RELATIVE PRICES OF OPTIONS, FORWARD CONTRACTS,
AND FUTURES CONTRACTS: THEORY AND EVIDENCE
BY
GAUTAM DHINGRA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1986
To my parents
ACNOWLEDGEMENTS
I thank the members of my dissertation committee, Robert
Radcliffe (chairman), Stephen Cosslett, Roger Huang, and
M.P.Narayanan, for their guidance and for helpful comments on
earlier drafts of this study. Thanks are also due to Young
Hoon Byun who generously gave his valuable time to discuss
various issues pertaining to this dissertation.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS . . . . . . . ... .. .iii
LIST OF SYMBOLS . . . . . . . . . .. . vi
ABSTRACT . . . . . . . . .. . . . vii
CHAPTERS
I INTRODUCTION . . . . . . . . . 1
Objective of the Study . . . . . . . 2
Tasks of the Study . . . . . . . . 2
II FORWARD CONTRACTS AND FUTURES CONTRACTS . . . 6
Introduction . . . . . . . . . 6
Previous Research . . . . . . . . 7
A New Approach to Comparing Forward Prices
and Futures Prices . . . . . ... .12
Selection of an Appropriate Asset . . . .. .16
Data . . . . . . . . .. . . . 17
Empirical Results . . . . . . . ... .20
Direction and Magnitude of the Difference
between Forward and Futures Prices .... .20
Test of Settlement Price Effect . . . .. .22
Test of Daily Resettlement Effect . . .. .27
QuasiArbitrage Opportunities in the
Futures Market . . . . . . ... .31
Transactions Costs and QuasiArbitrage . . .. .35
Time Trend in QuasiArbitrage Opportunities . . 40
Conclusion . . . . . . . ... . . 42
III OPTIONS AND FORWARD CONTRACTSTHE ISSUE OF
EARLY EXERCISE . . . . . . ... .43
Forward Contracts and The European
Options Portfolio . . . . . ... .43
Forward Contracts and The American
Options Portfolio ............. 45
Early Exercise of American Options . . . .. .46
Previous Research . . . . . . .. .46
Data . . . . . . . . . . . 49
Empirical Results . . . . . . . 51
Summary statistics . . . . . . 52
Test of dividend effect . . . . .. 55
Choice between closingout in secondary
market and exercisingRole of
transactions costs . . . . ... .57
Test of other early exercise propositions . 61
Conclusion . . . . . . . . . 64
IV FUTURES CONTRACT AND THE AMERICAN
OPTIONS PORTFOLIO . . . . ... .. .65
Introduction . . . . . . . . ... .65
Data . . . . . . . . . . . 66
Empirical Results . . . . . . . ... .67
Direction and Magnitude of the Difference
between Futures Prices and Inferred
Forward Prices . . . . . . ... 67
Test of NonSynchronicity Effect . . . . 69
Time Trend in the Difference Between Futures
Prices and Forward Prices . . . . .. .71
QuasiArbitrage Opportunities in the
Options Market . . . . . . ... 73
Transaction Costs and QuasiArbitrage ...... .76
Conclusion . . . . . . . .... . 79
V SUMMARY AND CONCLUSIONS . . . . . .. .80
BIBLIOGRAPHY . . . . . . . . ... . . 83
BIOGRAPHICAL SKETCH . . . . . . . . .. 85
LIST OF SYMBOLS
t = Initiation date of the forward, futures and option
contracts
s = Maturity date of the forward, futures and option
contracts, s>t
B(T) a Price, at time T, of a defaultfree discount bond
which pays one dollar at s, t
S(T) a Price at time T of the asset on which the contracts
are written
f(T) a Futures price at time T
g(T) E Forward price at time T
g (T) e Forward price inferred from option prices at time T
c(T) E Value of a European call option at time T
C(T) a Value of an American call option at time T
p(T) a Value of a European put option at time T
P(T) e Value of an American put option at time T
k(T) a Difference between the value of a European call and
a European put option, i.e., k(T)=c(T)p(T)
K(T) a Difference between the value of an American call and
an American put option, i.e., K(T)=C(T)P(T)
E a Exercise price of calls and puts
D(T) E Future value of all dividends to be paid from T to s
X(T) a Difference between futures prices and forward
prices, i.t X(T)af(T)g(T)
Z(T) = Difference between futures prices and inferred
forward prices, i.e, Z(T):f(T)g (T)
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
RELATIVE PRICES OF OPTIONS, FORWARD CONTRACTS,
AND FUTURES CONTRACTS: THEORY AND EVIDENCE
By
Gautam Dhingra
August 1986
Chairman: Robert C. Radcliffe
Major Department: Finance, Insurance, and Real Estate
Options, forward contracts and futures contracts are
traded independently in the market. However, there are some
common linkages among them. Some of these linkages are
analyzed in this dissertation.
