IMPLICATIONS OF OPTION MARKETS:
THEORY AND EVIDENCE
BY
THOMAS J. O'BRIEN
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
1980
ACKNOWLEDGMENTS
I would like to thank the following people for their
patience and technical assistance: Drs. Robert C. Radcliffe
(chairman), Richard H. Pettway, and W. Andrew McCollough
of the Finance Department; H. Russell Fogler of the
Management Department; and James T. McClave of the
Statistics Department.
TABLE OF CONTENTS
ACKNOWLEDGMENTS.
ABSTRACT
CHAPTER
I INTRODUCTION . . . . . . . .
Futures and Options . . . . . .
Use of Options in the U.S. . . . . .
Evolution Prior to 1973 . . . . .
The Post 1973 Option Market . . . .
The Moratorium and After. . . . .
Reasons for Listed Stock Option Usage .
Evidence of Reasons for Recent
Option Trading . . . . . .
Overview of the Study. . . . . . .
Notes . . . . . . . . .
II OPTIONS IN A SINGLE PERIOD CAPITAL MARKET
EQUILIBRIUM FRAMEWORK . . . . . . .
Equilibrium with One Option and No
Riskless Asset . . . . . .
Equilibrium with One Option and Unrestricted
Trading in the Riskless Security. . .
Equilibrium with One Option and a Margin
Constraint on Riskless Borrowing. . .
Notes . . . . . . . . .
III REVIEW OF ASSOCIATED LITERATURE . .
Introduction . . . . . .
Klemkosky and Maness . . . .
Hayes and Tennenbaum . . . .
Reilly and Naidu . . . . .
Trennepohl and Dukes . . . .
Implications . . . . . .
Notes . . . . . . .
IV PROPOSED ANALYSIS OF RETURNS OF ZERO
SYSTEMATIC RISK HEDGES. . . . . .
Excess Returns . . . . . . .
Zero Systematic Risk Returns . . .
Neutral Spread Returns. . . . .
Previous Empirical Results of Neutral
Spreads . . . . . . .
Neutral Hedge Returns . . . .
iii
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11
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71
Previous Empirical Results of Neutral
Hedges . . . . . . .
Direction of the Methodology. . .
Two Further Considerations. . . .
Selection of Time Span, Holding Period,
and Data . . . . . . .
Time Span: November 30, 1973August
29, 1975 . . . . . .
Holding Period Assumption: Monthly
Observations . . . . .
Description of the Data . . . .
Preliminary Procedure . . . .
Neutral Hedge Returns . . . .
Naive Diversification . . . .
The Hypothesis Test. . . . . .
Preliminary Test Procedure. . . .
Two Additional Procedures . . .
Notes . . . . . . . .
V EMPIRICAL FINDINGS . . . . . . .
Introduction and Summary of Results. . .
Naive Diversification . . . . .
Systematic Risk in Hedges?. . . . .
The Minimum Variance Zero Beta Portfolio?
Time Series Hypothesis Tests . . . .
First Version: Tests Without the Index
Second Version: Tests with Index . .
Notes . . . . . . . . .
VI REVIEW AND IMPLICATIONS OF THE RESEARCH
FOR OPTIONS AND THE CAPM. . . . . . .
A Test of the CAPM . . . . . . .
Systematic Risk of Neutral Hedges. . . .
Notes . . . . . . . . .
APPENDIX
BIBLIOGRAPHY
MONTHLY PORTFOLIO SIZEVARIANCE RELATION.
BIOGRAPHICAL SKETCH. .
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i
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
IMPLICATIONS OF OPTION MARKETS:
THEORY AND EVIDENCE
By
Thomas J. O'Brien
December, 1980
Chairman: Robert C. Radcliffe
Major Department: Finance, Insurance, and Real Estate
Many new markets for options have opened in recent
times. Financial research has been interested in the
experiences of option investors, and much empirical work
has been done in this regard. However, in previous studies
the issue of why options exist at all is rarely addressed.
The exploration of the role of options in important
theoretical models of capital market equilibrium leads to
the following conclusion concerning the effects of opening
new option markets: If margin ceilings impose effective
borrowing constraints on any investors, then the expansion
of the option market will cause the equilibrium intercept
of the riskreturn relation to shift downward. This theory
is tested in this study by an examination of a time series
of returns of optionstock hedges that theoretically have
zero systematic risk. The hedge returns, which empirically
did not appear to evidence significant systematic risk,
did behave in a manner consistent with the theory of the
riskreturn intercept change. However, the change in the
hedge returns was not statistically significant.
CHAPTER I
INTRODUCTION
The expansion of new option trading since April, 1973,
has been rapid and still continues. The purpose of this
study is to examine a potential effect that the expansion
may have had on the capital markets. The hypothesis is
that new option trading causes a change in the intercept
location of the theorized riskreturn relation. The
theoretical framework used for the development of the
hypothesis is the meanvariance capital asset pricing model
(MVCAPM).
The bases for the theory that options cause a change
in the intercept of the riskreturn relation are two alleged
circumstances (to be analyzed later): (a) options may be
employed by some traders to obtain leverage amounts that
are not possible through normal margin channels; and (b)
in a singleperiod, equilibriumtheoretic model where mar
gin constraints are effective, the introduction of new
options may cause a shift in the intercept of the riskreturn
relation. The first of these observations will be examined
in this chapter; the second observation is considered in
Chapter II. Chapter III is a discussion of other empirical
work that is potentially related to this study. In Chapter
IV a methodology for empirically testing the hypothesis is
described in detail. The results of our own tests are
presented in Chapter V. Chapter VI concludes the study by
reviewing the major results and suggesting various implica
tions and limitations. First, the nature of options is
discussed.
Futures and Options
In a treatise by Bachelier (1900), translated from the
French by Boness and included in Cootner's (1964) book,
options are described as being futures contracts but with
a limited liability feature for one of the contract parties.
Trading in a futures contract is similar to spot trading
in the commodity, except that in a futures contract the
exchange of money for the commodity is made at a specified
later date. The price at which the exchange is to be made
is agreed upon when the contract originates, however, and
not at the subsequent time of exchange. An option is
similar to a futures contract except that one of the ex
change parties has the option to cancel the future trade,
and payment for this privilege is made at the time the con
tract originates.
In all futures contracts there is a seller, who con
tracts to deliver the commodity for money, and a buyer,
who contracts to delivery money for the commodity. If the
buyer has an option to cancel the trade, then it is said
that the buyer owns a call option. In this case the seller
of the commodity is the writer of the call option. If the
commodity seller has the option to cancel, then it is said
that the commodity seller owns a put option. In this case
the commodity buyer is the writer of the put.
The above description of options applies to "European"
options. Alternatively a call or put option may be an
"American" option. An American option is the same as a
European option except in an American option the option
owner may call for the exchange (still at the agreed price)
anytime prior to the original settlement date. Since the
settlement date is the final time at which an option owner
may announce his option, this date is termed the expiration
date of the option.
It is from these characteristics that the following
definitions of call and put options have evolved: A call
option is a contract giving its owner the right to buy a
commodity at a specified price, at or before a specified
time (depending on whether it is European or American);
a put option is a contract giving its owner the right to
sell a commodity at a specified price, at or before a
specified time. In either case the right of contract is
granted by the writer of the option. If an option contract
is held beyond the expiration time, it becomes worthless.
In the case of either calls or puts, the option owner
must pay to the writer an amount for the limited liability
privilege; and in the case of an American option, the
option owner must sometimes pay to the writer an additional
amount for the right to activate an early exchange. Merton
(1973b) has argued that in an ideal market, this additional
payment for the early exercise privilege should be zero, if
either (a) the underlying commodity pays no dividends prior
to expiration time, or (b) the futures (settlement) price is
automatically adjusted for any dividend payments. In the
U.S. market for exchange listed common stock options there
is no dividend protection, and since many underlying stocks
do pay dividends, the early exercise privilege theoretically
has some value to the owners of some options traded on the
U.S. exchanges.2
In the case of futures contracts without options the
only thing to be negotiated is the future exchange price;
for options there are two amounts to be negotiated simul
taneously: (a) the future exchange price, and (b) the amount
that the holder of the option will give to the writer for
the limited liability and early exercise features. In the
present U.S. exchange listed option market, the future ex
change price for stock in dollars is contractstandardized.
What is negotiated is an amount called the premium. Thus,
the premium is consideration for three amounts: (a) the
price of the limited liability feature; (b) the price of
the early exchange feature; and (c) the difference between
what would have been the negotiated futures settlement price
and the contractstandardized settlement price.3
5
Use of Options in the U.S.
Evolution Prior to 1973
Although put and call options originated outside of
the U.S., the use of put and call options in the U.S. was
4
begun by financier Russell Sage in 1869. Sage and others
were in business to lend money to brokers, who would use
the borrowed funds (termed margin credit) to purchase
stocks. Experience showed that the brokers could not always
repay the loans when due, but it was impossible for the
lenders to charge enough interest to compensate for this
risk because of usury laws. Consequently, Sage devised a
way to employ call and put options to conduct his business,
and other lenders soon adopted Sage's method. Instead of
extending margin credit to a broker, the lender would write
a call option, and the broker would become the call option
owner. As part of this arrangement the broker would write
a put option back to the lender. Thus if the stock declined
in price, the lender could exercise his put option and sell
the stock to the broker at the put contract price. If the
broker could not pay, then this was the lender's loss. But
the lenders could charge what they wanted for the call
options, and presumably they charged enough to compensate
for the possibility of nonrepayment.5
The put and call business operated along these lines
until Sage died in 1906. Sage met all of his contractual
obligations, but his successors would, from time to time,
dishonor the contracts if the market went against them as
sellers. These reneges caused great resentment among
option traders, and the business was plagued by lawsuits
until the Securities and Exchange Act of 19331934. At
that time various dealers in puts and calls formed' the
Put and Call Brokers and Dealers Association, Inc. This
association brought two vital attributes to the business:
(1) uniform contracts and (2) endorsement by New York
Stock Exchange members.
The new market in options provided by the Puts and
Calls Brokers and Dealers Association extended option
trading to individuals as well as brokers. Individual
investors were permitted to own or write puts and calls
separately. However, the market involved the direct
matching of owners and writers. The large transaction
costs of the matching procedure for the separated contracts
provided the impetus for the first option exchange in 1973.
The Post 1973 Option Market
The first U.S. option exchange created for the purpose
of facilitating option trading was the Chicago Board Options
Exchange (CBOE). Other option exchanges subsequently began
operations following the unexpectedly huge success of the
CBOE. The activities of all option exchanges are presently
coordinated by the Options Clearing Corporation, which also
serves as guarantor to individuals on both writing and
owning sides of option contracts.
Exchange listed options are now standardized in the
following respects. Exchange listed option contracts are
for 100 shares. Premiums are specified in dollars per
option on one share, so that a contract quote of $4 would
mean that the buyer would pay a premium of $400 for the
standard contract on 100 shares. The standardized option
contracts which are traded on the organized exchanges are
quoted by the month of expiration and by the price at which
the future stock transaction will take placethe exercise
price. At present, these contracts expire at 11:50 p.m.
Eastern time on the Saturday following the third Friday of
the expiration month. The cutofftime for individual in
vestors to instruct brokers concerning exercise is 5:30 p.m.
Eastern time on the business day immediately preceding the
expiration time.
Example: An American Telephone and Telegraph Company
(AT&T) January 50 that is traded on the
Chicago Board Options Exchange (CBOE) is
an American call option contract to buy 100
shares of AT&T corporate common stock. The
price at which the 100 shares of stock may
be bought is $50 per share, and the stock
may be bought any time before the January
expiration time. If the price of a share
of AT&T exceeds $50 just prior to the
cutofftime, then it would be profitable
for the call owner to exercise the option
and resell the shares, assuming no trans
action costs. Net trading profit depends
upon the original cost of contract (the
premium).
If the price of a share of AT&T is
below $50 per share near the cutofftime,
then it would make no sense to exercise the
option and pay $50 per share, when the same
share could be purchased for a lower price
without an option. The trader in this case
should let the contract expire worthless.
The net loss will be the original purchase
price, or premium.
An AT&T January 50 P is a put contract
with the same specification, except a put
contract is a contract to sell 100 shares.
With puts, the trader should exercise if
the stock price, at cutofftime, is below
the exercise price. The trader will make a
net profit, if the stock price is below the
exercise price by more than the original
premium paid. If the stock price is above
the exercise price at cutofftime, the
trader should discard the worthless put.
End of Example.
The option market is organized in such a way that
individual investors can take profits without exercising
their options at the expiration time; the option owner, at
the expiration time, simply sells the contract back to a
writer for exactly the difference between the stock price
and the exercise price. This difference is the owner's
gross profit or loss. Whatever is the owner's gross profit
is the writer's loss, and vice versa.
