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 XT at time T . There exists Q, such that its price at 0 is PX = EQ [e −rT XT ] ∙ P (“physical measure”) and Q (“risk-neutral measure”) are two equivalent probability measures: ( ) dQ −rT 𝜉T = e , PX = EQ [e −rT XT ] = EP [𝜉T XT ]. 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 XT . This strategy depends on the financial market and is random. ˆ They want to maximize the expected utility of their final

wealth XT max (EP [U(XT )]) XT

U: utility (increasing because individuals prefer more to less). ˆ They want to minimize the cost of the strategy

cost at 0 = EQ [e −rT XT ] = EP [𝜉T XT ] Find optimal payoff XT

Carole Bernard

⇒ Optimal cdf F of XT

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 XT . This strategy depends on the financial market and is random. ˆ They want to maximize the expected utility of their final

wealth XT max (EP [U(XT )]) XT

U: utility (increasing because individuals prefer more to less). ˆ They want to minimize the cost of the strategy

cost at 0 = EQ [e −rT XT ] = EP [𝜉T XT ] Find optimal payoff XT

Carole Bernard

⇒ Optimal cdf F of XT

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Our Approach

ˆ Given the cdf F that the investor would like for his final wealth ˆ We give an explicit representation of the payoff XT such that

▶ XT ∼ F in the real world ▶ XT 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 XT at time T , and initial price at time 0 PX = EP [𝜉T XT ] ∙ F : XT ’s distribution under the physical measure P. The distributional price is defined as PD(F ) =

min

{YT ∣ YT ∼F }

{EP [𝜉T YT ]}

The “loss of efficiency” or “efficiency cost” is equal to: PX − PD(F )

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

A Simple Illustration Let’s illustrate what the “efficiency cost” is with a simple example. Consider : ˆ A market with 2 assets: a bond and a stock S. ˆ A discrete 2-period binomial model for the stock S. ˆ A strategy with payoff XT at the end of the two periods. ˆ 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 X2 , a payoff at T = 2 Real-world probabilities=p = 21 and risk neutral probabilities=q = 14 . S2 = 64 mm6 m m mmm S = 32 1 QQQ1−p p mm6 QQQ mm Q( mmm S0 = 16Q S2 = 16 QQQ1−p p mm6 m QQQ m m mm ( S1 = 8 Q QQQ1−p QQQ ( p

S2 = 4

1 4

1 16

X2 = 1

1 2

6 16

X2 = 2

1 4

9 16

X2 = 3

U(1) + U(3) U(2) 3 + , PD = Cheapest = 4 2 2 ( ) 1 6 9 = Price of X2 = + 2+ 3 , Efficiency cost = PX2 − PD 16 16 16 E [U(X2 )] =

PX2

Carole Bernard

Path-dependent inefficient strategies

10

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Y2 , a payoff at T = 2 distributed as X2 Real-world probabilities=p = 12 and risk neutral probabilities=q = 14 . S2 = 64 mm6 m m mmm S = 32 1 QQQ1−p p mm6 QQQ mm Q( mmm S0 = 16Q S2 = 16 QQQ1−p p mm6 m QQQ m m mm ( S1 = 8 Q QQQ1−p QQQ ( p

S2 = 4

1 4

1 16

Y2 = 3

1 2

6 16

Y2 = 2

1 4

9 16

Y2 = 1

U(3) + U(1) U(2) 3 + , PD = Cheapest = 4 2 2 (X and Y have the same distribution under the physical measure and thus the same utility) ( ) 1 6 9 P Price of X2 = + 2+ 3 , Efficiency cost = PX2 −strategies PD X2 = Carole Bernard Path-dependent inefficient E [U(Y2 )] =

11

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

X2 , a payoff at T = 2 Real-world probabilities=p = 12 and risk neutral probabilities=q = 14 . S2 = 64 mm6 m m mmm S = 32 1 QQQ1−q q mm6 QQQ mm Q( mmm S0 = 16Q S2 = 16 QQQ1−q q mm6 m QQQ m m mm ( S1 = 8 Q QQQ1−q QQQ ( q

