Predictive characteristics of standard deviations of stock price returns inferred from option prices

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Predictive characteristics of standard deviations of stock price returns inferred from option prices
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PREDICTIVE CHARACTERISTICS OF STANDARD DEVIATIONS
OF STOCK PRICE RETURNS INFERRED FROM OPTION PRICES










By

DONALD P. CHIRAS
















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA

1977























Q Copyright by
Donald P. Chiras
1977















ACKNOWLEDGMENTS


I would like to express my appreciation to the

members of my committee for their helpful comments and

suggestions.

Special thanks go to my chairman,Dr. William M.

Howard,for his decisive actions which permitted me to

achieve my goal without undue delay.

Substantial technical assistance was provided by

Steven Manaster to whom I am indebted for directing my

efforts in a productive manner. His research skills have

earned my deep respect and admiration.

I must also thank my wife, Gloria, for her encour-

agement and confidence in my ability. Her sacrifices

during the last few years have allowed me to accomplish

this objective.
















TABLE OF CONTENTS


ACKNOWLEDGMENTS . .

LIST OF SYMBOLS . .

ABSTRACT . .

CHAPTER:


Page

. iii

. vi

. viii


1 INTRODUCTION . .

Stock Purchase Options .

The Black-Scholes Model .

Extensions of the Model .

Other Related Studies .

Conclusions . .

2 PREDICTIVE CHARACTERISTICS OF
OPTION PRICES . .

Implied Standard Deviations .

Weighted Implied Standard
Deviations . .

The Hypothesis . .

Criteria for Selection of Data .

Data . .

Test Procedure . .

Observations on the Processing
of Data . .

Test Results . .

Conclusions . .










CHAPTER Page

3 A TEST OF THE EFFICIENCY OF THE
OPTION MARKET AND THE ACCURACY
OF THE EVALUATION MODEL ... 42

Introduction . .. 42

Discrepancies Between ISDs
for Options on the Same
Underlying Stock . .. 43

Procedure for Establishing
Risk-free Hedges . .. 44

Hedge Results . .. 49

Discussion of Results .. 52

Conclusions . .. 55

4 SUMMARY AND CONCLUSIONS .. 58

Objective of Study .. 58

Chapter 1 . .. 59

Chapter 2 . .. 60

Chapter 3 . .. 61

Conclusions . .. 62

APPENDICES

A VALUES OF WISD, SDHIST AND SDFUT
FOR EACH STOCK ADJUSTED TO AN
ANNUAL BASIS . .. 66

B OPTION DATA FOR HEDGED POSITIONS 79

REFERENCES . . 87

BIOGRAPHICAL SKETCH . .. 88















LIST OF SYMBOLS


Page*

CBOE Chicago Board Options Exchange 1

ELP Eligible long position for option hedge 45

ESP Eligible short position for option hedge 45

IMV Implied market value for option based on

the collective assessment of volatility

of all options (WISD) on the same under-

lying stock . ... 45

ISD Implied standard deviation of option

(expected stock standard deviation neces-

sary to justify option price) .. 7

SDFUT Sample future standard deviation of

stock returns (based on twenty monthly

observations after the observation

date). . . ... 29

SDHIST Sample historic standard deviation of

stock returns (based on twenty monthly

observations prior to the observation

date) . . ... 19



*Where first defined.













Page*

SE Standard error of intercept and slope

coefficient of linear regression models 31

WISD Weighted implied standard deviation

(the collective assessment of volatil-

ity of all options on the same under-

lying stock). . 11



Note: Symbols defined and used only within conse-

cutive headings are not included in this list.


*Where first defined.















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



PREDICTIVE CHARACTERISTICS OF STANDARD DEVIATIONS
OF STOCK PRICE RETURNS INFERRED FROM OPTION PRICES

By

Donald P. Chiras

August 1977

Chairman: William M. Howard
Major Department: Finance, Insurance and Real Estate

The major objective of this study is to obtain useful

information from observations of stock option prices. Tra-

ditional approaches have generally attempted to calculate

the price at which an option "should" trade. This study

attempts to determine the informational content of observed

option prices. The observed option prices are used as the

input to a model and the stock variability becomes the

output.

The secondary objective of this study is to use the

variability inferred from the option prices to test the

efficiency of the Chicago Board Options Exchange and also

the accuracy of the model. The Black-Scholes model, as

adjusted for dividends by Merton, is the evaluation model

for this study.


viii











An improved method of calculating a collective as-

sessment of stock price volatility from the options on each

underlying stock is developed and the change in predictive

characteristics of option prices over time is observed.

Option prices are found to be superior predictors of future

stock return variances than variances obtained from historic

stock price data for the period covered by this study.

A riskless trading strategy is developed which

allows anticipated gains to be separated into three com-

ponents:

1. Artifical gains created by option price

data which do not properly reflect closing

stock prices

2. Artificial gains anticipated as a result

of errors in the evaluation model, and

3. Real gains which result from inefficiencies

in the option market

The magnitude of each of the three above components are es-

timated. It is concluded that the option market was in-

efficient during the observation period (from June 29, 1973

until April 30, 1975) and that the model error was small

except for those options which have previously been identi-

fied by Merton as potential candidates for overvaluation.

There are several elements in this study which pro-

vide important new information:










1. The dividend adjustment and the improved

method of calculating a collective assess-

ment of volatility of stock price returns

provide better test results than those

achieved in previous studies

2. A trading strategy which approaches a

riskless process over time can be achieved

without requiring a position in the under-

lying stock

3. The hedging strategy can be implemented

without any net investment

4. Use of historic stock price data for con-

struction of option trading strategies

will be unproductive if the results of this

study are found to be valid in the future

The results obtained indicate that the procedural in-

novations introduced in this study permit substantial im-

provements in tests of the evaluation model and of option

market efficiency. The improved procedures will allow future

empirical research in the above areas to be more productive.















CHAPTER 1
INTRODUCTION



Stock Purchase Options


A stock purchase option is a security giving the

right to the owner or buyer of the option to purchase shares

of a designated stock at a fixed price until the day the

option expires. The price to be paid is called the "exer-

cise price" or the "striking price," and the day the option

is to expire is called the "expiration date" or "maturity

date." Stock purchase options are often referred to as

"call options." Call options are traded in units which

give the buyer of the option the right to purchase 100 shares

of stock. Listed call options are those which are traded

on a regulated option exchange such as the Chicago Board

Options Exchange (CBOE). The seller of a call option is

usually referred to as the "writer" of the option. A

specific option is referred to by expiration month, stock

name and exercise price. Thus a January IBM 280 refers to

a call option expiring in January which entitles the owner

to purchase 100 shares of IBM stock at a price of $280 per

share on or before the date of expiration. Call options

are originally written to expire in three, six or nine











months. A stock of a given company may have up to three

options for each of several different exercise prices.

Listed options have standard expiration dates which occur

in only four months of the year. All options in this study

expire on the last Monday of January, April, July or October.

The CBOE permits additional options at different exercise

prices to be traded when the underlying stock price ap-

proaches a new standard exercise price. Standard exercise

prices are at five dollars per share increments for securi-

ties selling up to seventy dollars per share, then ten

dollars per share on securities priced up to $200 per share

after which the increment becomes twenty dollars per share.

The new options created as a result of a stock approaching

a new standard exercise price expire in less than three,

six or nine months in order to coincide with the standard

expiration months. Currently the average number of listed

options on each stock is approximately ten. An example of

a stock purchase option follows:

On October 13, 1976, a January IBM 280

was selling for $837.50 while the stock was

selling for $272.50 per share. Thus some investors

were willing to pay $837.50 for the right to

purchase 100 shares of IBM stock at $280 per

share until the third Friday in January, 1977.

(All options expiring in 1976 or later can be









exercised only through the third Friday of

the expiration month.) The next day the

above option declined to $600, a drop of 29.7

percent while the price of one share of IBM

stock had fallen to $264.875, a decline of

2.8 percent. Thus the option buyer had invested

an amount of money equal to the price of an

option to purchase 100 shares of IBM stock or

$837.50. The IBM stock buyer had an investment

of $27,250. The loss to the stockholder was

$762.50 as compared to $237.50 for the buyer of

the option. The option buyer risked less money,

but the percentage loss on his investment was

greater.

An investor could adjust the number of options sold

per 100 shares of stock owned in order to attempt to neutral-

ize the effects of short-term market fluctuations. The gain

from the sale of the options sold would tend to offset the

loss resulting from the decline in price of his stock in-

vestment. In the above example the owner of 100 shares of

IBM stock would have had to sell more than three options to

offset the loss incurred due to the decline in the price of

IBM stock.










The Black-Scholes Model


Publication of the path-breaking papers by Fischer

Black and Myron Scholes [1,2] combined with the commencement

of option trading on the Chicago Board Options Exchange has

generated much interest in option price evaluation. Prior

to the introduction of the Black-Scholes model there had

been no explicit general equilibrium solution to the option

pricing problem. Previous attempts at option evaluation

were mostly based on "rule of thumb," graphical or economet-

ric approaches which attempted to predict option prices

from historic option price data. The Black-Scholes model

differs substantially from previous models. It is the first

to determine a "fair value" for an option based on capital

market theory.1

Black and Scholes [2, p. 643] assume an investor can

continuously adjust the number of options sold to offset

any instantaneous loss in his stock investment. Therefore

the investor can achieve a risk-free investment and should

not be able to create profits in excess of the risk-free

rate. Black and Scholes claim that even if the investor

does not continuously adjust the number of options sold he


1For the reader interested in the history and the
development of earlier option evaluation techniques I recom-
mend The Stock Options Manual by Gary L. Gastineau, pub-
lished by McGraw-Hill late in 1975. The book contains an
annotated bibliography. Each reference is augmented by a
concise summary indicating the level of difficulty and an
evaluation of the relative merit of the contents.











may still be able to achieve a risk-free investment by

diversifying his assets among many stocks and their options.

In our example using IBM stock and options with the

price change taking place in only one day a "theoretically

correct" hedge would have produced the risk-free rate of

return for a one-day period. The decline in price of the

options sold would have exceeded the loss on the IBM stock

investment by an amount necessary to achieve one day's in-

terest at the risk-free rate on the net dollar investment.

The assumptions underlying the Black-Scholes model

are as follows:

1. There are no penalties for short sales

2. Transactions costs and taxes are zero

3. The market operates continuously

4. The risk-free interest rate is constant

5. The stock price is continuous

6. The stock pays no dividends

7. The option can only be exercised at maturity

The option formula of Black and Scholes is:


W = XN(d) Ce-rtN(d2)


where: d n(X/C) + (r + v2)t
where: d vt -
1 vt


1The net dollar investment is the value of the stock
minus the proceeds from the sale of the options.










d2 = d, vt*


W = the current option price for a single

share of stock

X = the current stock price

C = the exercise price of the option

e = the base of natural logarithms

t = the time remaining until expiration of

the option

r = the continuous risk-free rate of

interest for the period t

v = the standard deviation of returns on

the stock during the period t (assumed

to be constant)

N(.) = the cumulative normal density function

of (.).

Thus one must observe only five parameters to compute the

equilibrium option price:

1. The stock price

2. The time to maturity

3. The exercise price

4. The risk-free interest rate

5. The standard deviation of future returns

on the stock


IBlack and Scholes [2, p. 640] assume the stock price
distribution is log-normal. The logarithms of the price











The last parameter cannot be observed. The past history

is of some help in the estimation of future standard devia-

tions, but changes over time do occur. The first three

parameters are directly observable and a proxy such as the

treasury bill rate may be used for the risk-free rate.

Since the option price can also be observed, the expected

future standard deviation of returns on the stock may be

inferred by calculating the value of v which is expected

in order to justify the option price. The value of v in-

ferred by calculation for each option price is called the

implied standard deviation (ISD). The predictive charac-

teristics of the implied standard deviations (ISDs) of

option prices are the major focus of this study.



Extensions of the Model


The assumptions of the Black-Scholes model appear to

create severe restrictions which limit its usefulness. How-

ever, the research of Merton [7, 8, 9], Cox and Ross [3]

and Ingersoll [4] has aided in indicating that no single

assumption appears to be crucial to the analysis.


ratios must be obtained before calculation of standard devia-
tions. The term "standard deviations" as used in this study
refers to those of the logarithms of the stock price ratios.









Dividend Adjustment

Dividends on some stocks may be substantial compared

to the risk-free rate and can have a significant effect on

the valuation of options whose stocks make such payments

during the life of the options. Merton [7] adjusts the

Black-Scholes model for a specific dividend policy. Divi-

dends are assumed to be paid continuously such that the

yield is constant. Smith [10; p. 26] presents an altered

version which corrects a minor discrepancy in Merton's

presentation. The form of the evaluation equation follows:


W = e-YtXN(dl) e-rtCN(d2)


where: d n(X/C) + (r y + v2)t
vt

d2 = d vt


y = the continuous yield on the stock or

the dollar value of the dividend

divided by the stock price.

The model above does not conform to actual dividend policies

of firms. To apply Merton's model it is necessary to con-

vert discrete payments to an equivalent continuous rate.

Although payments are not made at a constant rate,this

assumption appears more reasonable than ignoring dividends

entirely. At present there is no known solution to the

option pricing equation for a discrete dividend payment.









The existence of a discrete dividend payment may

cause the option holder to exercise his option prematurely.

