Thomas Knispel Leibniz Universität Hannover

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1 Optimal long term investment under model ambiguity Optimal long term investment under model ambiguity homas Knispel Leibniz Universität Hannover AnStAp0 Vienna, July 5th, 200 homas Knispel, Leibniz Universität Hannover

2 Optimal long term investment under model ambiguity 2 Outline Main problem: construction of optimal investment strategies under model ambiguity with a long term view hree problems: asymptotic maximization of robust expected power utility a robust outperformance criterion asymptotic minimization of robust downside risk homas Knispel, Leibniz Universität Hannover

3 Optimal long term investment under model ambiguity 3 Growth rates of robust utility homas Knispel, Leibniz Universität Hannover

4 Optimal long term investment under model ambiguity 4 Robust utility maximization Motivation: numerical representation of preferences (Gilboa&Schmeidler (89)): U(X) = inf Q Q E Q[u(X)] Utility maximization problem in the face of model ambiguity: - Fix an arbitrage-free financial market model. - X x 0,π wealth process of a self-financing strategy π with initial capital x 0 - finite time horizon - Find an investment strategy π with initial capital x 0 that maximizes inf E Q[u(X x 0,π )] among all admissible π. Q Q References: Quenez (2004), Schied&Wu (2005), Föllmer&Gundel (2006) homas Knispel, Leibniz Universität Hannover

5 Optimal long term investment under model ambiguity 5 Robust utility maximization (2) Power utility u(x) = λ xλ, λ (0, ): U (x 0 ) := sup π inf E Q[ Q Q λ (Xx 0,π ) λ ] grows exponentially as. Goal: maximize lim ln inf E Q[(X x 0,π ) λ ] among all strategies π Q Q Power utility u(x) = λ xλ, λ < 0: U (x 0 ) = λ inf π sup Q Q E Q [(X x 0,π ) λ ] converges to 0 exponentially fast as Goal: minimize lim ln sup Q Q E Q [(X x 0,π ) λ ] among all strategies π homas Knispel, Leibniz Universität Hannover

6 Optimal long term investment under model ambiguity 6 Asymptotic benchmark criteria homas Knispel, Leibniz Universität Hannover

7 Optimal long term investment under model ambiguity 7 Robust outperformance problem Finite maturity: maximize Q[X x 0,π H] among all strategies π quantile hedging (Föllmer&Leukert (999)) Special case: overperforming a bond with interest rate c H = exp(c ), L x 0,π := ln Xx 0,π Q[X x 0,π H] = Q[L x 0,π c] (portfolio s growth rate) Robust version: maximize inf Q Q Q[Lx 0,π c] among all strategies π Proposal for solution: asymptotic approach homas Knispel, Leibniz Universität Hannover

8 Optimal long term investment under model ambiguity 8 Robust outperformance problem (2) Asymptotic approach: If the family (L x 0,π ) 0 satisfies a large deviations principle with rate function IQ π, then Q[L x 0,π c] exp( I π Q(c) ) as. Problem (Pham (2003)): maximize lim ln Q[Lx 0,π c] among all π large deviations control problem Robust extension: maximize lim ln inf Q Q Q[Lx 0,π c] among all π homas Knispel, Leibniz Universität Hannover

9 Optimal long term investment under model ambiguity 9 Robust outperformance problem (3) Duality approach (heuristically): he rate function I π Q should be related to Λ Q (λ, π) := lim ln E Q[exp(λ L x 0,π )] }{{} logarithmic moment generating function = lim ln E Q[(X x 0,π ) λ ], }{{} growth rate of power utility i. e. lim ln Q[Lx 0,π c] = sup λ {λc Λ Q (λ, π)} (Gärtner-Ellis theorem) Duality formula: sup π with lim ln inf Q Q Q[Lx 0,π c] = sup λ>0 {λc Λ(λ)} Λ(λ) := sup π lim ln inf E Q[exp(λ L x 0,π Q Q )] = sup lim ln inf π E Q[(X x 0,π Q Q ) λ ] {z } optimal growth rate of expected power utility homas Knispel, Leibniz Universität Hannover