S2 = 4

E [U(X2 )] =

U(2) U(1) + U(3) + 4 2 (

PX2 = Price of X2 = Carole Bernard

1 4

1 16

X2 = 1

1 2

6 16

X2 = 2

1 4

9 16

X2 = 3

( ,

PD = Cheapest =

1 6 9 + 2+ 3 16 16 16

) =

5 2

,

) 1 6 9 3 3+ 2+ 1 = 16 16 16 2

Efficiency cost = PX2 − PD Path-dependent inefficient strategies

12

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Y2 , a payoff at T = 2 Real-world probabilities=p = 12 and risk neutral probabilities=q = 14 . S2 = 64 mm6 m m mmm S = 32 1 QQQ1−q q mm6 QQQ mm Q( mmm S0 = 16Q S2 = 16 QQQ1−q q mm6 m QQQ m m mm ( S1 = 8 Q QQQ1−q QQQ ( q

S2 = 4

E [U(X2 )] =

U(2) U(1) + U(3) + 4 2 (

PX2 = Price of X2 = Carole Bernard

,

1 6 9 + 2+ 3 16 16 16

1 4

1 16

Y2 = 3

1 2

6 16

Y2 = 2

1 4

9 16

Y2 = 1

(

1 6 9 3+ 2+ 1 16 16 16

PY2 = ) =

5 2

,

) =

3 2

Efficiency cost = PX2 − PD Path-dependent inefficient strategies

13

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

A simple illustration for X2 , a payoff at T = 2 Real-world probabilities=p = 21 and risk neutral probabilities=q = 14 . S2 = 64 mm6 m m mmm S = 32 1 QQQ1−q q mm6 QQQ mm Q( mmm S0 = 16Q S2 = 16 QQQ1−q q mm6 m QQQ m m mm ( S1 = 8 Q QQQ1−q QQQ ( q

S2 = 4

E [U(X2 )] =

U(1) + U(3) U(2) + 4 2

PX2 = Price of X2 = Carole Bernard

5 2

,

,

1 4

1 16

X2 = 1

1 2

6 16

X2 = 2

1 4

9 16

X2 = 3

PD = Cheapest =

3 2

Efficiency cost = PX2 − PD Path-dependent inefficient strategies

14

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

A simple illustration for X2 , a payoff at T = 2 Real-world probabilities=p = 21 and risk neutral probabilities=q = 14 . S2 = 64 mm6 m m mmm S = 32 1 QQQ1−q q mm6 QQQ mm Q( mmm S0 = 16Q S2 = 16 QQQ1−q q mm6 m QQQ m m mm ( S1 = 8 Q QQQ1−q QQQ ( q

S2 = 4

E [U(X2 )] =

U(1) + U(3) U(2) + 4 2

PX2 = Price of X2 = Carole Bernard

5 2

,

,

1 4

1 16

X2 = 1

1 2

6 16

X2 = 2

1 4

9 16

X2 = 3

PD = Cheapest =

3 2

Efficiency cost = PX2 − PD Path-dependent inefficient strategies

15

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

A simple illustration for X2 , a payoff at T = 2 Real-world probabilities=p = 21 and risk neutral probabilities=q = 14 . S2 = 64 mm6 m m mmm S = 32 1 QQQ1−p p mm6 QQQ mm Q( mmm S0 = 16Q S2 = 16 QQQ1−p p mm6 m QQQ m m mm ( S1 = 8 Q QQQ1−p QQQ ( p

S2 = 4

E [U(X2 )] =

U(1) + U(3) U(2) + 4 2

PX2 = Price of X2 = Carole Bernard

5 2

,

,

1 4

1 16

X2 = 1

1 2

6 16

X2 = 2

1 4

9 16

X2 = 3

PD = Cheapest =

3 2

Efficiency cost = PX2 − PD Path-dependent inefficient strategies

16

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Cost-efficiency ˆ The cost of a strategy (or of a financial investment

contract) with terminal payoff XT is given by: c(XT ) = E [𝜉T XT ] ˆ The “distributional price” of a cdf F is defined as

PD(F ) =

min

{Y ∣ Y ∼F }

{c(Y )}

where {Y ∣ Y ∼ F } is the set of r.v. distributed as XT is. We want to find the strategy that realizes this minimum.