Because of the efficiency of the options exchange there
is usually good liquidity in the sense that an option con
tract, once originated, can easily be resold to another
trader anytime prior to the expiration time. The considera
tion in any purchasesale of an option contract, the premium,
is negotiated virtually continuously according to supply and
demand. Thus option traders can incur profits and losses by
buying and selling option contracts over any time intervals,
without ever seriously considering taking delivery of the
stock.
The prices of the last contracts traded of each option
type on each stock are reported daily in financial and
metropolitan newspapers. Presently, transactions costs are
not included in the newspaper trade quotes. Transaction
costs vary with the size of the transaction in both dollar
and volume terms. Brokerage commission schedules are com
petitive and will vary from broker to broker. Transaction
costs are significant to nonexchange members who want to
trade a few contracts; however, transaction costs are less
significant to exchange members. The consequences of
present tax structures for studies of option trading are
complex to analyze, because there are different tax rates
for different investors. More detail on the arrangements
for common stock option trading may be obtained by consulting
Gastineau (1975), Golden (1975), the Chicago Board Options
Exchange Prospectus (1973), and the Options Clearing Cor
poration Prospectus (various dates).
The Moratorium and After
Trading volume in contracts of each underlying stock
and the number of underlying stocks had expanded rapidly
and steadily until July 1977, when the Securities and Ex
change Commission (SEC) halted the further expansion of
options trading on additional underlying securities. At
the time of the moratorium, call option markets on common
shares of 235 different companies were open, and put option
markets on 25 different companies were open. Volume of
underlying shares represented by option trading continued
to increase after the moratorium on expansion was imposed
by the SEC.
The purpose of the moratorium was to give the SEC a
chance to examine the effects of option trading and to
review the trading and selfregulating practices on the
exchanges. The result of the SEC study was The Report of
the Special Study of the Options Markets to the Securities
and Exchange Commission (hereafter called the SEC Study),
which was dated December 22, 1978.
In general the SEC Study (p. v) found that "options
can provide useful alternative investment strategies to
those who understand the complexities and risks of options
trading. But, since regulatory inadequacies in the options
markets have been found, the Options Study is making
specific recommendations needed to improve the regulatory
framework within which listed options trading occurs and to
increase the protection of public customers."
In 1980 the moratorium was lifted, and in May of 1980
the list of stocks underlying listed put options expanded
from the original 25 to 105. In July, 1980, the list of
new stocks underlying call and put options also began to
expand again.
Exchange listed options on physical commodities,
foreign currencies, or other securities (such as treasury
bills, bonds, mutual funds, and futures contracts) are not
presently traded in the U.S. However, options on physical
commodities are traded in other countries, including gold
futures options on the Winnepeg, Canada Exchange.
Reasons for Listed Stock Option Usage
Various reasons have been offered about why investors
and speculators trade options. Uses of options are ex
plained in passages and chapters of various investments
textbooks Cespecially in the more descriptive textbooks)
and are more thoroughly described in some of the less
7
technical books about options. Some typical rationales
for the use of options by individual (nonbroker)
investors are developed below. The interested reader is
advised to consult the sources in note 7 for more details.
Leverage. The use of borrowed funds to purchase in
vestments is referred to as financial leverage, or more
simply leverage. The use of puts and calls by brokers as
an alternative to margin arrangements has already been
described.
In modern markets margin credit may also be employed
by individuals, who borrow the funds from brokers or banks.
For individuals, the maximum amount of credit that can be
employed to purchase stocks is set by Regulations T and U
of the Federal Reserve. CCurrently the "margin ceiling"
is 50%, meaning no more than 50% of a stock's purchase
price may be paid with borrowed funds. This ceiling has
ranged from a low of 40% to a high of 100% in the last 40
years.) Investors who employ option contracts for leverage
instead of margin credit have three distinct advantages:
(1) Since an option contract premium will often be between
1% and 50% of the cost of the stock shares, option contracts
represent much higher degrees of leverage than are available
through the use of margin. (2) Option contracts offer a
limited liability feature which prevents loss of more than
the original premium paid. Such limited liability is not
available in margin trading. The limited liability feature
of knowing one's maximum loss in advance may be an induce
ment to individuals to employ the very high leverage. (3)
Standard brokerage policy requires a minimum equity of
$2,000 for margin trading in common stock. Thus option
trading may bring into the market some new investors, who
individually may have relatively small amounts of capital.
Hedging. If a stockholder becomes uncertain about the
future price volatility of a stock, or if the stockholder
anticipates a price decline, writing (selling) call options
would give the investor a position that is hedged against
unfavorable changes in those variables. The investor can
use his stock as collateral, thus temporarily changing the
strategy of his position, without incurring the transaction
costs of selling and buying stocks and without foregoing
dividends.
Income. Although it is theoretically possible for
investors to sell shares for current income purposes,
this practice is not normal. The costs of transactions
and the possibility of deferring taxes work to discourage
share turnover. One way for stock owners to receive cur
rent income is to write call options utilizing stock shares
as collateral. This technique is also useful for managers
of large institutional portfolios, for whom the sale of
shares is difficult without creating a significant stock
price decline, due to the sheer size of the average trans
action.
According to Paul Sarnoff C1968), a former options
broker for many years, the scenarios above describe the
primary nonbroker uses for options. Thus, in general,
option buyers are thought of as speculators who employ
options because there is more leverage than otherwise avail
able in the stock market (with limited liability), and who
are tolerant of risk and/or possess special information.
Option writers are primarily portfolio managers who desire
protection against large price drops, especially during
times of income need. Sarnoff's beliefs about why options
are utilized correspond to the evidence reported next.
Evidence of Reasons for Recent Option Trading
The SEC Study identified three categories of partici
pants in the options markets: (1) public nonprofessionals,
(2) professional money managers, and (3) professional traders
and arbitrageurs. The SEC Study also identified the basic
purposes served by the various common types of options
transactions. The basic purposes are: (a) to obtain
leverage, (b) to hedge positions in the underlying security,
(c) to increase current income from securities holdings,
(d) to arbitrage for profit, (e) to speculate or trade on
perceived overandundervalued situations, and Cf) to
facilitate the provision of brokerage and marketmaking
services in the underlying stocks.
The SEC Study (pp. 106107) describes the varying
perspectives of investors as they approach the market:
Traders, for example, attempt to capitalize
on undervalued and overvalued situations by using
complex mathematical models and computer techniques
to detect and arbitrage against perceived illogi
cal divergences in prices. Studies of option price
patterns, however, indicate that while price
divergences do occur which may provide profitable
trading opportunities for professionals the
divergences generally are too small for trading
opportunities by members of the public because of
transaction costs. Other, generally sophisticated,
investors perceive an opportunity to adjust the
riskreward mix of their portfolio of assets in a
more precise manner because of the additional com
binations of risk and potential return opened up
to them by the availability of exchange traded
options.
S Risk management and risk adjusted performance
have become basic criteria upon which professional
managerial ability is evaluated. Most individual
investors in options, however, are probably using
option purchases and sales as a substitute for
stock purchases and sales. Dealing in options
enables them to take shortterm positions in the
stock, or shift out of the stock in the shortterm
with lower transactions costs; and, for buyers,
it offers greater leverage than would be the case
if they were trading directly in the underlying
securities. (SEC Study, p. 107)
A survey released in 1976 and conducted by Louis Harris
Associates (1976) for the American Stock Exchange listed 10
strategies that appear to be most commonly employed by in
vestors. The 10 strategies listed by Harris are shown be
low:
Buying
1. Buying options in combination with stock ownership.
2. Buying options in combination with fixedincome
securities.
3. "Pure" buying of options without underlying stock
or fixedincome securities.
Mixed Strategies
4. Buying options against a short position in under
lying stock.
5. Buying options as a hedge against a short position
in securities related to the underlying security.
6. Selling options hedged against other related
securities.
7. Spreading options by buying and selling different
options in the same underlying securities.
Selling
8. Selling fully covered options.
9. Selling partially covered options.
10. Selling completely uncovered options.
The Harris survey found that among individual investors,
the largest percentage (58%) employed the pure buying of
options strategy (#3 above). Of the persons investing a
total of $2,500 or less, 49% employed the pure option buying
strategy. In contrast to individual investors, 79% of the
institutional investors surveyed concentrated their activities
in fully covered option writing strategies. Another survey,
by Robbins, Stobaugh, Sterling, and Howe (1979), sponsored
by the CBOE, also found that the two strategies followed
most frequently by investors were the simple buying and
covered writing of option contract strategies. The SEC
Study CP. 116) pointed out that: "Neither survey included
interviews with brokerdealers, a professional, but extremely
important group, using options in their activities. Block
positioning firms, marketmakers and other brokerdealers
make extensive use of options in providing dealer services
to the public market."
From the foregoing discussion it appears reasonable
to hypothesize that Ca) margin constraints are effective
in the U.S., and (b) one rationale for option owning is
that options offer a viable alternative to margin trading
as a means for more investors to obtain more leverage.
Overview of the Study
In the next chapter the effects of introducing options
into a theoretical model of risk and return are analyzed.
One assumption of the model is an effective margin con
straint. It is found in the model that as more options
are introduced, the equilibrium riskreturn relation is
altered; specifically, the expected return on securities
with zero systematic risk will decline toward the riskless
rate. This result is not surprising in light of the results
of Sharpe (1964), Lintner (1965), Black (1972), Fama (1976),
and Vasicek (1971), who originated and contributed heavily
to the construction of the theoretical model. Further de
tails are deferred to Chapter II.
In Chapter IV the methodology is developed to test the
hypothetical effect of options on the riskreturn relation.
The reader may be interested in a brief overview of the
procedure to be employed:
First are identified securities which are logical
candidates for a zero systematic risk portfolio. The rate
of return on a zero systematic risk securities portfolio
will be referred to as ECr ). Estimates of that expected
rate are denoted .
The time span of a time series of ra values covers an
extended period during which no new options began trading
on exchanges (Period 1) and another extended period during
which an ample number of new options began trading (Period 2).
During Period 1 the time series of observations for r
may be considered as estimates of an equilibrium rate,
assuming there are no changes in market equilibrium condi
tions. During Period 2 the time series observations of r
must be viewed, in light of the hypothesis, as including
two portions: (1) the new lower equilibrium rate of return
for zero systematic risk portfolios, and (2) the return
associated with the transition from the old equilibrium
state to the new one, following the theorized effect of the
new options.
The second portion may be significantly higher than
either of the equilibrium rates as is evident from the
following example: Assume that prior to the introduction
of new option trading the equilibrium rate of return on
a zero systematic risk asset is .08. Now suppose the
introduction of new option trading causes the equilibrium
rate for the zero systematic risk asset to decline to .06.
In order for this to occur, the price of the zero systema
tic risk portfolio must increase by 331/3%, say from 100
to 1331/3. The observed transition rate of return would
show up in the Period 2 time series and be relatively high
indeed.
This scenario establishes (qualitatively) what should
be expected from the empirical analysis if the following
assumptions are valid: (1) the theoretical framework
employed in Chapter II is valid; (2) other factors that
may affect E(r ) are properly accounted for; and (3) new
option trading has a significant enough impact to be ob
served.
In order to accomplish the purposes of the empirical
analysis, these general procedures are to be employed:
(1) The time series of observations of F are con
verted into "excess return" form by subtracting corresponding
observations of rf. This adjustment represents a method of
accounting for exogeneous shifts in the location of the risk
free rate, and simultaneously of the level of r Such
shifts could result from federal influence on interest rates,
for example. Whatever the exogeneous sources of disturbance,
it is the relative distance between r and rf values that is
being measured in this study, so the conversion to excess
returns is appropriate. This excess return variable is
e
referred to as r. Thus, abnormally high observations of
e re
r are expected to be found in Period 2 relative to Period 1.
(2) In order to gauge the significance of the values
e
of rE in Period 2 relative to Period 1, the following
statistical procedure is employed. The observations of
e
re are used as dependent variable observations in a multiple
regression on two variables called I1 and 12. The first
variable, Ii, takes on the value 1 for all observation
periods in Period 1 and Period 2. The second variable, 12,
takes on the value 1 for all observation periods in Period 2,
but has the value 0 for all observation periods in Period 1.
The resultant regression coefficient for II, will be the
mean of the e time series during Period The resultant
mean of the r. time series during Period 1. The resultant
regression coefficient of the variable 12 is the increase
e
in the mean of the r values from Period 1 to Period 2.
Thus, the tstatistic of the regression coefficient of I2
can be used to judge whether the values of the observations
e
of r are significantly higher in Period 2 than in Period 1.
