Consumption investment optimization with Epstein-Zin utility

Transcription

1 Consumpion invesmen opimizaion wih Epsein-Zin uiliy Hao Xing London School of Economics Dublin Ciy Universiy, March 6, /25

2 Wha are recursive uiliies? Given a consumpion sream c, V = W(c,m(V +1 )). Here m is he cerainy equivalence and W is he aggregaor. 2/25

3 Wha are recursive uiliies? Given a consumpion sream c, V = W(c,m(V +1 )). Here m is he cerainy equivalence and W is he aggregaor. Example (Kreps-Poreus 78, Epsein-Zin 89) V = (1 e δ )c 1 1 ψ +e δ E [ V 1 γ +1 ] 1 1 ψ 1 γ Here ψ: elasiciy of ineremporal subsiuion (EIS), γ: risk aversion EIS = ln(c ( +1/c ) u ) (c +1 ), where r = ln r u. (c ) ψ. 2/25

4 Wha are recursive uiliies? Given a consumpion sream c, V = W(c,m(V +1 )). Here m is he cerainy equivalence and W is he aggregaor. Example (Kreps-Poreus 78, Epsein-Zin 89) V = (1 e δ )c 1 1 ψ +e δ E [ V 1 γ +1 ] 1 1 ψ 1 γ Here ψ: elasiciy of ineremporal subsiuion (EIS), γ: risk aversion EIS = ln(c ( +1/c ) u ) (c +1 ), where r = ln r u. (c ) When γ = 1/ψ, i is he ime separable von Neumann-Morgensern uiliy [ ] V 1 γ = (1 e δ )c 1 γ +e δ E V 1 γ ψ. 2/25

5 Sochasic differenial uiliy [Duffie Epsein 92] considered ] T V = E [U(c T )+ f(c u,v u )du. Example (Kreps-Poreus, Epsein-Zin) Here 1 f(c,v) = δ c1 ψ 1 1 ((1 γ)v) 1 1 c 1 γ θ δθv, U(c) = 1 γ. ψ θ = 1 γ 1 1/ψ. When γ = 1/ψ, θ = 1. ] T V = E [e δt δs c1 γ s U(c T )+ δe 1 γ ds. 3/25

6 Applicaions Equiy premium puzzle, risk-free rae puzzle Excess volailiy puzzle Credi spread puzzle 4/25

7 Applicaions Equiy premium puzzle, risk-free rae puzzle: [Bansal-Yaron JF04] Excess volailiy puzzle: [Benzoni-Collin Dufresne-Goldsein JFE11] Credi spread puzzle: [Bhamra-Kuehn-Srebulaev RFS10] Three imporan ingrediens in all hese applicaions: Epsein-Zin uiliy wih γ > 1 and ψ > 1. Marke models where risky asses have unbounded price of risk. Sae price densiy (or he marginal uiliy of he opimal value). 4/25

8 Early resoluion of uncerainy When he represenaive agen has he Epsein-Zin uiliy wih γ > 1 and ψ > 1, She prefers early resoluion of uncerainy. 5/25

9 Early resoluion of uncerainy When he represenaive agen has he Epsein-Zin uiliy wih γ > 1 and ψ > 1, She prefers early resoluion of uncerainy. V 1 1/ψ = U = (1 e δ ) c1 1/ψ [ ] 1 1 1/ψ +e δ E U θ θ +1 (1 e δ ) c1 1/ψ 1 1/ψ +e δ { E [U +1 ]+ 1 } θ 1 2E [U +1 ] var [U +1 ]. 5/25

10 Early resoluion of uncerainy When he represenaive agen has he Epsein-Zin uiliy wih γ > 1 and ψ > 1, She prefers early resoluion of uncerainy. V 1 1/ψ = U = (1 e δ ) c1 1/ψ [ ] 1 1 1/ψ +e δ E U θ θ +1 (1 e δ ) c1 1/ψ 1 1/ψ +e δ { E [U +1 ]+ 1 } θ 1 2E [U +1 ] var [U +1 ]. Hence asks a sizeable risk premium o compensae fuure uncerainy. 5/25

11 Lieraure Uiliy [Duffie-Epsein 92]: Lipschiz aggregaor [Duffie-Lions 92]: Markovian seing [Schroder-Skiadas 99]: θ = 1 γ 1 1/ψ > 0 [Kraf-Seifried-Sefensen 13]: special relaion beween γ and ψ [Kraf-Seiferling-Seifried working paper]: special relaion is removed for model wih bounded marke price of risk. 6/25

