Minimal Sufficient Conditions for a Primal Optimizer in Nonsmooth - PowerPoint PPT Presentation

Minimal Sufficient Conditions for a Primal Optimizer in Nonsmooth Utility Maximization Harry Zheng Imperial College (Joint work with Nicholas Westray, Humboldt Univ. Berlin) Workshop on Stochastic Analysis and Finance Hong Kong City University 2 July 2009 1

Classical Utility Maximization • Assume a financial market consists of one bank account and one stock. • Assume the bank account is equal to 1. (interest rate r = 0) • Discounted stock price S is modelled by dS ( t ) = αS ( t ) dt + σS ( t ) dW ( t ) , S (0) = s > 0 where α > 0 excess return, σ > 0 stock volatility, W ( t ) Brownian motion. • Let H ( t ) be number of shares of asset S ( t ). Then portfolio wealth X ( t ) satisfies SDE dX ( t ) = H ( t ) dS ( t ) • Denote by U a utility function which is strictly increasing, strictly concave, continuously differentiable, U ′ (0) = ∞ and U ′ ( ∞ ) = 0. • Expected terminal utility maximization problem: H ∈A ( x ) E [ U ( X ( T ))] max (1) 2

Martingale Approach • Define set of random variables B ( x ) = { B : B ≥ 0 , F T − measurable , E [ H ( T ) B ] ≤ x } where � t � t � θ ( s ) dW ( s ) − 1 � θ ( s ) 2 ds H ( t ) = exp − 2 0 0 is stochastic density process and θ ( t ) = α/σ market price of risk. • Find optimal solution B ∗ to a static optimization problem B ∈ B ( x ) E [ U ( B )] max (2) • Find an admissible process H ∗ for a representation of B ∗ , i.e., X H ∗ ( T ) = B ∗ , a.s. 3

Dual Problem • Define dual function of U : ˜ U ( y ) = sup ( U ( x ) − xy ) . x ∈ R + • From budget constraint E [ H ( T ) B ] ≤ x and definition of dual function, we have the relation E [ ˜ � � B ∈ B ( x ) E [ U ( B )] ≤ min max U ( yH ( T ))] + xy . y ∈ R + • Dual problem is defined by E [ ˜ � � U ( yH ( T ))] + xy . min y ∈ R + • If there exist a B ∗ ∈ B ( x ) and y ∗ ≥ 0 such that E [ U ( B ∗ )] = E [ ˜ U ( y ∗ H ( T ))] + xy ∗ then B ∗ solves primal problem and y ∗ solves dual problem. 4

Construction of Optimal Solutions • For y ≥ 0 define B ∗ = I ( yH ( T )), where I ( y ) = ( U ′ ) − 1 ( y ) = − ˜ U ′ ( y ). Duality relation implies E [ U ( B ∗ )] = E [ ˜ U ( yH ( T ))] + y E [ H ( T ) B ∗ ] . • Find y ∗ to budget equation E [ H ( T ) I ( yH ( T ))] = x. Then y ∗ is optimal dual solution and B ∗ optimal primal solution. • B ∗ is replicated by martingale representation theorem. 5

Conditions for Martingale Approach Success of martingale approach above is based on following conditions: • There is no trading constraint. • Filtration is generated by diffusion asset price process. • Market is complete, i.e, stochastic density process H ( t ) is unique. • Utility function U is differentiable, strictly concave, which is crucial for definition of function I and for existence of y ∗ . 6

Wealth Process T finite time horizon, a market consisting of one bond, equal to 1, and d stocks, S 1 , . . . , S d modelled by an R d -valued semimartingale on a filtered probability space (Ω , F , ( F t ) 0 ≤ t ≤ T , P ), satisfying the usual conditions. For a predictable S -integrable process, the wealth processes is defined by X x,H = x + H ∙ S, (3) where x > 0 initial endowment. The set for optimization is following, for some X x,H as defined in (3) � � + ( P ) : X ≤ X x,H X ∈ L 0 X ( x ) := . T 7