10 Optimal long term investment under model ambiguity 0 Asymptotic minimization of robust downside risk Goal: minimize lim ln sup Q[L x 0,π Q Q c] among all strategies π References: Hata, Nagai & Sheu (2008) (solution for the non-robust case) Duality approach: It is natural to expect that where inf π lim ln sup Q[L x 0,π Q Q c] = sup{λc Λ(λ)}, λ<0 Λ(λ) = inf π lim ln sup Q Q E Q [exp(λ L x 0,π) )] = inf π lim ln sup E Q [(X x 0,π ) λ ]. Q Q dual problem related to the optimal growth rates of robust expected power utility with negative parameters homas Knispel, Leibniz Universität Hannover

11 Optimal long term investment under model ambiguity Duality approach to asymptotic benchmark criteria involves growth rates of robust expected utility. homas Knispel, Leibniz Universität Hannover

12 Optimal long term investment under model ambiguity 2 Financial market model primary products: bond S 0, risky asset S prices affected by an economic factor Y (non-tradable) (Ω, F, (F t ) t 0, Q 0 ) path space of a 2-dim. Brownian motion (Wt, Wt 2 ) t 0 price dynamics under the reference measure Q 0 : - dy t = g(y t ) dt + ρ(y t ) dw t = g(y t ) dt + ρ (Y t ) dwt + ρ 2 (Y t ) dwt 2 - dst 0 = St 0 r(y t ) dt, S0 0 = - dst = St (m(y t ) dt + σ(y t ) dwt ) prices S 0, S and the factor dynamics are subject to model ambiguity homas Knispel, Leibniz Universität Hannover

13 Optimal long term investment under model ambiguity 3 Financial market model (2) probabilistic models: Q := {Q η η = (η t ) t 0 C} - Γ R 4 convex and compact - C set of all progressively measurable processes η with η t = (ηt, ηt 2, ηt 2, ηt 22 ) Γ - densities: D η dqη := F dq 0 := E( η 0 t Y t + ηt 2 dw t ) Girsanov s theorem ((W,η, W 2,η ) two-dimensional Q η -Brownian motion): dy t = [g(y t ) + (ρ(y t ), η t Y t + η 2 t )] dt + ρ(y t ) dw η t dst = St ([m(y t ) + σ(y t )(ηt Y t + ηt 2 )] dt + σ(y t ) dw η t Q perturbation of the drift terms homas Knispel, Leibniz Universität Hannover

14 Optimal long term investment under model ambiguity 4 Special cases i) Black-Scholes model with uncertain drift: under Q 0 : St 0 = exp(rt), dst = St (m dt + σ dwt ) factor process Y plays no role Γ = {(0, 0)} [a, b] {0} dst = St ([m + σηt 2 ] dt + σ dw,η t ) under Q η Q Schied (2005): least favorable measure for robust utility maximization explicit solutions for all three problems available homas Knispel, Leibniz Universität Hannover

15 Optimal long term investment under model ambiguity 5 Special cases (2) ii) Geometric Ornstein-Uhlenbeck model with uncertain rate of mean reversion: under Q 0 : dy t = η 0 Y t dt + σ dw t S t := exp(y t + αt) ds t = S t (( η 0 Y t + 2 σ2 + α) dt + σ dw t ) S 0 t = exp(rt) Γ = [ η 0 b σ, η 0 a σ ] {(0, 0, 0)} dy t = [η 0 σηt ]Y t dt + σ dw,η t under Q η Q random rate of mean reversion taking values in [a, b] explicit solutions for all three problems available homas Knispel, Leibniz Universität Hannover

16 Optimal long term investment under model ambiguity 6 Problem formulation Basic notation: π = (π t ) t 0 proportions of current wealth invested in S (Xt π ) t 0 wealth process of a self-financing trading strategy π Asymptotic maximization of robust power utility u(x) = λ xλ, λ (0, ): maximize lim ln inf Q η Q E Q η [(X π ) λ ] among all π Goal: Determine the optimal growth rate Λ(λ) := sup π lim an optimal long term trading strategy π (λ), an asymptotic worst-case measure Q η (λ). ln inf Q η Q E Q η [(X π )λ ], homas Knispel, Leibniz Universität Hannover