(3) The possibility exists that equilibrium shifts
will occur as a result of factors other than new option
trading, such as changes in expected inflation or in multi
period preferences. One indication of this possibility is
for the residuals in the regression to exhibit serial
correlation. If this correlation occurs, then the meaning
fulness of the regression is in question. A wellknown
statistical procedure (CochraneOrcutt) may be employed
to counteract serial correlation in the residuals and what
ever effects this correlation may have on the tstatistic
of the coefficient of 12.
The next chapter presents the theoretical basis for
the hypothesis.
Notes
There is only one delivery date, which is decided upon
at the time the contract originates. The delivery date is
also called the settlement date.
2
Empirical examinations of option pricing models in
dicate that models which adjust for dividends are more valid
than option models with no dividend adjustment. See, for
example, Galai (1977) and Chiras (1977). Option pricing
models are discussed in Chapter IV.
For example, consider a call option, and let the prices
of the limited liability feature and the early exercise
feature be $.50 and $.25, respectively. Assume the contract
standardized settlement price is $50.00, but that two traders
would have preferred to negotiate a futures price of $52.00.
The buyer of the call option must pay $2.00 more for the
contract than he would have if the settlement price were
$52.00. Thus the call contract premium should be $2.00 +
.50 + .25 = 2.75 in this case.
This account of option trading in the U.S. prior to
1973 is paraphrased from Sarnoff (1968).
As long as risk conditions dictated that lenders re
quired a rate of interest less than the usury law ceiling,
lenders could have used either the option method or the
direct lending method. In order to obtain higher "interest"
compensation if the option method was chosen, lenders would
build this compensation into a higher price of the call
option. For example, a call option would sell for more if
conditions dictated 10% interest than if conditions dictated
6%.
See Phillips and Smith C1980) for a discussion of
transactions costs of options exchanges.
Examples of less technical investments textbooks which
discuss option strategies are Johnson (1978), Mendelson and
Robbins (1976), and Wright (1977). The more recent the text,
the more detail about options can usually be found. Recent
technical investments texts, like Sharpe (1978) and Francis
(1976), usually have a formal chapter about stock options.
Examples of nontechnical books about options include Clasing
(1975), Cloonan (1973), Dadekian (1968), Filer (1966), and
Sarnoff (1968). In addition, Malkiel and Quandt (1969) is
written for a wider audience than only options experts and
is detailed in its explanation of various option trading
strategies.
22
An interesting example of the use of options by bro
kers to arbitrage for profit and acquire capital appeared in
an article of the Wall Street Journal (August 7, 1980, p. 30).
Brokers went long 1000 calls and 1000 puts on the shares of
Tandy Corporation. Since the prices were $9 per call and $3
per put, the "spread" was $6. Brokers are required to outlay
only the amount of the spread for this position. The brokers
then sold 1000 shares of the stock short at $66 to create a
perfect hedge position that would return $6 per share, or
$600.00. The brokers could then use the short sale proceeds
to invest elsewhere. Since the short sale proceeds were
$6.6 million, brokers had $6.0 million in "free" capital,
upon which no interest had to be paid. In this case the
brokers were exploiting a riskless arbitrage situation.
CHAPTER II
OPTIONS IN A SINGLE PERIOD
CAPITAL MARKET EQUILIBRIUM FRAMEWORK
This chapter introduces options into the analysis of a
single period capital market model. Equilibrium conditions
are derived in the usual fashion, except that investors are
permitted to hold options in their portfolios as well as
stocks. A primary distinction between stocks and options in
this context is that options are not issued by corporations,
whereas stocks are. The aggregate market value of all
option positions in the market is zero; for every investor
who owns an option, there is another investor who is short
an option. The aggregate market value of each stock in the
market is not constrained, but must be positive, of course,
to be realistic.
Three circumstances will be examined: (i) the case
where no riskless security exists; (ii) the case where
investors can go long or short the riskless security in any
amount; (iii) the case of margin restrictions on trading
the riskless security. The riskless security is defined to
be a security with a fixed and knowninadvance nominal
return. As with options, the aggregate holdings of the
riskless security are presumed to be zero. In other words,
the economy as a whole cannot have an excess of borrowing
over lending or an excess of lending over borrowing.
The models to be derived are single period equilibrium
models based upon the meanvariance criterion of Markowitz
(1959). In addition, it will be assumed that (a) all
investors have identical probability beliefs;1 (b) all
investors make their portfolio decisions at the same dis
crete points in time; (c) the market is perfectly void of
indivisibilities, taxes, transactions costs, and monopoly
influence by any investor; and (d) no consumption price
inflation exists.2 Thus, this chapter applies an elementary
SharpeLintner CAPM framework to a setting that includes
options.
The analyses in the three sections of the chapter
correspond to three distinct circumstances mentioned earlier
in connection with the trading of the riskless security.
In all three sections only one option is assumed at first.
Generalization is subsequently made to a portfolio of
options with the following result: if the riskless security
is either unavailable or restricted (and if investors are
constrained by the latter circumstance), then the equilib
rium expected rate of return on zero systematic risk secu
rities declines as the number of options in the market
increases.
In the proofs to follow investors are assumed to employ
in their portfolios any of n risky assets (stocks), the
option3 and, when specified, a riskless security. The
random rates of return on the assets are denoted rl, r2, ..,
rn, while the rates of return on the option and the riskless
security are denoted by ro (a random variable) and rf, res
pectively. Similarly, the proportionate investments by an
individual into each of the n risky assets are denoted by
X1, x2, .., Xn; xo and Xf denote the proportionate holdings
of the option and the riskless security, respectively. For
any investor the total of his portfolio proportions must be
equal to 1, i.e.,
n
xo + Xf + E xi = 1. (1)
i=l
By definition, an investor's portfolio expected rate of
return is given by:
n
E(r) = xiE(ri) + xoE(ro) + xfrf. (2)
i=l
An investor's portfolio variance is given by:
2 n n n
2 = xx2cov(ri + 2x x cov(ro,ri). (3)
i=lj=1 i=1
The above relationships will be employed in all three
of the sections to follow.
Equilibrium with One Option and No Riskless Security
In the first case to be considered, xf is constrained
to be zero, since trading in the riskless security is not
permitted. Under this circumstance, the only securities
available for trading are the n risky assets and the option.
Substituting xf = 0 into equation (1) and rearranging, we
find that:
n
xo = 1 xi, (4)
i=l
if there is no riskless security. Now equations (2) and (3)
can be applied to the no riskless security case by the sub
stitution for x from equation (4) to get (a superscript k
has been added to denote the kth investor):
k n k n k
E = xiE(ri) + (1 Z xi)E(r0); (5)
i=l i=l
and
k 2 n n k k n k 2 2
(a ) = x x.jcov(r ,r) + (1 x.) a
1 i=lj= 1 i=l1
n k n k
+ 2(1 Z x.) Z x.cov(r ,r ), (6)
i=l I i=l1 o
where the numerical subscripts in the terms to the left of
the equal signs in equations (5) and (6) are references to
the section of the chapter.
Thus the investor's portfolio problem of minimizing
variance for each level of return can be solved by mini
mizing the following Lagrange function constructed out of
equations (5) and (6):
k k 2 kk n n k
L = (or ) + X [E(r) x1E(ri) (1 Z xi)E(ro) (6a)
1 1 i=1 i i=1
where Xk is the Lagrange multiplier for investor k.4
Substitute equation (6) for the expression (a )2 in equa
k r
tion (6a), differentiate L1 with respect to each of the n
portfolio weights of the n risky assets, and set the
derivatives equal to zero. The result is equation system
(7.1) through (7.n) below:
dLk nk
= 2[ E x.cov(r ,rj)
k 1=1 k
dx n
+ (1 E x
i=l
n k 2
(1 Z x.)o
i=l 1
k n k
i)cov(r ,r1) jcov(ro,rj)]
o 1 j=1
 k[E(rl) E(ro)] = 0;
k
dL n
1 = 21 E x.cov(r ,rj)
Sk j=1
dx2 n
+ (1 Z x
i=l
n
(1 Z xk)o2
i=l
k n k
.)cov(r ,r2) x cov(r ,r.)]
1 j=l j o j
 [E(r2) E(ro)] = 0
(7.2)
k
dL1 n k n k 2
S 2[ Z x.cov(r,rj) (1 xi)o
Sj=li=l o
dx n k n k
n + (1 xi)cov(ro,rn) Z x cov(r,r)]
i=l j=1 j 0
[E ) E(r )] =
 X [E(r ) E(r)] = 0
(7.1)
0. (7.n)
It is from equations (7.1) through (7.n) that market
equilibrium conditions may be derived as follows:
It is useful to first expand equation (7.1) as (7.1a)
below:
k k k
2[(x1cov(r,r) + x2cov(r,r) + ... + xnov(rr
k k k2 2n
k2cov(r(,r7)
(1 xk x ... x
1 2 n o
 xcov(rr) x kcov(r,r2) ... xncov(r,r (7.la)
k k k
+ (1 I X  .. x )cov(ro,rl)]
k[E(rl) E(ro)] = 0.
Assume there are a total of P investors. Let w be
the proportion of total market wealth represented by the kth
investor's wealth. Multiply equation (7.1a) by wk for each
of the investors to get:
2w [x1cov(rl,rl) + x2cov(rl,r2) + ... + xncov(r,r)
k k k k 2
2w [1 x x2  x ] (7.1b)
kk k k
2w [x cov(ro,rl) + x cov(r ,r2) + ... + x cov(ro,rn)]
wk k k k wkk[
+ 2w [1 xl x2 ... ]cov(r r ro,rl) w E(r ) E(r0)]
= 0.
Next sum the P weighted first order conditions over all
investors in the market to get an aggregated version of the
first first order condition:
P P
P k k P k k
2cov(rl,rl) Z w x + 2cov(rl,r2) w 2+ ...
k=l k=l
Pk k
+ 2cov(rlrn) w n
k=l
Pk P kk P P k k k 2
2[ Z w w x I wx ... X w x
k=l k=l k=l 2 k=l o
P k P k (7.1c)
2[cov(r ,rl) Z wkx1 + cov(r ,r2) wx + .. 7.
k=l k=l
k k
+ cov(ro,rn) w x
o n k=l
2ck P k kk k P k k
+2cov(r rl)[ Z w Z w xI w x21 ... Z w Xn
k=l k=l k=l k=l
k k
[E(rl) E(r )] w = 0.
k=l
P P
By definition, wk= 1. Also, Z w xl is the propor
k=l k=l
tionate weight of the first security in the market port
k k
folio. Similarly, Z w x2 is the weight of the second
k=l
security in the market portfolio; and so on for all risky
assets. Let these weights be denoted xm, x2, ..., xm
Thus, the terms in the first and third brackets of equation
(7.1c) are both equal to:
m m m
1 x x2 x
which is equal to zero. Make these substitutions into equa
tion (7.1c) to obtain:
2cov(rl,rl)xm + 2cov(rl,r2)x2 + ... + 2cov(rl,rn)x
2cov(r,rl)xm 2cov(ro,r)x ... 2cov(r,rn)xm
P kk
[E(rl) E(r )] Z w X = 0. (7.1d)
Sk=l
P
Define Am = wkk Thus Xm is a weighted average
k=l
of the individuals' Lagrange multipliers, where the weights
are proportions of aggregate wealth. By substituting Xm
into equation (7.1d) and collecting terms, we get:
n n m m
2 Z x cov(rl,r.) 2 Z x cov(r ,r) [E(r) E(ro)]
j=l j j=l 3 o j 1 0
= 0 (7.1e)
Now divide (7.1e) by 2 and rearrange terms to get:
n n m
E x.cov(rl,r) x.cov(r ,r = [E(r) E(r )]
j=l i' j=l 3 J 2 1o "
(7.1f)
Equation (7.1f) is the result of aggregating the first
of the first order conditions across all investors. The
other first order conditions may be similarly aggregated
to obtain the following equation system:
n m n m m
Z x.cov(r ,r.) x.cov(r r) [E(r)E(r)]
j=1 1 I j= 1 2 0
m
n m n m
Z x cov(r ,rj) Z x.cov(r rj) [E(r )E(r )];
j=12 Ij=l o3 2 2
n
x .cov(r ,r )
j=l n
In order to
m
n m A
 xjcov(r,r) = [E(r )E(r,)],
j=1l 2 o
complete the proof the reader should
(8.n)
recognize the following relations:
n n
m m
Z x.cov(ri,r) = cov(r., Z x.r ) = cov(r ,r )
j=l j=l 3 m
(9)
m m m
where r = x1r + x 2r + ... + x r the return on the
market portfolio (of risky assets only).