Carole Bernard

Path-dependent inefficient strategies

17

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Minimum Cost-efficiency Given a payoff XT with cdf F . We define its inverse F −1 as follows: F −1 (y ) = min {x / F (x) ≥ y } . Theorem Define

XT★ = F −1 (1 − F𝜉 (𝜉T )) then XT★ ∼ F and XT★ is a.s. unique such that PD(F ) = c(XT★ ) Consider a strategy with payoff XT distributed as F . The cost of this strategy satisfies: ∫ 1 −1 PD (F ) ⩽ c(XT ) ⩽ E [𝜉T F (F𝜉 (𝜉T ))] = F𝜉−1 (v )F −1 (v )dv 0

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

The least efficient payoff Theorem Let F be a cdf such that F (0) = 0. Consider the following optimization problem: max

{Z ∣ Z ∼F }

{c(Z )}

The strategy ZT★ that generates the same distribution as F with the highest cost can be described as follows: ZT★ = F −1 (F𝜉 (𝜉T ))

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

One needs E [𝜉T FX−1 (1 − F𝜉 (𝜉T ))] ⩽ E [𝜉T XT ] ⩽ E [𝜉T FX−1 (F𝜉 (𝜉T ))] It comes from the following property. Let Z = FZ−1 (U), then E [FZ−1 (U)FX−1 (1 − U)] ⩽ E [FZ−1 (U)X ] ⩽ E [FZ−1 (U)FX−1 (U)]

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Path-dependent payoffs are inefficient Corollary To be cost-efficient, the payoff of the derivative has to be of the following form: XT★ = F −1 (1 − F𝜉 (𝜉T )) It becomes a European derivative written on ST as soon as the state-price process 𝜉T can be expressed as a function of ST . Thus path-dependent derivatives are in general not cost-efficient. Corollary Consider a derivative with a payoff XT which could be written as XT = h(𝜉T ) Then XT is cost efficient if and only if h is non-increasing. Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Black and Scholes Model Under the physical measure P, dSt = 𝜇dt + 𝜎dWtP St Under the risk neutral measure Q, dSt = rdt + 𝜎dWtQ St St has a lognormal distribution. 𝜉T = e where a = exp Carole Bernard

(1

−rT

(

dQ dP

) =e

−rT

( a

T

) 2 ) − rT b = Tb(r + 𝜇 − 𝜎 2

ST S0

)−b

𝜇−r . 𝜎2 Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Black and Scholes Model Any path-dependent financial derivative is inefficient. Indeed 𝜉T = e

−rT

(

dQ dP

) =e T

−rT

( a

ST S0

)−b

( ) where a = exp 12 Tb(r + 𝜇 − 𝜎 2 ) − rT b = 𝜇−r . 𝜎2 To be cost-efficient, the payoff has to be written as ( ( ( ) )) ST −b ★ −1 X =F 1 − F𝜉 a S0 It is a European derivative written on the stock ST (and the payoff is increasing with ST when 𝜇 > r ). Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Put option in Black and Scholes model Assume a strike K . The payoff of the put is given by LT = (K − ST )+ . The payoff that has the lowest cost and is distributed such as the put option is given by YT★ = FL−1 (1 − F𝜉 (𝜉T )) .

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Put option in Black and Scholes model Assume a strike K . The payoff of the put is given by LT = (K − ST )+ . The cost-efficient payoff that will give the same distribution as a put option is ⎛ YT★ = ⎝K −

S02 e

) ( 2 2 𝜇− 𝜎2 T

ST

⎞+ ⎠ .

This type of power option “dominates” the put option.