12 Lieraure Uiliy [Duffie-Epsein 92]: Lipschiz aggregaor [Duffie-Lions 92]: Markovian seing [Schroder-Skiadas 99]: θ = 1 γ 1 1/ψ > 0 [Kraf-Seifried-Sefensen 13]: special relaion beween γ and ψ [Kraf-Seiferling-Seifried working paper]: special relaion is removed for model wih bounded marke price of risk. Marke [Schroder-Skiadas 99]: bounded marke price of risk [Schroder-Skiadas 03]: facor model, ψ = 1 [Kraf-Seifried-Sefensen 13]: Heson model, special relaion beween γ and ψ [Kraf-Seiferling-Seifried working paper]: bounded price of risk 6/25

13 Sae price densiy The sae price densiy is imporan for applicaions [ ] D exp v f(cs,vs )ds c f(c,v ). 0 7/25

14 Sae price densiy The sae price densiy is imporan for applicaions [ ] D exp v f(cs,vs )ds c f(c,v ). 0 Uiliy gradien [Duffie-Skiadas 94], [El Karoui-Peng-Quenez 01]: [ ] T V 0 V0 = E f(c s,v s ) f(cs,v s )ds E 0 [ T 0 c f(c s c s )+ vf(v s V s )ds ]. Define he adjoin process Γ = exp ( 0 vf(c s,v s )ds ). [ ] T V 0 V0 E Γ s c f(c s cs ) /25

15 Sae price densiy The sae price densiy is imporan for applicaions [ ] D exp v f(cs,vs )ds c f(c,v ). 0 Uiliy gradien [Duffie-Skiadas 94], [El Karoui-Peng-Quenez 01]: [ ] T V 0 V0 = E f(c s,v s ) f(cs,v s )ds E 0 [ T 0 c f(c s c s )+ vf(v s V s )ds ]. Define he adjoin process Γ = exp ( 0 vf(c s,v s )ds ). [ ] T V 0 V0 E Γ s c f(c s cs ) 0. 0 However, when γ,ψ > 1, f is neiher Lipschiz nor joinly concave. 7/25

16 Our conribuions A verificaion resul for Epsein-Zin uiliy wih γ,ψ > 1, exending [Hu-Imkeller-Müller 05] and [Cheridio-Hu 11]. Mehods specially designed for unbounded price of risk (risk sensiive conrol lieraure, see [Guasoni-Roberson 12]). Verificaion of he sae price densiy. 8/25

17 Epsein-Zin uiliy via BSDE Recall ] T V c = E [U(c T )+ f(c u,vu c )du, where 1 f(c,v) = δ c1 ψ 1 1 ((1 γ)v) 1 1 c 1 γ θ δθv, U(c) = 1 γ. ψ Consider Y = e δθ (1 γ)v c which saisfies Y = e δθt c 1 γ T + T F(,c,y) = δθe δ c 1 1 ψ y 1 1 θ. F(s,c s,y s )ds T Then F saisfies he monooniciy condiion [Pardoux 99] Z s db s, where 9/25

18 Epsein-Zin uiliy via BSDE Recall ] T V c = E [U(c T )+ f(c u,vu c )du, where 1 f(c,v) = δ c1 ψ 1 1 ((1 γ)v) 1 1 c 1 γ θ δθv, U(c) = 1 γ. ψ Consider Y = e δθ (1 γ)v c which saisfies Y = e δθt c 1 γ T + T F(,c,y) = δθe δ c 1 1 ψ y 1 1 θ. 0 e δs cs 1 1/ψ F(s,c s,y s )ds T Then F saisfies he monooniciy condiion [Pardoux 99] { [ ] T Consider C a : c : E ds <,E[c 1 γ T ] < Theorem (Exisence) Z s db s, where Suppose γ,ψ > 1. For any c C a, V c exiss and is unique among class D processes. }. 9/25

19 Concaviy c V c is concave, if f(c,v) is joinly concave and Lipshciz in v, [Duffie-Epsein 92]. 10/25

20 Concaviy c V c is concave, if f(c,v) is joinly concave and Lipshciz in v, [Duffie-Epsein 92]. An orderly equivalen ransformaion 1 Y = e δt c1 ψ T T 1 1 ψ (Y,Z) = (Y 1/θ, 1 θ Y 1/θ 1 Z)/(1 1/ψ). 1 δe δs c1 ψ s 1 1 ψ This driver is joinly concave when θ < T 2 (θ 1)Z2 s ds Z s db s. Y s Theorem (Concaviy) When γ,ψ > 1, C a c V c is concave. 10/25

21 Consumpion invesmen problem Financial marke: S 0 : risk free asse, S = (S 1,...,S n ): risky asses. ds 0 = S0 r(x )d, ds = diag(s )[(r(x )+µ(x ))d +σ(x )dw ρ ], dx = b(x )d +a(x )dw, d W ρ,w = ρ(x )d. The wealh process saisfies dw = W [(r +π µ )d +π σ dw ρ ] c d. Problem: V c 0 Max! 11/25

