# State-dependent utility maximization in Lévy markets

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1 State-dependent utility maximization in Lévy markets José Figueroa-López 1 Jin Ma 2 1 University of California at Santa Barbara 2 Purdue University KENT-PURDUE MINI SYMPOSIUM ON FINANCIAL MATHEMATICS, 2007

2 Outline 1 Introduction The portfolio optimization problem Relation to the replication of contingent claims 2 What has been done? Solution using convex duality 3 What is being done here? An explicit" solution for Lévy-driven markets 4 Conclusions

3 The portfolio optimization problem Formulation 1 Market: A risky asset with price process S t. A bond with value process B t. Frictionless with continuous trading. 2 Goal: Allocate an initial wealth w so that to maximize the agent s expected final utility during a finite time horizon [0, T ]. 3 State-dependent utility: U(w, ω) : R + Ω R s.t. Increasing and concave with the wealth w, for each state of nature ω Ω. Constant for wealths w above certain threshold H(ω) > 0. 4 Problem: Maximize E {U(V T, ω)} s.t. the agent s wealth process {V t } 0 t T satisfies V 0 w and V 0. Utility function graph

4 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

5 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

6 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

7 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

8 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

9 Relation to the replication of contingent claims Super-replication 1 Setting The discounted asset price {B 1 t S t } is a (locally bounded) semimartingale. The class M of Equivalent Martingale Measures (EMM) is non-empty. 2 The hedging problem An agent needs to deliver the payoff H of a claim at time T. He wishes to hedge away the risk by investing in the market 3 Super-replication: Any liability H can be hedged away completely, when starting with large enough wealth. The Fundamental Theorem (Kramkov 97) [ For any w w := sup Q M E Q B 1H ], there exists an T admissible {V t } t T such that V 0 = w and V T H

10 Relation to the replication of contingent claims Optimal partial replication 1 Motivation: w is typically too high". Initial wealth < w = Shortfall risk (sometimes, V T 2 The shortfall minimization problem: < H). Minimize E { L ( (H V T ) +)} s.t. V 0 w and V t 0, where L is increasing, convex, and null at 0. 3 Equivalent formulation: (Föllmer & Leukert, 2000) Maximize E {U (V T, ω)} s.t. V 0 w and V t 0, with U(w; ω) := L(H(ω)) L((H(ω) w) + ). Utility function graph

11 Solution using convex duality The dual problem 1 Primal problem: p (w) := sup E {U(V T, ω)}. V :V 0 w 2 The dual domain Γ: Nonnegative supermartingales {ξ t } t 0 ; 0{ ξ 0 1; } ξ t B 1 t V t is a supermartingale for all {V t } t T. t T 3 The dual problem: d (λ) := inf ξ Γ } E {Ũ(λξT B 1, ω), T where Ũ is the convex dual function of U.

12 Solution using convex duality Relationship between the dual and primal problems Theorem ([KrSch 99], [FllmLkrt, 2000]) 1 Weak duality: p (w) d (λ) + λw, for all λ > 0. 2 Strong duality: p (w) = d (λ ) + λ w, for some λ > 0. 3 The dual problem is attainable at some ξ. 4 The primal problem is attainable at some admissible V. 5 Dual characterization of the optimal final wealth: ( ) V = I λ ξ B 1, T T T where I(z, ω) := (U ) 1 (z) H.

13 Solution using convex duality Refinements for concrete models A natural problem For a specific market model and a given utility function, Can one narrow down the dual domain Γ Γ where to search ξ? 1 Itô incomplete market: (Karatzas et. al. 91) ds t = S t {{b t dt + σ t dw t }. t ξ t = exp 0 G(s)dW s 1 t 2 ds} 0 G(s) 2 = E( 0 G(s)dW s). 2 Lévy market: (Kunita 03) ds t = S t {b t dt + σ t dw t + } h(t, R ( z)ñ(dt, dz) d ξ t = E 0 G(s)dW s + ) F (t, 0 R z)ñ(dt, dz). d

14 Solution using convex duality Refinements for concrete models A natural problem For a specific market model and a given utility function, Can one narrow down the dual domain Γ Γ where to search ξ? 1 Itô incomplete market: (Karatzas et. al. 91) ds t = S t {{b t dt + σ t dw t }. t ξ t = exp 0 G(s)dW s 1 t 2 ds} 0 G(s) 2 = E( 0 G(s)dW s). 2 Lévy market: (Kunita 03) ds t = S t {b t dt + σ t dw t + } h(t, R ( z)ñ(dt, dz) d ξ t = E 0 G(s)dW s + ) F (t, 0 R z)ñ(dt, dz). d

