Path-dependent Inefficient Strategies and How to Make Them Efficient - - PowerPoint PPT Presentation

Path-dependent Inefficient Strategies and How to Make Them Efficient Frankfurt MathFinance Conference - March 2010

Carole Bernard (University of Waterloo) & Phelim Boyle (Wilfrid Laurier University)

Carole Bernard Path-dependent inefficient strategies 1

Cost-Efficiency Main result Example 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

Cost-Efficiency Main result Example 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

Cost-Efficiency Main result Example 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

Cost-Efficiency Main result Example 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 often not optimal! ▶ Examples: the put option and the geometric Asian option.

Carole Bernard Path-dependent inefficient strategies 3

Cost-Efficiency Main result Example 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 often not optimal! ▶ Examples: the put option and the geometric Asian option.

Carole Bernard Path-dependent inefficient strategies 3

Cost-Efficiency Main result Example 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 often not optimal! ▶ Examples: the put option and the geometric Asian option.

Carole Bernard Path-dependent inefficient strategies 3

Cost-Efficiency Main result Example 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 often not optimal! ▶ Examples: the put option and the geometric Asian option.

Carole Bernard Path-dependent inefficient strategies 3

Cost-Efficiency Main result Example 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 4

Cost-Efficiency Main result Example Preferences Limits

Some Assumptions ∙ Consider an arbitrage-free market. ∙ Given a strategy with payoff XT at time T. There exists Q, such that its price at 0 is PX = EQ[e−rTXT] ∙ P (“real measure”) and Q (“risk-neutral measure”) are two equivalent probability measures: 휉T = e−rT (dQ dP )

T

, PX = EQ[e−rTXT] = EP[휉TXT].

Carole Bernard Path-dependent inefficient strategies 5

Cost-Efficiency Main result Example Preferences Limits

Some Assumptions ∙ Consider an arbitrage-free market. ∙ Given a strategy with payoff XT at time T. There exists Q, such that its price at 0 is PX = EQ[e−rTXT] ∙ P (“real measure”) and Q (“risk-neutral measure”) are two equivalent probability measures: 휉T = e−rT (dQ dP )

T

, PX = EQ[e−rTXT] = EP[휉TXT].

Carole Bernard Path-dependent inefficient strategies 5

Cost-Efficiency Main result Example Preferences Limits

Some Assumptions ∙ Consider an arbitrage-free market. ∙ Given a strategy with payoff XT at time T. There exists Q, such that its price at 0 is PX = EQ[e−rTXT] ∙ P (“real measure”) and Q (“risk-neutral measure”) are two equivalent probability measures: 휉T = e−rT (dQ dP )

T

, PX = EQ[e−rTXT] = EP[휉TXT].

Carole Bernard Path-dependent inefficient strategies 5

Cost-Efficiency Main result Example Preferences Limits

Motivation Investors have a strategy that will give them a final wealth XT. This strategy depends on the financial market and is random.

❼ They want to maximize the expected utility of their final

wealth XT max

XT

(EP[U(XT)]) U: utility (increasing because individuals prefer more to less).

❼ They want to control the cost of the strategy

cost at 0 = EQ[e−rTXT] = EP[휉TXT]

Carole Bernard Path-dependent inefficient strategies 6

Cost-Efficiency Main result Example Preferences Limits

Motivation Investors have a strategy that will give them a final wealth XT. This strategy depends on the financial market and is random.

❼ They want to maximize the expected utility of their final

wealth XT max

XT

(EP[U(XT)]) U: utility (increasing because individuals prefer more to less).

❼ They want to control the cost of the strategy

cost at 0 = EQ[e−rTXT] = EP[휉TXT]

Carole Bernard Path-dependent inefficient strategies 6

Cost-Efficiency Main result Example Preferences Limits

Motivation Investors have a strategy that will give them a final wealth XT. This strategy depends on the financial market and is random.

❼ They want to maximize the expected utility of their final

wealth XT max

XT

(EP[U(XT)]) U: utility (increasing because individuals prefer more to less).

