Explicit Representation of Cost-efficient Strategies Carole Bernard (University of Waterloo) joint work with Phelim Boyle (Wilfrid Laurier University, Waterloo). Carole Bernard Explicit Representation of Cost-efficient Strategies 1
Introduction Cost-Efficiency Examples Preferences Conclusions Contributions ▶ Deriving explicitly the cheapest and the most expensive strategy to achieve a given distribution under general assumptions on the financial market. ▶ Extension of the work by Cox, J.C., Leland, H., 1982. “ On Dynamic Investment Strategies ,” Proceedings of the seminar on the Analysis of Security Prices , U. of Chicago. (published in 2000 in JEDC ). Dybvig, P., 1988a. “ Distributional Analysis of Portfolio Choice ,” Journal of Business . Dybvig, P., 1988b. “ Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market ,” RFS . ▶ Suboptimality of path-dependent contracts in Black Scholes model Carole Bernard Explicit Representation of Cost-efficient Strategies 2
Introduction Cost-Efficiency Examples Preferences Conclusions Some Assumptions ∙ Consider an arbitrage-free and complete market. ∙ Given a strategy with payoff X T at time T . There exists Q , such that its price at 0 is P X = E Q [ e − rT X T ] ∙ P (“physical measure”) and Q (“risk-neutral measure”) are two equivalent probability measures: ( dQ ) 휉 T = e − rT P X = E Q [ e − rT X T ] = E P [ 휉 T X T ] . , dP T Carole Bernard Explicit Representation of Cost-efficient Strategies 3
Introduction Cost-Efficiency Examples Preferences Conclusions Motivation: Traditional Approach to Portfolio Selection Investors have a strategy that will give them a final wealth X T . This strategy depends on the financial market and is random. ❼ For example they want to maximize the expected utility of their final wealth X T max ( E P [ U ( X T )]) X T U : utility (increasing because individuals prefer more to less). ❼ for a given cost of the strategy cost at 0 = E Q [ e − rT X T ] = E P [ 휉 T X T ] Find optimal payoff X T ⇒ Optimal cdf F of X T Carole Bernard Explicit Representation of Cost-efficient Strategies 4
Introduction Cost-Efficiency Examples Preferences Conclusions Motivation: Traditional Approach to Portfolio Selection Investors have a strategy that will give them a final wealth X T . This strategy depends on the financial market and is random. ❼ For example they want to maximize the expected utility of their final wealth X T max ( E P [ U ( X T )]) X T U : utility (increasing because individuals prefer more to less). ❼ for a given cost of the strategy cost at 0 = E Q [ e − rT X T ] = E P [ 휉 T X T ] Find optimal payoff X T ⇒ Optimal cdf F of X T Carole Bernard Explicit Representation of Cost-efficient Strategies 4
Introduction Cost-Efficiency Examples Preferences Conclusions Cost-efficient strategies ❼ Given the cdf F that the investor would like for his final wealth ❼ We derive an explicit representation of the payoff X T such that ▶ X T ∼ F in the real world ▶ X T corresponds to the cheapest strategy (=cost-efficient strategy) ▶ What is cost-efficiency? ▶ Explicit construction of cost-efficient strategies. Carole Bernard Explicit Representation of Cost-efficient Strategies 5
Introduction Cost-Efficiency Examples Preferences Conclusions Cost-efficient strategies ❼ Given the cdf F that the investor would like for his final wealth ❼ We derive an explicit representation of the payoff X T such that ▶ X T ∼ F in the real world ▶ X T corresponds to the cheapest strategy (=cost-efficient strategy) ▶ What is cost-efficiency? ▶ Explicit construction of cost-efficient strategies. Carole Bernard Explicit Representation of Cost-efficient Strategies 5
Introduction Cost-Efficiency Examples Preferences Conclusions A Simple Illustration Let’s illustrate what the “efficiency cost” is with a simple example. Consider : ❼ A market with 2 assets: a bond and a stock S . ❼ A discrete 2-period binomial model for the stock S . ❼ A strategy with payoff X T at the end of the two periods. ❼ An expected utility maximizer with utility function U . Carole Bernard Explicit Representation of Cost-efficient Strategies 6
� � � � � � Introduction Cost-Efficiency Examples Preferences 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 16 4 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 Explicit Representation of Cost-efficient Strategies 7
� � � � � � Introduction Cost-Efficiency Examples Preferences 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 16 4 E [ U ( Y 2 )] = U ( 3 ) + U ( 1 ) + U (2) P D = Cheapest = 3 , 4 2 2 ( X and Y have the same distribution under the physical measure and thus the same utility) ( 1 16 + 6 162 + 9 ) P X 2 = Price of X 2 = 163 Efficiency cost = P X 2 − P D , Carole Bernard Explicit Representation of Cost-efficient Strategies 8
� � � � � � Introduction Cost-Efficiency Examples Preferences 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 P X 2 = Price of X 2 = 163 Efficiency cost = P X 2 − P D , 2 Carole Bernard Explicit Representation of Cost-efficient Strategies 9
� � � � � � Introduction Cost-Efficiency Examples Preferences 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 P Y 2 = 161 , 4 2 2 ( 1 16 + 6 162 + 9 ) = 5 P X 2 = Price of X 2 = 163 Efficiency cost = P X 2 − P D , 2 Carole Bernard Explicit Representation of Cost-efficient Strategies 10
� � � � � � Introduction Cost-Efficiency Examples Preferences 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 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 E [ U ( X 2 )] = U (1) + U (3) + U (2) P D = Cheapest = 3 , 4 2 2 P X 2 = Price of X 2 = 5 Efficiency cost = P X 2 − P D , 2 Carole Bernard Explicit Representation of Cost-efficient Strategies 11
� � � � � � Introduction Cost-Efficiency Examples Preferences 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 P X 2 = Price of X 2 = 5 Efficiency cost = P X 2 − P D , 2 Carole Bernard Explicit Representation of Cost-efficient Strategies 12
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