Substitute these relations into the aggregate equations
(8.1) through (8.n) to obtain the following equation
system:
cov(rl,rm) cov(ro,rm) = [E(rl) E(ro)];
2
m
cov(r2,r ) cov(rorm) [E(r2 E(r )];
2) r m 2 2o)
m
cov(rh,r ) cov(ro,r ) [E(r ) E(r )].
n m o m 2 o
(10.1)
(10.2)
(10.n)
(8.1)
(8.2)
To eliminate the unknown Am/2 factor from the equation
system, first multiply (10.1) by x then (10.2) by x, and
so forth.
x1cov(rl
These multiplications result in:
m
,r) xmcov(ro,r) = E(r x o
m 1 0 m 2 1 1 E(1
(11.1)
x2E(r )];
xm cov(rr
2 2 rm),r, x 2 cov~ro~rm)
2 [x2E(r2)
2
m ov(rnrm) m m
xcv(r ,r) xmcov(r,rm 2 [XnE(rn) xnE(ro )
(11.n)
Add the n equations (11.1) through (11.n) together to
obtain:
n n
Z x.cov(r,r ) cov(r ,r ) i x.
i=l i i o m
m n mn m
[ x.E(ri) E(r ) E xi
2 i= 1 1 i=
n
m
Using equation (9) and the fact that Z x. = 1,
i=l 1
equation (12) may be simplified to:
Am
cov(r ,r) cov(r ) [E(r ) E(r )]
m m o m 2 m o
(12)
(13)
(11.2)
Since cov(rm,rm) in equation (13) is simply am, the
variance of the market portfolio, equation (13) may be
rearranged to yield:
A U2 cov(ro,rm)
= m14)
2 E(rm) E(ro)
Now substitute the results for Am/2 in equation (14)
into each of the equations (11.1) through (11l.n). For any
security, i, the market equilibrium relation for returns is:
2 cov(r ,rm)
cov(ricov( ro,,rm) = m )o r E(r )E(r ) (15)
E(rm)E(ro)
which can be rearranged into more familiar form:
cov(ri,r ) cov(ro,r )
E(r) = E(ro) + cov( [E(r )E(ro)].(16)
a cov(r rm)
m m
Equation (16) represents the market equilibrium
relation that must hold for all securities if a single
option and n risky assets (but no riskless security) are
available for trading by investors.
Equilibrium with One Option and Unrestricted Trading
in the Riskless Security
In this section a riskless security is introduced into
the framework. It is assumed that there are no constraints
on the level of holding of the riskless security for any
investor. Therefore, equations (1), (2) and (3) hold in
full. From equation (1) it is known that for any investor:
n
x = 1 xi x. (17)
o i=1 f
Substitute the relation above into equations (2) and (3),
the expressions for an investor's portfolio mean and
variance, to obtain:
n n
E (r) = x E(r ) + (1 x xf)E(r ) + xfrf; (18)
i=l 1 i=l o
2 n n n 2
a Z E xixcov(ri,rj) + (1 x Xf) o
r2 i=lj=l i=l o (19)
n n
+ 2(1 E x. Xf) Z x.cov(r ,ri).
i=l i=l
Thus the investor's portfolio problem of minimizing
variance for each level of return can be solved by mini
mizing the following Lagrange function:
2 n n
L = + [E2 (r) Z x E(r) (1 Z xi )E(r )x fr
L2 i=l i= 1f
(20)
Substitute expression (19) into expression (20).
Differentiating L2 in (20) with respect to each of the port
folio weights of the first n risky securities and setting
the derivatives equal to zero, we get:
dL2 n n 2
2[ E x.cov(r,rj) (1 Z x xp)
dxI j= i=l
n
(1 Z xixf)cov(ro,r)
i=1
n
i x.cov(r rj)]
j=1J J
 X[E(rl) E(r)] = 0;
n n 2
S2[ Z x.cov(r2r) (1 xx ) +
j=1 i=l f
n n
(1 x,xf)cov(r r ) E x.cov(r ,r.)]
i=l I o 2 j=1 3 o
 X[E(r2) E(ro)] = 0;
dL n n 2
2= 2[ x.cov(r,r) (1 x.x) +
dx j=1 i=l f 0
n
n n
(1 Z xix )cov(r ,r ) Z x.cov(r ,r.)]
i=l n j= J
 X[E(rn) E(ro)] = 0.
nO
(21.1)
dL2
dx2
(21.2)
(21.n)
Next take the derivative of L2 in equation (20) with
respect to xf and set equal to zero to get:
dL n 2 n
 2= 2[(1 Z x x )a + Z x cov(r ,r )] + A[E(r )rJ =0.
dxf i=l 1 f o i=l o
(22)
Each investor has an equation set (21.1) through (21.n)
and (22). Each equation in the set may be aggregated over
all investors to obtain the following aggregate demand
relations:
n m n m m 2 n m m
Z x.cov(r ,r) ( Z x.x )a + (1 Z x )cov(r ,r)
j=l J j i=l 1 f o i=l 1 f 1
j=1 i=1 i=1
(23.1)
n n n
jZ xcov(r,rj) = r [E(rl)E(ro)]'
n m nm m 2 n m
Z x cov(r ,r) (1 x fx )o + (1 Z x. x)cov(r ,r)
j=l 2 j i=l i f 0 i=l f o 2
(23.2)
n m m
Z x cov(r ,rj) = [E(r2)E(ro)];
j=1 J 2
n m m m 2 n m m
Z x.cov(r ,r ) (1 Z x xf ) + (1 x. x )cov(rr n)
j=1 J n J i=l1 i=l 1 f n
(23.n)
m
Z xcov(r ,r.) = E(r )E(ro)]
j=l J 1 2
where xm is the weighted sum of the individual investors'
1 p
proportions for the ith stock (i.e., x. = wkxk, where
1 k=l 1
wk is the kth investor's proportion of total market wealth).
Thus x. is the proportion of the wealth of the market port
1
folio represented by the ith asset. The term xm is a
similarly weighted sum of the individual investors' propor
tions for the riskless security.
By the use of a similar technique, equation (22) may be
aggregated over all investors to obtain:
n n m m
(1 Z xix)m 2 + E xcov(r,r) = [E(r )r (24)
i=l i=l 2
n
Of course, xf = 1 Z xi = 0. Using that fact, along
i=l
with equation (24), in simplifying equations (23.1) through
(23.n) we get:
m
n m n m
j x cov(rl,r.) i xcov(ro,r ) [E(r)E(r )
j=1 1 2 1
(25.1)
m
n n m
Z x.cov(r r ) Z x.cov(ro,r ) = [E(r )E(r )]
j=l J j=l J j 2 2
(25.2)
m
n m n m
Sxcov(r ,r.) E x.cov(r ,r ) [E(r )E(r )];
j=1 n j=l o j 2 n
(25.n)
nm A
Z x.cov(r ,r) = [E(r )r f]
i=1 1 o 1 2
(26)
Now substitute equation (26) into each of the equations
of the (25.1) to (25.n) set to get:
n m
E x cov(rl,r.)
j=1 J
n m
E x.cov(r2,r )
j=l J
m
A
+ [rfE(ro)
+ [ff.(roJ]
m
= [E(rl) E (r)]
2
n m
E x.cov(r ,r.) + rf E(r )] [E(r )E(r )];
j=1 3 n 2fo 2n
(27.n)
The equation set (27.1) through (27.n) may now be
easily rearranged to obtain:
n
j x.cov(r ,rj)
n m
j x.cov(r ,r.)
j=l J J
= [E(rl)rf]
= 2[E(r2)r ];
m
n m A
E x.cov(r ,r.) = [E(rn)rfj.
j=1 3 n 2 n
(28.1)
(28.2)
(28.n)
and
(27.1)
(27.2)
It may be seen from the equation set (28.1) through
(28.n) that no traces of the option remain in the first
order conditions. Equations (28.1) through (28.n) may be
solved in a manner similar to the familiar SharpeLintner
capital asset pricing model that has no options. To do
this first recall equation (9). Thus, from equations (28.1)
through (28.n):
m
cov(rl,rm) = L[E(rl) r ]; (29.1)
m
n 2 f
cov(r2,r m f[E(r2) r^]; (29.2)
cov(rn9 ) 2 [E(r^ rJf. (29.n)
Now to eliminate the Xm/2 factor from the equation sys
m m
tem, first multiply (29.1) by xl, (29.2) by x2, and so
forth. These multiplications result in:
m
m X m m
xlcov(rl,rm) = i xlE(r x1rf]; (30.1)
m
m 1 m m
x2cov(r2,rm) = [x2E(r2) x2r ]; (30.2)
m
m 1 m m
xncov(rn,rm) = nE(rn) xnrf]. (30.n)
41
Next sum equations (30.1) through (30.n) vertically to
get:
n m m n nm
E x cov(r,r ) I E(r.) r E x (31)
i=l 1 1 m i=l 1 1 f
m
Using equation (9) again and noting that E x. = 1
i=l 1
(since xf=O), equation (31) is equivalent to:
m
cov(rm ,r) = [E(r ) rf] (31a)
2
Note that cov(rm,rm) in equation (31a) is simply am,
the variance of the market portfolio, it follows easily
that:
Xm a2
m *(31b)
2 E(rm)rf
Now substitute from equation (31b)into equation system
(27.1) through (27.n), and after rearranging, the familiar
SharpeLintner capital asset pricing model results:
cov(ri,r )
E(ri) = rf + 1 [E(r )r ]. (32)
1 f 2 m f
m
Thus the familiar SharpeLintner model applies even in
a world with an option, as long as a riskless security is
available for unrestricted trading.
Equilibrium with One Option and a Margin Constraint6
on Riskless Borrowing
In the case to be derived here xf is constrained to be
above some constant negative amount, denoted C. For
example, if C is .50, then an investor may borrow up to 1/3
of the value of his portfolio of risky securities. If C =
1, then fully onehalf of the risky asset portfolio may be
financed with borrowed funds, and so on. In this case equa
tions (18) and (19) again represent the investor's portfolio
expected return and variance; they are repeated here as
equations (33) and (34) below, with the margin constraint
(35):
n n
E3(r) = x.E(r ) + (1 Z x E(r ) + xfrf; (33)
i=l i f) r
2 n n n 22
a = i x.xjcov(ri,rj) + (1 i xx f) ao
r3 i=lj=l 1 1 i
(34)
n n
+ 2(1 Z xix ) x.cov(r ,ri);
i=l i=l 1 o
C < xf. (35)
Thus the investor's portfolio problem of minimizing
variance for every level of expected return, given the mar
gin constraint in (35), can be solved by minimizing the
following Lagrange function:
2 n n
L = a + [E 3(r) x E(r ) (1 E x x )E(r )
3 i i=l f o
1=1 i=l
(36)
xfrf] + X[C x ],
where the second Lagrange multiplier, X1, relates to the
inequality constraint.
Now substitute the portfolio variance expression from
(35) into equation (36) to get:
n n n 2
L3 =Z Z xx cov(r ,r) + (1 x.x ) a +
i=lj=l 1 J i=l 1
n n n
2(1 Z xix ) xicov(r ,ri) + X[E3(r) xiE(ri)
i=l i=l
n
(1 Z x xf)E(ro) xfrf] + 1[C xf]. (37)
i=l1
Differentiate L3 in (37) with respect to each of the
portfolio weights of the n risky assets, and set the
derivatives equal to zero to obtain equations (38.1)
through (38.n) below:
dL n
3 2[ E x cov(r ,r.)
dx j=l 1 3
n
(1 C xx )a +
i=l 0
(1 Z x x coverr r )
i=1 1 f o 1
n
Sx.cov(r o r.)] 
j=1j 3 0
X[E(rl) E(ro)] = 0;
dL3 n n
= 2[ Z x.cov(r r ) (1 xixf ) +
dx2 j=l 2 2 0 i=l o
n n
(1 x.xf)cov(r,r2) x.cov(r ,r)]
i=l 1j=l 3
[E(r2) E(ro)] = 0;
dL3 n n 2
2[ Z x.cov(r ,r) (1 E x.x )o +
dx j=l 3 n ji=l o
n n
(1 Z xx )cov(r ,r) xcov(r ,r )]
i=l i f) n j=1 (r 0
X[E(rn) E(ro)] = 0.
(38.1)
(38.2)
(38.n)
Next take the derivative of L3 in (37) with respect to
xf and set it equal to zero:
dL n n
= 2[(1 Z x.x )2 + E x.cov(r r)] +
dxf i=l i 1 o' i
(39)
[E(ro) rf] 1 = 0.
Equation (39) is different from its counterpart equa
tion, (22), in the second section, because of the constraint
and the appearance of X.