❼ They want to control the cost of the strategy

cost at 0 = EQ[e−rTXT] = EP[휉TXT]

Carole Bernard Path-dependent inefficient strategies 6

Cost-Efficiency Main result Example Preferences Limits

Efficiency Cost ∙ Given a strategy with payoff XT at time T, and initial price at time 0 PX = EP [휉TXT] ∙ F : XT’s distribution under the physical measure P. The distributional price is defined as PD(F) = min

{YT ∣ YT ∼F} {EP [휉TYT]}

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

Carole Bernard Path-dependent inefficient strategies 7

Cost-Efficiency Main result Example 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 XT at the end of the two periods. ❼ An expected utility maximizer with utility function U.

Carole Bernard Path-dependent inefficient strategies 8

Cost-Efficiency Main result Example Preferences Limits

A simple illustration for X2, a payoff at T = 2 Real probabilities=p = 1

2 and risk neutral probabilities=q = 1 4.

S2 = 64

1 4 1 16

X2 = 1 S1 = 32

p

• 1−p
• S0 = 16

p

• 1−p
• S2 = 16

1 2 6 16

X2 = 2 S1 = 8

p

• 1−p
• S2 = 4

1 4 9 16

X2 = 3 E[U(X2)] = U(1) + U(3) 4 + U(2) 2 , PD = Cheapest = 3 2 PX2 = Price of X2 = ( 1 16 + 6 162 + 9 163 ) , Efficiency cost = PX2 − PD

Carole Bernard Path-dependent inefficient strategies 9

Cost-Efficiency Main result Example Preferences Limits

Y2, a payoff at T = 2 distributed as X2 Real probabilities=p = 1

2 and risk neutral probabilities=q = 1 4.

S2 = 64

1 4 1 16

Y2 = 3 S1 = 32

p

• 1−p
• S0 = 16

p

• 1−p
• S2 = 16

1 2 6 16

Y2 = 2 S1 = 8

p

• 1−p
• S2 = 4

1 4 9 16

Y2 = 1 E[U(Y2)] = U(3) + U(1) 4 + U(2) 2 , PD = Cheapest = 3 2 (X and Y have the same distribution under the physical measure and thus the same utility) PX2 = Price of X2 = ( 1 16 + 6 162 + 9 163 ) , Efficiency cost = PX2 − PD

Carole Bernard Path-dependent inefficient strategies 10

Cost-Efficiency Main result Example Preferences Limits

X2, a payoff at T = 2 Real probabilities=p = 1

2 and risk neutral probabilities=q = 1 4.

S2 = 64

1 4 1 16

X2 = 1 S1 = 32

q

• 1−q
• S0 = 16

q

• 1−q
• S2 = 16

1 2 6 16

X2 = 2 S1 = 8

q

• 1−q
• S2 = 4

1 4 9 16

X2 = 3 E[U(X2)] = U(1) + U(3) 4 + U(2) 2 , PD = Cheapest = ( 1 163 + 6 162 + 9 161 ) = 3 2 PX2 = Price of X2 = ( 1 16 + 6 162 + 9 163 ) = 5 2 , Efficiency cost = PX2 − PD

Carole Bernard Path-dependent inefficient strategies 11

Cost-Efficiency Main result Example Preferences Limits

Y2, a payoff at T = 2 Real probabilities=p = 1

2 and risk neutral probabilities=q = 1 4.

S2 = 64

1 4 1 16

Y2 = 3 S1 = 32

q

• 1−q
• S0 = 16

q

• 1−q
• S2 = 16

1 2 6 16

Y2 = 2 S1 = 8

q

• 1−q
• S2 = 4

1 4 9 16

Y2 = 1 E[U(X2)] = U(1) + U(3) 4 + U(2) 2 , PY2 = ( 1 163 + 6 162 + 9 161 ) = 3 2 PX2 = Price of X2 = ( 1 16 + 6 162 + 9 163 ) = 5 2 , Efficiency cost = PX2 − PD

Carole Bernard Path-dependent inefficient strategies 12

Cost-Efficiency Main result Example Preferences Limits

A simple illustration for X2, a payoff at T = 2 Real probabilities=p = 1

2 and risk neutral probabilities=q = 1 4.