22 Dynamic equaion The homoheic propery of Epsein-Zin uiliy implies V = W1 γ where Y saisfies he following BSDE Y = T 1 γ ey, T H(s,Y s,z s )ds Z s dw s. (1) We expec ha V + 0 f(c s,v s )ds is a maringale. H(,y,z) =quadraic in z +θ δψ ψ e ψ θ y +(1 γ)r(x)+ 1 γ 2γ µ Σ 1 µ(x) δθ. 12/25

23 Dynamic equaion The homoheic propery of Epsein-Zin uiliy implies V = W1 γ where Y saisfies he following BSDE Y = T 1 γ ey, T H(s,Y s,z s )ds Z s dw s. (1) We expec ha V + 0 f(c s,v s )ds is a maringale. H(,y,z) =quadraic in z 0 +θ δψ ψ e ψ θ y 0 +(1 γ)r(x)+ 1 γ 2γ µ Σ 1 µ(x) =: h(x) 0 δθ. 12/25

24 Exisence Thanks o previous bounds on H, soluion o (1) can be consrucion via he localizaion echnique in [Briand-Hu 06]. Theorem [ ] T Suppose γ,ψ > 1 and E 0 h(x s)ds >. Then (1) admis a soluion (Y,Z) such ha [ [ ( T )] T E h(x s )ds ] C Y C +loge exp h(x s )ds. In paricular, since h 0, Y is bounded from above. 13/25

25 Exisence Thanks o previous bounds on H, soluion o (1) can be consrucion via he localizaion echnique in [Briand-Hu 06]. Theorem [ ] T Suppose γ,ψ > 1 and E 0 h(x s)ds >. Then (1) admis a soluion (Y,Z) such ha [ [ ( T )] T E h(x s )ds ] C Y C +loge exp h(x s )ds. In paricular, since h 0, Y is bounded from above. The candidae opimal consumpion and invesmen sraegies: π = 1 γ σ 1 (µ +σ ρ Z ) and c W = δ ψ e ψ θ Y. 13/25

26 Verificaion [Hu-Imkeller-Muller] + comparison (π,c) is permissible when (W π ) 1 γ e Y is of class D and c C a. The same definiion as in [Hu-Cheridio 11], when marke price of risk is bounded. 14/25

27 Verificaion [Hu-Imkeller-Muller] + comparison (π,c) is permissible when (W π ) 1 γ e Y is of class D and c C a. The same definiion as in [Hu-Cheridio 11], when marke price of risk is bounded. For any permissible (π,c), le V = (Wπ ) 1 γ 1 γ e Y. V (Wπ T )1 γ 1 γ T T + f (c s,v s )ds Z s db s. Therefore (V, Z) is a supersoluion. Consider he soluion (V c,z). Then comparison implies w 1 γ 1 γ ey0 = V 0 V c 0, c. 14/25

28 Verificaion con Need V + 0 f(c s,v s )ds is a maringale: Assumpion There exiss a Lyapunov funcion φ C 2 (E) such ha i) φ(x), as x E; ii) F[φ] is bounded from below on E, where F is associaed o (1). Time separable uiliy: [Guasoni-Roberson 12], [Roberson-X. 14] 15/25

29 Verificaion con Need V + 0 f(c s,v s )ds is a maringale: Assumpion There exiss a Lyapunov funcion φ C 2 (E) such ha i) φ(x), as x E; ii) F[φ] is bounded from below on E, where F is associaed o (1). Time separable uiliy: [Guasoni-Roberson 12], [Roberson-X. 14] Need o show c C a. Assumpion Some momen condiion of he marke price of risk under a maringale measure (usually we choose he minimal maringale measure). 15/25

30 Verificaion con Need V + 0 f(c s,v s )ds is a maringale: Assumpion There exiss a Lyapunov funcion φ C 2 (E) such ha i) φ(x), as x E; ii) F[φ] is bounded from below on E, where F is associaed o (1). Time separable uiliy: [Guasoni-Roberson 12], [Roberson-X. 14] Need o show c C a. Assumpion Some momen condiion of he marke price of risk under a maringale measure (usually we choose he minimal maringale measure). When he marke price of risk is bounded, neiher condiion is needed. 15/25

