Path-dependent inefficient strategies and how to make them efficient. Illustrated with the study of a popular retail investment product Carole Bernard (University of Waterloo) & Phelim Boyle (Wilfrid Laurier University) Carole Bernard Path-dependent inefficient strategies 1
Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Outline of the presentation ◮ What is cost-efficiency? ◮ Path-dependent payoffs are not cost-efficient. ◮ Consequences on the investors’ preferences. ◮ Illustration with a popular investment product: the locally-capped globally-floored contracts ( highly path-dependent ). ◮ Why do retail investors buy these contracts? ◮ Provide some explanations & evidence from the market. - Investors can overweight probabilities of getting high returns. - Locally-capped products are complex ◮ Provide a simple model Carole Bernard Path-dependent inefficient strategies 2
Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Efficiency Cost Dybvig (RFS 1988) explains how to compare two strategies by analyzing their respective efficiency cost. It is a criteria independent of the agents’ preferences. What is the “efficiency cost”? Carole Bernard Path-dependent inefficient strategies 3
Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Efficiency Cost • Given a strategy with payoff X T at time T . • Its no-arbitrage price P X . • F : X T ’s distribution under the physical measure. The distributional price is defined as: PD ( F ) = { Y T | Y T ∼ F } { No-arbitrage Price of Y T } min The “loss of efficiency” or “efficiency cost” is equal to: P X − PD ( F ) Carole Bernard Path-dependent inefficient strategies 4
Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Toy 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 financial contract with payoff X T at the end of the two periods. ❼ An expected utility maximizer with utility U . Let’s illustrate what the “efficiency cost” is and why it is a criteria independent of agents’ preferences. Carole Bernard Path-dependent inefficient strategies 5
� � � � � � Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Toy Example for X 2 , a payoff at T = 2 Real 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 = e − rT 3 , 4 2 2 � 1 16 + 6 162 + 9 � P X = Price of X = e − rT 163 , Efficiency cost = P X − P D Carole Bernard Path-dependent inefficient strategies 6
� � � � � � Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Y 2 , a payoff at T = 2 distributed as X Real 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 = e − rT 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 = Price of X = e − rT 163 , Efficiency cost = P X − P D Carole Bernard Path-dependent inefficient strategies 7
� � � � � � Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision X 2 , a payoff at T = 2 Real 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 � 1 − q 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 P D = Cheapest = e − rT , 161 = 4 2 � 1 � 16 + 6 162 + 9 = 5 P X = Price of X = e − rT 2 e − rT 163 Efficiency cost = P X − P , Carole Bernard Path-dependent inefficient strategies 8
� � � � � � Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Y 2 , a payoff at T = 2 Real 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 � 1 − q 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 = e − rT 2 e − rT , 161 4 2 � 1 � 16 + 6 162 + 9 = 5 P X = Price of X = e − rT 2 e − rT 163 Efficiency cost = P X − P , Carole Bernard Path-dependent inefficient strategies 9
� � � � � � Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Toy Example for X 2 , a payoff at T = 2 Real 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 2 e − rT , 4 2 P X = Price of X = 5 2 e − rT , Efficiency cost = P X − P D Carole Bernard Path-dependent inefficient strategies 10
� � � � � � Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Toy Example for X 2 , a payoff at T = 2 Real 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 2 e − rT , 4 2 P X = Price of X = 5 2 e − rT , Efficiency cost = P X − P D Carole Bernard Path-dependent inefficient strategies 11
� � � � � � Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Toy Example for X 2 , a payoff at T = 2 Real 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 2 e − rT , 4 2 P X = Price of X = 5 2 e − rT , Efficiency cost = P X − P D Carole Bernard Path-dependent inefficient strategies 12
Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Cost-efficiency in a general arbitrage-free model ❼ In an arbitrage-free market , there exists at least one state price process ( ξ t ) t . We choose one to construct a pricing operator. ❼ The cost of a strategy (or of a financial investment contract) with terminal payoff X T is given by: c ( X T ) = E [ ξ T X T ] ❼ The “distributional price” of a cdf F is defined as: PD ( F ) = { Y | Y ∼ F } { c ( Y ) } min where { Y | Y ∼ F } is the set of r.v. distributed as X T is. ❼ The efficiency cost is equal to: c ( X T ) − P D ( F ) Carole Bernard Path-dependent inefficient strategies 13
Cost-Efficiency Main result Example Preferences Retail Market Overweighting Impact on Decision Minimum Cost-efficiency Given a payoff X T with cdf F . We define its inverse F − 1 as follows: F − 1 ( y ) = min { x / F ( x ) ≥ y } . Theorem Define T = F − 1 (1 − F ξ ( ξ T )) X ∗ then X ∗ T ∼ F and X ∗ T is unique a.s. such that: PD ( F ) = c ( X ∗ T ) Carole Bernard Path-dependent inefficient strategies 14
Recommend
More recommend