7
Together with the generalized KuhnTucker conditions
and equations (40) and (41) below, equations (38.1) through
(38.n) and (39) form the first order conditions to be
satisfied if there is a solution to the investor's decision
problem;
A (C x ) = 0; (40)
f
A1 > 0. (41)
From the complementary slackness condition, (equation
1
(40)), it is obvious that either A = 0 or xf = C, or both.
If XA = 0 and xf i C, then the margin constraint, while it
is publicized, is not effectively binding the investor's
1
decision; if A i 0 and x = C, then the margin constraint
is effective.
Aggregate each of conditions (38.1) through (38.n) and
(39) across all investors (in a manner similar to that of
the first section) to obtain equations (42.1) through (42.n)
and (43) below:
n m n n
Sx cov(r,r) (1 x xm) + (1 x x)cov(rr)
j=l J 3 i=l i xf o 0 i=l i (ro'
n mm
xmcov(r ,r ) = [E )E(r )] (42.1)
j=1 6 5 2 1 0
n n n
Sx cov( r ) (1 xx )2 + (1 m x xm)cov(rr
j=l J 2' i=l 1 f i=o i 0 o2
m
n
x.cov(r,' ) = x [E(r2)E(ro)]; (42.2)
j=j2 0
n m n m m 2 n
j xjcov(r ,rj) (1 ) x.x )o + (1 i xixf )cov(ro,rn)
j=l n j i=l 1 i=l
n m
m X
x.cov(r ,r ) = [E(r )E(r )]; (42.n)
j=1 o J (ro
n m n m m
(1 x xf)a E x.cov(r r ) = [rfE(r )]+ x1 ,(43)
i=l i=l 2
m m Im
where x. and xf are as before, and X is the weighted
1
aggregate of all the individual investors' X 's divided by
1m 1 P k Ik 1k
two, i.e., Z w X where X is the kth investor's
2k=1
1 k
X and w is the proportion of market wealth held by the
kth investor.
Im
If X is equal to zero, i.e., if the margin ceiling is
not restrictive on anyone, then equation (43) reduces to
equation (24). Under that circumstance the problem is no
different than the case of unrestricted trading in the risk
im
less security. If X is not equal to zero, then the margin
constraint is binding on at least one investor. In this
case the path to establishing equilibrium relations from
equations (42.1) through (42.n) and (43) is different than
if Xlm = 0. To continue, assuming Xlm / 0, first recognize
that x = 0. Consequently, the aggregate amount of the
riskless security is assumed to be zero in the same manner
as the option.
mn m
Using the fact that x = 0 = 1 E x. and using equa
i=l 1
tion (9), equations (42.1) through (42.n) and (43) may be
reexpressed as:
m
cov(rl,rm) cov(r0,r) = 2[E(rl) E(ro)]; (44.1)
m
cov(r ,r ) cov(r ,r) = [E(r) E(r )]; (44.2)
cov(r ,r ) cov(r,r ) [E(r ) E(r )]; (44.n)
cov(rn,rm) cov(ro r ) = [E(r ) E(r )]; (44.n)
0 mm 2 n o
m
cov(ro,rm) = 2[E(ro) rf] X (45)
Equation set (44.1) through (44.n) and (45) must hold
in market equilibrium. Equation (45) is the relation
between the equilibrium expected rate of return on the
option and the riskless rate. Equations (44.1) through
(44.n) may be used, ignoring (45), to derive the same
equilibrium conditions as those in the first section of the
chapter. Equation (16) would hold in the case just des
cribed, since:
m 2 cov(r m) (46)
2 E(r ) E(ro)
Substitute equations (46) and (45) into the ith equa
tion of (44) to get another equilibrium expression for
the margin ceiling case, equation (47) below:
2
lm m cov(r ,rm)
cov(ri,rm) + A = m [E(ri)f]. (47)
E(r ) E(r )
By rearranging equation (47) it is found that:
Im
cov(ri,r ) + X
E(r.) = rf + m[E(r )E(r )]. (48)
S2 cov(r ,r m 0
m o 'm
Consider now equation (48) applied to a zero beta
security:
1m
E(rc) = rf + 2 [E(rm)E(r)]. (49)
am cov(r ,rm)
If one assumes that options are used by many investors in
lieu of margined stock, then an obvious connection exists
between an increase in the number of options being traded and
a decrease in A As is evident from equation (49), a de
Im
crease in X should create a decrease in E(r ). This
establishes the dissertation's hypothesis that:
As the quantity of options being traded increases,
the equilibrium expected zero beta rate of return
decreases.
The model in (49) and this chapter is not sufficiently
detailed in assumptions enough for one to establish a precise
mathematical relation between the number of options in the
option market and the aggregate margin constraint multiplier,
Im
A Perhaps a direct link could be established under some
assumptions involving investor heterogeneity, but this task
is not undertaken at the present time.
Notes
All investors are assumed to have perfect information.
2
The noinflation assumption is relaxed in some CAPM's
(e.g., Solnik's [1978]), but such models are beyond the scope
of this study.
Alternatively, the option will eventually be viewed as
a portfolio of options.
4The interpretation of Xk is that Xk is the amount of
additional expected return the investor must get if he is
to accept a small amount of additional variance in his port
folio.
The superscript k has been dropped for convenience. The
proofs in this section and the next are abbreviated somewhat
from the detail of the previous section.
Black (1972) and Vasicek (1971) previously provided re
sults in this area. Black considered no riskless borrowing
but allowed riskless lending, a circumstance that has been
assumed here to be impossible. Vasicek looked at the margin
constraint idea, but assumed the riskless rate to be zero.
7ee Lueberger (1973)
See Luenberger (1973).
CHAPTER III
REVIEW OF ASSOCIATED LITERATURE
Introduction
The literature about the effects of new option trading
includes both theoretical as well as empirical papers. The
theoretical contributions have been made in frameworks other
than the one employed in Chapter II; however, the author is
not aware of any other work that formally considers the
role of any kind of futures contract, let alone options
specifically, in the singleperiod meanvariance framework.
Since the other theoretical studies employ other frameworks,
a detailed review of those studies is omitted here. The
interested reader is referred to the works of Hirshleifer
(1975), Danthine (1978), Ross (1976), Schrems (1973),
Townsend (1978), Breeden (1978), Rubinstein (1976a), Friesen
(1979), and Long (1974). The roles of futures (and
especially options) identified by the above theoreticians
do vary depending upon which framework is employed. While
the papers are interesting, they are too complex to adequately
review here.
On the empirical side attention has been focused on the
effects of options on the underlying securities, rather than
on marketbased variables. In particular, no study has been
concerned with the impact of options on market factor interest
rates. Despite this, several of these studies are reviewed
in this chapter, since the studies do contain evidence that
is interesting in light of the present topic and methodology.
There are two categories of empirical studies associated
with effects of options. The first category concerns the
shortrun effects of option expirations. The second category
concerns the effects of the advent of new option trading in
the longrun as well as the shortrun. Only the second cate
gory of these studies is of direct interest to review here.
Klemkosky and Maness
In a study published in 1980 by Klemkosky and Maness
(KM), the results of an extensive investigation of the
impacts of new option trading on the underlying stocks were
reported. The major conclusions at which KM arrived were:
(a) that the options had a negligible impact on the risk of
the underlying stocks; and (b) that excess returns which
had existed in underlying stocks before the commencement of
new option trading had been bid out of the stock prices
subsequent to the option listing.
The KM study examined two risk measures for the under
lying stocks, beta and standard deviation, and one performance
measure, Jensen's (1969) alpha (c). The KM methodology of
measuring alphas, betas, and changes in alphas and betas is
basically the same as one portion of the methodology proposed
in the next chapter: the Gujarati (1970) interactive dummy
variable technique applied to the excess returns version of
the linear market model. In the form shown by KM, the
linear excess returns market model is given below in equation
(31):
it t = :i + i(rt
r.t rft 1 rmt
 rf) + eit
ri = the monthly holding period return, including
dividends as well as price appreciation of
security i in month t.
rft = the 30day Tbill yield on a bond equivalent
basis in month t.
rmt = the CRSP Investment Performance Index, including
dividends, in month t.
= the intercept term representing Jensen's
performance measure.
6i = beta of security i, and
eit = a random error term.
To the above market model, KM applied the Gujarati
interactive dummy variable technique using time series data.
Thus KM estimated the parameters in the following model:
rit rft = + D + 6i (r rft) +
it f i mt rt
1
f3 (rm
i mt
 ft) D + eit
ft it
(2)
where
D = 0 for the period subsequent to the option
listing ("postlisting")
D = 1 for the period prior to the option listing
("prelisting").
Thus .i and Bi are measures of the postperiod alpha and beta,
1 1
while + and B + B are measures of the preperiod
alpha and beta.
where
The KM study analyzed three "waves" of stocks that be
came underlying securities for options. Group 1 stocks
consisted of the 32 stocks which had options listed from
April 1973 through October 1973. (These were the stocks
from which came the ones used as underlying stocks for this
study.) The prelisting period data for Group 1 consisted
of monthly security returns for the 36 months from January
1970 through December 1972. The postlisting period was
from January 1974 through December 1976. KM dropped the
1973 period so as to avoid any problems in testing that
might be associated with the effects of the announcement
of option listing on the underlying securities.
Group 2 stocks consisted of the 32 securities which had
options listed on the CBOE beginning December 1974 and ending
June 1975. (No new options were listed between October 1973
and December 1974, but many were listed continuously after
June 1975.) KM omitted December 1974 through June 1975
from their analysis of Group 2 stocks. The prelisting
period was January 1971 through November 1974 for Group 2.
The postlisting period was July 1975 through June 1978.
Group 3 consisted of the 39 stocks that had options
listed on the American Stock Exchange CASE) from January 1974
through June 1975. This time period was omitted from the
analysis. The preASE period was from January 1972 to
December 1974, and the postASE period was from July 1975
to June 1978.
KM utilized two different market indicesthe CRSP
equalweighted (EW) and the CRSP market valueweighted (VW)
indices. KM noted that the 103 securities, because of their
large market values, will dominate or greatly influence in
the aggregate any market valueweighted index, so the authors
used both indices. The empirical results of the KM study
are summarized next.
KM observed that the performance measure (=) decreased
for most securities in the postlisting period (83 out of 103
for the EW index and 68 out of 103 for the VW index). This
result was also observed for the "portfolio of all underlying
stocks" in each group. In all 6 cases the performance measure
dropped; 5 of the 6 cases were significant. KM also noted
that the performance measure had been significantly positive
in 5 of the prelisting cases, and not significantly dif
ferent from zero in any of the postlisting periods. KM
concluded that excess returns appeared to have been bid out
of the underlying security returns with the advent of option
trading. This conclusion is somewhat consistent with the
theory and hypothesis of this dissertation; however, a major
caveat for the KM study is the Roll (1977a) critique.1
The changes in the betas (measures of systematic risk)
for the stocks were not so consistent. Only a few stocks
had significant beta changes. More stocks showed beta de
clines than increases. Viewing the stocks in each group as
portfolios the beta changes were also not consistent: the
portfolio beta for the Groups 1 and 2 stocks declined, and
the Group 3 portfolio beta increased. Anyway, the changes
were insignificant in all 6 cases.
KM also observed that the coefficient of determination,
R2, did not change significantly from the pre to the post
2
listing periods. The R was calculated as the proportion
of the total variation of stock returns explained by the
linear relationship with the market portfolio.
KM finally looked at changes in total risk (measured
by variance) for the stocks and the portfolios. They found
that the total risk of the Group 1 stocks went up after
option listing, while the total risk of the Groups 2 and 3
stocks went down in the postoption listing period. In
terestingly, the variance of the market index behaved in
the same direction. The change in the variance of the mar
ket index and in a large proportion of the stocks was
significant.
The changes in the variances of the portfolios were
not consistent. The Group 1 portfolio variance increased,
but not significantly. The Group 2 and Group 3 portfolios
experienced a decline in variance; the decline was significant
for Group 2, but not for Group 3. Also KM reported the
variance comparisons using "deflated" returns. Deflated
returns were defined to be (rit rft) divided by (rmt rft)
and were used to account directly for shifts in the market
index variability of returns. When the alternative method
was used the results for the individual stocks were about
the same. However, the change in portfolio variance for
Group 1 became significant in the case of the EW index. The
change in the variance of the Group 2 stocks switched from
being significant to insignificant. The Group 3 portfolio
variance changes switched from being insignificant to
significant, but again only in the case of the EW index.
This concludes the review of the KlemkoskyManess
study.
Hayes and Tennenbaum
A study that was conducted by Hayes and Tennenbaum (HT)
was published in 1979, and it analyzed the impact of option
trading on the volume of trading in the underlying shares.
The authors' statistical tests indicated that an effect of
listed options was to increase the volume of trading in the
underlying shares. HT theorized that this effect occurred
because the availability of the options increased the number
of ways that the underlying stock can be used in investors'
portfolios.