17 Optimal long term investment under model ambiguity 7 Ansatz Solution: optimization problem involves both maximizing controls π (trading strategies) minimizing controls η (probabilistic models) problem corresponds to a stochastic differential game on an infinite time horizon alternative approach: duality approach to robust utility maximization for a varying time horizon methods from risk-sensitive control (see, e. g., Fleming & McEneaney (95)) dynamic programming techniques characterization of Λ(λ), π (λ), η (λ) in terms of an ergodic Bellman equation homas Knispel, Leibniz Universität Hannover

18 Optimal long term investment under model ambiguity 8 Robust utility maximization heorem (Schied&Wu (2005)): U (x 0 ) := sup π where inf E Q[u(X x 0,π )] = inf { inf inf E Q [v(yy )] + x 0 y} Q Q y>0 Q Q Y Y } Q {{} dual value function - Y Q := {Y 0 Y 0 = and XY is a Q-supermartingale for all X X ()}, - v(y) := sup x>0 {u(x) xy} (convex conjugate function of u) here exist an optimal wealth process X x 0,π and a worst-case measure Q. robust extension of Kramkov&Schachermayer (99) homas Knispel, Leibniz Universität Hannover

19 Optimal long term investment under model ambiguity 9 Heuristic derivation i) parametrization of the class Y Qη ={Y 0 Y 0 = und (Y t X π t /S 0 t ) t Q η -supermartingale for all π} I. P Q 0 martingale measure on (Ω, F ) Z ν t := dp dq 0 F = E( 0 θ(y s ) dw s (θ market price of risk function, ν progressively measurable) 0 ν s dw 2 s ) t II. M := {ν = (ν t ) t 0 ν progressively measurable with 0 ν2 t dt <, > 0} {( dp Ft dq ) η t P martingale measure} {((D η t ) Zt ν ) t ν M} Y Qη homas Knispel, Leibniz Universität Hannover

20 Optimal long term investment under model ambiguity 20 Heuristic derivation (2) ii) Duality approach for a finite time horizon : U (x 0 ) = λ xλ 0 ( inf inf E Q 0 [(Z ν (S 0 ) ) λ λ (D η ) λ ]) λ ν M η C =... = λ xλ 0 ( inf inf E R ν M η C η,ν [exp( l(λ, η t, ν t, Y t ) dt)] 0 }{{} =:v(λ,y, ) ) λ Girsanov s theorem: dy t = h(λ, η t, ν t, Y t ) dt + ρ(y t ) dw η,ν t, W η,ν R η,ν -BM exponential of integral criterion homas Knispel, Leibniz Universität Hannover

21 Optimal long term investment under model ambiguity 2 Heuristic derivation (3) iii) Dynamic programming approach to identify v(λ, y, ): Hamilton-Jacobi-Bellman equation: v t = 2 ρ 2 v yy + inf ν R inf η Γ {l(λ, η, ν, )v + h(λ, η, ν, )v y}, v(, 0) iv) heuristic Ansatz: ( λ) ln v(λ, y, ) ln U λ (x 0) Λ(λ) + ϕ(λ, y) ergodic Bellman equation Λ(λ)= 2 ρ(y) 2 [ϕ yy (λ,y) + λ ϕ2 y(λ,y)] + inf ν R inf η Γ {( λ)l(λ,η,ν,y) + ϕ y(λ,y)h(λ,η,ν,y)} References: Bensoussan&Frehse (992), Nagai (996), Kaise and Sheu (2006),... ) he control η (λ) = η (λ, Y ) defined by the minimizing function η (λ, ) should define an asymptotic worst-case measure Q η (λ). homas Knispel, Leibniz Universität Hannover

22 Optimal long term investment under model ambiguity 22 Heuristic derivation (4) v) Computation of an optimal trading strategy under Q η (λ) : alternative version of the ergodic Bellman equation: Λ(λ)= 2 ρ(y) 2 [ϕ yy (λ,y) + ϕ 2 y he maximizing function π (λ, y) = λ should provide an (λ,y)] + sup{( λ) e l(λ,π,η (λ, y),y) + ϕ y (λ,y) e h(λ,π,η (λ, y),y)} π R σ(y) (ϕ y(λ, y)ρ (y) + θ(y) + η, (λ, y)y + η 2, (λ, y)). optimal long term trading strategy π (λ) = π (λ, Y ). homas Knispel, Leibniz Universität Hannover