An option which is most likely to be exercised before

maturity would have a high dividend yield, an ex-dividend

date just prior to its expiration date and an underlying

stock price well above the option's exercise price. Con-

sider the holder of an option who wishes to exchange the

asset for cash. He may either sell the option or exercise

it to receive the dividend before selling the stock. When

the investor exercises instead of selling he loses the

premium in excess of the intrinsic value.1 Therefore in

order to benefit from early exercise the option holder

must receive a dividend on the stock which exceeds the loss

of the excess premium plus the cost of the transactions.2

Because of the above factors which tend to minimize the

occurrence of early exercise, the errors resulting from this

source are not expected to be significant.

The evaluation model for this study is the Black-

Scholes model as adjusted for dividends by Merton. The

processing of initial data indicates that the model as

adjusted for dividends produces more consistent and real-

istic standard deviations of stock returns than the model

as presented by Black and Scholes.


iIntrinsic value of an option is the excess of the
stock price relative to the option exercise price.
In addition, the stock price will decline on the ex-
dividend date.









Discontinuous Stock Price Movements

Cox and Ross [3] and Merton [8, 9] examine discon-

tinuous stock price movements. Their findings are consist-

ent with observations from this study. The violation of

this basic assumption of the model does not result in

significant errors except for options which have short

remaining lives or for those on stocks with prices far

below the exercise price [9, p. 343]. A more detailed

discussion of the effects of discontinuous stock price

movements is presented in Chapter 3.


Tax Effects

Ingersoll [4] considers the effect of taxes in his

paper which applies option evaluation techniques to closed-

end mutual funds. He finds that better results can be

obtained by adjusting the option evaluation model for

income taxes. He is also one of the first to use dividends

as a parameter in the model enabling him to examine the

pricing of dividend and capital shares of dual purpose

funds.1

The above studies all show the Black-Scholes model

to remain useful under the relaxation of several basic

assumptions.


1Dual purpose funds have two classes of shares.
Dividend shares are entitled to the total net dividends
received on all assets of the fund. Capital shares are
entitled to receive the total of all net capital gains.










Other Related Studies


Latane and Rendleman

Latand and Rendleman [6] have presented results

based on the implied standard deviations of stock returns

as obtained by use of the Black-Scholes model and CBOE

options. They calculated a weighted average of the ISDs

as a measure of market forecasts of return variability.

The rationale for their use of a particular weighting

function is explained in the following manner:

It would be unreasonable to expect option
prices for a given company to reflect the
arithmetic average of implied standard deviations
from all options on its stock which are traded
in a particular point in time. The ISDs on those
options whose prices are the least sensitive to
a precise specification of the standard devia-
tion are likely to be unrepresentative of the
market's underlying expectation. Implied stand-
ard deviations on such options could take on
a wide range of values within a narrow range of
option prices, and accordingly, should not be
given as much weight as ISDs of options in which
the standard deviation is a more important
factor. Therefore, we use a weighted average
implied standard deviation (WISD) in which the
ISDs for all options on a given underlying
stock are weighted by the partial derivative of
the B-S equation with respect to each implied
standard deviation. [6, p. 371]

However, the weighting formula presented was found to con-

flict with the above explanation. The following formula

was described by Latand and Rendleman [6, p. 371]:


N N
WISD = ( ISD d2t)( di
j= t ijt = ijt











where: WISDit = weighted average implied standard

deviation for stock i in period t

ISDijt = implied standard deviation for

option j of stock i in period t

dijt = partial derivative of the price

of option j of stock i in period t

with respect to its implied standard

deviation using the Black-Scholes

model.

A simple example illustrates the error in the use of the

above weighting function:

Given two options on a single stock, both options

having the same implied standard deviation of monthly re-

turns (ISDijt) and partial derivative (dijt) of 0.10 and

1.0 respectively, the following WISD for period t would be

calculated:


WISD = (0.01xl.0+0.01X1.0)(1.0+l.0)-1


= (0.02)(2)-1


= 0.0707


Although both options implied the same standard deviation

of returns on the stock, the weighting formula of Latan4

and Rendleman produced an implied standard deviation of









returns which is biased toward zero. If a larger number

of options were traded on a particular stock the standard

deviation implied by the formula would approach zero as a

limit.

Using the biased weighting formula Latan6 and Rendle-

man explored the usefulness of the WISDs in predicting

future standard deviations of stock returns. The WISDs and

two sample historic standard deviations (a four-year period

prior to the WISD observation period and a 38-week sample

of the observation period) were compared to a two-year

sample taken after the WISD observation period. They also

constructed weekly hedges between stocks and their options.

First they used "underpriced" and then "overpriced" options

and compared the returns from both types of hedges with

the risk-free rate to obtain a measure of "excess returns."2

A critical assumption in their study and in this study is

that option prices behave as if they are generated by the

relevant valuation model.

Latand and Rendleman concluded that the WISD is

generally a better predictor of future return variability

than sample historic standard deviations. They also re-

ported that tests of their trading model "suggest" that the

Black-Scholes model can be used in determining whether


iThe values of ISDit and dijt are always equal to
or greater than zero.
2The return on a treasury bill was used as the stand-
ard for the risk-free rate.










individual options were properly priced for short-term

riskless hedging [6, p. 381].

Although WISDs were calculated for a 38-week period,

there was no attempt to test the predictive characteristics

of options observed at any one point in time. All the

WISDs were averaged together for each stock during the 38-

week period of the study.1 There were 24 stocks in the

sample. The aggregation of the data permitted only one

comparison between each of the predictors of future standard

deviations of returns for the 24 stocks.

Latan6 and Rendleman state that many observations of

option prices produced non-meaningful standard deviations

as solutions to the Black-Scholes formula and that such

data was discarded. They found that a standard deviation

of zero was not low enough to enable one to calculate

the observed option price. Merton's dividend-adjusted

model frequently provides realistic standard deviations for

those options.

The current study examines the predictive character-

istics of WISDs for each observation date. A different

weighting formula for obtaining WISDs is used. The improved

formula is discussed in Chapter 2.


IBeginning October 5, 1973 and ending June 28, 1974.









Trippi

A recent study by Robert R. Trippi [11] pursued an

investment strategy to test the efficiency of option trading

on the Chicago Board Options Exchange. The ISDs of option

prices were calculated by a process similar to that used

by Latane and Rendleman. Ninety-day certificates of deposit

were used as a proxy for the risk-free rate instead of

rates of U.S. Treasury bills maturing closest to each op-

tion expiration date. Trippi calculated an arithmetic

average of the ISDs to obtain a "collective assessment of

volatility" for each stock. Latand and Rendleman had

previously labeled the use of such an arithmetic average

as "unreasonable." Selection of options was narrowed by

the following exclusions:

(a) Options with premiums of less than $1.00

(b) Options with less than three weeks of

remaining life, and

(c) Options with premiums less than 1.3 times

the stock price less the exercise price

[11, p. 95]

The reason for the above exclusions follows:

the valuation model does not work well
with very cheap options (transactions costs be-
come substantial), with those having very short
remaining lives, or with those that are very
deep in the money. [11, p. 95]

An implied market value was computed by insertion of

the collective assessment of volatility into the Black-











Scholes equation and calculating the resulting option price.

The calculated price must differ from the observed price

in order to permit a profit opportunity. The following

decision rule was used to construct portfolios of options

for one-week holding periods [11, p. 95]:

Long portfolios are to include all options

whose prices are more than 15 percent below

the implied market value and short portfolios

are to include all options whose prices are

15 percent above the implied market value.

Trippi assumed that within one week the options would ap-

proach values which would substantially reduce the differ-

ences between their prices and their implied market values.

Four hundred three options were selected for the port-

folios during the test period, 202 long positions and 201

short positions. The average weekly return on the total

market value of the long and short positions was 10.8 per-

cent.

Trippi also conducted tests to determine whether it

was possible to have executed the profitable transactions.

Option prices were examined in detail from April 7 through

April 25, 1975. The following day's opening prices were


iGiven no initial position in an option the buyer is
considered "long" the option and the seller considered
"short" the option.










observed to determine whether favorable option prices con-

tinued to exist. Trippi states:

it appears that the initiation of options
positions at the closing prices or their equiva-
lents would have been frequently possible. [11,
p. 97]

A more detailed discussion of Trippi's test is presented in

Chapter 3. Trippi concluded that the option market (CBOE)

could have allowed one to realize large profits after com-

mission costs during the period covered by his study.



Conclusions


Previous studies by Latan6 and Rendleman and Trippi

have investigated option prices for predictive characteris-

tics which permit construction of profitable trading strate-

gies. Weaknesses in the studies have been noted and reme-

dies have been suggested to improve the test procedures in

order to increase the potential rewards of further research.

A test of the predictive characteristics of option

prices as related to the price movements of their underlying

stocks is presented in Chapter 2. Better results are

achieved by the addition of three major factors:

1. The use of an improved formula for the

calculation of weighted implied standard

deviations

2. The use of an evaluation model which per-

mits an adjustment for dividends










3. The use of a test procedure which allows

observations of the changes in predictive

characteristics of option prices over

time

A trading strategy similar in principle to that of

Trippi is developed in Chapter 3. However, there are four

major differences in procedure which improve the returns

and expand the domain of eligible options beyond the boun-

daries of Trippi's exclusion rules:

1. Dividends are included as a variable in

the evaluation equation

2. The formula for weighting ISDs is improved

3. The trading rule creates risk-free hedges

in lieu of portfolios of options with

unknown risk

4. Many options for which Trippi claimed ". .

the valuation model does not work well .. ."

are included

Chapter 4 contains a summary and the conclusions of

this work.














CHAPTER 2
PREDICTIVE CHARACTERISTICS OF OPTION PRICES



Implied Standard Deviations


One popular method of option price evaluation re-

quires an estimate of the historic standard deviation of

returns on a stock (SDHIST). That estimate is then used

to calculate an equilibrium option price from an option

evaluation formula.1 An implicit assumption underlying

the above procedure is that the SDHIST is the best avail-

able estimate of the future standard deviation of stock

returns. If equilibrium market prices reflect a more ac-

curate estimate of future standard deviations of stock

returns,the above method of option evaluation would not be

useful in the calculation of "fair" values for stock options.

The difference between the observed and the calculated op-

tion price would reflect the difference between the esti-

mated SDHIST and the aggregate market measure of expected

future standard deviation of returns and not, as some option

traders believe, a profit opportunity. The use of historic

price data to construct trading rules for options would be


1As stated in Chapter 1, returns are expressed as
price ratios and the standard deviations of those price
ratios are used in option evaluation formulas.










ineffective (except for those individuals who received the

commissions generated by such an unproductive procedure).

If option prices contain information which can be used to

calculate better indicators of future stock return variances

than the estimates obtained from historic stock price data,

that information may be of substantial value to stock or

option traders. The procedure for obtaining estimates of

future standard deviations of stock returns would be simpli-

fied. One would need only to examine the current option

price instead of processing numerous historic stock price

data. The current option price when entered into the evalua-

tion equation would permit the calculation of the standard

deviation inferred from the market price. In this study

that value obtained by use of the current option price is

called the "implied standard deviation" (ISD).



Weighted Implied Standard Deviations


During the period encompassed by this study there

was an average of 6.3 option prices recorded per stock for

each observation date. Each of those options had a differ-

ent ISD. The ISDs must be combined in order to produce a

single estimate of future standard deviation of returns for

each stock. In Chapter 1 we examined the attempt of

Latan& and Rendleman to combine ISDs to produce WISD values.

Although they presented a biased weighting function,their










reasoning for not accepting an arithmetic average of the

ISDs is sound. There are considerable differences in the

sensitivities of option prices to changes in expected stock

return variances. An example follows:

For a nine-month option on a stock selling

at 70 percent of the option's exercise price

with an implied standard deviation of monthly

returns equal to 0.03626, a 50-percent increase

in the implied standard deviation to 0.05439

would result in a calculated 487-percent increase

in the option price. Under the same circum-

stances but with the stock price at 130 percent

of the option's exercise price the calculated

option price obtained by use of the model will

increase only 4 percent.1

Latand and Rendleman had intended to weight the ISDs

by the partial derivatives of the Black-Scholes model with

respect to each implied standard deviation. That is equiva-

lent to weighting ISDs according to the sensitivity of the

dollar price change for the options relative to the incre-

mental change in the implied standard deviation. A rational

investor measures returns as the ratio of the dollar price

change to the size of the investment. The reasoning of

Latan4 and Rendleman emphasizes the total dollar return


1A risk-free rate of 6 percent is assumed.










without regard to the size of the investment. In order to

be consistent with a rational measure of returns the price

elasticity of options with respect to their ISDs must be

considered. We must be concerned with the percentage

change in the price of an option with respect to the per-

centage change in its ISD.

The equation used to obtain the weighted implied

standard deviation of the options on one stock for each

observation date is


N dWj vj
WISD = l dvj W j = 1,2...N
N dWj vj
E N = The number of op-
j=l dvj Wj tions recorded on
a particular stock
for the observa-
tion date


where: WISD = the weighted implied standard deviation

for a particular stock on the observa-

tion date

ISDj = the implied standard deviation of op-

tion j for the stock

dW v = the price elasticity of option j with
dvj Wj
respect to its implied standard devia-

tion (v).


1A one-dollar price change on a one-dollar stock is
considered equivalent to the same price change on a fifty-
dollar stock.











The Hypothesis


A test of the following hypothesis will aid in the

determination of the predictive characteristics of stock

option prices:


STANDARD DEVIATIONS INFERRED FROM OPTION

PRICES HAVE BEEN BETTER PREDICTORS OF STANDARD

DEVIATIONS OF FUTURE STOCK RETURNS THAN STANDARD

DEVIATIONS OBTAINED FROM HISTORIC STOCK RETURNS.


