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Suboptimality of Asian Executive Indexed

Options

Carole BernardPhelim Boyle

Jit Seng Chen

Actuarial Research Conference

August 13, 2011

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Outline

1. Options Preliminaries

2. Assumptions

3. Asian Executive Indexed Option

4. Cost-Efficiency

5. Constructing a Cheaper Payoff

6. True Cost Efficient Counterpart

7. Numerical Results

8. Stochastic Interest Rates

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Options Preliminaries

1 2 3 470

75

80

85

90

95

100

105

110

115

120

125

Time

Price

S1 = 100

H1 = 110

S2 = 75

H2 = 80

S3 = 120

H3 = 100

S4 = 80

H4 = 110

K= 90

Sample Price Paths

Stock, S

Benchmark, H

Strike, K

S4 = 4

S1S2S3S4 = 92.12, H4 =4

H1H2H3H4 = 99.19

European Call Option Payoff = max(S4 K, 0) = 0

Asian Option Payoff = max(S4

K, 0) = 2.12

Asian Indexed Option Payoff = max(S4 H4, 0) = 0

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Assumptions

1. Black-Scholes market: Extension to Vasicek short rate

2. Stock St and benchmark Ht driven by Brownian motions

3. Existence of state-price process t

4. Agents preferences depend only on the terminal distribution ofwealth

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Asian Executive Indexed Option

Asian Executive Indexed Option (AIO) proposed by Tian (2011):

Averaging: Prevent stock price manipulation Indexing: Only reward out-performance More cost-effective than traditional stock options

Provide stronger incentives to increase stock prices

Construct a better payoff:

Same features as the AIO

Strictly cheaper Use the concept of cost-efficiency

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Cost-Efficiency

From Bernard, Boyle and Vanduffel (2011):

Definition (1)

The cost of a strategy with terminal payoff XT is given by

c(XT) = EP[TXT]

where the expectation is taken under the physical measure P.

Intuition: T represents the price of a particular state

Definition (2)

A payoff is cost-efficient (CE) if any other strategy that generatesthe same distribution costs at least as much.

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Cost-Efficiency

Theorem (1)Let T be continuous. Define

YT = F1XT

(1 FT(T))

as the cost-efficient counterpart (CEC) of the payoff XT. Then,YT is a CE payoff with the same distribution as XT and is almostsurely unique.

Intuition: CEC is achieved by reshuffling the outcome of XT in

each state in reverse order with T while preserving the originaldistribution

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Constructing a Cheaper Payoff

1. Apply Theorem 1 to each term of the AIO

AT = max(ST HT, 0)

to get

AT

= maxdS

S1/

3

T dH

H1/

3

T, 0

2. It can be shown that: AT

d=AT

AT costs strictly less than AT

AT inherits the desired features of AT, but comes at a cheaperprice

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True Cost Efficient Counterpart

True CEC

AT = F

1

AT (1 FT(T))is estimated numerically

Examples:

1. Empirical cumulative distribution functions (CDFs) for eachpayoff in the base case 1

2. Reshuffling of AT to AT and AT

3. Order of AT, A and AT vs T

4. Price of each payoff and the efficiency loss

1K = 100, S0 = 100, r = 6%, S = 12%, I = 10%, S = 30%, I = 20%, = 0.75, qS = 2%,

qI = 3%, T = 1

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Numerical Results

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Payoffs

Probability

distribution

Empirial CDFs ofAT, A

T and AT

AT

A

T

AT

Figure: Comparison of the CDFs of AT, A

T andAT.

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Numerical Results

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

AT

A T

Plot of AT vs A

T

Figure: Reshuffling of outcomes ofAT to A

T

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Numerical Results

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

AT

AT

Plot of AT vs AT

Figure: Reshuffling of outcomes ofAT to AT

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Numerical Results

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

T

AT

Plot of AT vs T

Figure: Plot of outcomes ofAT vs T

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Numerical Results

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

80

T

A T

Plot of AT

vs T

Figure: Plot of outcomes of A

T vs T

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Numerical Results

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

T

AT

Plot of AT vs T

Figure: Plot of outcomes of AT vs T

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Numerical Results

CaseAT

AT

AT

VT VT Eff Loss VT Eff Loss

Base Case 3.26 4.34 33% 4.36 34%r = 4% 2.96 4.37 48% 4.40 49%S = 8% 3.97 4.35 10% 4.36 10%

I = 13% 3.26 4.34 33% 4.36 34%S = 35% 3.97 5.04 27% 5.07 28%I = 15% 3.27 4.34 33% 4.36 33% = 0.9 2.28 2.86 25% 2.87 26%

qS = 1.5% 3.27 4.35 33% 4.37 34%

qI = 2% 3.25 4.34 33% 4.36 34%

Table: Prices and efficiency loss of AT and AT compared against ATacross different parameters.

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Stochastic Interest Rates

Extension to a market with Vasicek short rate:

1. State price process expressed as a function of market variables2. Pricing formula for the AIO

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Summary

Reviewed the use of averaging and indexing in the context of

executive compensation

Constructed a strictly cheaper payoff with the same featuresas the AIO using cost-efficiency

Numerical examples that illustrate reshuffling of payoffs and

loss of efficiency Extension to the case of stochastic interest rates

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