HT conducted 2 different types of tests. The first
was a crosssectional analysis. HT compared a 43company
sample of optioned stocks with a control group of 21 stocks
of similar size, but for which there were no options. Using
a system of dummy variables for the option group and the
control group, and for a preoption trading period and a
postoption trading period, HT analyzed the percentage
trading volume compared to the total NYSE volume. HT found
that the control group's trading volume was about 17% of the
NYSE total in the pre and postoption periods. (The pre
option period was May 1972 to April 1973; the postoption
period was May 1973 to September 1977.) HT also found that
the optioned stocks had a mean percentage volume of 25% in
the preoption period and that the percentage jumped to
almost 34% in the postoption period.
The second analysis that HT performed was a longitudinal
analysis. The authors examined the stock volume for at least
a year before and a year after options began trading on that
stock. For the 43 companies, stock volume was the dependent
variable in a multiple regression on the 2 independent
variables: option volume and NYSE volume. In 1 version of
the longitudinal test the 43 stocks' volume data were
aggregated, and so was the option volume data. In the
second version of the longitudinal test the stocks' volumes
and option volumes were analyzed individually.
HT found that in the first version of the longitudinal
analysis there was a significant association between stock
trading volume and option trading volume. The results of
the second version, the longitudinal test with each indi
vidual stock, showed corroborating results.
Hayes and Tennenbaum concluded that they had provided
evidence of a linkage between option trading and price
"continuity" in the underlying shares. That is, HT linked
the volume increases with price continuity in the underlying
stock. This is an interesting finding and appears to be
evidence that options improve market efficiency, at least
for the underlying stocks. Since increased volume of under
lying stock trading has not been predicted by this disserta
tion's theory, such an increase does not contradict the
hypothesis here. Increased underlying stock trading volume
could (intuitively) be a manifestation of the relocating
process of the riskreturn intercept. Further analysis of
this point here, however, is beyond the scope of this review.
Reilly and Naidu
In a paper that has been professionally presented, but
not as yet published, Reilly and Naidu added their analysis
of option trading impacts on underlying stock volume and
volatility to the existing evidence. In addition, Reilly
and Naidu (RN) examined the impact of option trading on
the market liquidity of the underlying stocks. RN employed
2 types of measures of liquidity. The first was the bidask
spread; the second was a version of the Amivest Liquidity
Index.
The Amivest Liquidity Index attempts to relate the
average dollar amount of trading to a 1% change in the price
as follows:
n
E Pt Vt
t=1
Amivest Index. = t(3
1 n ( '
E %APt
t=l
where
Pt = Closing price for stock i, on day t.
Vt = Share volume of trading for stock i on day t.
%APt = The percent change in price for stock i on day t.
The higher the dollar volume of trading is, per 1% of price
change, the higher will be the liquidity of the stock, in
the opinion of the users of the Amivest Index. Reilly and
Naidu modified the Amivest Index so that interday price
ranges were accounted for as follows:
n
E P V
t=t
Modified Liquidity Index = t (4)
n fHL
SH+L/2 t
t=l
where
H is the high price for the day, and
L is the low price for the day.
The RN analysis focused on effects surrounding the
listing dates of options on the CBOE and the ASE. The 5
days before and after the listing dates were excluded.
Activity in the 20day periods before and after the 5day
periods were examined. In all, 12 stocks listed on the CBOE
on May 22, 1975, and May 23, 1975, and 10 stocks listed on
the ASE May 30, 1975, were tested. Also, control groups
of 12 and 10 randomly selected NYSE stocks were studied.
The results of the RN tests for effects of options
on the market spreads indicated that the percentage spread
for the optioned stocks was lower than that for the random
stocks. In addition, in going from the prelisting period
to the postlisting period, the market spread declined
slightly for the optioned stocks and increased slightly for
the random stocks. The change was not significant, but RN
remarked that the market for the optioned stocks was still
superior in terms of liquidity to that for the random stocks.
With regard to both the Amivest Index and the Modified
Liquidity Index, however, RN observed virtually no change in
market liquidity. Thus Reilly and Naidu observed no signifi
cant changes in any measures of liquidity.
RN used the same data in their analysis of changes in
underlying stock volatility and relative trading volume. In
addition, RN examined an aspect of price performance of the
stocks in the 20day periods on either side of the listing
time. Five nonsystematic volatility measures were examined
for the optioned stocks, the random stocks, and the S & P
400 Industrial Index. For all 5 measures the stock price
volatility of the optioned stocks declined in the period
after listing. The range of the decline was from 25% to
35%. The volatility of the random stocks declined by a
smaller amount, from 3% to about 8%. Relative to the market,
the volatility of the option stocks also declined in the
postlisting period. Since the RN stocks are contained in
Klemkosky and Maness' Group 2 and Group 3 stocks, RN's
findings are basically consistent with those of KM, dis
cussed earlier.
For the volume of trading, RN found no significant
change in the relative volume of trading of either the
optioned stock groups or the random stock groups. RN
concluded that there was almost no shortrun impact on the
volume of trading for the underlying stocks as a result of
option listing. This conclusion is not necessarily con
tradictory to the HayesTennenbaum report, since HT looked
at a much longer term.
Reilly and Naidu also looked at price performance from
the following perspective: they measured the ratio of the
average price of an optioned stock to the average price of
a random stock. In the prelisting period the price ratio
was stable in the 1.73 to 1.80 range. In the postlisting
period this ratio jumped to 1.98 and remained in this range
for about 7 days. Then the ratio gradually declined back
to the prelisting period range. No possible reasons were
offered for this finding.
Trennepohl and Dukes
An analysis of the effect of option listing on under
lying stock betas was the focus of a paper, published in
1979, by Trennepohl and Dukes (TD). TD examined the
original 32 stocks listed on the CBOE from April 1973 to
October 1973. Weekly holding period returns on each of the
32 stocks and on each of 18 nonoptioned stocks (assumed to
be in a control group) were examined. The nonoptioned
stocks were selected as a stratified random sample, repre
senting the same industries present in the optioned stock
sample.
The betas were calculated for all of the 32 optioned
and 18 nonoptioned stocks for a 21/2 year period from
October 1970 through April 1973 ("before") and a 21/2 year
period from October 1973 through April 1975 ("after").
The betas were calculated using the following (no
excess returns) market model:
R. = a. + b. R + e (35)
where
R = weekly holding period returns of stock i.
a. = Yaxis intercept.
Cov (Ri,RM)
bi = beta of security i, i.e. ( ar(RM) ).
RM = the rate of return on the Standard and Poors
500 Index.
e. = random error term.
1
The betas obtained from the regressions were analyzed
by 3 methods. The first method was a paired differences
test. The mean of all 32 optioned stock betas was 1.22
before 1973 and .873 after 1973, a mean difference of .347.
Since the t value associated with this mean difference was
3.574, TD claimed that the observed change was significant.
However, a similar result was observed in connection with
the nonoptioned stocks. The average beta changed from 1.137
to .934 (a change of .203), with a t value of 2.317. TD
observed that it appeared that the reduction in the betas
had been caused by "general market influences" rather than
the option trading.
However, in a ttest of the difference in the mean
change of the betas, the authors found that the hypothesis
of no mean difference change, between optioned and non
optioned stocks, could be rejected at a confidence level
slightly over 89%. A nonparametric ChiSquare test of
64
directional changes in the betas essentially confirmed the
results of this ttest. Thus, it was concluded by KD that
the betas for the option stocks decreased more than the
betas for the nonoptioned stocks, but with a statistical
level of confidence that may be considered to be marginal.
Implications
There is no finding in any of the studies reviewed that
is contrary to the hypothesis of this dissertation at the
level of theory presented. In fact, the conclusions that
excess returns have been bid out of underlying stock prices
2
by KM, and that trading in the underlying stocks has
become more continuous by'HT, tend to support the theory
of this study.
65
Notes
Roll (1977a) called into question any studies of
empirical estimates of systematic risk obtained by time
series regressions of returns on a market index; he showed
how far off results could be if one doesn't know the "true"
market portfolio.
2Roll's caveat notwithstanding.
CHAPTER IV
PROPOSED ANALYSIS OF RETURNS OF ZERO
SYSTEMATIC RISK HEDGES
Excess Returns
The previous chapters established the general hypothesis
that as more options are traded, the equilibrium expected
zero systematic risk rate of return, E(r,), will theoretically
decline toward the riskless rate of interest. Since addi
tions of new options to the market have occurred over a
period of time, the empirical analysis here should involve
an examination of time series. Specifically, time series
values of rf and rZ are to be examined and tested during
periods when new options did and did not begin trading. In
a time series analysis of riskfree and zero systematic risk
rate estimates, the possibility that rf and E(r,) could
change, for reasons other than new option trading, is a
problem. This problem may be easily overcome by focusing
the analysis on excess return; an excess return is defined
to be an observed return minus the corresponding time series
e
observation for rf. Let r denote the time series of
differences between each observation for r and the corres
ponding observation for rf.
Zero Systematic Risk Returns
Neutral Spread Returns
An obvious candidate for a zero systematic risk port
folio is a neutral option spread of the kind identified by
Galai (1977). In order to simulate the performance of a
neutral spread one must determine dCl/dS and dC2/dS, the
first derivative values of the 2 call prices with respect
to the stock price. The neutral spread is created by going
long dC2/dS times the first option and going short dCl/dS
times the second option.2 The neutral spread is riskless
for the instant during which it is created; neutral spreads
have zero systematic risk over discrete short time intervals,
assuming normally distributed underlying stock prices.3
In order to employ the neutral spread method for
simulating r returns, one must know or assume a differen
tiable option pricing function of the underlying stock price.
Fortunately, several alternative option models are suitable
for usage in the method of neutral spreads. The BlackScholes
(1973) model, the most popular option model in finance
research at present, is:
C = SN(dl) XerN(d2)
where S BlackScholes
In + + (r + C /2)T Model
dI (1)
and
d2 = d1 a T
In equation (1) C is the price of the call option; S is
the price of the underlying stock; X is the option's exercise
price; r is the continuously compounded riskfree interest rate,
2
a constant over time; a is the continuous variance rate, also
a constant over time; T is the time until expiration of the
option; and N(d ) is the cumulative unit normal distribution
function value at d If equation (1) is assumed as a valid
call option pricing model, then dC/dS, for use in construct
ing neutral spreads, is N(d1).
Significantly, equation (1) holds only for options that
are dividendprotected or for options whose underlying stocks
pay no dividends.4 A dividendprotected option is one whose
exercise price automatically is adjusted, without loss or
gain in the value of the position of the call owner, for cash
dividend payments made to the holders of the underlying
stock. Since U.S. exchangelisted options are not dividend
protected, and since underlying stocks commonly pay dividends,
some extension of equation (1) is desirable; a popular can
didate for a nondividendprotected option model is the Merton
(1973b) model:
C = ed S N(d{) Xer N Cd2)
where Merton
In + (r d + 2/2)T Model (2)
d =
and
d = d' a
2 1
In equation (2) d is the continuously compounded dividend
yield, based on the current stock price, S. The Merton model
assumes d is constant and known in advance; the other vari
ables in his model are the same as those in the BlackScholes
model. In fact, the Merton model is the same as the Black
Scholes model except for the dividend assumption. The first
derivative of C with respect to S, using equation (2) is:
dC/dS = edTNCd), (2')
which is an input necessary to the method of neutral spreads.
The interest rate, r, in either of the equations, (1)
or (2), is essentially the same concept as the riskfree rate
described in Chapter II. However, as was indicated in this
chapter, the r in the option models is a continuoustime
instantaneous riskless rate, which is assumed to be constant
over time. Of course, in reality the short term interest
rate appears to be stochastic rather than constant. However,
Merton (1973b) has argued that the continuously compounded
equivalent to the discretetime treasury bill rate for the
next T years is suitable as an interpretation for r in
equations (1) and (2).
Four of the variables in equation (2) are directly ob
servable; they are the stock price, S; the exercise price,
X; the time to maturity of the option, T; and the dividend
yield, d. The dividend yield is not usually known in ad
vance with certainty, but educated forecasts will very often
be correct, since most companies follow "stable" dividend
policies.
The variable in equation (3) which is not directly ob
servable is a Option researchers once thought this vari
able could be estimated with reasonable accuracy using
historical data; however, Geske (1979) has discussed and
pointed out some potential inadequacies with using his
torical variance estimates in option models. Since some
value for a2 must be assumed in order to calculate the
derivative in equation (2'), and since historical estimates
of 02 are potentially inadequate, the implied variance method
of Latane' and Rendleman (1976), Chiras and Manaster (1978),
and Trippi (1977) must be employed. The implied variance
method yields a value for a2 by the researcher (a) observing
an actual option price, (b) assuming the option model holds
true, and (c) calculating the value of a2 that equates the
option formula price to the actual option price.