If the hypothesis is accepted one may suspect that

option prices can provide a valuable source of information

relative to the variance of future stock returns. If the

latter fact can be confirmed by the results of future re-

search,the use of historic stock price data to implement

profitable trading rules for options will be discredited

and many unsuspecting investors may be spared the associated

expenses and futility.

Historic stock returns are readily available and it

is likely that such information is part of the data assimi-

lated by the market. It is possible that other current in-

formation may aid in the estimate of future standard devia-

tions of stock returns. One may suspect that market prices

reflect future standard deviations more accurately than

historic sample data. If not, use of historic models would

allow one to obtain greater than "normal" returns in the










option market. In the latter case the option market would

be suspected of being inefficient. The alternative conclu-

sion would be that the evaluation model was not sufficiently

accurate to allow useful calculations of option values.

The model may be used to discover inefficiencies even if

the hypothesis is accepted. A trading rule is developed in

Chapter 3 to test for discrepancies in pricing among indi-

vidual options on the same underlying stock.



Criteria for Selection of Data


There are considerations which necessitate restric-

tions in the selection of data:

1. Observations of option and stock prices

which are not well correlated in time.

Data are recorded at the close of the last

trading day of each month; it is likely

that some option prices do not accurately

reflect the closing stock price because

of the lack of a transaction after the

last stock price change.

2. The process by which options attain

equilibrium price may be inhibited by

transactions costs. The effect is more

pronounced for the lowest priced options

as transactions costs are a substantial











percentage of their market price. Options

with little time remaining to maturity will

have large transactions costs relative to

their premiums which may prevent them from

attaining equilibrium prices

In order to minimize the errors created by the above

factors the following selection criteria will be used:

1. Options which trade below their intrinsic

values are deleted from this study.1 These

options are probably not reflecting the ac-

tual closing price of the underlying stock

for if they were the discount would be arbi-

traged away by a trader who owns a seat on

the exchange

2. Options with a remaining life of less than

twenty-four days are not observed due to

the method of selecting the observation dates

for option prices. The options expire on

the last Monday of the exercise month and

observations are taken as of the last

trading day of each month

3. Options selling below $0.50 whose under-

lying stock price is more than $5.00 below


1Intrinsic value is the excess of the stock price rela-
tive to the option exercise price.










the option's exercise price are deleted

from this study. The Chicago Board Op-

tions Exchange prohibits establishing a

new position either as a buyer or seller

in such options



Data


There are 23 monthly observation periods (t) begin-

ning June, 1973 and ending April, 1975. Data are recorded

for the last trading day of each month for each option on

stocks whose options were traded on the Chicago Board Options

Exchange as of June 29, 1973. Data recorded on each date

are


1. The discounts on the U.S. Treasury bills

which mature closest to the standard op-

tion exercise dates

2. The closing price of each underlying

stock

3. The number of days to the standard exer-

cise dates

4. The standard exercise prices

5. Values unique to each option:

a. The closing price

b. The dividend that the under-

lying stockholder is entitled


S










to receive as a result of

his ownership up to the

exercise date

6. The monthly price ratios of each underlying

stock for each twenty-month period preceding

and following each observation date.l

For 1, 2, 4 and 5a above the data are recorded directly

from the Wall Street Journal. For number 3 the standard

exercise month is recorded and the number of days are cal-

culated. The U.S. Treasury bill discount rate is converted

to an equivalent continuous interest rate. The values for

5b are taken fromMoody's Handbook of Common Stocks pub-

lished by Moody's Investors Service, Inc.2 The dividends

are converted to an equivalent continuous rate. For number

6 above the stock return data are taken from the computer

tapes available from the Center for Research in Security

Prices at the University of Chicago. The logarithms of

the monthly price ratios are computed and the sample stand-

ard deviations of the logarithms of monthly ratios are

calculated for the twenty-month period proceeding and fol-

lowing each observation date. From the data in 1 through 5


iThe monthly price ratio is the price at the end of
the month divided by the price at the end of the previous
month. Ratios have been corrected for stock distributions
other than dividends.
It is assumed that the market's forecast of dividends
is perfect during the life of each option. Analysis of final
data indicates that such an assumption is not crucial to
this study.









above an implied standard deviation (ISD) for each option

can be calculated by the use of an iterative search process.1

The weighted average of the ISDs for each stock on each ob-

servation date is calculated by use of the equation pre-

sented on page 22 which eliminates the bias present in

the formula used by Latan4 and Rendleman. The weighted

average of the implied standard deviations (WISD) of a

particular stock is assumed to be the collective assessment

of the expected future standard deviation of returns on the

stock.



Test Procedure


The following graph and linear regression equations

aid the explanation of test procedures for Hypothesis 1.


SDHISTit WISDit SDFUTi,t i = 1,2,...,23
1 t 1 t
t-20 t t+20 t = 1,2,...,23


SDFUTi,t = ah + BhSDHISTi,t


SDFUTit = a + B WISDit


lit is impossible to solve the evaluation equation
directly for the ISD. By use of the partial derivative of
the option price with respect to its implied standard devia-
tion the value of the ISD is altered until the calculated
option price is within $0.001 of the observed option price.
Convergence is rapid, sometimes requiring only three itera-
tions.











SDFUT = a + B hSDHIST. + B WISD.
i,t c ch 1,t co i,t


where:


ac, Bc


WISDi,t = the weighted implied standard

deviation of returns for stock

i at time t

:DHISTit = the sample historic standard

deviation of returns for stock

i from time t-20 to time t

SDFUT. = the sample standard deviation

of returns for stock i from

time t to time t+20

h and Bh = coefficients of the simple

regression model for SDFUTi,t

versus SDHISTit

o and B = coefficients of the simple

regression model for SDFUTi,t

versus WISDi,t

hand B = coefficients of the multiple
Sco
regression model for SDFUTi,t

versus SDHISTit and WISD.
i,t i,t


The SDHISTs and the WISDs are compared to the SDFUTs

to determine which predictor was superior during the test

period. The period examined began two months after the start

of listed option trading and lasts for 22 months (23 observa-

tions). The data allows one to compare the predictive abili-

ties of option traders at various stages of the development


S


a


a










of the listed option market. The combination of SDHISTs

and WISDs are also compared to the SDFUTs to determine

whether each predictor provides unique information or con-

tains only information already captured by the other. The

values of SDHIST, WISD and SDFUT are presented in Appendix

A.



Observations on the Processing of Data


While calculating ISDs to obtain the WISD values for

the regression models, several items of interest were dis-

covered. The use of the dividend adjusted model increased

the value of the ISD calculated for each option on which

the dividend adjustment was relevant. Most option data

which previously had produced no solution for an ISD by

insertion into the Black-Scholes model now enabled one to

produce meaningful ISD values. For options on stocks with

high yields and low WISD values the effect was most pro-

nounced. The ISDs now had less variation and the method

used to discover and to correct erroneous data was simpli-

fied. By scanning ISD values for each stock significant

data errors could often be detected by selection of those

options whose ISDs were inconsistent with those of other

options on the same stock. The most inconsistent ISDs

were discovered on options which violated the CBOE rule for

the establishment of a new position (short or long). Those









options have been eliminated from this study. The second

most common inconsistency was found on options whose stocks

were selling substantially above the option exercise price.

Those options, as noted in a previous example, are not very

sensitive to changes in expected standard deviation. The

weighting function effectively eliminates their impact on

the WISD values calculated. Thus it is possible to base

this study on an improved evaluation formula and a better

weighting function for WISDs which do not require the ex-

clusion of as much option data as was necessary in the re-

search of Trippi and that of Latan4 and Rendleman.



Test Results


The estimate of the intercept (a) and slope coeffi-

cient (B) and their standard errors (SEs) are presented in

Tables 1 and 2 for the linear regression models SDFUT =

a + B(SDHIST) and SDFUT = a + B(WISD). Values of R2 and the t

statistic for B are also included. Subscripts have been

deleted to simplify presentation of data. Values are re-

corded for each model for every month. Data were observed

for twenty-three stocks on each date. The values of SDHIST,

WISD and SDFUT were calculated for each stock. The SDHIST

and the WISD values were compared to the SDFUT values to

determine which of the two was the better predictor of the

SDFUT values.











Table 1

Results for the Regression
SDFUT = a + B(SDHIST)


3.00**
3.08**
3.23**
2.98**
2.70**
1.84
1.92
2.08
1.80
2.04
2.14
2.47
2.51**
2.71**
2.63**
3.69**
2.51**
2.59**
2.30
3.45**
3.90**
3.70**
2.74**


level (one-tail test).
(one-tail test).


Month


a(SE)


B(SE)


.205(.07)
.232(.06)
.224(.06)
.226(.06)
.235(.07)
.284(.06)
.277(.06)
.269(.06)
.288(.06)
.277(.06)
.271(.06)
.268(.05)
.264(.05)
.238(.06)
.236(.06)
.181(.04)
.177(.06)
.174(.05)
.185(.05)
.158(.04)
.127(.04)
.123(.04)
.145(.04)


.776(.25)
.622(.20)
.653(.20)
.662(.22)
.640(.24)
.348(.19)
.357(.19)
.377(.18)
.326(.18)
.368(.18)
.386(.18)
.402(.16)
.413(.16)
.443(.16)
.430(.16)
.440(.12)
.352(.14)
.353(.14)
.311(.13)
.306(.09)
.363(.09)
.374(.10)
.291(.11)


*All t values are significant at the .05
**Indicates significance at the .01 level










Table 2

Results for the Regression
SDFUT = a + B(WISD)


Month a(SE) B(SE) t* R2


1 .170(.08) .579(.19) 3.07** .31
2 .238(.09) .368(.18) 2.00 .16
3 .199(.08) .435(.17) 2.60** .25
4 .166(.09) .575(.20) 2.86** .28
5 .207(.11) .405(.21) 1.96 .15
6 .244(.08) .292(.16) 1.79 .13
7 .276(.08) .354(.18) 1.98 .16
8 .218(.08) .439(.19) 2.28 .20
9 .208(.08) .538(.22) 2.40 .22
10 .115(.07) .718(.17) 4.15** .45
11 .098(.09) .833(.25) 3.33** .35
12 .153(.08) .617(.19) 3.20** .33
13 .156(.09) .570(.20) 2.83** .28
14 .146(.07) .575(.17) 3.44** .36
15 .186(.06) .410(.12) 3.29** .34
16 .184(.05) .321(.11) 3.05** .31
17 .112(.05) .391(.10) 3.75** .41
18 .101(.06) .345(.09) 3.78** .40
19 .137(.06) .290(.11) 2.76** .27
20 .135(.04) .351(.08) 4.16** .45
21 .092(.04) .403(.09) 4.56** .50
22 .054(.05) .499(.11) 4.52** .49
23 .058(.05) .481(.10) 4.59** .50


level (one-tail test).
(one-tail test).


*All t values are significant at the .05
**Indicates significance at the .01 level










The parameters estimated in Table 1 indicate that

historic standard deviations (SDHISTs) explained approxi-

mately 26 percent (the average value of R2 is .26) of the

future standard deviations (SDFUTs) of stock returns. From

Table 2 the corresponding value of R2 for the weighted

implied standard deviations (WISDs) obtained from option

prices is .32. The increase is 23 percent and the trend

in the increase is even more striking. During the first

several months of the study trading of listed options was

a relatively new experience for investors. Option trading

on the Chicago Board Options Exchange began in April 1973.

Until February 1974 the data indicate little difference

between the predictive characteristics of historic standard

deviations and those of standard deviations implied by

option prices. Beginning in March 1974, less than a year

after the start of listed option trading, the option im-

plied standard deviations showed a sudden increase in predic-

tive ability. They then begin to explain more of the

future standard deviation of stock returns. The average

value of R2 for the remainder of the study increases to .39

as compared to .21 in the prior period. The regression

using SDHIST does not indicate a significant trend in pre-

dictive ability over time. The evidence suggests that

the predictive abilities of option implied standard devia-

tions were continuing to improve during the entire period








under study. That evidence supports the hypothesis that

the option market became more efficient as traders gained

more experience. The t values in the tables relate to the

testing of the hypothesis that the true values of B are

greater than zero. Negative values of B may be ruled out

on theoretical grounds. The higher the t value the lower is

the probability that the sample could have been obtained

from a distribution whose actual value of B was zero. The

null hypothesis, that B equals zero, is rejected at the .05

significance level in each month for both regression models.

The t values in Table 2 show a tendency to increase over

time similar to the trend in the corresponding values of R2.

Table 3 lists the estimates of the parameters B1 and

B2, the corresponding t values and R2 for the regression

model SDFUT = a + B1(WISD) + B2(SDHIST). The tabulated

values support the results obtained from the analysis of

the first two regression models. The R2 values are sub-

stantially higher than corresponding values in Table 2

for only three months (July, August and October 1973). After

October 1973, the fifth month of twenty-three in this

study, the multiple regression model does not produce

substantially better values of R2 than those obtained by

use of the option implied standard deviations. The t values

do not indicate consistently high levels of confidence as

in the previous models. However, the t values for the











Table 3


Results for the Regression
SDFUT = a + B1(WISD) + B2(SDHIST)


Month B1 B2 t(Bl) t(B2) R2


1 0.035 0.038 0.91 0.74 .33
2 -0.035 0.654 -0.14 2.10* .31
3 0.048 0.601 0.17 1.62 .34
4 0.256 0.419 0.67 0.98 .31
5 -0.035 0.674 -0.11 1.67 .26
6 0.116 0.225 0.29 0.48 .14
7 0.221 0.172 0.73 0.54 .17
8 0.339 0.106 0.87 0.30 .20
9 0.612 -0.068 1.47 -0.21 .22
10 0.993 -0.295 3.58** -1.26 .49
11 0.918 -0.070 2.28* -0.27 .35
12 0.502 0.133 1.88* 0.64 .34
13 0.416 0.157 1.29 0.62 .29
14 0.648 -0.075 1.79* -0.23 .36
15 0.401 0.013 1.68 0.05 .34
16 0.110 0.340 0.72 1.85* .41
17 0.425 -0.051 2.40* -0.24 .40
18 0.370 -0.042 2.35* -0.20 .41
19 0.250 0.061 1.35 0.27 .27
20 0.290 0.072 1.88* 0.48 .46
21 0.317 0.099 1.87* 0.60 .51
22 0.381 0.128 2.19* 0.88 .51
23 0.482 -0.001 3.08** -0.01 .50


*Indicates significance at the .05 level (one-tail test).
**Indicates significance at the .01 level (one-tail test).









coefficient of WISD are at a substantially higher value

than those of the coefficients of SDHIST. After February

1974 the t values associated with the coefficients of

SDHISTs deteriorate so markedly relative to those of the

coefficients of WISDs that the use of the SDHIST values

appears to add no information that is not already contained

in the values of WISD.