23 Optimal long term investment under model ambiguity 23 Verification theorem necessary: verification theorem existence of a solution ( Λ(λ), ϕ(, λ)) of the ergodic Bellman equation has to be shown solution not necessarily unique specific solution yields the optimal growth rate and the optimal strategy existence of such a specific solution can be shown for linear Gaussian factor models and under strong assumptions in the non-linear case Assumption: here exists a regular solution Λ(λ) R +, ϕ(, λ) C 2 (R) to eλ(λ)= 2 ρ(y) 2 [ϕ yy (λ,y) + λ ϕ2 y (λ,y)] + inf inf {( λ)l(λ,η,ν,y) + ϕ y(λ,y)h(λ,η,ν,y)} ν R η Γ homas Knispel, Leibniz Universität Hannover

24 Optimal long term investment under model ambiguity 24 Verification theorem (2) heorem he solution Λ(λ) coincides with Λ(λ) = sup π lim ln inf Q η Q E Q η [(X π )λ ] (optimal growth rate of robust power utility) Λ Q η (λ)(λ) := sup π A lim ln E Q η (λ)[(x π )λ ] (optimal growth rate with respect to Q η (λ) ) he strategy π (λ) = π (λ, Y ) is optimal: Λ(λ) = lim ln inf E Q η Q Q η [(X π (λ) ) λ ] = lim ln E Q (λ)[(x π (λ) η ) λ ]. homas Knispel, Leibniz Universität Hannover

25 Optimal long term investment under model ambiguity 25 Case study: Geometric Ornstein-Uhlenbeck-model bond: S 0 t = exp(rt) stock: S t := exp(y t + αt) dy t = (η 0 σηt )Y t dt + σ dw,η t under Q η Q random rate of mean reversion taking values in [a, b] Solution to the asymptotic robust utility maximization problem: optimal growth rate: Λ(λ) = 2 ( λ)a + λ(r + 2σ 2 ( 2 σ2 + α r) 2 ) optimal strategy: π t (λ)=π (λ, Y t ) with π (λ, y)= λ a σ 2 y+ σ 2 ( 2 σ2 +α r) worst-case measure Q η (λ) : Y admits the minimal rate of mean reversion a. homas Knispel, Leibniz Universität Hannover

26 Optimal long term investment under model ambiguity 26 Summary asymptotics of robust utility maximization described by ergodic Bellman equations duality approach to asymptotic robust benchmark criteria via growth rates of robust expected power utility ergodic Bellman equations are an efficient tool to analyze long term investment problems Open problems and future research existence and properties of solutions to ergodic Bellman equations asymptotic utility maximization via ergodic BSDE s theory of robust large deviations homas Knispel, Leibniz Universität Hannover

27 Optimal long term investment under model ambiguity 27 References Fleming, W. H., McEneaney, W. M.: Risk-sensitive control on an infinite time horizon. In: SIAM J. Control Optim., volume 33(6):pp , 995. Föllmer, H., Schachermayer, W.: Asymptotic arbitrage and large deviations: In: Math. Financ. Econ., volume (3-4):pp , Hata, H., Nagai, H., Sheu, S.-J.: Asymptotics of the probability minimizing a downside risk. In: Ann. Appl. Probab., volume 20():pp , 200. Hernández-Hernández, D., Schied, A.: Robust utility maximization in a stochastic factor model. In:Statist. Decisions, volume 24():pp , Knispel,.: Asymptotics of robust utility maximization. Submitted, 200. Knispel,.: Asymptotic minimization of robust downside risk. Submitted, 200. Pham, H.: A large deviations approach to optimal long term investment. In: Finance Stoch., volume 7(2):pp , Schied, A., Wu, C.-.: Duality theory for optimal investments under model uncertainty. In: Statist. Decisions, volume 23(3):pp , homas Knispel, Leibniz Universität Hannover

28 Optimal long term investment under model ambiguity 28 hank you for your attention! homas Knispel, Leibniz Universität Hannover