15 Solution using convex duality Refinements for concrete models A natural problem For a specific market model and a given utility function, Can one narrow down the dual domain Γ Γ where to search ξ? 1 Itô incomplete market: (Karatzas et. al. 91) ds t = S t {{b t dt + σ t dw t }. t ξ t = exp 0 G(s)dW s 1 t 2 ds} 0 G(s) 2 = E( 0 G(s)dW s). 2 Lévy market: (Kunita 03) ds t = S t {b t dt + σ t dw t + } h(t, R ( z)ñ(dt, dz) d ξ t = E 0 G(s)dW s + ) F (t, 0 R z)ñ(dt, dz). d

16 Solution using convex duality Refinements for concrete models A natural problem For a specific market model and a given utility function, Can one narrow down the dual domain Γ Γ where to search ξ? 1 Itô incomplete market: (Karatzas et. al. 91) ds t = S t {{b t dt + σ t dw t }. t ξ t = exp 0 G(s)dW s 1 t 2 ds} 0 G(s) 2 = E( 0 G(s)dW s). 2 Lévy market: (Kunita 03) ds t = S t {b t dt + σ t dw t + } h(t, R ( z)ñ(dt, dz) d ξ t = E 0 G(s)dW s + ) F (t, 0 R z)ñ(dt, dz). d

17 Solution using convex duality What is being used and assumed? Key result Kunita-Watanabe (1967) ξ is a positive local martingale iff ξ t = ξ 0 E(X) with X t := t 0 t G(s)dW s + 0 F(s, z)ñ(ds, dz), F > 1. ξ is a positive supermartingale iff ξ t = ξ 0 E(X A) where X is as above and A is increasing predictable s.t. A < 1. Assumptions on the utility function Strictly increasing and concave. Inada conditions: U (0 + ) = and U ( ) = 0. U is unbounded.

18 An explicit" solution for Lévy-driven markets Set-up 1 The model: ds t = S t (b dt + d Z t ), where Z is a Lévy process. B t 1. 2 Primal problem: p (w) := sup V :V 0 w E {U(V T, ω)}. where U is a bounded state-dependent utility function. 3 Dual problem: } d (λ) := inf {Ũ(λξT E, ω), Γ ξ Γ where Γ is a suitable subclass of Γ to be chosen so that the dual theorem" holds.

19 An explicit" solution for Lévy-driven markets The Dual Theorem Let Γ be a convex subclass of Γ such that (i) w Γ := sup ξ Γ E {ξ T H} < (ii) Γ is closed under Fatou convergence. Then, for each 0 < w < w Γ, there exist λ > 0 and ξ Γ s.t. } 1 d (λ ) := inf Γ ξ Γ E {Ũ(λ ξ T, ω) is attainable at ξ 2 E { ξ I ( λ ξ )} T T = w, where I(z) := (U ) 1 (z) H 3 p (w) E [ U ( I ( λ ξ ))] T Furthermore, if (iii) Γ contains ξ t := E [ ] dq dp F t for any EMM Q M then I ( λ ξ T ) is super-replicable by an admissible V s.t. V 0 = w. Hence, V solves the primal problem.

20 An explicit" solution for Lévy-driven markets Construction of the dual class Γ 1 Let S be a family of loc. bounded local martingales s.t. S is predictable convex S is closed under Émery distance Then, the class Γ := {ξ := ξ0 E (X A) : X S, A increasing, and ξ 0}, is convex and closed under Fatou convergence". 2 The class S := {X t := t 0 t G(s)dW s + 0 meets the above conditions. F(s, z)ñ(ds, dz) : F 1}, 3 The class Γ := Γ Γ fulfills the conditions necessary for the Dual Theorem.

21 Conclusions The method here is more explicit in the sense that the dual domain enjoys an explicit parametrization. Such a parametrization could lead to certain discrete time approximations. It can accommodate more general jump-diffusion models driven by Lévy processes such as ds i (t) = S i (t ){bt idt + d j=1 σij t dw j t + R h(t, z)ñ(dt, dz)}. d What about optimal portfolio problems with consumption, { } sup E V :V 0 w & V 0 U 1 (V T ) + T where dv t := rv t dt + β t ds t c t dt? 0 U 2 (t, c(t))dt,

22 Graphs Bibliography Utility function and its convex dual function Ũ(z, ω) := sup {U(w, ω) z w} = U(I(z)) z I(z), 0 w H I(z) := (U ) 1 (z) H = Ũ (z). Return 1 Return 2 Return 3

23 Graphs Bibliography Bibliography Figueroa-Lopez and Ma. State-dependent utility maximization in Lévy markets, Preprint. Available at Kramkov and Schachermayer. The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Finance and Stochastics, Föllmer and Leukert. Efficient hedging: Cost versus shortfall risk. Finance and Stochastics, Karatzas, Lehoczky, Shreve, and Xu. Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control and Optimization, Kunita. Variational equality and portfolio optimization for price processes with jumps. In Stoch. Proc. and Appl. to Mathem. Fin., Kramkov. Optional decomposition of supermartingales and pricing of contigent claims in incomplete security markets. Prob. Th. and Rel. fields, 1996.

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