Table 4 presents the regression parameters esti-

mated by pooling all the observations into one "grand

regression" for each model. First the regression parameters

are presented for the "uncorrected" model. Results are

distorted by possible violations of the basic assumptions

of the classical normal linear regression model. The

regression parameters presented in the lower half of Table

4 indicate the substantial improvement in the t values for

each regression coefficient obtained by correcting each

model for heteroskedasticity (variance of disturbances

not consistent for all observations). The results are ob-

tained by normalizing the data (subtracting the mean from

each monthly value of WISD, SDHIST and SDFUT) and dividing

the remainders by the standard error of their associated

monthly regression. Mutual correlation (standard devia-

tion of returns correlated between stocks) and autoregres-

sion (disturbances at one point in time carrying over into







































.ri
In

0






0


a)
a) I

o












4)

0
U)




4






o
.0











,-4
40
H)

























4J .4
0

44
MO
4)









0









) 4J
4J





















0
U)







40



0 C
0 )
U)
U)
































0>
Ea
0 0





0 0


0 U)






H 0
+-

0ia

U)
'0
) S







-4 0)


'01

a,-4
4I

0
ct
0
u
I




a.0
U)









1
another period) are ignored. Data in Table 4 support

the conclusions previously obtained from the estimates

of the monthly regression parameters. The t values are

substantially higher for the coefficients of WISD values

than for those of the SDHIST values. The analysis of

the transformed multiple regression model results in an Rz

of .66, an increase of only .03 over the value of R2 ob-

tained using the WISDs alone.- That tends to confirm the

previous conclusion that the WISDs have been a better pre-

dictor of the future standard deviation of stock returns.

Although the t statistic for the coefficient of SDHIST is

significant, any additional information contained does

not appear to adequately reward the extra effort required

to include the SDHIST values in the analysis.



Conclusions


The information contained in Tables 1, 2, 3 and 4

provides substantial support for the hypothesis for the

period of this study after February, 1974. The change in


lIt is also possible to correct for mutual correlation
and autoregression by applying modified Aitkin's estimation
formulas to the data after transformation as indicated by
Kmenta [5, pp. 512-514]. However one of the steps would
involve calculation of a variance-covariance matrix of the
order 506X506. The analysis of data up to this point has
provided very satisfactory results and this problem is left
for future research.




41




the predictive characteristics over time is also an impor-

tantdiscovery. The results of this study indicate that

during the first five months covered by the Latan6 and

Rendleman study the WISDs and the SDHISTs were both

relatively poor indicators of the SDFUTs. However, during

the last four months covered by their study the WISDs were

clearly the superior predictors of SDFUTs.

The evidence presented in this chapter is strong.

The conclusion of this researcher is that the WISDs have

been substantially better predictors of SDFUTs than have

the SDHISTs.















CHAPTER 3
A TEST OF THE EFFICIENCY OF THE OPTION MARKET
AND THE ACCURACY OF THE EVALUATION MODEL



Introduction


In this chapter a trading strategy is developed

which permits the testing of the efficiency of the CBOE

and the accuracy of the evaluation model. A crucial under-

lying assumption is that the aggregate market assessment

of volatility is the same for options of different matur-

ities.

A hedge ratio must be calculated for each option in

order to employ the trading strategy. To calculate the

hedge ratio, the reciprocal of the derivative of the

evaluation model with respect to the stock price ( E)-1 is
dX
computed as follows:1


W = e-YtXN(d)- e-rtCN(d2)


dW e-YtN(dl)



dW -1 eyt
dX N(d,)



iThe parameters of the equation are defined in Chap-
ter 1, pages 6 and 8.










The derivative, dW above, represents the rate of
aX
price change in the option for an instantaneous price change

in the underlying stock. If dW were equal to 0.2, a $1.00
dX
instantaneous increase in stock price would result, in

theory, in a $0.20 price increase in its option. There-
dW -1
fore, if one were to sell () options for each share of
dX
stock held, the $1.00 gain in the stock price would be off-

set by the loss of $0.20 on each of the five options. Thus

the hedge ratio is 5.0. The above example demonstrates the

underlying principle of the evaluation model.



Discrepancies Between ISDs
for Options on the Same Underlying Stock


The standard deviation of a stock's return is unique,

therefore if the option market is perfectly efficient and

the evaluation model is precise, all options on the same

stock at a given time must have identical ISD values. How-

ever it may not be possible to record all option prices and

their underlying stock prices simultaneously. Data avail-

able for this study contain values of closing transactions

only.1 A reported closing option price may not properly

reflect the closing stock price because of the lack of a


1Exceptions occur when the last transaction of the
option is outside the range of its closing bid and asked
prices. In those cases the closing bid or asked price
nearest the last transaction price is reported at the close
of trading.









transaction following the last change in the stock price.

The above factor may cause the calculated ISDs to differ

on options having the same underlying stock. If the model

is not properly specified and the market is inefficient,

both those factors will create additional differences in

the ISDs for options on the same stock. The above com-

ponents of differences in the ISDs may be classified by

type:

Type 1. The component due to the closing

stock price not being properly re-

flected in the closing option price

Type 2. The component due to specification

errors in the model, and

Type 3. The component due to inefficiencies

in the option market

It is possible to separate the combined effects of Type 1

and Type 3 components from those of Type 2. The procedure

is developed in the following section.



Procedure for Establishing Risk-free Hedges


In Chapter 2 WISDs are calculated from the ISD values

for all options on the same underlying security on a given

observation date. In order to evaluate each of those

options relative to the others and to create a profitable

trading strategy the following steps are taken for each

observation date:










1. An "implied market value" (IMV) is cal-

culated for each option by using the value

of its associated WISD in the evaluation

equation

2. The observed option price is compared to

the IMV

3. Options whose price exceeds the IMV by

at least 10 percent are selected as "eligible

short positions" (ESPs). Options for which

the IMV exceeds their price by 10 percent

or more are selected as "eligible long posi-

tions" (ELPs)

4. A risk-free hedge is created between an ESP

and an ELP for each stock having at least

one of each type of option. For stocks hav-

ing several ESPs or ELPs the two options

with the maximum percentage difference from

the IMV are selected for the hedge

5. The amount of each option included in the

hedged position is determined by the value

of its hedge ratio so that each pair of

options will produce offsetting gains and

losses for an instantaneous movement in the

underlying stock prices


IThe hedge ratio is the reciprocal of the derivative
of the evaluation formula with respect to the stock price.
It represents the number of option contracts which must











6. All option positions are closed one month

later

An example of the above procedure follows:

A stock has five options with the following

prices, IMVs and hedge ratios.

Option Price IMV Price/IMV Hedge Ratio

A $10.00 $7.50 1.33 2.0

B 6.00 6.00 1.00 1.7

C 4.00 5.00 0.80 4.3

D 1.50 2.00 0.75 6.0

E 1.00 0.80 1.25 11.2

The A option has the highest price/IMV ratio. The D option

has the lowest price/IMV ratio. A hedge is constructed in

which six D options are long and two A options are short.

Option D has a hedge ratio of 6.0. Its rate of price

change relative to that of the stock is 0.16 (the recipro-

cal of its hedge ratio). Option A has a rate of price

change half that of its underlying stock. If the stock

were to decline instantaneously by $1.00 the above hedge,

in theory, would produce offsetting gains and losses. The

value of the position in A options would decline $1.00,

the value of the D options would also decline $1.00.


be sold per 100 shares of stock held in order for the in-
stantaneous total dollar change in the value of the options
sold to be equal to the instantaneous total dollar change
in the stock position.











However since the A options are short, the $1.00 decline

represents a $1.00 profit offsetting the loss incurred

on the D options. If both options converged to their

respective IMVs instantaneously the net profit would be

$2.50 for each A option and $0.50 for each D option. The

total gain from the hedge would be $8.00, $5.00 from the

two A options and $3.00 from the six D options. Equal

dollar hedges may be constructed while maintaining the

latter ratio.1

The above strategy is used to create risk-free

portfolios of hedges which eliminate the effect of stock

price movements.2 The anticipated gain results from the

tendency of the price of each pair of options to approach

a common ISD. The holding period for options in this

study is one month which is required by the interval be-

tween observation dates. During the holding period the

hedge ratios may change and thereby alter the risk

characteristics of each hedge. That induced risk is di-

versified away by selection of many hedged positions over

time.


In equal dollar hedges the sum of the long and the
short positions is constant.
ZEach hedge is risk-free only if traders price op-
tions according to the model and the positions are con-
tinously adjusted.










If the evaluation model is correct and the options

return to equilibrium prices before the end of the month,

the expected gain from the above hedge is $8.00, the same

as for an instantaneous convergence to the IMV values.

If the model is correct the total gain achieved from all

the hedges should equal the sum of all expected gains.1

If the gain is larger than expected the options

must have (on average) prices which differ from equilibrium

more than the model indicates. If the gain is smaller than

expected the opposite is true. The selection process favors

the latter condition. Option prices which differ most

from their IMVs will have a tendency to be those for which

the model is most inaccurate. Therefore the options

appearing to have the greatest discrepancy from their IMV

will also have a tendency to be evaluated with maximum

error.

The anticipated gains may be separated into two

categories as illustrated in Figure 1:

1. Gains resulting from observed prices

returning (on average) to equilibrium

(Type 1 plus Type 3 components), and

2. Additional gains anticipated as a result

of errors in the model (Type 2 component)


1Assuming sufficient diversification.











Category 1 Category 2
r--------*------- >i--~
S i
Type 1 Component Type 3 Component Type 2 Component



OBSERVED PRICE AT WHICH ACTUAL "FAIR" IMPLIED
OPTION OPTION COULD MARKET PRICE MARKET
PRICE BE PURCHASED VALUE


Figure 1. Categories of anticipated gains as
related to components of differences
in ISD values



By determining the gains expected from options return-

ing to common ISD values and comparing them to realized gains,

the above components can be identified. The difference is

the Type 2 component. The realized gains represent the sum

of the Type 1 and Type 3 components.



Hedge Results


The returns on each monthly portfolio of hedges are

presented in Table 5. Data for individual hedges are

presented in Appendix B. The percentage returns are re-

lated to the total dollar value of the hedge (the sum of

the values of the long and the short positions). In theory

the returns are infinite because there is no net invest-

ment. The hedges may be adjusted so that the total dollar

value of the long positions is exactly equal to the total










Table 5

Monthly Hedge Returns


Month Number of Hedges Portfolio Return (%)


1 1 19.28
2 0 --
3 0 ---
4 1 10.30
5 4 8.43
6 8 17.19
7 4 -8.96
8 7 12.97
9 8 24.93
10 1 15.60
11 10 3.29
12 7 9.74
13 3 8.63
14 10 9.12
15 11 6.45
16 9 0.70
17 9 7.77
18 8 15.13
19 4 4.21
20 5 3.18
21 6 24.76
22 2 1.16


Total 118 Mean 9.70











dollar value of the short positions. The proceeds from

the sale of options are received by the seller and may be

used to purchase other options. However, one must (due

to institutional restrictions) deposit collateral. An in-

vestor having sufficient stock as collateral can hedge by

using options without being required to make any addi-

tional cash deposit with a brokerage firm. The return

on his stock portfolio is unaffected by the option trans-

actions. The marginal return to his portfolio due to any

gain on option transactions is infinite since the marginal

investment is zero.

There are 22 holding periods between the 23 obser-

vation dates. During the first several months of the study

the trading volume and the number of option contracts are

low and hedging opportunities are few. During hedge periods

2 and 3 there are no selections. During each of periods 1,

4 and 10 there is only one hedge selected. Out of twenty

periods for which one or more hedges are selected only one

period showed a negative return. The return for equal

initial dollar portfolios averaged 9.70 percent per month.

Other observations of interest from data in Appen-

dix B follow:

1. Of 118 hedges 93 were profitable

2. The geometric rate of return resulting

from a constant dollar position in each

hedge is 9.69 percent per month










3. The average rate of return resulting

from hedges in which the number of each

option contract selected is equal to

its hedge ratio is 9.96 percent per month

4. The anticipated return from the above

hedging strategy is 17.45 percent per month

5. The portionof the anticipated gain

which is realized is 57 percent, leaving

43 percent attributable to Type 2 error

(error in the model)

6. The gain on the short positions is 134

percent of their anticipated returns

7. The gain on the long positions is minus

24 percent of their anticipated returns

8. There are 92 hedges between options of

different maturities. For 72 of those

hedges the option with the shortest time

to maturity is selected as being the

most overvalued



Discussion of Results


The returns are positive for 78.8 percent of the

hedges and for 95 percent of the monthly portfolios. This

indicates the power of the trading strategy. The various











methods of applying the hedge strategy (equal dollar hedges,

equal dollar portfolios, etc.) produce similar returns.