Previous Empirical Results of Neutral Spreads
Empirical analyses of neutral option spread returns
for daily and monthly holding periods have been reported
by Galai (1977) and Chiras (1977), respectively. Galai,
who used the nowsuspect hlis.rical variance approach, re
ported the appearance of skewness in the frequency distri
bution of neutral spread returns (see note 3). In addition,
Galai reported that the neutral spread returns had variances
that were too large to permit statistical inferences to be
made. c.his avoided the historical variance problem by
using the implied variance method. Chiras found, on a
selected basis, some strikingly high returns; however, he
did not report any analysis of the statistical properties
of the neutral spread returns. In fact, neither Galai nor
Chiras considered the important empirical question of
whether the neutral spreads contained any systematic risk.
Neutral Hedge Returns
A second possible candidate for a model of a zero
systematic risk security is very similar to a neutral
option spread. However, rather than a neutral spread of
2 options, the second method involves a neutral hedge of
1 option and the underlying stock. A neutral hedge is
created by going long a share of the stock, and simultan
dC
eously going short 1/ga options. Neutral hedges neverthe
less have zero systematic risk under the same conditions
as neutral spreads.5
Previous Empirical Results of Neutral Hedges
Returns of neutral hedges have been examined by Black
and Scholes (1972), Galai (1977), and Finnerty (1978). None
of these 3 studies employed the adjustedfordividends model,
equation (2): Black and Scholes and Finnerty used equation
C1) exclusively; Galai used equation (1) primarily, and
later considered the effects of dividends by a different
means than equation (2). All 3studies employed the proble
matic historical variance approach. Both Black and Scholes
and Galai analyzed daily holding periods; they claimed to
have found no evidence of significant systematic risk in
their neutral hedge positions. Finnerty looked at weekly
holding periods; he claimed he did find some significant
systematic risk. These researchers' opinions are highly
regarded, and their findings are not necessarily illogical.
However, all 3 studies measured systematic risk by the
commonplace method of calculating the regression coefficient
in a least squares regression of hedge returns on a market
index, and this method may be invalid, as Roll (1977a) has
argued.6
Direction of the Methodology
A choice should be made between neutral spreads and
neutral hedges. For this study neutral hedges are chosen,
because of the report by Galai that neutral spreads had
skewness and large variances. To calculate neutral hedge
returns, the method of implied variance will be used; thus,
the "historical variance problem" of the 3 previous studies
of neutral hedges will be avoided. In addition to the
improved technology in estimating 2, this study will
utilize the adjustedfordividends model, equation (2);
since equation (2) was not used in any of the 3 previous
studies of neutral hedges, the use of equation (2) here
represents an improvement over previous work. Finally,
since Roll has cast doubts about the meaningfulness of the
systematic risk estimates of the 3 earlier neutral hedge
studies, some effort will be made in this study to assess
systematic risk through an alternative method. The alter
native method will be elaborated upon later in the chapter.
Two Further Considerations
1. There is a potential problem in employing neutral
hedge returns to model zero systematic risk security returns
for the purpose of testing the dissertation's hypothesis.
The problem is that the riskreturn slope may shift during
the time span being studied. If the slope of the risk
return relation were to change over the time span studied,
then there would be some movement of the underlying stock
prices to new equilibrium levels. This movement in the
underlying stock prices could cause some abnormalities in
hedge returns that may obscure the direct effect of the
option trading on ECrz). The abnormalities would most
logically be expected to have an impact on the results if
the underlying stocks used to construct the neutral hedges
in this study were imbalanced by being comprised of either
too many high systematic risk stocks or too many low sys
tematic risk stocks. In order to account for this problem,
an empirical analysis must be employed to check for the
presence of the potential effects of a shifting riskreturn
slope. The procedure is described later in the chapter.
2. The other consideration in connection with neutral
hedges is whether a portfolio may be viewed as the minimum
variance zero systematic risk portfolio in the capital mar
ket. If investors could continuously readjust their hedge
*positions, a portfolio of neutral hedges would surely be
the minimum variance zero systematic risk portfolio, since
each of the neutral hedge returns would have no uncertainty
at all. However, the problem is that investors cannot
continuously readjust hedge positions, and so a positive
variance must be a property of neutral hedges held over dis
crete time intervals. As has been pointed out by Boyle and
Emanuel (1980), it is possible to reduce the variance of
individual neutral hedge positions by about 3/4 by con
structing a portfolio of a large number of neutral hedges.
This dramatic variance reduction is made possible by the
low correlation between neutral hedges relative to the
correlation between (positively correlated) underlying stock
returns. Thus an empirical question, that is important to
this research study, arises: whether the variance of the
neutral hedges can be reduced by enough through diversifi
cation to permit us to consider a portfolio of neutral
hedges as having the minimum variance of all zero systematic
8,9
risk portfolios.89
Selection of Time Span, Holding Period, and Data
Time Span: November 30, 1973August 29, 1975
The empirical study should extend over a period of
approximately 2 years in order that the predicted effects
of new option trading be given ample time to show up. The
21month time span from the end of November, 1973, through
August, 1975, was selected. This time span is divided into
2 contiguous segments. The first segment, Period 1, is the
12month term ending November, 1974; during this segment no
options began trading on any new underlying stocks. The
second segment, Period 2, is the 9month segment beginning
at the end of November, 1974; during this time options began
trading on a total of 82 new underlying stocks. Table 41
shows the frequency distribution for the number of new under
lying stocks over the months of both Period 1 and Period 2.
Holding Period Assumption: Monthly Observations
The next decision is the assumed holding period for
which to simulate returns. A holding period of longer than
1 month, given the 21month span of the study, would not be
feasible, because too few time series observations would
result to perform any meaningful statistical analysis. For
a shorter holding period, 2 weeks or 1 week, more time
series observations would lie within the time span; in
addition, there should be less liability of systematic risk
in the neutral hedge returns.
Although these arguments in favor of shorter holding
periods are reasonable, given the time span, holding periods
shorter than 1 month would entail high data gathering costs.
To see why the data costs would be so high under those
circumstances, consider the portion of the study concerned
with the reduction of hedge variance via naive diversifi
cation. To facilitate the best possible naive diversifi
cation analysis it is necessary to use the price observations
of all options quoted at any one point in time. The ob
servation of all options at the end of each of the 21 months
in the timespan will result in 2985 usable neutral hedge
returns for the study. Since all of the data must be gathered
TABLE 41
Frequency Distribution of
New Underlying Stocks
December 1973  August 1975
Month Number of New Underlying Stocks
1. December 1973 0
2. January 1974 0
3. February 1974 0
4. March 1974 0
5. April 1974 0
6 6. May 1974 0
7. June 1974 0
8 July 1974 0
S9. August 1974 0
10. September 1974 0
11. October 1974 0
12. November 1974 0
13. December 1974 8
14. January 1975 20
15. February 1975 0
16. March 1975 6
17. April 1975 0
.18. May 1975 12
S19. June 1975 31
 20. July 1975 5
21. August 1975 0
Total Added In Period 2 82
by hand, the use of weekly or even biweekly data would
significantly increase the data costs over those for monthly
holding periods. Thus, monthly holding periods have been
chosen. The problem of potential systematic risk in the
hedge returns should be addressed, since some possibility
exists that monthly holding periods are not "short enough"
(in the BlackScholes sense) to validate the assumption of
zero systematic risk.
Description of the Data
All available options at the end of each month from
November, 1973, through August, 1975, are employed in this
study. For this study an option is "available," if it
satisfies the following criteria: (a) The option's under
lying stock must have been on the CBOE's list of underlying
stocks throughout the entire 21month time span; (b) two
consecutive monthend price quotations must have been ob
servable; (c) neither monthend price observation was
allowed to be below the option's intrinsic value at that
point in time, with intrinsic value defined as:
I = S + D X
where
I = Intrinsic value
S = Stock price
D = Dividend of the stock over the life of the option
X = Exercise price of the option.
Options whose observed prices are below their intrinsic
values are excluded from the study, because their price
observations obviously violate the logic that riskless
arbitrage opportunities have already been eliminated by
professional traders.10
Only 32 stocks comprised the list of underlying stocks
as of November 30, 1973. (This remained the entire list
until new options began being added in Period 2.) The
options of 2 of these stocks, Great Western Financial, and
Gulf and Western, were not used in the study, due to data
gathering complications in both cases. The list of the 30
underlying stocks used in the study is given in Table.42.
For various reasons the number of available options
will vary from month to month; therefore, the number of
neutral hedges from which to calculate r values will vary
as well. This situation must and will be considered in the
statistical analysis.
The option closing prices were observed for each month
from the Wall Street Journal at the beginning of the fol
lowing month. Thus, the firstofthemonth Wall Street
Journals were consulted from December 1, 1973, through
September 1, 1975. Stock prices were also observed as
monthly closing quotes from the same Wall Street Journals.
Dividends were obtained from Moody's Dividend Record for
the period; these amounts were first converted into con
tinuously compounded dividend yields and then employed in
equation (2') for the computation of the "hedge ratios."ll
Two proxies for the riskless rate are used in the study.
For use in equation (2'), r was assumed to be the midpoint
TABLE 42
Underlying Stocks Used in the Study
AT&T
Atlantic Richfield
Bethlehem Steel
Brunswick
Eastman Kodak
Exxon
Ford
INA
International Harvester
Kresge
Loews
McDonalds
Merck
NW Airlines
Pennzoil
Poloroid
RCA
Sperry Rand
Texas Instruments
Upjohn
Weyerhauser
Xerox
Avon
Citicorp
IBM
ITT
Kerr McGee
4MM
Monsanto
Sears
of the two continuous rates of interest implied by the bid
and ask prices of U.S. Government treasury bills maturing
at the (approximate) time the option expires. For use in
the time series hypothesis test, rf is the midpoint of the
2 onemonth rates of interest implied by the bid and ask
prices of treasury bills with 1 month to maturity. Thus
for each month t, 1 observation for rft is available. The
observations for r and rf are from the same Wall Street
Journals as the option prices and stock prices.
Preliminary Procedure
For each month, t, the number of available options is
necessarily the number of different neutral hedge position
returns, Nt. For example, in month 1, the number of options
and thus hedge returns, is 119. The number of "available"
neutral hedge positions in each of the 21 months of the
study is provided in Table 43. From Table 43 one can see
that 1381 total observations are included in Period 1, and
1604 in Period 2, for a combined total of 2985. Since the
returns in any month are representative of no more than 30
underlying stocks, there is often more than 1 hedge return
for each of the underlying stocks for each month.
Neutral Hedge Returns
Consider the jth neutral hedge for month t, out of the
total of Nt available neutral hedges for that month. Let
Sjt represent the price at the beginning of month t of the
underlying stock of the jthneutral hedge. Similarly, C
underlying stock of the j neutral hedge. Similarly, C.
Jt
TABLE 43
The Number of Options Employed in the
Study
Month Number of Options
Number Month Employed in the Study
1 December 1973 119
2 January 1974 116
3 February 1974 115
4 March 1974 157
5 April 1974 112
6 May 1974 110
7 June 1974 133
8 July 1974 97
9 August 1974 137
10 September 1974 163
11 October 1974 125
12 November 1974 197
13 December 1974 201
14 January 1975 127
15 February 1975 156
16 March 1975 182
17 April 1975 126
18 May 1975 159
19 June 1975 136
20 July 1975 136
21 August 1975 181
Total 2,985
denotes the price at the beginning of month t of the option
.th
of the j neutral hedge of month t. The associated hedge
ratio, defined from equation (2'), is 1/(ed N(dj)). The
corresponding ending prices of the stock and the option are
denoted as Sjt+ and Cjt+1, respectively. The realized
return on the jth neutral hedge over month t is thus given
by: S 1 (C C.)
jt+l Sjt dT jt+ Cjt
Xjt= N(d)jt (4)
1 C.
t ed *.N(di)jt t
The denominator in equation (4) is the amount of equity
assumed to be invested in the jth neutral hedge at the be
ginning of month t. The numerator in (4) represents the
change in the dollar position of the hedge from the beginning
of the month to the end of the month.
Naive Diversification
From the Nt options in period t, portfolios of neutral
hedges of size 5, 10, 15, 20, 25, 30, 35, 40, 45 and 50 will
be examined. Portfolios of the various sizes will be con
structed for each month of the study; for any of these port
folios the neutral hedges to be included will be randomly
selected (with replacement). First, the computer selects a
random number representing one of the 30 stocks; next,
another random number determines which of the options to
choose for that stock. Then another stock is chosen randomly,
and so on. This process is repeated until the desired number
of neutral hedges has been randomly drawn from the Nt
available positions. The portfolio return is calculated as
an arithmetic average of the returns on each of the neutral
hedges assumed to be in the portfolio. Therefore, equal
amounts of invested equity are simulated in each of the
neutral hedges of the portfolio.
