Optimal Investment with State-Dependent Constraints Carole Bernard SAFI 2011, Ann Arbor, May 2011. Carole Bernard Optimal Investment with State-Dependent Constraints 1/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions ◮ This talk is joint work with Phelim Boyle (Wilfrid Laurier University, Waterloo, Canada) and with Steven Vanduffel (Vrije Universiteit Brussel (VUB), Belgium). ◮ Outline of the talk: Characterization of optimal investment strategies for an 1 investor with law-invariant preferences and a fixed investment horizon Extension to the case when investors have state-dependent 2 constraints . Carole Bernard Optimal Investment with State-Dependent Constraints 2/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions ◮ This talk is joint work with Phelim Boyle (Wilfrid Laurier University, Waterloo, Canada) and with Steven Vanduffel (Vrije Universiteit Brussel (VUB), Belgium). ◮ Outline of the talk: Characterization of optimal investment strategies for an 1 investor with law-invariant preferences and a fixed investment horizon Extension to the case when investors have state-dependent 2 constraints . Carole Bernard Optimal Investment with State-Dependent Constraints 2/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Part I: Optimal portfolio selection for law-invariant investors Characterization of optimal investment strategies for an investor with law-invariant preferences and a fixed investment horizon • Optimal strategies are “cost-efficient”. • Cost-efficiency ⇔ Minimum correlation with the state-price process ⇔ Anti-monotonicity • Explicit representations of the cheapest and most expensive strategies to achieve a given distribution. • In the Black-Scholes setting, ◮ Optimality of strategies increasing in S T . ◮ Suboptimality of path-dependent contracts. Carole Bernard Optimal Investment with State-Dependent Constraints 3/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Main Assumptions • Consider an arbitrage-free market. • Given a strategy with payoff X T at time T . There exists Q , such that its price at 0 is c ( X T ) = E Q [ e − rT X T ] • P (“physical measure”) and Q (“risk-neutral measure”) are two equivalent probability measures: � dQ � ξ T = e − rT c ( X T ) = E Q [ e − rT X T ] = E P [ ξ T X T ] . , dP T We assume that all market participants agree on the state-price process ξ T . Carole Bernard Optimal Investment with State-Dependent Constraints 4/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Cost-efficient strategies A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. • Given a strategy with payoff X T at time T and cdf F under the physical measure P . The distributional price is defined as PD ( F ) = { Y | Y ∼ F } { E [ ξ T Y ] } = min { Y | Y ∼ F } c ( Y ) min • The strategy with payoff X T is cost-efficient if PD ( F ) = c ( X T ) Carole Bernard Optimal Investment with State-Dependent Constraints 5/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Cost-efficient strategies A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. • Given a strategy with payoff X T at time T and cdf F under the physical measure P . The distributional price is defined as PD ( F ) = { Y | Y ∼ F } { E [ ξ T Y ] } = min { Y | Y ∼ F } c ( Y ) min • The strategy with payoff X T is cost-efficient if PD ( F ) = c ( X T ) Carole Bernard Optimal Investment with State-Dependent Constraints 5/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Cost-efficient strategies A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution under P costs at least as much. • Given a strategy with payoff X T at time T and cdf F under the physical measure P . The distributional price is defined as PD ( F ) = { Y | Y ∼ F } { E [ ξ T Y ] } = min { Y | Y ∼ F } c ( Y ) min • The strategy with payoff X T is cost-efficient if PD ( F ) = c ( X T ) Carole Bernard Optimal Investment with State-Dependent Constraints 5/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Literature ◮ Cox, J.C., Leland, H., 1982. “On Dynamic Investment Strategies,” Proceedings of the seminar on the Analysis of Security Prices , 26 (2), U. of Chicago (published in 2000 in JEDC ), 24 (11-12), 1859-1880. ◮ Dybvig, P., 1988a. “Distributional Analysis of Portfolio Choice,” Journal of Business , 61 (3), 369-393. ◮ Dybvig, P., 1988b. “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market,” Review of Financial Studies , 1 (1), 67-88. Carole Bernard Optimal Investment with State-Dependent Constraints 6/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Simple Illustration Example of • X T ∼ Y T under P • but with different costs in a 2-period binomial tree. ( T = 2) Carole Bernard Optimal Investment with State-Dependent Constraints 7/41
� � � � � � Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions A simple illustration for X 2 , a payoff at T = 2 Real-world probabilities: p = 1 2 and risk neutral probabilities= q = 1 4 . 1 1 S 2 = 64 X 2 = 1 4 16 p S 1 = 32 p 1 − p 1 6 S 0 = 16 S 2 = 16 X 2 = 2 2 16 1 − p p S 1 = 8 1 − p 1 9 S 2 = 4 X 2 = 3 4 16 E [ U ( X 2 )] = U (1) + U (3) + U (2) P D = Cheapest = 3 , 4 2 2 � 1 � 16 + 6 162 + 9 P X 2 = Price of X 2 = 163 Efficiency cost = P X 2 − P D , Carole Bernard Optimal Investment with State-Dependent Constraints 8/41
� � � � � � Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Y 2 , a payoff at T = 2 distributed as X 2 Real-world probabilities: p = 1 2 and risk neutral probabilities: q = 1 4 . 1 1 S 2 = 64 Y 2 = 3 4 16 p S 1 = 32 p 1 − p 1 6 S 0 = 16 S 2 = 16 Y 2 = 2 2 16 1 − p p S 1 = 8 1 − p 1 9 S 2 = 4 Y 2 = 1 4 16 E [ U ( Y 2 )] = U (3) + U (1) + U (2) P D = Cheapest = 3 , 4 2 2 X 2 and Y 2 have the same distribution under the physical measure � 1 16 + 6 162 + 9 � P X 2 = Price of X 2 = 163 Efficiency cost = P X 2 − P D , Carole Bernard Optimal Investment with State-Dependent Constraints 9/41
� � � � � � Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions X 2 , a payoff at T = 2 Real-world probabilities: p = 1 2 and risk neutral probabilities: q = 1 4 . 1 1 S 2 = 64 X 2 = 1 4 16 q S 1 = 32 q 1 − q 1 6 S 0 = 16 S 2 = 16 X 2 = 2 2 16 1 − q q S 1 = 8 1 − q 1 9 S 2 = 4 X 2 = 3 4 16 � 1 E [ U ( X 2 )] = U (1) + U (3) + U (2) 163 + 6 162 + 9 � = 3 P D = Cheapest = 161 , 4 2 2 � 1 16 + 6 162 + 9 � = 5 c ( X 2 ) = Price of X 2 = 163 Efficiency cost = P X 2 − P D , 2 Carole Bernard Optimal Investment with State-Dependent Constraints 10/41
� � � � � � Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Y 2 , a payoff at T = 2 Real-world probabilities: p = 1 2 and risk neutral probabilities: q = 1 4 . 1 1 S 2 = 64 Y 2 = 3 4 16 q S 1 = 32 q 1 − q 1 6 S 0 = 16 S 2 = 16 Y 2 = 2 2 16 1 − q q S 1 = 8 1 − q 1 9 S 2 = 4 Y 2 = 1 4 16 � 1 E [ U ( X 2 )] = U (1) + U (3) + U (2) 163 + 6 162 + 9 � = 3 c ( Y 2 ) = 161 , 4 2 2 � 1 16 + 6 162 + 9 � = 5 c ( X 2 ) = Price of X 2 = 163 Efficiency cost = P X 2 − P D 2 Carole Bernard Optimal Investment with State-Dependent Constraints 11/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Traditional Approach to Portfolio Selection Consider an investor with increasing law-invariant preferences and a fixed horizon. Denote by X T the investor’s final wealth. • Optimize a law-invariant objective function max X T ( E P [ U ( X T )]) where U is increasing. 1 Minimizing Value-at-Risk 2 Probability target maximizing: max X T P ( X T > K ) 3 ... 4 • for a given cost (budget) cost at 0 = E Q [ e − rT X T ] = E P [ ξ T X T ] Find optimal strategy X ∗ ⇒ Optimal cdf F of X ∗ T T Carole Bernard Optimal Investment with State-Dependent Constraints 12/41
Introduction Cost-Efficiency Examples State-Dependent Constraints Conclusions Traditional Approach to Portfolio Selection Consider an investor with increasing law-invariant preferences and a fixed horizon. Denote by X T the investor’s final wealth. • Optimize a law-invariant objective function max X T ( E P [ U ( X T )]) where U is increasing. 1 Minimizing Value-at-Risk 2 Probability target maximizing: max X T P ( X T > K ) 3 ... 4 • for a given cost (budget) cost at 0 = E Q [ e − rT X T ] = E P [ ξ T X T ] Find optimal strategy X ∗ ⇒ Optimal cdf F of X ∗ T T Carole Bernard Optimal Investment with State-Dependent Constraints 12/41
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