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Cost-efficient payoff of a put cost efficient payoff that gives same payoff distrib as the put option 100

80 Put option

Payoff

60

Y* Best one

40

20

0 0

100

200

300

400

500

ST

With 𝜎 = 20%, 𝜇 = 9%, r = 5%, S0 = 100, T = 1 year, K = 100. Distributional price of the put = 3.14 Price of the put = 5.57 Efficiency loss for the put = 5.57-3.14= 2.43 Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Geometric Asian contract in Black and Scholes model Assume a strike K . The payoff of the Gemoetric Asian call is given by ( 1 ∫T )+ GT = e T 0 ln(St )dt − K )+ (( )1 ∏n n −K . which corresponds in the discrete case to k=1 S kT n

The efficient payoff that is distributed as the payoff GT is given by ( √ ) K + 1/ 3 ★ − GT = d ST d 1− √1 S0 3 e

(

√ )( ) 2 1 𝜇− 𝜎2 T 3

1 − 2

where d := . This payoff GT★ is a power call option. If 𝜎 = 20%, 𝜇 = 9%, r = 5%, S0 = 100. The price of this geometric Asian option is 5.94. The payoff GT★ costs only 5.77. Similar result in the discrete case. Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Example: the discrete Geometric option 120 100

Payoff

80 60

Z*T

40 Y*T

20 0 40

60

80

100 120 140 160 180 200 220 240 260 Stock Price at maturity ST

With 𝜎 = 20%, 𝜇 = 9%, r = 5%, S0 = 100, T = 1 year, K = 100, n = 12. Price of the geometric Asian option = 5.94. The distributional price is 5.77. The least-efficient payoff Z ★ costs 9.03. T

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Utility Independent Criteria Denote by ˆ XT the final wealth of the investor, ˆ V (XT ) the objective function of the agent, Assumptions 1

2

3

4

Agents’ preferences depend only on the probability distribution of terminal wealth: “law-invariant” preferences. (if XT ∼ ZT then: V (XT ) = V (ZT ).) Agents prefer “more to less”: if c is a non-negative random variable V (XT + c) ⩾ V (XT ). The market is perfectly liquid, no taxes, no transaction costs, no trading constraints (in particular short-selling is allowed). The market is arbitrage-free and complete.

For any inefficient payoff, there exists another strategy that these agents will prefer. Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Utility Independent Criteria Denote by ˆ XT the final wealth of the investor, ˆ V (XT ) the objective function of the agent, Assumptions 1

2

3

4

Agents’ preferences depend only on the probability distribution of terminal wealth: “law-invariant” preferences. (if XT ∼ ZT then: V (XT ) = V (ZT ).) Agents prefer “more to less”: if c is a non-negative random variable V (XT + c) ⩾ V (XT ). The market is perfectly liquid, no taxes, no transaction costs, no trading constraints (in particular short-selling is allowed). The market is arbitrage-free and complete.

For any inefficient payoff, there exists another strategy that these agents will prefer. Carole Bernard

Path-dependent inefficient strategies

29

Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Link with First Stochastic Dominance Theorem Consider a payoff XT with cdf F , 1

Taking into account the initial cost of the derivative, the cost-efficient payoff XT★ of the payoff XT dominates XT in the first order stochastic dominance sense : XT − c(XT )e rT ≺fsd XT★ − PD (F )e rT

2

The dominance is strict unless XT is a non-increasing function of 𝜉T . Thus the result is true for any preferences that respect first stochastic dominance.

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

A Very Different Approach Theorem Any payoff XT which cannot be expressed as a function of the state-price process 𝜉T at time T is strictly dominated in the sense of second-order stochastic dominance by HT★ = E [XT ∣ 𝜎(𝜉T )] = g (𝜉T ), which is a function of 𝜉T . Consequently in the Black and Scholes framework, any strictly path-dependent payoff is dominated by a path-independent payoff. ˆ Same cost. ˆ Different distribution.