The proportion of the reported gain relative to

that inferred from the calculated IMVs is substantial.

However, the test is unable to separate the returns due to

Type 1 error from the returns which could have been earned

from execution of transactions at the actual prices that

could have been obtained in the option market.

To test whether it was possible to execute profit-

able transactions, Trippi (11, p. 97), in a similar study,

examined returns obtained from simulated purchase of op-

tions at the opening prices on the day following the

selection of potential profit opportunities. He selected

71 options as being potential profit opportunities during

the period from April 7 through April 25, 1975. On the

following days 37 options had opening prices which he con-

sidered favorable. The returns from simulated purchase at

the opening prices of the "favorable" and "unfavorable"

priced options were 80 to 90 percent of those reported

from his hedges which used closing prices. The "favorable"

group had one-week returns of 8.7 percent while the "un-

favorable" group had one-week returns of 10.3 percent

Trippi did not calculate the anticipated rate of return

for his selection process. However, the return from his

trading strategy represented 66 percent of the minimum










potential.1 For the trading rule developed in this study

the returns are 100 percent of the minimum potential.

That represents a 51-percent increase in calculated returns

over that achieved by Trippi. The difference in the hold-

ing period should not bias the reported returns as the

optimal holding period is expected to be less than one

week. A holding period in excess of that necessary for

options to return to equilibrium will only increase the

induced risk which must be diversified away.

The substantial difference in the returns for the

long and the short positions are primarily a result of the

underlying stocks declining an average of 9.4 percent dur-

ing the observation period. There is not sufficient infor-

mation to determine whether options were overpriced or

underpriced during the period.

Over 78 percent of the hedges included options hav-

ing different maturities. Of those the option closest to

maturity was selected for the short position 78 percent of

the time. Thus for 61 percent of the hedges a shorter

maturity option was selected as the most overpriced. If

the selection process was totally random only 67 percent

of hedges would be expected to have options of different


iThe minimum potential is the difference from an
option's IMV necessary for it to meet the minimum require-
ment for selection. For Trippi's study that value averages
16.3 percent. For this study it is 10 percent.










maturities. Then only 33 percent of the hedges would have

the shortest maturity option selected as the most over-

priced. Therefore the actual number observed was nearly

twice the expected number. That result is consistent

with evidence presented by Merton (9). He examined the

impact of an error in the assumption (crucial to the evalua-

tion model) that stock returns are continuous.1 He found:

the effect of specification error in the
underlying stock returns on option prices will
generally be rather small. However, there
are some important exceptions: short maturity
options can have significant discrepancies
if a significant fraction of the total varia-
bility comes from the jump component [9,
p. 3451

He also warns:

deep out of the money options which are
greatly "overvalued" in percentage terms may not
be overvalued at all if the underlying stock
process includes jumps. [9, p. 343]



Conclusions


Trippi's follow-up test of the probability of ini-

tiating profitable hedge positions gives a strong indication

that the majority of the gains from his trading strategy

and of that presented in this study could be realized in


iThis implies that over a short period of time the
stock price cannot change by very much. If the stock price
"jumps," the instantaneous variance rate may be infinite.










practice. Thus the information available at present, al-

though limited, leads to the conclusion that the CBOE was

inefficient during the period covered by this study. The

"captured" gains exceed commission costs. An individual

may pay less than 4 percent if transactions are made

through a discount brokerage firm.1 Those in the best

position to profit from inefficiencies in the option

market are the members of the individual option exchanges.

The magnitude of the error in the evaluation model

has been estimated. The anticipated return indicated by

the model is 17.45 percent for the observation period.

The simulated trading strategy "captured" more than half

the discrepancy indicated by the model. The error attribut-

able to the model is estimated at 43 percent of the antici-

pated gain or an average of a 7.49 percent price error.

However, the options selected for inclusion in the trading

strategy are expected to be those for which the model tends

to be most inaccurate. Therefore the actual error is esti-

mated to be much less than 7.49 percent.

The most overvalued options tended to be those which

Merton has singled out for potential error in his evalua-

tion. The error due to the existence of a jump component

may cause those options to appear overvalued relative to


iCommissions are a function of the option price and
the amount invested. Discounts are offered by many firms
and rates may often be negotiated.




57





other options. The tendency for options with shorter matur-

ities to dominate the "overvalued" positions is consistent

with the assumption that stock returns have a jump compon-

ent.

The source and magnitude of inefficiencies in the

option market and the errors in the evaluation model have

been investigated in this work. Although the test results

are not all conclusive other research supports the estimates

of the magnitude of the various types of errors and ineffi-

ciencies defined in this study.















CHAPTER 4
SUMMARY AND CONCLUSIONS



Objective of Study


The major objective of this study is to obtain useful

information from observations of stock option prices. Tra-

ditional approaches have generally attempted to calculate

the price at which an option "should" trade. This study

attempts to determine the informational content of observed

option prices. Initially, it is assumed that the market

prices of options may be more representative of "fair"

values than prices calculated from evaluation models using

historic stock variability as an input. Therefore the ob-

served option prices are used as the input to the model and

the stock variability becomes the output.

The secondary objective of this study is to use the

variability inferred from the option prices to test the

efficiency of the CBOE and also the accuracy of the evalua-

tion model.











Chapter 1


Choice of Model

Evidence is presented which explains the rationale

for the selection of the Black-Scholes model, as adjusted

for dividends by Merton, for the option evaluation model

in this study. The primary benefit of the dividend

adjusted model is that it provides meaningful solutions of

ISDs for many options for which the unadjusted Black-Scholes

model has no solution. In addition the calculated ISDs for

options on the same underlying stock are more consistent

than ISDs calculated without correcting for dividends.


Related Studies

Weaknesses in the studies by Latane and Rendleman

and by Trippi are noted and are corrected. Four major inno-

vations are used to improve the results of the study by

Latan6 and Rendleman;

1. An adjustment for dividends is included

2. A correction for the weighting of ISDs

is developed

3. Observations of changes in the predictive

characteristics of option prices over

time are included

4. Many option data which have been excluded

(due to the lack of a dividend adjustment)

are included










To improve the results for the study by Trippi:

1. An adjustment for dividends is included

2. A correction for the weighting of ISDs is

developed

3. A risk-free trading strategy is developed

in lieu of portfolios of options with

unknown risk, and

4. Ad hoc exclusion rules for certain classes

of options are eliminated



Chapter 2


In Chapter 2 an improved weighting formula for ob-

taining WISDs is presented. The predictive characteristics

of the WISDs versus those of the SDHISTs are tested. The

method of ordinary least squares is used and the estimates

of parameters for regression models are presented in tabular

form. Test results indicate that the predictive character-

istics of WISDs varied substantially during the observation

period. Those of the SDHISTs did not appear to have a sig-

nificant trend. During the period covered by the first nine

months of this study there was no significant difference

between the predictive characteristics of the WISDs and

the SDHISTs. However, for the period covered by the last

fourteen months of this study, the WISDs are clearly the

superior predictors of the SDFUTs.










Chapter 3


In Chapter 3 a trading strategy is developed to test

the efficiency of the CBOE and the accuracy of the evalua-

tion model. Hedges are constructed between options on the

same underlying stock such that (after diversification

across stocks and over time) a riskless trading process

is approached. The gain anticipated (assuming a perfect

model) is divided into three components. The first com-

ponent (Type 1) is created by option price data which does

not properly reflect closing stock prices. The second

component (Type 2) is a function of errors in the evaluation

model. The third component (Type 3) is the result of in-

efficiencies in the option market (CBOE). The sum of the

Type 1 and 3 components is estimated directly from the por-

tion of expected gain "captured" by the trading strategy.

Trippi's test of the magnitude of the gain achieved as a

result of simulated purchases of options at opening prices

is used to estimate the Type 1 component. Thus all three

components can be estimated since the Type 2 component

creates the remainder of the anticipated gain. The conclu-

sion is that the CBOE was inefficient during the observation

period and that the model error was small except for those

options which had previously been identified by Merton as

potential candidates for overvaluation.











Conclusions


There are several elements in this study which pro-

vide important new information:

1. The dividend adjustment and the improved

weighting formula for ISDs provide better

results than those achieved in previous

studies

2. A trading strategy which approaches a

riskless process over time can be achieved

without requiring a position in the under-

lying stock

3. The hedging strategy can be implemented

without any net investment

4. Use of historic stock price data for con-

struction of option trading strategies

will be unproductive if the WISDs continue

to be the superior predictors in the future

5. Results of the trading strategy employed

in this work may be improved by elimination

of those options for which Merton indicates

the model is most inaccurate

6. The Type 1 component can only be estimated

with a high degree of accuracy if the

trading strategy is employed by using











actual purchases and sales in the option

market

There are further implications which are suggested

from the results of this study:

1. The characteristics of predictive abili-

ties of option prices may be better defined

by

a. use of SDHISTs and SDFUTs for

other than twenty-month periods

(i.e., three, six or nine months),

and

b. isolating classes of options (low

priced, short maturity, etc.)

which indicate, after testing,

that their ISDs are not reliable

indicators of SDFUTs

2. The gains "captured" by the trading strategy

may be improved by

a. using calculations of WISDs which

exclude options as indicated in l(b)

above

b. requiring a greater difference

between the option price and

the IMV for selection of hedged

positions





64





3. Errors in the model may be better defined

by constructing greater numbers of hedges

and examining "captured" gains for various

classes of options.

The results obtained indicate that the procedural

innovations introduced in this study permit substantial

improvements in tests of the evaluation model and of option

market efficiency. The improved procedures will allow

future empirical research in the above areas to be more

productive.



