31 Verificaion con Need V + 0 f(c s,v s )ds is a maringale: Assumpion There exiss a Lyapunov funcion φ C 2 (E) such ha i) φ(x), as x E; ii) F[φ] is bounded from below on E, where F is associaed o (1). Time separable uiliy: [Guasoni-Roberson 12], [Roberson-X. 14] Need o show c C a. Assumpion Some momen condiion of he marke price of risk under a maringale measure (usually we choose he minimal maringale measure). When he marke price of risk is bounded, neiher condiion is needed. Theorem (Verificaion) Under above assumpions, π and c maximize he Epsein-Zin uiliy among all permissible sraegies. 15/25

32 Sae price densiy Recall [ D = cexp v f(cs,v s ] )ds c f(c,v ). 0 16/25

33 Sae price densiy Recall [ D = cexp v f(cs,v s ] )ds c f(c,v ). 0 Theorem The following saemens hold: i) W π D + 0 D s c sds is a supermaringale for any (π,c); ii) W D + 0 D s c s ds is a maringale. Therefore [ ] [ E W π TD T + T 0 D s c s ds w = E W TD T + T 0 D sc s ds ]. 16/25

34 Example Consider ds = S [(r +σλx )d +σ X dw ρ ], dx = b(l X )d +a X dw. Assume b,l,r 0, a,σ,λ > 0, and bl > 1 2 a2. 17/25

35 Example Consider ds = S [(r +σλx )d +σ X dw ρ ], dx = b(l X )d +a X dw. Assume b,l,r 0, a,σ,λ > 0, and bl > 1 2 a2. The Lyapunov funcion can be chosen as φ(x) = C log(x)+cx, for sufficienly small C,C > 0. The momen condiion is saisfied when [ bλρ (ψ 1) a + 1 ] 2 (ψ (ψ 1)ρ2 )λ 2 < b2 2a 2. 17/25

36 Example Consider ds = S [(r +σλx )d +σ X dw ρ ], dx = b(l X )d +a X dw. Assume b,l,r 0, a,σ,λ > 0, and bl > 1 2 a2. The Lyapunov funcion can be chosen as φ(x) = C log(x)+cx, for sufficienly small C,C > 0. The momen condiion is saisfied when [ bλρ (ψ 1) a + 1 ] 2 (ψ (ψ 1)ρ2 )λ 2 < b2 2a 2. This condiion is saisfied in he empirically relevan specificaions [Liu-Pan 03]: λ = 0.47, σ = 1, b = 5, a = 0.25, ρ = 0.5, ψ = /25

37 Numeric resuls: Opimal consumpion-wealh raio 0.12 Opimal consumpion wealh raio ψ=0.2 ψ=1.5 ψ= Volailiy c W = δ ψ e ψ θ Y. 18/25

38 Opimal invesmen sraegies γ=2 γ=5 γ=8 Opimal porfolio Volailiy π = 1 γ σ 1 (µ +σ ρ Z ). 19/25

39 Log-linear ransformaion Campbell-Shiller approximaion (infinie horizon) e ψ θ y 1 ψ y, when θ 0. θ 0.45 Opimal consumpion wealh raio Exac soluion Campbell Shiller approximaion Time [] 20/25

40 Convergence o saionary limi ψ < opimal consumpion wealh raio δ=0.06 δ=0.08 δ=0.1 δ= Time [] 21/25

41 Convergence o saionary limi ψ > 1 Opimal consumpion wealh raio δ=0.06 δ=0.08 δ=0.1 δ= Time [] 22/25

42 Localizaion echnique of Briand and Hu Consider a quadraic BSDE wih unbounded erminal condiion: Y = ξ + T f(,y,z )d T dw. Suppose ha we have an a priori bounds on Y: Y Y Y, for locally bounded Y,Y. 23/25

43 Localizaion echnique of Briand and Hu Consider a quadraic BSDE wih unbounded erminal condiion: Y = ξ + T f(,y,z )d T dw. Suppose ha we have an a priori bounds on Y: Y Y Y, for locally bounded Y,Y. 1. Wih runcaed ξ n, BSDE admis soluion (Y n,z n ) wih Y n+1 Y n [Kobylanski 00]. 2. For a reducing sequence (σ m ), consruc (Y m ) σm := lim n (Y n) σm. 3. Since (Y k ) σm = (Y k 1 ) σm for k > m, define Y σm := (Y m ) σm. 4. Send m and verify he erminal condiion. 23/25

44 Conclusion This paper covers hree aspecs, which are imporan for applicaions: γ,ψ > 1 unbounded marke value of risk sae price densiy When ψ > 1, finie horizon problem converges slowly o is infinie horizon limi Fuure sudies: Dualiy Variaional formulaion of he recursive uiliy [Geoffard 95], [El Karoui-Peng-Quenez 97] Equilibrium [Dumas-Uppal-Wang 00], [Bank-Riedel 01] 24/25

45 Thanks for your aenion! 25/25