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

A Very Different Approach Theorem Any payoff XT which cannot be expressed as a function of the state-price process 𝜉T at time T is strictly dominated in the sense of second-order stochastic dominance by HT★ = E [XT ∣ 𝜎(𝜉T )] = g (𝜉T ), which is a function of 𝜉T . Consequently in the Black and Scholes framework, any strictly path-dependent payoff is dominated by a path-independent payoff. ˆ Same cost. ˆ Different distribution.

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Example: the Lookback Option Consider a lookback call option with strike K . The payoff on this option is given by ( )+ LT = max {St } − K . 0⩽t⩽T

The cost efficient payoff with the same distribution YT★ = FL−1 (1 − F𝜉 (𝜉T )) . The payoff that has the highest cost and has the same distribution as the payoff LT is given by ZT★ = FL−1 (F𝜉 (𝜉T )) .

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Example: the Lookback Option 180 160 140

Payoff

120 100 80 Y*T

60 40

Z*T

20 0 40

60

80

100 120 140 160 Stock Price at maturity ST

180

200

220

With

𝜎 = 20%, 𝜇 = 9%, r = 5%S0 = 100, T = 1 year, K = 100. Distributional Price of the lookback = 18.85 Price of the lookback call = 19.17 Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Example: the Lookback Option Y*T

120

HT 100

Payoff

80 60 40 20 0

50

100 150 Stock Price at maturity ST

200

With 𝜎 = 20%, 𝜇 = 9%, r = 5%S0 = 100, T = 1 year, K = 100. Comparison of the two payoffs

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Example: the Lookback Option 0.7 0.6

CDF

0.5 0.4 0.3 0.2 cdf of Lookback = cdf of Y*T

0.1

cdf of HT 0 0

5

10

15 Payoff

20

25

30

With 𝜎 = 20%, 𝜇 = 9%, r = 5%S0 = 100, T = 1 year, K = 100. Comparison of the cdf of the two payoffs

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Explaining the Demand for Inefficient Payoffs 1

State-dependent needs ˆ Background risk: ˆ Hedging a long position in the market index ST (background risk) by purchasing a put option PT , ˆ the background risk can be path-dependent. ˆ Stochastic benchmark or other constraints: If the investor

wants to outperform a given (stochastic) benchmark Γ such that: P {𝜔 ∈ Ω / WT (𝜔) > Γ(𝜔)} ⩾ 𝛼. ˆ Intermediary consumption. 2

3

Other sources of uncertainty: the state-price process is not always a monotonic function of ST (non-Markovian interest rates for instance) Transaction costs, frictions: Preference for an available inefficient contract rather than a cost-efficient payoff that one needs to replicate.

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

Conclusion ˆ A preference free framework for ranking different investment

strategies. ˆ For a given investment strategy, we derive an explicit analytical expression 1 2

for the cheapest strategy that has the same payoff distribution. for the most expensive strategy that has the same payoff distribution.

ˆ There are strong connections between this approach and

stochastic dominance rankings. This may be useful for improving the design of financial products.

Carole Bernard

Path-dependent inefficient strategies

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Introduction

Cost-Efficiency

Main result

Examples

Preferences

Limits

References ▶ Bernard, C., Maj, M., and Vanduffel, S., 2010. “Improving the Design of Financial Products in a Multidimensional Black-Scholes Market,” NAAJ, forthcoming. ▶ Cambanis S., Simons J. and Stout W. 1976. “Inequalities for Ek(X , Y ) when the marginals are fixed”. Z. Wahrscheinlichkeitstheorie Verw. Geb., 36:285-294. ▶ Cox, J.C., Leland, H., 1982. “On Dynamic Investment Strategies,” Proceedings of the seminar on the Analysis of Security Prices, 26(2), University of Chicago. (published in 2000 in Journal of Economic Dynamics and Control, 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,” RFS. ▶ Goldstein, D.G., Johnson, E.J., Sharpe, W.F., 2008. “Choosing Outcomes versus Choosing Products: Consumer-focused Retirement Investment Advice,” Journal of Consumer Research, 35(3), 440-456. Carole Bernard

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