APPENDIX A
VALUES OF WISD, SDHIST AND SDFUT
FOR EACH STOCK ADJUSTED TO AN ANNUAL BASIS










STOCK 1
AMERICAN TELEPHONE


MONTH WISD SDHIST SDFUT


1 0.1436 0.1152 0.2102


0.1860

0.2256

0.1421

0.2648

0.2229

0.1497

0.1125

0.1142

0.1227

0.2077

0.2019

0.2312

0.2457

0.2199

0.2464

0.1900

0.2739

0.2965

0.1931

0.2037

0.2862

0.2819


0.1156

0.1153

0.1337

0.1441

0.1471

0.1545

0.1541

0.1535

0.1604

0.1644

0.1570

0.1552

0.1665

0.1635

0.1644

0.1894

0.1947

0.1957

0.2078

0.2102

0.2103

0.2057


0.2103

0.2057

0.1926

0.1863

0.1867

0.1844

0.1843

0.1913

0.1886

0.1832

0.1900

0.1900

0.1674

0.1674

0.1713

0.1489

0.1349

0.1350

0.1227

0.1210

0.1186

0.1178


STOCK 2
ATLANTIC RICHFIELD


MONTH WISD SDHIST SDFUT


0.3762 0.2950

0.4139 0.2930

0.3911 0.2692

0.2921 0.2668

0.4864 0.2720

0.3953 0.2603

0.4042 0.2641

0.4041 0.2963

0.2577 0.2842

0.3206 0.2898

0.2905 0.2709

0.3093 0.2726

0.2830 0.2712

0.2978 0.2677

0.2528 0.2709

0.3221 0.2801

0.2619 0.3089

0.3397 0.2971

0.3776 0.2975

0.3549 0.2985

0.3766 0.2993

0.3349 0.2989

0.3193 0.3094


0.2993

0.2989

0.3094

0.3047

0.2997

0.2977

0.2904

0.2607

0.2552

0.2514

0.2440

0.2433

0.2517

0.2579

0.2589

0.2477

0.1830

0.1841

0.1845

0.1984

0.1947

0.1940

0.1836










STOCK 3
BETHLEHEM STEEL


MONTH WISD SDHIST SDFUT


0.4006

0.4087

0.4038

0.4658

0.5812

0.5344

0.4819

0.3699

0.2707

0.4209

0.3138

0.3712

0.3171

0.3617

0.4230

0.4403

0.3799

0.4579

0.4134

0.3022

0.3396

0.3560

0.3478


0.2809

0.2802

0.2664

0.3058

0.3062

0.3286

0.3466

0.3466

0.3320

0.3399

0.3365

0.3459

0.3403

0.2996

0.3010

0.3067

0.3218

0.3241

0.3271

0.3794

0.3781

0.3926

0.4015


0.3781

0.3926

0.4015

0.3820

0.3827

0.3640

0.3518

0.3574

0.3579

0.3517

0.3521

0.3818

0.3825

0.3869

0.3787

0.3669

0.3605

0.3643

0.3623

0.3164

0.3167

0.3007

0.2912


MONTH WISD SDHIST SDFUT


0.6403

0.5986

0.6893

0.6651

0.6559

0.8691

0.7398

0.7075

0.5858

0.5908

0.5118

0.4777

0.5681

0.6338

0.7441

0.7464

0.7520

0.8205

0.7948

0.6300

0.6305

0.5407

0.5949


0.4611

0.5910

0.5913

0.5810

0.5755

0.7161

0.7264

0.7251

0.7364

0.7235

0.7249

0.7229

0.7178

0.7127

0.7071

0.6914

0.6939

0.7162

0.7162

0.7203

0.7199

0.6293

0.6249


0.7199

0.6293

0.6249

0.6286

0.6278

0.4368

0.4341

0.4415

0.4339

0.4463

0.4428

0.4745

0.4815

0.4432

0.4258

0.4044

0.4051

0.3889

0.3784

0.3448

0.3530

0.3380

0.3442


STOCK 4
BRUNSWICK











STOCK 5
EASTMAN KODAK


MONTH WISD SDHIST SDFUT


0.2295

0.2778

0.1970

0.2835

0.3134

0.3181

0.2720

0.2982

0.2486

0.3127

0.2958

0.3623

0.3119

0.2757

0.3694

0.3397

0.4624

0.5597

0.4804

0.3352

0.3459

0.3210

0.3211


0.1604

0.1594

0.1528

0.1586

0.1482

0.1778

0.1777

0.1801

0.1744

0.1728

0.1679

0.1603

0.1586

0.1906

0.1865

0.2268

0.2458

0.2481

0.2479

0.2838

0.3295

0.3329

0.3513


0.3295

0.3329

0.3513

0.3508

0.3506

0.3398

0.3396

0.3373

0.3454

0.3491

0.3477

0.3499

0.3516

0.3350

0.3341

0.2926

0.2889

0.2769

0.2754

0.2620

0.2177

0.2150

0.1883


STOCK 6
EXXON


MONTH WISD SDHIST SDFUT


0.2709

0.2895

0.2968

0.2849

0.3415

0.3040

0.2918

0.3067

0.2549

0.2843

0.2642

0.2683

0.3052

0.2622

0.3498

0.3024

0.3547

0.4077

0.3494

0.2524

0.2484

0.2801

0.2722


0.1472

0.1492

0.1585

0.1606

0.1606

0.1545

0.1560

0.1661

0.1684

0.1697

0.1666

0.1731

0.1720

0.1769

0.1927

0.1991

0.2395

0.2394

0.2405

0.2649

0.2661

0.2671

0.2735


0.2661

0.2671

0.2735

0.2787

0.2818

0.2830

0.2801

0.2724

0.2713

0.2719

0.2690

0.2630

0.2642

0.2655

0.2475

0.2168

0.1911

0.1695

0.1711

0.1526

0.1605

0.1568

0.1473










STOCK 7
FORD


MONTH WISD SDHIST SDFUT


0.3893

0.3664

0.3619

0.3493

0.4395

0.4421

0.4203

0.3198

0.2884

0.2992

0.2655

0.2807

0.4095

0.3066

0.3131

0.3257

0.4544

0.5034

0.4441

0.2963

0.4912

0.4085

0.4241


0.2141

0.2154

0.2064

0.2131

0.2302

0.2685

0.2687

0.2885

0.2902

0.2935

0.2905

0.2945

0.2962

0.2808

0.2773

0.2777

0.2810

0.2811

0.2946

0.3084

0.3150

0.3317

0.3318


STOCK 8
GULF AND WESTERN


MONTH WISD SDHIST SDFUT


0.3150

0.3317

0.3318

0.3239

0.3258

0.2836

0.2821

0.2721

0.2852

0.2870

0.2882

0.3019

0.2997

0.2950

0.2818

0.2826

0.2548

0.2533

0.2530

0.2491

0.2324

0.2252

0.2338


0.5147

0.5209

0.5092

0.4671

0.5741

0.5505

0.5034

0.3615

0.3595

0.4230

0.3793

0.4487

0.4570

0.3948

0.4346

0.3945

0.4581

0.6336

0.5490

0.3820

0.4328

0.4611

0.4779


0.3327

0.3448

0.3198

0.2772

0.2780

0.2839

0.2765

0.2837

0.2791

0.2758

0.2740

0.2822

0.2825

0.2881

0.2891

0.2794

0.2899

0.2922

0.3016

0.3007

0.3004

0.2945

0.3021


0.3004

0.2945

0.3021

0.2934

0.2999

0.2715

0.2719

0.2728

0.2733

0.2731

0.2694

0.2749

0.2727

0.2298

0.2491

0.2400

0.2362

0.2509

0.2436

0.2480

0.2526

0.2457

0.2402










STOCK 9
INA


MONTH WISD SDHIST SDFUT


0.4357

0.4578

0.4334

0.4486

0.4509

0.4449

0.3819

0.2709

0.2693

0.3102

0.2774

0.2980

0.4297

0.4016

0.4534

0.5251

0.4685

0.4876

0.6111

0.3735

0.4470

0.4386

0.3247


0.3084

0.3116

0.3116

0.3104

0.3111

0.3207

0.3122

0.3183

0.3047

0.2849

0.2979

0.2999

0.2933

0.2906

0.2929

0.2899

0.3376

0.3364

0.3436

0.3334

0.3067

0.3041

0.3084


0.3067

0.3041

0.3084

0.3121

0.3232

0.3349

0.3319

0.3412

0.3585

0.3606

0.3421

0.3464

0.3288

0.3116

0.3035

0.3054

0.2728

0.2771

0.2678

0.2687

0.2687

0.2723

0.2797


STOCK 10
INTERNATIONAL HARVESTER


MONTH WISD SDHIST SDFUT


0.4086

0.4697

0.5102

0.5150

0.5538

0.5484

0.5107

0.3961

0.3607

0.3629

0.3687

0.3699

0.4483

0.3703

0.4129

0.3876

0.4658

0.5070

0.4485

0.3877

0.4049

0.4062

0.4409


0.2949

0.2944

0.2936

0.2947

0.2933

0.3480

0.3232

0.3231

0.3249

0.3255

0.3081

0.3080

0.3114

0.3052

0.3072

0.2958

0.2981

0.2994

0.2647

0.2716

0.3296

0.3252

0.3063


0.3296

0.3252

0.3063

0.3246

0.3301

0.2795

0.2782

0.2844

0.2848

0.2889

0.3045

0.3412

0.3329

0.3320

0.3260

0.3221

0.3449

0.3486

0.3451

0.3469

0.3033

0.3064

0.3117











STOCK 11
KRESGE


MONTH WISD SDHIST SDFUT


1 0.3722 0.2165 0.4835

2 0.6027 0.2270 0.4825

3 0.4905 0.2155 0.4887

4 0.4981 0.2357 0.4767

5 0.4566 0.2518 0.4783

6 0.4674 0.2905 0.4569

7 0.4053 0.2986 0.4539

8 0.3142 0.2954 0.4538

9 0.3000 0.2899 0.4637

10 0.4176 0.3021 0.4532

11 0.3087 0.3013 0.4538

12 0.3382 0.3186 0.4440

13 0.4157 0.3170 0.4443

14 0.3969 0.3452 0.4143

15 0.4401 0.3403 0.4117

16 0.4534 0.4138 0.2990

17 0.5585 0.4727 0.2388

18 0.6720 0.4682 0.2284

19 0.6125 0.4673 0.2236

20 0.4162 0.4841 0.2039

21 0.5441 0.4835 0.1764

22 0.5315 0.4825 0.1698

23 0.4849 0.4887 0.1670


STOCK 12
LOEW'S


MONTH WISD SDHIST SDFUT


0.5081

0.6049

0.5539

0.4870

0.5416

0.5329

0.5737

0.4310

0.4358

0.4845

0.3800

0.5583

0.5602

0.4485

0.6211

0.6881

0.5949

0.6509

0.7331

0.4286

0.5130

0.5752

0.4856


0.2828

0.2928

0.2890

0.3038

0.2798

0.3020

0.3035

0.2986

0.2993

0.3054

0.3043

0.3403

0.3427

0.3241

0.3265

0.3176

0.3703

0.3765

0.3702

0.3916

0.3942

0.3939

0.3832


0.3942

0.3939

0.3832

0.4022

0.4037

0.3775

0.3804

0.3841

0.3840

0.3944

0.3941

0.4006

0.4006

0.3973

0.3854

0.3576

0.3384

0.3457

0.3505

0.3462

0.3517

0.3602

0.3715










STOCK 13
MCDONALD'S


MONTH WISD SDHIST SDFUT


1 0.4749 0.2842 0.5903

2 0.3564 0.2919 0.5832

3 0.4084 0.2829 0.6114

4 0.3760 0.2811 0.6080

5 0.4542 0.2770 0.6102

6 0.5629 0.3868 0.5659

7 0.4818 0.3920 0.5568

8 0.4160 0.3860 0.5559

9 0.4264 0.3874 0.5638

10 0.4301 0.3876 0.5686

11 0.3843 0.3880 0.5686

12 0.4727 0.3879 0.5704

13 0.4073 0.3851 0.5595

14 0.4749 0.4177 0.5228

15 0.5527 0.3945 0.5179

16 0.5249 0.4631 0.4311

17 0.5780 0.5587 0.3450

18 0.8034 0.5559 0.3440

19 0.7220 0.5589 0.3207

20 0.4765 0.5727 0.3049

21 0.5390 0.5903 0.2823

22 0.4664 0.5832 0.2793

23 0.3986 0.6114 0.2242


STOCK 14
MERCK


MONTH WISD SDHIST SDFUT


0.2541

0.3053

0.3655

0.3645

0.3704

0.3416

0.2985

0.2542

0.2403

0.2485

0.2315

0.3151

0.3129

0.2981

0.3068

0.3787

0.4074

0.4930

0.5001

0.3767

0.3581

0.3309

0.2928


0.1295

0.1298

0.1753

0.1732

0.1753

0.1748

0.1786

0.1810

0.1748

0.1633

0.1614

0.1619

0.1579

0.1980

0.1941

0.2484

0.3425

0.3428

0.3392

0.3393

0.3677

0.3664

0.3535


0.3677

0.3664

0.3535

0.3562

0.3547

0.3672

0.3687

0.3686

0.3768

0.3786

0.3919

0.3939

0.3973

0.3854

0.3837

0.3320

0.2589

0.2586

0.2590

0.2606

0.2298

0.2622

0.2660










STOCK 15
NORTHWEST AIRLINES


MONTH WISD SDHIST SDFUT


0.5981

0.6468

0.7208

0.5965

0.7244

0.6801

0.6076

0.5085

0.4405

0.4921

0.4295

0.4486

0.5744

0.5169

0.6619

0.6146

0.7129

0.8549

0.7873

0.5298

0.6173

0.5371

0.5252


0.4105

0.4225

0.4267

0.4523

0.4413

0.4487

0.4437

0.4351

0.4541

0.4470

0.4475

0.4336

0.4271

0.4273

0.4332

0.4569

0.4504

0.4415

0.4509

0.4888

0.4951

0.4770

0.4760


0.4951

0.4770

0.4760

0.4488

0.4459

0.4325

0.4319

0.4377

0.4136

0.4358

0.4395

0.4856

0.4826

0.4777

0.4617

0.4062

0.3970

0.3791

0.3586

0.3091

0.3156

0.3141

0.3096


STOCK 16
PENNZOIL


MONTH WISD SDHIST SDFUT


1 0.5100 0.3790 0.4545

2 0.6377 0.3768 0.4529

3 0.6026 0.3515 0.4506

4 0.5091 0.3862 0.4184

5 0.6232 0.3836 0.4198

6 0.6080 0.3754 0.4158

7 0.4544 0.3903 0.3961

8 0.4648 0.3882 0.3996

9 0.3772 0.3881 0.3998

10 0.4204 0.3834 0.3927

11 0.4391 0.3531 0.3878

12 0.5118 0.4431 0.3076

13 0.5052 0.4268 0.3164

14 0.5341 0.4267 0.3290

15 0.6265 0.4273 0.3262

16 0.6183 0.4327 0.2955

17 0.5902 0.4663 0.2524

18 0.6161 0.4712 0.2571

19 0.5871 0.4689 0.2728

20 0.4940 0.4498 0.2733

21 0.4478 0.4545 0.2739

22 0.5003 0.4529 0.2685

23 0.4782 0.4506 0.2695











STOCK 17
POLAROID


MONTH WISD SDHIST SDFUT


0.3621

0.4184

0.4424

0.4368

0.5865

0.5356

0.5183

