 0. Then ˆκ (p) If S is coninuous, T uniformly in, P-a.s. ˆπ(p) λ in L 2 loc (M). S coninuous + NFLVR srucure condiion: R = M + d M λ, some λ L 2 (M). loc Limi is he log-opimal sraegy. Marcel Nuz (ETH) Risk Aversion Asympoics 5 / 15

11 Asympoics p 0 for D 1 Propensiy o consume κ := Theorem (p 0) c X (π,c). Assume u p 0 (x 0) < for some p 0 > 0. Then ˆκ (p) If S is coninuous, T uniformly in, P-a.s. ˆπ(p) λ in L 2 loc (M). S coninuous + NFLVR srucure condiion: R = M + d M λ, some λ L 2 (M). loc Limi is he log-opimal sraegy. Marcel Nuz (ETH) Risk Aversion Asympoics 5 / 15

12 Asympoics p 0 for general D Theorem (p 0) Assume u p 0 (x 0) < for some p 0 > 0. Then ˆκ (p) D η uniformly in, P-a.s. where η := E [ T D s μ (ds) F ]. If S is coninuous, ˆπ(p) λ + Z η η in L 2 loc (M) wih he KW decomposiion η = η 0 + A η + Z η M + N η. Noe: Asympoically, Z η η is he hedging demand caused by D. Marcel Nuz (ETH) Risk Aversion Asympoics 6 / 15

13 Ouline 1 Inroducion 2 Asympoics p 0 3 Asympoics p (0h order) 4 Asympoics p (1s order) Marcel Nuz (ETH) Risk Aversion Asympoics 6 / 15

14 Asympoics p As he relaive risk aversion (1 p), he agen will be oo afraid o rade. If here is no rading, consumpion κ = (1 + T ) 1 is opimal. Theorem (p ) ˆκ (p) (1 + T ) 1 P-a.s., for each. If (F ) is coninuous, ˆπ(p) 0 in L 2 loc (M). Marcel Nuz (ETH) Risk Aversion Asympoics 7 / 15

15 Describing Consumpion via he Opporuniy Process D 1 for simpliciy. Dual problem for xed p < 0: [ T ] inf E U (Y ) μ (d), Y Y 0 where U (y) = sup x>0 ( U(x) xy ) = q 1 y q, q := p p 1. Y is a se of supermaringales Y > 0, conaining he EMMs. Dual opporuniy process L = L (p): [ L T ] (p) = ess sup Y Y E (Y s /Y ) q μ (ds) F Proposiion (N.'09a) ˆκ = 1/L Marcel Nuz (ETH) Risk Aversion Asympoics 8 / 15

16 Describing Consumpion via he Opporuniy Process D 1 for simpliciy. Dual problem for xed p < 0: [ T ] inf E U (Y ) μ (d), Y Y 0 where U (y) = sup x>0 ( U(x) xy ) = q 1 y q, q := p p 1. Y is a se of supermaringales Y > 0, conaining he EMMs. Dual opporuniy process L = L (p): [ L T ] (p) = ess sup Y Y E (Y s /Y ) q μ (ds) F Proposiion (N.'09a) ˆκ = 1/L Marcel Nuz (ETH) Risk Aversion Asympoics 8 / 15

17 ˆκ (p) (1 + T ) 1 is equivalen o L (p) 1 + T. Trivial esimae L (p) 1 + T. Fix densiy process Z of an EMM, hen T L (p) = ess sup Y Y E [ (Y s /Y ) q ] F μ (ds) = T T T E [ (Z s /Z ) q F ] μ (ds) E [ (Z s /Z ) F ] μ (ds) 1 μ (ds) = 1 + T. Marcel Nuz (ETH) Risk Aversion Asympoics 9 / 15

18 Ouline 1 Inroducion 2 Asympoics p 0 3 Asympoics p (0h order) 4 Asympoics p (1s order) Marcel Nuz (ETH) Risk Aversion Asympoics 9 / 15