S2 = 64

1 4 1 16

X2 = 1 S1 = 32

q

• 1−q
• S0 = 16

q

• 1−q
• S2 = 16

1 2 6 16

X2 = 2 S1 = 8

q

• 1−q
• S2 = 4

1 4 9 16

X2 = 3 E[U(X2)] = U(1) + U(3) 4 + U(2) 2 , PD = Cheapest = 3 2 PX2 = Price of X2 = 5 2 , Efficiency cost = PX2 − PD

Carole Bernard Path-dependent inefficient strategies 13

Cost-Efficiency Main result Example Preferences Limits

A simple illustration for X2, a payoff at T = 2 Real probabilities=p = 1

2 and risk neutral probabilities=q = 1 4.

S2 = 64

1 4 1 16

X2 = 1 S1 = 32

q

• 1−q
• S0 = 16

q

• 1−q
• S2 = 16

1 2 6 16

X2 = 2 S1 = 8

q

• 1−q
• S2 = 4

1 4 9 16

X2 = 3 E[U(X2)] = U(1) + U(3) 4 + U(2) 2 , PD = Cheapest = 3 2 PX2 = Price of X2 = 5 2 , Efficiency cost = PX2 − PD

Carole Bernard Path-dependent inefficient strategies 14

Cost-Efficiency Main result Example Preferences Limits

A simple illustration for X2, a payoff at T = 2 Real probabilities=p = 1

2 and risk neutral probabilities=q = 1 4.

S2 = 64

1 4 1 16

X2 = 1 S1 = 32

p

• 1−p
• S0 = 16

p

• 1−p
• S2 = 16

1 2 6 16

X2 = 2 S1 = 8

p

• 1−p
• S2 = 4

1 4 9 16

X2 = 3 E[U(X2)] = U(1) + U(3) 4 + U(2) 2 , PD = Cheapest = 3 2 PX2 = Price of X2 = 5 2 , Efficiency cost = PX2 − PD

Carole Bernard Path-dependent inefficient strategies 15

Cost-Efficiency Main result Example Preferences Limits

Cost-Efficiency

❼ The cost of the payoff XT is c(XT) = E[휉TXT]. ❼ The “distributional price” of a cdf F is defined as

PD(F) = min

{Y ∣ Y ∼F} {c(Y )}

We want to find the strategy Y that realizes this minimum. 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 X★

T = F −1 (1 − F휉 (휉T))

then X★

T ∼ F and X★ T is a.s. unique such that

PD(F) = c(X★

T)

Carole Bernard Path-dependent inefficient strategies 16

Cost-Efficiency Main result Example Preferences Limits

Path-dependent payoffs are inefficient Corollary To be cost-efficient, the payoff of the derivative has to be of the following form: X★

T = F −1 (1 − F휉 (휉T))

It becomes a European derivative written on ST as soon as the state-price process 휉T 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(휉T) Then XT is cost efficient if and only if h is non-increasing.

Carole Bernard Path-dependent inefficient strategies 17

Cost-Efficiency Main result Example Preferences Limits

Black and Scholes Model Under the physical measure P, dSt St = 휇dt + 휎dW P

t

Under the risk neutral measure Q, dSt St = rdt + 휎dW Q

t

St has a lognormal distribution. 휉T = e−rT (dQ dP )

T

= e−rTa (ST S0 )−b where a = exp ( 1

2Tb(r + 휇 − 휎2) − rT

) b = 휇−r

휎2 .

Carole Bernard Path-dependent inefficient strategies 18

Cost-Efficiency Main result Example Preferences Limits

Black and Scholes Model Any path-dependent financial derivative is inefficient. Indeed 휉T = e−rT (dQ dP )

T

= e−rTa (ST S0 )−b where a = exp ( 1

2Tb(r + 휇 − 휎2) − rT

) b = 휇−r

휎2 .

