Explicit Representation of Cost Efficient Strategies Suboptimality of Path-dependent Strategies Carole Bernard (University of Waterloo) & Phelim Boyle (Wilfrid Laurier University) Carole Bernard Path-dependent inefficient strategies 1
Introduction Cost-Efficiency Main result Examples Preferences Limits Motivation / Context ▶ Starting point: work on popular US retail investment products. How to explain the demand for complex path-dependent contracts? ▶ Met with Phil Dybvig at the NFA in Sept. 2008. ▶ Path-dependent contracts are not “efficient” (JoB 1988, “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market” in RFS 1988). Carole Bernard Path-dependent inefficient strategies 2
Introduction Cost-Efficiency Main result Examples Preferences Limits Motivation / Context ▶ Starting point: work on popular US retail investment products. How to explain the demand for complex path-dependent contracts? ▶ Met with Phil Dybvig at the NFA in Sept. 2008. ▶ Path-dependent contracts are not “efficient” (JoB 1988, “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market” in RFS 1988). Carole Bernard Path-dependent inefficient strategies 2
Introduction Cost-Efficiency Main result Examples Preferences Limits Motivation / Context ▶ Starting point: work on popular US retail investment products. How to explain the demand for complex path-dependent contracts? ▶ Met with Phil Dybvig at the NFA in Sept. 2008. ▶ Path-dependent contracts are not “efficient” (JoB 1988, “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market” in RFS 1988). Carole Bernard Path-dependent inefficient strategies 2
Introduction Cost-Efficiency Main result Examples Preferences Limits 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 Path-dependent inefficient strategies 3
Introduction Cost-Efficiency Main result Examples Preferences Limits 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 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 They want to minimize the cost of the strategy cost at 0 = E Q [ e − rT X T ] = E P [ 휉 T X T ] ⇒ Optimal cdf F of X T Find optimal payoff X T Carole Bernard Path-dependent inefficient strategies 4
Introduction Cost-Efficiency Main result Examples Preferences Limits 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 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 They want to minimize the cost of the strategy cost at 0 = E Q [ e − rT X T ] = E P [ 휉 T X T ] ⇒ Optimal cdf F of X T Find optimal payoff X T Carole Bernard Path-dependent inefficient strategies 4
Introduction Cost-Efficiency Main result Examples Preferences Limits Our Approach ❼ Given the cdf F that the investor would like for his final wealth ❼ We give an explicit representation of the payoff X T such that ▶ X T ∼ F in the real world ▶ X T corresponds to the cheapest strategy Carole Bernard Path-dependent inefficient strategies 5
Introduction Cost-Efficiency Main result Examples Preferences Limits Outline of the presentation ▶ What is cost-efficiency? ▶ Path-dependent strategies/payoffs are not cost-efficient. ▶ Explicit construction of efficient strategies. ▶ Investors (with a fixed horizon and law-invariant preferences) should prefer to invest in path-independent payoffs: path-dependent exotic derivatives are usually not optimal! ▶ Examples: the put option and the geometric Asian option. Carole Bernard Path-dependent inefficient strategies 6
Introduction Cost-Efficiency Main result Examples Preferences Limits Efficiency Cost Dybvig (RFS 1988) explains how to compare two strategies by analyzing their respective efficiency cost. What is the “efficiency cost”? It is a criteria for evaluating payoffs independent of the agents’ preferences. Carole Bernard Path-dependent inefficient strategies 7
Introduction Cost-Efficiency Main result Examples Preferences Limits Efficiency Cost ∙ Given a strategy with payoff X T at time T , and initial price at time 0 P X = E P [ 휉 T X T ] ∙ F : X T ’s distribution under the physical measure P . The distributional price is defined as PD ( F ) = { Y T ∣ Y T ∼ F } { E P [ 휉 T Y T ] } min The “loss of efficiency” or “efficiency cost” is equal to: P X − PD ( F ) Carole Bernard Path-dependent inefficient strategies 8
Introduction Cost-Efficiency Main result Examples Preferences Limits 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 Path-dependent inefficient strategies 9
� � � � � � Introduction Cost-Efficiency Main result Examples Preferences Limits 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 Path-dependent inefficient strategies 10
� � � � � � Introduction Cost-Efficiency Main result Examples Preferences Limits 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 Path-dependent inefficient strategies 11
� � � � � � Introduction Cost-Efficiency Main result Examples Preferences Limits 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 Path-dependent inefficient strategies 12
� � � � � � Introduction Cost-Efficiency Main result Examples Preferences Limits 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 Path-dependent inefficient strategies 13
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