19 1s Order Asympoics No inermediae consumpion. Addiional assumpions: S is coninuous. There exiss an [ EMM( Q wih )] nie enropy dq H(Q P) := E log dq dp dp Exponenial hedging problem: Le B L (F T ) be a coningen claim. Consider: [ max E exp ( ) ] B θ R T. θ Θ There exiss an opimal ˆθ(exp) Θ, Θ := {θ L(R) : θ R is a Q-supermaringale for all EMM Q wih H(Q P) < }. Marcel Nuz (ETH) Risk Aversion Asympoics 10 / 15

20 Theorem (p, no inermediae consumpion) Le (F ) be coninuous. For he claim B := log(d T ), (1 p) ˆπ(p) ˆθ(exp) in L 2 loc (M). Here ˆπ(p) denoes fracions of wealh and ˆθ(exp) denoes moneary amouns. Marcel Nuz (ETH) Risk Aversion Asympoics 11 / 15

21 Ouline of Proof Opporuniy process L: analogue of L for primal problem. KW decomposiion L(p) = L 0 (p) + A L(p) + Z L(p) M + N L(p). Exponenial opporuniy process L exp = L exp 0 + A Lexp + Z Lexp M + N Lexp (1 p)ˆπ(p) θ(exp) Z L(p) L(p) Z L exp L exp (1) Monooniciy of p L (p). (2) Convergence L (p) L exp P-a.s. for all. (3) Convergence of maringale pars M L(p) M Lexp in H 2 loc. Marcel Nuz (ETH) Risk Aversion Asympoics 12 / 15

22 Ouline of Proof Opporuniy process L: analogue of L for primal problem. KW decomposiion L(p) = L 0 (p) + A L(p) + Z L(p) M + N L(p). Exponenial opporuniy process L exp = L exp 0 + A Lexp + Z Lexp M + N Lexp (1 p)ˆπ(p) θ(exp) Z L(p) L(p) Z L exp L exp (1) Monooniciy of p L (p). (2) Convergence L (p) L exp P-a.s. for all. (3) Convergence of maringale pars M L(p) M Lexp in H 2 loc. Marcel Nuz (ETH) Risk Aversion Asympoics 12 / 15

23 (3) Convergence of Maringale Pars Wan: L(p) converge M L(p) converge Requires conrol over he semimaringale decomposiion Here via BSDE: Proposiion (Bellman BSDE, N.'09b) The riple (l, z, n) := ( L(p), Z L(p), N L(p)) is he minimal soluion of ( ) ( ) dl = 1 q l 2 λ + z l d M λ + z l + z dm + dn, l T = D T. L exp saises he BSDE for q = 1 and p q 1. Marcel Nuz (ETH) Risk Aversion Asympoics 13 / 15

24 Convergence of Bellman BSDEs (non-uniformly) quadraic BSDEs wih generaor f (p) (, y, z) (1 + y) λ 2 + (1 + y 1 ) z 2. Uniform localizaion such ha L(p), L 1 (p) C and λ d M λ C. Carry ou Kobylanski's (AoP'00) argumen: {M L(p) } p<0 is bounded in H 2 weak convergence in H 2. Use a es funcion argumen and he BSDE o ge srong convergence. Marcel Nuz (ETH) Risk Aversion Asympoics 14 / 15

25 Talk based on Risk Aversion Asympoics for Power Uiliy Maximizaion (preprin) [a] The Opporuniy Process for Opimal Consumpion and Invesmen wih Power Uiliy [b] The Bellman Equaion for Power Uiliy Maximizaion wih Semimaringales Seleced relaed lieraure: Coninuiy of opimal EMMs: Grandis&Rheinländer (AoP'02), Mania&Tevzadze (GeorgMJ'03) log-uiliy: Goll&Kallsen (AAP'03) Approximaion of uiliy on R: Schachermayer (AAP'01) Sabiliy of Meron problem: Jouini&Napp (FS'04), Larsen (MF'09), Kardaras&šikovi (MF'10),... Thanks for your aenion! Marcel Nuz (ETH) Risk Aversion Asympoics 15 / 15