To be cost-efficient, the payoff has to be written as X★ = F −1 ( 1 − F휉 ( a (ST S0 )−b)) It is a European derivative written on the stock ST (and the payoff is increasing with ST when 휇 > r).

Carole Bernard Path-dependent inefficient strategies 19

Cost-Efficiency Main result Example Preferences Limits

The Least Efficient Payoff Theorem Let F be a cdf such that F(0) = 0. Consider the following

• ptimization problem:

max

{Z ∣ Z∼F} {c(Z)}

The strategy Z★

T that generates the same distribution as F with

the highest cost can be described as follows: Z★

T = F −1 (F휉 (휉T))

Consider a strategy with payoff XT distributed as F. The cost of this strategy satisfies PD(F) ⩽ c(XT) ⩽ E[휉TF −1(F휉(휉T))] = ∫ 1 F −1

휉

(v)F −1(v)dv

Carole Bernard Path-dependent inefficient strategies 20

Cost-Efficiency Main result Example 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 Y ★

T = F −1 L

(1 − F휉 (휉T)) .

Carole Bernard Path-dependent inefficient strategies 21

Cost-Efficiency Main result Example 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 Y ★

T =

⎛ ⎝K − S2

0e2 ( 휇− 휎2

2

) T

ST ⎞ ⎠

+

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

Carole Bernard Path-dependent inefficient strategies 22

Cost-Efficiency Main result Example Preferences Limits

Cost-efficient payoff of a put

100 200 300 400 500 20 40 60 80 100 ST Payoff cost efficient payoff that gives same payoff distrib as the put option Y* Best one Put option

With 휎 = 20%, 휇 = 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 23

Cost-Efficiency Main result Example Preferences Limits

Geometric Asian contract in Black and Scholes model Assume a strike K. The payoff of the Gemoetric Asian call is given by GT = ( e

1 T

∫ T

0 ln(St)dt − K

)+ which corresponds in the discrete case to ((∏n

k=1 S kT

n

) 1

n − K

)+ . The efficient payoff that is distributed as the payoff GT is given by G★

T = d

( S1/

√ 3 T

− K d )+ where d := S

1− 1

√ 3

e

(

1 2 −

√

1 3

)( 휇− 휎2

2

) T

. This payoff G★

T is a power call option. If 휎 = 20%, 휇 = 9%,

r = 5%, S0 = 100. The price of a geometric Asian option is 5.94. The payoff G★

T costs only 5.77.

Similar result in the discrete case.

Carole Bernard Path-dependent inefficient strategies 24

Cost-Efficiency Main result Example Preferences Limits

Geometric Asian contract in Black and Scholes model Assume a strike K. The payoff of the Gemoetric Asian call is given by GT = ( e

1 T

∫ T

0 ln(St)dt − K

)+ which corresponds in the discrete case to ((∏n

k=1 S kT

n

) 1

n − K

)+ . The efficient payoff that is distributed as the payoff GT is given by G★

T = d

( S1/

√ 3 T

− K d )+ where d := S

1− 1

√ 3

e

(

1 2 −

√

1 3

)( 휇− 휎2

2

) T

. This payoff G★

T is a power call option. If 휎 = 20%, 휇 = 9%,

r = 5%, S0 = 100. The price of a geometric Asian option is 5.94. The payoff G★

T costs only 5.77.

Similar result in the discrete case.

Carole Bernard Path-dependent inefficient strategies 24

Cost-Efficiency Main result Example Preferences Limits

Example: the discrete Geometric option

40 60 80 100 120 140 160 180 200 220 240 260 20 40 60 80 100 120 Stock Price at maturity ST Payoff YT

*

ZT

*

With 휎 = 20%, 휇 = 9%, r = 5%, S0 = 100, T = 1 year, K = 100, n = 12. Price of a geometric Asian option = 5.94. The distributional price is 5.77. The payoff Z★

T costs 9.03. Carole Bernard Path-dependent inefficient strategies 25

Cost-Efficiency Main result Example Preferences Limits

Utility Independent Criteria Denote by

❼ XT the final wealth of the investor, ❼ V (XT) the objective function of the agent,

Assumptions (adopted by Dybvig (JoB1988,RFS1988))

1 Agents’ preferences depend only on the probability

distribution of terminal wealth: “law-invariant” preferences. (if XT ∼ ZT then: V (XT) = V (ZT).)