For each portfolio size for each month, 30 portfolios
will be constructed according to the random selection method
described above. The variance of the 30 portfolio returns
will serve as an estimate of the portfolio variance of a
naivelydiversified portfolio of that number of neutral
hedges. 'Although the simulated portfolios could easily
contain some of the same hedge positions, the returns of
each of the 30 portfolios for a given size will be assumed
to be independent. Thus, the variance of the 30 portfolio
returns for a given portfolio size serves as the variance
estimate of a portfolio of that number of neutral hedges.
The results of this naive diversification analysis are
reported in the next chapter.
The Hypothesis Test
Preliminary Test Procedure
Let each of the neutral hedge returns be transformed
into "excess return" formdefine Xe as
jt
X = Xt r (5)
jt jt ft
for all j in the month, and for every month, t. For any
month t, j may take on a value from 1 to Nt.
Let PI stand for Period 1 and P2 stand for Period 2.
The hypothesis test is a test for the difference between
e e
the mean of the Xe values in P1 and the mean of the Xe
jt jt
values in P2. Let U stand for the mean of the Xe in P2.
"1 jt
To test the hypothesis, specify the following relationship:
Xe P1 = U + e. (6)
jt 1 j
Xe P2 = U + ej (7)
jt 2 e
where e is an error term.
In order to test the hypothesis, the mean return in
each month t must be calculated as follows:
Nt
e = EX N (8)
t j=t t
If (a) the Nt are equal for all t; (b) the standard
e
deviations around the X are equal for all t (that is, if
Nt
(Xe t e 2
S jl jt
SN 1
are equal for all t); and (c) there is no systematic risk in
the neutral hedges, then the test procedure outlined in
Chapter I may be employed. Thus the following regression
would be performed:
e
X 1 0 e
S: 1
P1
e U
X12 1 0 1 e2 (9)
e U2 U1 e
13
P2
e 1 1 e
21 21
Equation (9) may also be expressed as follows:
e 12 U1 + e (9')
{ 1 1 21 U2 U1 e t
where I1 and 12 are defined in Chapter I.
However, since Nt is not the same value for all t, this
problem must be solved. In addition, there is potential
,
heteroskedasticity in the X even without the problem of
the varying Nt values. Both of these problems may be
simultaneously overcome by converting the ordinary least
squares regression in (9) into generalized least squares
form. This procedure involves weighting each of the Xe
and each of the observations of the independent regression
variables. The weights, Wt, are defined below:
W= (10)
t
where
S (11)
xt vTN
t
and where
rNt
Nt e e 2
E (X. X)2
3t t
S j=l (12)
t N 1 (2)
1e
t
The multiplication of the Xe and the independent
variables vectors by the Wt values accounts statistically
for the information value of each of the X~. The following
new regression is specified:
el__
1e 1X
1X 1 0 S
.xl 1
S 1
1 012
e U
W2X2 12 1 S(13)
W ye 1 1 U U e3
1313 2 1 13
_S =
x13 x13 x13
e21
W e 1 1
W X S
21 21 S S 21
x21 x21
Another statistical problem is the potential correla
tion over time of the error terms. Correlation over time
could be indicative that some factor other than option
trading is nonrandomly affecting the equilibrium expected
excess returns on zero systematic risk securities. Among
e
the factors which could affect r returns in that manner
are changes in expected inflation and changes in the
interest rate term structure. Although these factors are
exogeneous to the capital asset pricing model used in the
theory of Chapter II, movements of these variables to new
equilibrium levels could nevertheless cause trends in the
e
X time series values. In order to overcome this potential
problem, the regression in (13) above should be repeated
using the CochraneOrcutt (1949) procedure of accounting
for correlation over time.
Two Additional Procedures
Two additional problems were discussed earlier, and
must now be considered. The first problem is the possibility
of systematic risk appearing in the 1month holding period
returns of the neutral hedges. The second difficulty is the
possibility of a change in the riskreturn slope during the
time period of the study. This effect could be due to the
new option trading, although this idea has not been
specifically hypothesized here.
As a possible way to overcome the first of these
potential problems a new variable, representative of the
12
market index, will be introduced into the regression.1 The
CRSP13 Valueweighted index of stock returns, with dividends
reinvested, will be employed as the index variable. Since
new option trading could have a simultaneous effect on the
behavior of the index, an interactive variable is also
added to the regression.
Let Mt denote the value of the market index return for
month t; then define the new interactive variable to be MIt
0 in Period 1, MIt = Mt in Period 2. As before, these 2 new
independent variables must be adjusted for the impact of
heteroskedasticity (differences in the S ). The full re
14
gression model is:
WXe W 0 e
1 1 1 1 0 e
c 2 1
e
W X W WM M e
1212 12 0 212 0+ 12 (14)
13X3 W13 13 13 13 13 13 13
S2 1
e
W ye W W WM W M e
21 21 21 21 21 21 21 21 21
Equation (14) may be rewritten in vector form as
equation (14') below:
c1
SI al(14
W j = Wt 12 Mt MIt + et (14')
2 1
Assuming for the moment that Mt measures the true mar
ket portfolio, the coefficient of Mt in equation (14) is the
systematic risk measure for the hedges for Period 1.
Similarly, the coefficient for MIt in equation (14) is the
change in systematic risk from Period 1 to Period 2. The
coefficient of 12, a2 al in equation (14), would no longer
be interpreted as the change in the mean neutral hedge
return from Period 1 to Period 2. More importantly though,
a2 a" could still be interpreted as the change in the mean
of the zero systematic risk portion of the neutral hedge
returns from Period 1 to Period 2.15 In equation (14) as in
equation (13), the potential serial correlation in the error
terms is to be accounted for by the use of the CochraneOrcutt
procedure.
The second problem concerns the possible impact of a
change in the slope of the (singleperiod) riskreturn re
lation during the time period of the studya neutral hedge
return may be affected if the underlying stock is shifting
to a new equilibrium level of expected return. Thus, if
the underlying stocks were, on the whole, imbalanced in terms
of their systematic risk, then the Xt could carry effects
other than changes in the riskreturn intercept. Thus, if
the underlying stocks consist of too many high systematic
risk stocks, or too many low systematic risk stocks, and if
the potential riskreturn relation shift is significant
enough, then the test outlined above may not capture the
effects of the options on E(r).
The following procedure will be employed to see if
this problem is present: A set of 7 or 8 underlying stocks
with the highest risk is formed; similarly, a set of 7 or 8
underlying stocks with the lowest systematic risk is
assembled. For each of these 2 sets, the neutral hedge
returns are examined by rerunning the basic tests previously
described. If the neutral hedges of the high systematic
risk stocks exhibit similar behavior to the neutral hedges
of the low systematic risk stocks, then the possible shifting
slope of the riskreturn relation is most likely not a
significant problem for this study.
The most common systematic risk measure is "beta," which
is proportional to the covariance between a stock's return
and the market return.16 The betas for all of the 30 under
lying stocks were gathered from Value Line Investment Survey
for the periods included in this study.17 Since Value Line
updates its beta estimates every quarter, the betas on all
of the underlying stocks will generally vary from quarter
to quarter. However, all 8 stocks in Table 44a had betas
among the highest 10 of the 30 underlying stocks, each
quarter. Also shown are the betas for each of the 8 stocks
for the beginning, middle and ending quarters of the study.
Similarly, 7 stocks were consistently in the bottom
10 of the underlying stocks, ranked by beta. These 7 stocks
and their betas are shown in Table 44b. The empirical
results of the hypothesis tests are presented in Chapter V
along with the naive diversification analysis findings.
TABLE 44a
High Beta Stocks
4th 4th 4th
Quarter Quarter Quarter
1973 1974 1975
1. Brunswick 1.73 1.85 1.65
2. Loews 1.78 1.60 1.40
3. McDonalds 1.34 1.55 1.55
4. NW Airlines 1.71 1.70 1.55
5. Pennzoil 1.40 1.40 1.35
6. Poloroid 1.23 1.45 1.40
7. Sperry Rand 1.42 1.30 1.30
8. Texas Instruments 1.32 1.25 1.20
Average "HighBeta" 1.491 1.513 1.425
Average of all 30 stocks 1.178 1.197 1.178
TABLE 44b
Low Beta Stocks
4th 4th 4th
Quarter Quarter Quarter
1973 1974 1975
1. ATT .77 .75 .75
2. Merck 1.00 1.00 1.05
3. Exxon 1.12 .85 .95
4. Bethlehem Steel 1.00 1.05 1.05
5. Minn. Mining Manuf. 1.00 1.00 1.05
6. Sears .95 1.00 1.05
7. IBM 1.04 1.05 1.05
Average "Low Beta" .983 .957 .993
Average of all 30 stocks 1.178 1.197 1.178
Notes
A popular example of a direct exogeneous influence on
rf is federal monetary policy. Monetary policy which in
fluences rf could indirectly influence E(r ).
2See Galai (1977).
Black and Scholes claimed this result for neutral
hedges consisting of stock and option positions. However,
the BlackScholes argument carries over to neutral spreads.
Neutral spreads and hedges are riskless (zero variance)
over the instant of creation and would remain riskless if
investors could continuously revise portfolio positions as
stock prices change. The inability of investors to con
tinuously rebalance is the source of the variance to neutral
spreads and hedges held for discrete intervals; systematic
risk is presumably not present, however.
The other assumptions from which equation (1) is de
rived are discussed elsewhere. See, for example, Black and
Scholes (1973), Cox and Ross (1976), Rubinstein (1976b),
Brennan (1979), Cox, Ross, and Rubinstein (1979), and
Rendleman and Bartter (1979).
The conditions are small holding period intervals and
normallydistributed underlying stock prices. See note 3.
Black and Scholes, Galai, and Finnerty also found
neutral hedge returns to be larger than treasury bill yields.
Assuming the meanvariance framework is appropriate, and
assuming the neutral hedges really did not have any systema
tic risk, the positive excess returns support the Black
equilibrium model over the SharpeLintner model. Thus, the
positive excess neutral hedge returns would, by implication,
also support the contention that, prior to the vast expan
ion of option trading in the late 1970's, E(rZ) exceeded rf.
To some extent these results are immune to the Roll critique,
since neutral hedge returns are theoretically of zero sys
tematic risk, Instead of discussing the implications of
their findings in terms of comparing the Black and Sharpe
Lintner models, the option empiricists considered the
positive excess returns to be evidence of option market
inefficiency. For a different point of view see Phillips
and Smith (1980), who discuss the role of trading costs.
7Boyle and Emanuel show that, if the correlation
between underlying stocks is p, then the correlation between
two neutral hedges is p2
8
The portfolio variance asymptote found in this study
may be compared to the variance estimates of E(r,) found by
other methods by researchers such as Fama and MacBeth (1974).
9
It is assumed at this point that "empirical" neutral
hedges, held for one month without rebalancing, actually
have zero systematic risk. The procedure outlined here in
the text suggests a possible way of gauging the amount of
systematic risk in the neutral hedges. Under the procedure
outlined and assuming a large number of different option
hedges are available for portfolio simulation, the portfolio
variance should approach an asymptote of about 1/4 of the
variance of individual hedges. If portfolio variance cannot
be reduced by 3/4, then some systematic risk is probably
in the neutral hedge returns. If the variance, on the other
hand, drops by 3/4, then little or no systematic risk is
probably in the hedges. The reader should be aware that
although each neutral hedge is uncorrelated with the market
portfolio, each neutral hedge does have some correlation
with each other neutral hedge generally. (See note 7.)
1Some reasons why option prices have occasionally been
observed to be below intrinsic value have been pointed out
by Galai (1977, p. 172).
A hedge ratio is the reciprocal of dC/dS. The hedge
ratio is the number of option contracts to short against
100 shares of the stock to create a neutral hedge.
12
The results of this procedure must be viewed in the
same light as the systematic risk measures obtained by Black
and Scholes, Galai, and Finnerty, because of the implications
of Roll's critique. However, the time series of mean neutral
hedge returns is theoretically uncorrelated with a market
index. Whether (or not) the time series of neutral hedge
returns has significant correlation with respect to the
index may be information that will cause us to have less
(or more) confidence that neutral hedges are really un
correlated with the "true" market portfolio. See note 9,
also, for the primary way by which the hedges will be
examined for systematic risk.
13
Center for Research in Security Prices at the Univer
sity of Chicago.
1This model has been credited to Gujarati (1970).
1This method of checking for systematic risk is
supplementary to the method discussed in note 9.