0.4142

0.3597

0.5440

0.4877

0.6264

0.5727

0.6441

0.7486

0.6920

0.7968

0.9723

0.9193

0.6053

0.7291

0.6314

0.6089


0.3132

0.3173

0.3305

0.3216

0.3009

0.3573

0.3381

0.3699

0.3657

0.3896

0.3926

0.4994

0.4762

0.4940

0.4961

0.5426

0.6208

0.6175

0.6151

0.6230

0.6214

0.6444

0.6946


STOCK 18
RCA


MONTH WISD SDHIST SDFUT


0.6214

0.6444

0.6946

0.6994

0.7134

0.6930

0.6939

0.6754

0.6812

0.6684

0.6739

0.6103

0.6059

0.5733

0.5503

0.4447

0.3799

0.3830

0.3620

0.3658

0.4055

0.3972

0.3518


0.5089

0.5377

0.5242

0.4695

0.5128

0.6497

0.5739

0.4631

0.4384

0.3613

0.4358

0.4632

0.5227

0.5321

0.5027

0.6662

0.6193

0.7009

0.6872

0.5415

0.4647

0.5309

0.5522


0.2436

0.2577

0.2413

0.2555

0.2528

0.3203

0.3213

0.3258

0.3306

0.3288

0.3318

0.3320

0.3284

0.3409

0.3466

0.3526

0.3511

0.3516

0.3504

0.4014

0.4088

0.4092

0.4072


0.4088

0.4092

0.4072

0.4468

0.4465

0.3934

0.3952

0.3950

0.3925

0.3999

0.3873

0.4408

0.4379

0.3916

0.4032

0.3608

0.3594

0.3611

0.3628

0.3421

0.3505

0.3462

0.3435










STOCK 19
SPERRY RAND


MONTH WISD SDHIST SDFUT


1 0.4178 0.2809 0.3355

2 0.4374 0.2660 0.3206

3 0.5198 0.2633 0.3476

4 0.4417 0.2423 0.3508

5 0.5419 0.2397 0.3598

6 0.4555 0.2436 0.3691

7 0.4189 0.2445 0.3664

8 0.4009 0.2093 0.3614

9 0.3284 0.2090 0.3676

10 0.3881 0.2112 0.3654

11 0.3387 0.2074 0.3705

12 0.3898 0.2088 0.3891

13 0.3609 0.2073 0.3900

14 0.4178 0.2312 0.3734

15 0.4869 0.2385 0.3615

16 0.4824 0.2694 0.3200

17 0.5277 0.3181 0.2973

18 0.6162 0.3185 0.3056

19 0.5479 0.3085 0.3015

20 0.3708 0.3303 0.2884

21 0.3711 0.3355 0.2895

22 0.3445 0.3206 0.2935

23 0.3996 0.3476 0.2600


STOCK 20
TEXAS INSTRUMENTS


MONTH WISD SDHIST SDFUT


0.4681

0.4349

0.5635

0.3637

0.6303

0.4987

0.4747

0.3527

0.3193

0.4210

0.3549

0.3973

0.3839

0.3616

0.3553

0.3297

0.3918

0.5250

0.5172

0.3447

0.4230

0.4605

0.3777


0.2047

0.2518

0.2394

0.2465

0.2450

0.2614

0.2557

0.2526

0.2546

0.2774

0.2780

0.2892

0.3172

0.3322

0.3280

0.3864

0.4123

0.4096

0.4129

0.4145

0.4248

0.3912

0.4167


0.4248

0.3912

0.4167

0.4075

0.4082

0.4148

0.4128

0.4119

0.4161

0.4043

0.4035

0.4262

0.4060

0.3896

0.3952

0.3178

0.3022

0.3124

0.3096

0.3081

0.3125

0.3073

0.2741










STOCK 21
UPJOHN


MONTH WISD SDHIST SDFUT


1 0.4839 0.2557 0.5618

2 0.4648 0.2727 0.5475

3 0.5112 0.2905 0.5555

4 0.4502 0.2920 0.5589

5 0.4616 0.2678 0.5542

6 0.4456 0.2692 0.5757

7 0.4585 0.3190 0.5653

8 0.4534 0.3367 0.5659

9 0.3760 0.3122 0.5748

10 0.4633 0.3039 0.5812

11 0.3380 0.3130 0.5730

12 0.4170 0.3153 0.5766

13 0.4316 0.3188 0.5835

14 0.3938 0.3193 0.5869

15 0.4628 0.3282 0.5832

16 0.4652 0.4235 0.5207

17 0.6147 0.4245 0.5263

18 0.6563 0.4157 0.5246

19 0.5318 0.4114 0.5252

20 0.7044 0.5412 0.3369

21 0.5781 0.5618 0.3252

22 0.5197 0.5475 0.3251

23 0.5539 0.5555 0.3194


STOCK 22
WEYERHAEUSER


MONTH WISD SDHIST SDFUT


0.3763

0.4683

0.4275

0.3496

0.5208

0.5124

0.5281

0.3585

0.3061

0.2796

0.2750

0.3664

0.3641

0.3462

0.3615

0.3640

0.4956

0.5417

0.5518

0.3532

0.3936

0.3978

0.3712


0.2092

0.2147

0.2175

0.2127

0.2053

0.2121

0.2152

0.2009

0.1916

0.1916

0.1903

0.2337

0.2381

0.2418

0.2523

0.3150

0.3163

0.3067

0.3091

0.3241

0.3250

0.3183

0.3263


0.3250

0.3183

0.3263

0.3333

0.3303

0.3348

0.3267

0.3379

0.3400

0.3390

0.3337

0.3376

0.3369

0.3480

0.3394

0.2647

0.2623

0.2638

0.2656

0.2507

0.2507

0.2506

0.2387










STOCK 23
XEROX


MONTH WISD SDHIST SDFUT


1 0.2577 0.1633 0.4412

2 0.2762 0.1633 0.4389

3 0.3002 0.1457 0.4439

4 0.3568 0.1593 0.4438

5 0.2978 0.1575 0.4419

6 0.3486 0.1904 0.4388

7 0.3520 0.1910 0.4390

8 0.2768 0.1743 0.4408

9 0.2558 0.1751 0.4457

10 0.3085 0.1692 0.4491

11 0.3555 0.1724 0.4511

12 0.2731 0.1867 0.4960

13 0.3574 0.1858 0.4973

14 0.3796 0.2314 0.4931

15 0.3776 0.2442 0.4900

16 0.3674 0.2927 0.4589

17 0.4405 0.3138 0.4615

18 0.6333 0.3362 0.4360

19 0.5795 0.3458 0.4188

20 0.3618 0.4417 0.3380

21 0.5057 0.4412 0.3363

22 0.4602 0.4389 0.3359

23 0.4948 0.4439 0.3329







































APPENDIX B
OPTION DATA FOR HEDGED POSITIONS
















APPENDIX B
OPTION DATA FOR HEDGED POSITIONS


DAYS OPT. INIT.
OPT. EXER. OPT.
MO. STK. LIFE PRICE PRICE


1 15 207 25
207 30

4 19 119 45
210 55

5 6 86 100
177 100

5 7 86 60
177 60

5 17 86 110
177 130

5 19 86 40
177 55

6 5 56 140
147 140

6 7 56 50
147 50

6 8 56 30
147 30

6 10 56 30
238 35

6 12 56 25
147 25

6 19 56 55
238 55

6 22 56 60
56 70


2.38
1.00

10.13
5.25

4.63
6.00

2.13
3.25

7.63
5.50

14.63
6.25

0.75
2.00

0.94
1.63

0.63
1.19

0.75
1.63

0.50
1.00

1.25
4.00

15.50
5.00


IMPL.
MKT.
VAL.

2.10
1.18

8.96
5.90

4.13
6.98

1.90
3.65

6.46
6.79

13.18
7.02

0.42
2.67

0.63
2.03

0.45
1.52

0.54
2.17

0.34
1.16

1.04
4.71

13.14
6.58


END INIT. PROFIT RETURN
OPT. $ OR ON
PRICE POSIT. LOSS HEDGE(%)

3.75 5.72 -3.31
2.38 3.73 5.13 19.28

9.75 13.31 0.49
6.25 10.07 1.92 10.30

1.25 11.34 8.27
3.75 12.71 -4.76 14.59

0.19 7.05 6.43
0.69 8.27 -6.52 -0.61

0.56 19.53 18.09
1.50 16.92 -12.30 15.87

8.00 16.66 7.55
2.75 11.53 -6.46 3.86

0.13 10.78 8.98
1.63 8.98 -1.68 36.95

0.13 5.32 4.61
0.88 5.15 -2.38 21.33

0.13 3.51 2.81
0.75 3.54 -1.30 21.33

0.44 3.66 1.52
1.00 4.58 -1.76 -2.91

0.31 2.93 1.10
0.88 3.12 -0.39 11.67

0.31 5.40 4.05
2.25 8.90 -3.89 1.11

18.50 18.40 -3.57
10.00 8.43 8.43 18.13












DAYS OPT. INIT. IMPL. END INIT. PROFIT RETURN
OPT. EXER. OPT. MKT. OPT. $ OR ON
MO. STK. LIFE PRICE PRICE VAL. PRICE POSIT. LOSS HEDGE(%)


6 23 56 160
147 160

7 4 208 20
208 15

7 5 117 140
208 140

7 7 117 40
208 60

7 17 117 70
208 110

8 4 176 20
176 25

8 10 176 30
176 25

8 11 85 40
176 40

8 12 85 25
85 20

8 13 85 70
176 70

8 17 85 110
176 110

8 23 85 140
176 140

9 4 57 20
148 20

9 5 57 120
239 120

9 6 57 90
148 100


0.69
2.13

1.50
3.50

1.63
3.00

4.13
0.63

10.50
1.75

1.38
0.88

1.19
3.50

0.56
1.13

0.50
1.50

0.88
1.50

0.81
2.38

1.50
3.13

0.63
1.31

0.63
3.25

1.75
1.06


0.06
1.25

1.38
2.88

0.63
1.50

6.13
0.63

15.00
2.38

1.31
0.63

1.19
3.38

0.50
1.31

0.25
1.25

0.19
1.25

0.13
0.88

0.38
1.63

0.25
1.06

0.56
5.38

0.38
0.75


12.12 11.02
10.39 -4.28 29.93

3.68 -0.31
5.74 1.02 7.59

10.32 6.35
11.29 -5.64 3.27

7.59 -3.67
4.72 -0.00 -29.86

17.90 -7.67
8.86 3.16 -16.83

3.56 -0.16
3.83 1.10 12.65

3.49 -0.00
5.84 0.21 2.22

3.12 0.35
3.66 0.61 14.11

2.59 1.30
2.58 -0.43 16.72

6.27 4.93
5.57 -0.93 33.75

8.13 6.88
10.51 -6.64 1.29

10.02 7.51
11.20 -5.38 10.08

2.34 1.41
3.22 -0.61 14.23

7.23 0.72
9.96 6.52 42.13

6.11 4.80
5.90 -1.73 25.55












DAYS OPT. INIT. IMPL. END INIT. PROFIT RETURN
OPT. EXER. OPT. MKT. OPT. $ OR ON
MO. STK. LIFE PRICE PRICE VAL. PRICE POSIT. LOSS EDGEE()


9 9 57 40
148 40

9 12 57 25
57 20

9 17 57 95
148 110

9 18 57 25
148 20

9 21 57 75
148 75

10 7 117 60
117 50

11 4 87 20
87 15

11 5 87 120
87 100

11 9 87 35
87 30

11 10 87 30
87 25

11 11 87 35
178 35

11 12 87 20
178 25

11 13 87 70
87 50

11 14 87 90
87 80

11 17 87 80
178 80


0.69
1.25

0.25
1.25

0.94
0.88

0.38
2.00

1.38
3.25

0.63
2.13

0.44
1.31

1.63
6.75

0.50
1.94

0.56
1.81

1.06
1.75

1.25
0.75

0.63
5.88

1.25
4.00

0.88
2.25


0.55
1.44

0.18
1.39

0.76
1.01

0.22
2.44

1.23
3.57

0.49
2.72

0.32
1.66

1.30
7.76

0.44
2.15

0.49
2.06

0.96
1.94


0.13 2.74 2.24


0.94

0.13
1.31

0.06
0.50

0.06
1.38

0.75
3.63

0.44
2.13

0.25
1.00

1.31
9.25

0.19
0.88

0.25
1.25

2.63
3.75


3.30 -0.76 24.63

2.14 1.07
2.31 0.11 26.58

6.67 6.23
7.14 -3.06 22.92

2.75 2.29
3.39 -1.06 20.07

5.93 2.70
8.67 1.00 25.30

4.79 1.44
4.42 -0.00 15.60

2.48 1.06
2.23 -0.53 11.33

9.26 1.78
10.73 3.97 28.77

2.45 1.53
3.11 -1.70 -3.09

2.67 1.48
3.11 -0.97 8.97

3.23 -4.74
4.02 4.59 -2.13

2.39 -1.91
3.13 2.35 7.89

5.67 0.56
8.22 -5.07 -32.46

5.16 1.76
6.69 0.00 14.04

6.87 5.89
8.73 -7.52 -10.45











DAYS OPT. INIT. IMPL. END INIT. PROFIT RETURN
OPT. EXER. OPT. MKT. OPT. $ OR ON
MO. STK. LIFE PRICE PRICE VAL. PRICE POSIT. LOSS HEDGE(Z)


1.75 1.50 3.75 9.94 -11.35


11 20 87 120
178 100

12 6 147 70
238 80

12 9 147 35
147 30

12 10 56 30
56 25

12 12 56 20
147 20

12 16 147 20
238 25

12 18 56 20
147 20

12 22 56 35
56 40

13 11 117 30
208 35

13 13 117 60
208 60

13 17 117 40
208 45

14 3 177 35
268 30

14 6 86 80
268 70

14 8 86 20
177 25

14 11 86 30
268 30


9.75

6.00
4.13

0.69
1.06

0.25
1.25

0.25
0.69

1.13
0.94

0.25
0.63

5.00
1.25

6.50
4.00

1.50
2.38

3.25
2.75

1.19
3.75

1.63
10.00

1.06
0.69

1.69
3.25


10.83

6.74
3.75

0.48
1.66

0.20
1.45

0.20
0.77

1.25
0.84

0.18
0.70

4.39
1.64

5.86
4.44

1.29
2.76

2.73
3.09

1.30
3.23

1.89
9.09

1.19
0.60

1.52
3.61


18.00

4.88
3.00

0.25
0.63

0.06
0.56

0.13
0.50

1.13
0.75

0.13
0.38

2.75
0.50

1.69
1.31

0.19
0.75

0.63
0.81

0.81
2.88

0.69
7.00

1.31
0.81

1.00
3.25


16.55 14.00

9.22 -1.73
10.49 2.86

3.80 2.42
2.30 -0.95

2.07 1.55
2.34 -1.29

1.93 0.97
2.44 -0.66

2.89 -0.00
3.74 0.75

1.89 0.94
2.21 -0.88

6.41 2.89
2.86 -1.72

8.41 6.23
6.90 -4.63

6.66 5.83
7.13 -4.88

7.98 6.45
7.28 -5.13

3.76 -1.19
6.93 1.62

4.95 -2.86
14.85 4.45

2.24 0.52
3.04 -0.55

3.93 1.60
5.89 0.00


10.01


5.76


24.15


5.99


6.95


11.23


1.53


12.60


10.40


6.88


8.62


4.03


8.07


-0.49


16.34












DAYS OPT. INIT.
OPT. EXER. OPT.
MO. STK. LIFE PRICE PRICE


14 12 86 20
86 15

14 13 86 50
177 50

14 14 86 80
177 80

14 17 86 25
177 40

14 20 86 100
268 90

14 23 86 120
268 100

15 3 147 35
238 30

15 6 56 80
238 80

15 7 56 45
238 45

15 10 56 25
147 20

15 11 56 30
147 35

15 13 56 35
238 40

15 15 56 20
147 25

15 16 147 15
238 20

15 17 56 20
147 40


0.25
1.13

0.75
1.75

0.88
1.63

5.25
1.50

1.56
8.00

1.50
10.13

0.81
2.88

0.69
3.50

0.63
2.13

0.25
2.00

1.00
0.75

3.88
4.38

1.06
0.81

3.50
1.75

4.00
0.63


IMPL.
MKT.
VAL.