2 Agents prefer “more to less”: if c is a non-negative

random variable V (XT + c) ⩾ V (XT).

3 The market is perfectly liquid, no taxes, no transaction costs,

no trading constraints (in particular short-selling is allowed).

4 The market is arbitrage-free.

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

Carole Bernard Path-dependent inefficient strategies 26

Cost-Efficiency Main result Example Preferences Limits

Utility Independent Criteria Denote by

❼ XT the final wealth of the investor, ❼ V (XT) the objective function of the agent,

Assumptions (adopted by Dybvig (JoB1988,RFS1988))

1 Agents’ preferences depend only on the probability

distribution of terminal wealth: “law-invariant” preferences. (if XT ∼ ZT then: V (XT) = V (ZT).)

2 Agents prefer “more to less”: if c is a non-negative

random variable V (XT + c) ⩾ V (XT).

3 The market is perfectly liquid, no taxes, no transaction costs,

no trading constraints (in particular short-selling is allowed).

4 The market is arbitrage-free.

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

Carole Bernard Path-dependent inefficient strategies 26

Cost-Efficiency Main result Example 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 X★

T of the payoff XT dominates XT in the

first order stochastic dominance sense : XT − c(XT)erT ≺fsd X★

T − PD(F)erT

2 The dominance is strict unless XT is a non-increasing function

• f 휉T.

Thus the result is true for any preferences that respect first stochastic dominance.

Carole Bernard Path-dependent inefficient strategies 27

Cost-Efficiency Main result Example Preferences Limits

Explaining the Demand for Inefficient Payoffs

1 State-dependent needs

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

wants to outperform a given (stochastic) benchmark Γ such that: P {휔 ∈ Ω / WT(휔) > Γ(휔)} ⩾ 훼.

❼ Intermediary consumption.

2 Other sources of uncertainty: the state-price process is not

always a monotonic function of ST (non-Markovian interest rates for instance)

3 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 28

Cost-Efficiency Main result Example Preferences Limits

Explaining the Demand for Inefficient Payoffs

1 State-dependent needs

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

wants to outperform a given (stochastic) benchmark Γ such that: P {휔 ∈ Ω / WT(휔) > Γ(휔)} ⩾ 훼.

❼ Intermediary consumption.

2 Other sources of uncertainty: the state-price process is not

always a monotonic function of ST (non-Markovian interest rates for instance)

3 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 28

Cost-Efficiency Main result Example Preferences Limits

Explaining the Demand for Inefficient Payoffs

1 State-dependent needs

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

wants to outperform a given (stochastic) benchmark Γ such that: P {휔 ∈ Ω / WT(휔) > Γ(휔)} ⩾ 훼.

❼ Intermediary consumption.

2 Other sources of uncertainty: the state-price process is not

always a monotonic function of ST (non-Markovian interest rates for instance)

3 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 28

Cost-Efficiency Main result Example Preferences Limits

Explaining the Demand for Inefficient Payoffs

1 State-dependent needs

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

wants to outperform a given (stochastic) benchmark Γ such that: P {휔 ∈ Ω / WT(휔) > Γ(휔)} ⩾ 훼.

❼ Intermediary consumption.

2 Other sources of uncertainty: the state-price process is not

always a monotonic function of ST (non-Markovian interest rates for instance)

3 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 28

Cost-Efficiency Main result Example Preferences Limits

Conclusion

❼ A preference free framework for ranking different investment

strategies.

❼ For a given investment strategy, we derive an explicit

analytical expression

1

for the cheapest strategy that has the same payoff distribution.

2

for the most expensive strategy that has the same payoff distribution. ❼ 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 29