Smooth Utility Maximization • U : R + → R is strictly increasing, strictly concave, C 1 , and U ′ (0) = ∞ and U ′ ( ∞ ) = 0. • Primal problem in incomplete market. � � u ( x ) := sup U ( X ) E X ∈X ( x ) • Domain of dual problem Y ∈ L 0 ( R + , F T ) : E [ XY ] ≤ xy for all X ∈ X ( x ) � � Y ( y ) := . • Dual problem � � Y ∈Y ( y ) E [ ˜ w ( x ) := inf inf U ( Y )] + xy y> 0 8

Kramkov and Schachermayer (1999) Result Theorem 1. Let M ( S ) � = ∅ (set of equivalent local martingale measures for S ) and xU ′ ( x ) AE ( U ) = lim sup U ( x ) < 1 x →∞ (asymptotic elasticity condition). Let x > 0 be such that u ( x ) < ∞ , then (i) There exists a unique solution y ∗ > 0 and Y ∗ ∈ Y ( y ∗ ) to dual problem � ˜ � w ( x ) = E U ( Y ∗ )] + xy ∗ . (ii) There exists a unique solution X ∗ to primal problem u ( x ) = E [ U ( X ∗ )] . (iii) E [ X ∗ Y ∗ ] = xy ∗ and X ∗ = − ˜ U ′ ( Y ∗ ) . (i)-(iii) imply u ( x ) = w ( x ). In classical case, Y + ( y ) = { yH ( T ) } , which implies Y ∗ = y ∗ H ( T ) and y ∗ being determined from (iii). 9

Nonsmooth Utility Functions Assumption 1 (Inada Condition) . � � inf ∂U ( x ) = 0 , sup ∂U ( x ) = ∞ . x ∈ R + x ∈ R + Assumption 2 (Concave Increasing Condition) . U (0) = 0 , U ( ∞ ) = ∞ and U is concave and increasing on R + . Assumption 3 (Asymptotic Elasticity Condition) . | q | y AE( ˜ U ) := lim sup sup U ( y ) < ∞ . ˜ y → 0 q ∈ ∂ ˜ U ( y ) Assumption 4 (No Arbitrage Condition) . M ( S ) � = ∅ . 10

Deelstra, Pham, Touzi (2001) Result DPT prove following theorem using quadratic inf-convolution method. Theorem 2. Let Assumptions 1-4 hold. Let x > 0 be such that u ( x ) < ∞ , then (i) There exists some y ∗ > 0 and Y ∗ ∈ Y ( y ∗ ) to dual problem � ˜ � w ( x ) = E U ( Y ∗ ) + xy ∗ . (ii) There exists some X ∗ ∈ X ( x ) to primal problem u ( x ) = E [ U ( X ∗ )] . (iii) E [ X ∗ Y ∗ ] = xy ∗ and X ∗ ∈ − ∂ ˜ U ( Y ∗ ) . 11

Deelstra-Pham-Touzi Conjecture (2001) For conjugate functions U and ˜ U , U ( x ) ≤ ˜ U ( y ) + xy for all y ≥ 0 , U ( x ) = ˜ U ( y ) + xy if and only if x ∈ − ∂ ˜ U ( y ) . Suppose ( y ∗ , Y ∗ ) are optimal for dual problem and ˆ X satisfies ˆ X ∈ − ∂ ˜ U ( Y ∗ ) and E [ ˆ XY ∗ ] = xy ∗ . Then E [ U ( ˆ X )] = E [ ˜ U ( Y ∗ ) + ˆ XY ∗ ] = E [ ˜ U ( Y ∗ )] + xy ∗ ≥ sup E [ U ( X )] . X ∈X ( x ) It is clear that ˆ X is optimal if and only if it is an element of X ( x ). Conjecture . If ˆ X ∈ − ∂ ˜ U ( Y ∗ ) and E [ ˆ XY ∗ ] = xy ∗ , then ˆ X ∈ X ( x ). Note that if U strictly concave or if a market is complete, then conjecture is trivially positive. To make conjecture nontrivial, U must be not strictly concave and market not complete. 12