0.19
1.41

0.66
1.97

0.67
2.06

4.61
1.74

1.31
8.89

1.04
11.91

0.92
2.61

0.60
3.98

0.52
2.39

0.18
2.27

0.84
0.94

3.26
5.21

0.88
0.97

3.15
2.03

3.60
0.74


END INIT. PROFIT RETURN
OPT. $ OR ON
PRICE POSIT. LOSS HEDGE(Z)


0.25
0.88

0.38
1.44

0.13
0.88

1.81
0.63

0.38
7.13

0.13
5.25

0.25
1.56

0.06
0.63

0.06
0.88

0.06
1.44

2.13
1.75

0.06
0.63

0.06
0.13

1.38
0.75

0.13
0.19


1.98 0.00
1.97 -0.44 -10.98

4.71 2.36
5.99 -1.07 12.02

6.10 5.23
6.02 -2.78 20.22

7.73 5.06
5.26 -3.07 15.35

8.66 6.58
15.43 -1.69 20.30

11.57 10.61
18.12 -8.72 6.34

3.32 -2.30
6.16 2.81 5.39

5.04 4.58
9.62 -7.90 -22.65

3.10 2.79
5.33 -3.13 -4.08

2.07 1.55
3.40 -0.96 10.88

3.23 -3.64
3.08 4.10 7.33

6.82 6.71
8.34 -7.15 -2.90

3.12 2.94
3.01 -2.55 6.36

5.25 3.19
3.88 -2.22 10.61

5.82 5.64
3.76 -2.63 31.37












DAYS OPT. INIT. IMPL. END INIT. PROFIT RETURN
OPT. EXER. OPT. MKT. OPT. $ OR ON


MO. STK. LIFE PRICE PRICE


15 21 56 75
147 85

15 23 147 80
238 100

16 5 207 100
207 70

16 9 116 25
207 20

16 11 116 25
207 25

16 13 116 30
207 40

16 19 116 25
207 30

16 20 116 80
207 80

16 21 116 50
207 65

16 22 207 30
207 35

16 23 116 90
207 90

17 3 85 30
267 30

17 5 85 90
176 100

17 8 85 20
176 25

17 9 176 20
267 30


1.56
1.31

12.13
5.25

0.75
4.25

1.00
4.25

0.56
0.81

1.63
0.63

2.38
2.25

0.63
1.38

3.13
1.31

1.25
0.63

0.75
1.19

1.19
3.00

1.75
1.88

2.25
1.63

8.00
2.38


VAL.

1.22
2.02

10.92
5.97

0.54
5.59

1.23
3.79

0.48
1.08

1.34
0.80

2.70
2.05

0.53
1.62

2.77
1.51

1.39
0.56

0.45
1.52

1.35
2.54

1.47
2.29

2.56
1.40

7.14
2.83


PRICE POSIT. LOSS HEDGE(%)


0.06
0.13

1.38
0.75

1.88
10.50

3.00
8.00

2.63
3.63

7.25
4.00

6.50
4.38

3.38
6.50

4.25
2.25

3.00
1.56

1.88
4.00

0.81
2.75

0.50
1.38

3.50
2.38

8.50
3.00


7.53 7.23
5.95 -5.38 13.69

18.25 16.18
13.35 -11.44 15.00

9.37 -14.04
8.57 12.60 -8.03

2.87 5.75
6.63 -5.84 -0.99

2.84 -10.42
2.63 9.10 -24.16

4.97 -17.20
3.47 18.76 18.51

4.25 7.38
5.69 -5.38 20.10

5.94 -26.13
6.51 24.27 -14.91

7.97 -2.87
6.40 4.57 11.85

3.57 5.00
3.64 -5.45 -6.36

9.34 -14.01
6.61 15.65 10.30

2.91 -0.92
6.42 0.54 -4.12

9.31 6.65
9.03 -2.41 23.13

3.39 1.89
4.41 -2.04 -1.93

9.40 -0.59
5.15 1.36 5.29












DAYS OPT. INIT.
OPT. EXER. OPT.
MO. STK. LIFE PRICE PRICE


17 10 85 20
267 20

17 11 85 20
176 30

17 16 85 15
267 15

17 21 85 65
176 75

17 22 176 35
267 25

18 5 56 70
238 80

18 6 56 70
238 70

18 11 56 25
147 25

18 13 56 30
147 40

18 17 56 30
238 25

18 19 147 35
238 30

18 22 56 30
147 30

18 23 56 80
147 90

19 5 115 90
206 60

19 7 115 40
206 35


IMPL. END INIT. PROFIT RETURN


MKT.


1.75
3.50

5.75
1.94

2.88
4.88

1.19
1.25

1.56
8.38

4.38
6.25

2.00
4.88

2.00
3.00

7.25
4.50

1.06
4.75

2.25
5.88

2.25
3.25

1.25
1.88

0.88
12.25

0.88
3.75


VAL.

1.94
3.15

5.23
2.28

3.23
4.39

0.97
1.62

1.98
6.81

3.84
7.27

1.74
5.54

1.74
3.37

6.42
5.11

0.97
5.52

2.50
5.34

2.05
3.65

1.10
2.60

0.96
10.94

1.05
3.32


OPT. $ OR ON
PRICE POSIT. LOSS HEDGE(%)

1.69 3.05 -0.11
3.38 5.89 0.21 1.12

4.50 7.06 1.53
1.88 4.73 -0.16 11.67

4.25 3.85 1.84
5.75 6.85 -1.23 5.71

1.06 7.60 0.80
1.50 6.72 1.34 14.93

1.88 4.35 0.86
7.38 11.51 1.37 14.12

0.75 10.43 8.64
4.38 14.35 -4.31 17.50

0.50 6.74 5.06
4.63 10.92 -0.56 25.44

0.50 4.51 3.38
2.25 5.62 -1.40 19.53

1.88 10.09 7.48
1.75 9.08 -5.55 10.07

0.06 4.40 4.14
3.00 8.03 -2.96 9.50

1.13 5.74 -2.87
3.75 10.04 3.63 4.82

0.38 4.65 3.87
2.88 5.94 -0.68 30.14

0.06 7.96 7.56
0.31 8.29 -6.91 4.03

0.69 6.83 -1.46
15.25 18.83 -4.62 -23.69

0.88 3.51 0.00
3.63 7.69 0.26 2.32











DAYS OPT. INIT. IMPL.
OPT. EXER. OPT. MKT.
MO. STK. LIFE PRICE PRICE VAL.


19 20 115 90
206 60

19 21 115 75
206 45

20 1 266 50
266 45

20 3 175 30
175 25

20 16 175 20
175 15

20 21 84 40
175 50

20 23 84 90
84 70

21 4 56 15
147 10

21 5 56 100
238 90

21 7 56 35
56 30

21 17 56 25
238 25

21 21 56 45
147 50

21 23 56 90
147 80

22 7 116 40
207 35

22 16 116 20
116 15


1.38
17.25

0.63
12.38

2.00
5.50

2.63
6.88

1.63
4.63

2.13
1.13

0.75
4.88

0.38
2.25

1.50
8.00

1.38
4.50

1.31
3.38

0.75
1.25

1.88
6.50

1.75
5.75

1.56
5.13


END
OPT.


2.18
15.13

0.87
11.15

2.27
4.66

2.88
6.14

1.79
4.17

1.46
1.58

0.56
5.42

0.23
2.56

1.17
9.52

1.58
3.91

1.14
3;84

0.66
1.45

1.42
7.47

2.03
5.04

1.78
4.58


PRICE

0.81
17.25

0.06
1.94

3.63
6.75

2.38
5.25

2.25
5.50

1.63
1.25

1.88
8.25

0.25
3.50

0.88
11.00

2.50
7.13

1.13
4.38

0.56
1.63

0.13
4.38

1.31
4.75

1.50
5.13


INIT. PROFIT RETURN
$ OR ON
POSIT. LOSS HEDGE(%)

6.22 -2.54
23.87 -0.00 -8.45

4.79 -4.31
17.38 14.66 46.67

4.54 3.69
8.00 -1.81 14.94

4.43 -0.42
7.94 1.88 11.73

3.49 1.35
5.97 -1.13 2.28

7.80 1.84
4.87 0.54 18.73

7.28 -10.91
8.51 5.89 -31.79

2.22 0.74
3.15 1.76 46.53

7.86 3.28
14.45 5.42 38.98

3.48 2.84
6.39 -3.73 -9.00

3.95 0.56
6.42 1.90 23.73

4.36 1.09
5.40 1.62 27.78

9.74 9.09
13.34 -4.36 20.51

4.44 -1.11
9.33 1.62 3.74

3.15 -0.13
6.07 -0.00 -1.42















REFERENCES


1. Black, F., and M. Scholes. The Valuation of Option
Contracts and a Test of Market Efficiency. Journal
of Finance, May, 1972.

2. Black, F., and M. Scholes. The Pricing of Options and
Corporate Liabilities. Journal of Political Economy,
May/June, 1973.

3. Cox, J. C., and S. A. Ross. A Survey of New Results in
Financial Option Pricing Policy. Journal of Finance,
May, 1976.

4. Ingersoll, J. A Theoretical and Empirical Investigation
of the Dual Purpose Funds: An Application of Contin-
gent Claims Analysis. Journal of Financial Economics,
Vol. 3, Nos. 1/2, 1976.

5. Kmenta, Jan. Elements of Econometrics. New York: Mac-
millan, 1971.

6. Latand, H., and R. Rendleman, Jr. Standard Deviation
of Stock Price Ratios Implied in Option Prices.
Journal of Finance, May, 1975.

7. Merton, R. Theory of Rational Option Pricing. The Bell
Journal of Economics and Management Science, Spring,
1973.

8. Merton, R. Option Pricing When Underlying Stock Re-
turns Are Discontinuous. Journal of Financial Eco-
nomics, Vol. 3, Nos. 1/2, 1976.

9. Merton, R. The Impact on Option Pricing of Specifica-
tion Error in the Underlying Stock Price Returns.
Journal of Finance, May, 1976.

10. Smith, C., Jr. Option Pricing: A Review. Journal of
Financial Economics, Vol. 3, Nos. 1/2, 1976.

11. Trippi, R. A Test of Option Market Efficiency Using a
Random-Walk Valuation Model. Journal of Economics
and Business, Winter, 1977.


87













BIOGRAPHICAL SKETCH


Donald P. Chiras was born in Worcester, Massachusetts,

on November 8, 1938. He graduated from Classical High School

there in 1957. He attended the United States Naval Academy

and was commissioned an Ensign upon graduation in 1961.

After completion of air navigation training he was

assigned to Project Magnet, an airborne geomagnetic survey,

which charts the earth's magnetic field. While serving in

this capacity, Mr. Chiras developed a method of reducing

gyro-compass drift which increased navigational accuracy.

That led to his role as a navigator in planning and execut-

ing the pioneering single-aircraft flight across the un-

charted wastes of the Antarctic Continent and the treacher-

ous waters of the South Atlantic Ocean. He received the

Navy Commendation Medal for the flight which was the first

in history to prove the feasibility of air travel between

the Eastern and Western Hemispheres via trans-Antarctic

routing. He was then selected as the first in his gradua-

tion class to return to the Naval Academy to instruct mid-

shipmen. There he received the highest fitness report

rating of any officer-instructor.










After leaving the Navy in 1967 he was awarded a

College of Engineering Fellowship at the University of

Florida where he received a master's degree in mechanical

engineering in 1968. He accepted a position with Shell

Oil Co. as a project engineer responsible for the design,

construction and profitability analysis of offshore oil

production projects. After becoming the offshore division

electrical engineer he proved the economic feasibility of

the introduction of gas turbine generators (modified jet

engines) to provide electric power generation for offshore

oil production platforms.

Mr. Chiras then joined Merrill Lynch, Pierce,

Fenner and Smith, Inc. for five years as an account execu-

tive where he became research liaison and a member of the

Executive's Club. Upon his departure in 1974 he became

a registered investment advisor and financial consultant

and accepted a Graduate Council Fellowship at the Univer-

sity of Florida to pursue his doctorate in business ad-

ministration with a major in finance.

In 1977, he accepted a position as a visiting as-

sistant professor in the Department of Finance, Insurance

and Real Estate at the University of Florida where he

will teach investments. He is currently managing invest-

ment portfolios and is active in personal financial con-

sulting.




90





He has received notice of his inclusion in the 1977

edition of Who's Who in Business and Finance.

He is married to the former Gloria L. Perez, who

is an assistant professor of nursing at the University of

Florida. They have one child, Jennifer, age 5.




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