Utility Function 2 √ x  x ∈ [0 , 1]  x + 1 x ∈ (1 , 5) U ( x ) := . 2 √ x − 4 + 4 x ∈ [5 , ∞ )  U is C 1 and satisfies Assumptions 1 and 2, but not strictly concave. Dual function is given by � 4 − 4 y + 1 y ∈ (0 , 1) ˜ y U ( y ) = y ∈ [1 , ∞ ) . 1 y U satisfies Assumption 3, but ˜ ˜ U is not continuously differentiable at y = 1. Subdifferential of ˜ U is given by  − 4 − 1 y ∈ (0 , 1) y 2   ∂ ˜ [ − 5 , − 1] y = 1 U ( y ) = . (4) − 1 y ∈ (1 , ∞ )   y 2 13

Probability Space Next we describe our market. Let (Ω , F , ( G t ) t ≥ 0 , P ) be a filtered probability space on which a process W and a random variable ξ are defined. (i) W is a standard Brownian motion and ( G t ) t ≥ 0 is augmented filtration generated by W . (ii) ξ is independent of entire path of W and valued in { 0 , 1 } with probabilities P ( ξ = 0) = 1 P ( ξ = 1) = 2 3 , 3 . To construct the filtration for our example we take ( G t ) t ≥ 0 and define F t := σ ( G t , σ ( ξ )) . 14

Asset Process We introduce stopping time τ , defined by � � t > 0 : W t + t ∈ ( − log 4 , log 4) τ := inf 2 / , and set terminal time horizon for utility maximization T := τ. A calculation based on first exit time of Brownian motion with drift shows that W τ + τ = 4 W τ + τ = 1 � � � � 2 = log 4 5 , 2 = − log 4 5 . P P Asset process S is defined by � � W t ∧ τ + 1 2( t ∧ τ ) S t := 1 { ξ =0 } + exp 1 { ξ =1 } . We observe that S has continuous paths, is uniformly bounded, nonnegative and at time T is valued in set { 1 / 4 , 1 , 4 } . 15

Equivalent Martingale Measures We may construct a large family of martingale measures simply by changing distribution of ξ . This is equivalent to determining pair ( q 0 , q 1 ) where Q ( ξ = 0) = q 0 , Q ( ξ = 1) = q 1 . For consistency we set p 0 = 1 3 and p 1 = 2 3 . Let q = ( q 0 , q 1 ) and consider processes Z q , defined for t ≥ 0 by t := q 0 q 1 Z q 1 { ξ =0 } + S − 1 1 { ξ =1 } . (5) t p 0 p 1 Using independence of W and ξ one can verify that Z q is a P -martingale for nonnegative q such that q 0 + q 1 = 1. Moreover, for any such q , Z q S is a P -martingale (it is constant). This implies that if we define d Q q d P = Z q T , then Q q ∈ M and therefore market is incomplete. 16

Primal Solution with Constant Investment Strategies We focus on u (1) and consider X const (1) = X ∈ L 0 � � + ( P ) : X ≤ 1 + H ( S T − S 0 ) , for H ∈ R � � − 1 3 , 4 �� X ∈ L 0 = + ( P ) : X ≤ 1 + H ( S T − S 0 ) , for H ∈ . 3 Define maximization problem u const (1) := � � sup U ( X ) E X ∈X const (1) which, in present setting, is following � 8 � � � 15 U (1 + 3 H ) + 2 1 − 3 + 1 u const (1) = sup 15 U 4 H 3 U (1) . H ∈ [ − 1 3 , 4 3 ] Since U is C 1 , one finds maximum solution H ∗ = 85 64 as optimal replicating strategy with optimal terminal wealth X ∗ := 1 + ( H ∗ ∙ S ) T = 1 { ξ =0 } + 319 64 1 { S T =4 ,ξ =1 } + 1 256 1 { S T = 1 4 ,ξ =1 } . 17