First, equilibrium forward prices are compared to observed
futures prices to determine the impact of daily resettlement
feature of futures contracts. Using the data on Major Market
Index futures, significant differences between futures prices
and forward prices are observed during 1985. The differences
are too Ilpje to be explained by daily resettlement. In fact,
the differences are large enough to allow for the existence
of "quasiarbitrage" opportunities between spot and futures
markets. However, such opportunities seem to have declined
over time, presumably as a result of the actions of the
arbitrageurs.
Second, a specific portfolio of options is compared to a
forward contract. It has been shown that if the options are
European, this portfolio is equivalent to a forward contract.
It is argued in this study that this portfolio closely
approximates a forward contract even when the options are
American. The argument is even stronger for the case of index
options. These options are used to infer forward prices which
are then compared with observed futures prices. It is found
that the difference between futures prices and inferred
forward prices is substantially smaller than the difference
between futures prices and equilibrium forward prices. This
implies that during 1985, quasiarbitrage opportunities in
options and futures markets, if they existed at all, were
less abundant than similar opportunities in spot and futures
markets. Partial evidence regarding the existence of
quasiarbitrage opporunties within the options market is
also uncovered.
As a byproduct of the main analysis, a number of
propositions regarding early exercise of American options are
analyzed. It is found that dividends do not influence early
exercises of index options in the manner suggested by theory
for individual stock options.
viii
CHAPTER I
INTRODUCTION
Financial markets have experienced a proliferation of
securities in recent years. This propagation process is
nowhere more evident than in the options and futures markets
where a multitude of novel contracts have been introduced
since the mid1970s. Some of these contracts have gained
widespread acceptance in the short span of time that they
have been in existence. This is evident from the following
table which provides an overview of the trading activity in
some of the most popular financial futures and options.
Table 1
Number of Contracts Traded on June 2, 1986
Underlying Options Futures
Instrument Volume Volume
1. NYSE Composite Index 7514 15400
2. Treasury Bonds 586 310718
3. Major Market Index 77591 7648
4. S&P 500 Index 6300 99929
5. S&P 100 Index 338823 NC
6. Value Line Index 9091 5316
7. Swiss Franc 14592 24516
8. W.German Mark 6924 26978
NC: no contracts available
(Source: The Wall Street Journal, June 3, and June 4, 1986)
Forward contracts do not find a place in this table
because of lack of reported data about their trading
activity. Nevertheless, they are an important part of the
discussion that follows.
In recent years it has been common to find a number of
assets on which all three types of contractsoptions,
futures and forwardsare traded simultaneously. It should
not be surprising that some strong interrelationships exist
among them stemming directly from the fact that they have the
same underlying asset. Some of these interrelationships are
anlayzed in this dissertation.
Objective of the Study
The objective of this study is twofold:
(1) To consolidate our knowledge regarding the inter
relationships among the following three types of
contracts
(a) Options
(b) Forward Contracts
(c) Futures Contracts
(2) To empirically test the interrelationships among these
contracts.
Tasks of the Study
The following tasks are undertaken in order to achieve the
objectives listed above.
The first issue addressed is the relationship between
forward contracts and futures contracts. The major economic
difference between the two contracts is that futures
contracts are settled daily (i.e., are markedtomarket),
whereas forward contracts are settled only at maturity.
Several researchers have shown that when interest rates are
nonstochastic, forward prices must be equal to futures
prices. However, in the presence of stochastic interest
rates, daily resettlement can cause futures prices to differ
from forward prices.
A number of researchers have empirically investigated the
effect of daily resettlement. A drawback of these studies is
that they are unable to isolate the economic cause of the
difference between futures prices and forward prices, namely,
the daily resettlement feature, from the institutional causes
such as higher liquidity and guaranteed performance of
futures contracts. Moreover, forward contracts present a
problem in testing, first, because there is a general lack of
good quality data regarding their trading activity, and
secondly, because the maturities of forward and futures
contracts are difficult to match.
The approach used in this study to solve these problems is
to compare futures prices observed in the market to the
forward prices based on a simple, yet powerful, arbitrage
model. In doing so the effect of daily resettlement can be
analyzed by abstracting from the institutional reasons that
may cause forward prices to differ from futures prices. A
second advantage of this approach is that a large data set
can be obtained so that more reliance can be placed on the
statistical results. An empirical study is carried out to
answer the following questions.
(1) Is there a difference between futures prices and forward
prices for stock index contracts?
(2) Is the difference related to markingtomarket?
(3) Is the difference a function of
(a) Time to maturity of the contracts?
(b) Procedure for establishing the settlement price of
futures contracts?
(4) Does the difference follow a systematic trend over time?
(5) Are there any arbitrage opportunities in the spot and
futures markets related to the difference between futures
prices and forward prices?
The detailed analysis and empirical results are presented
in Chapter II.
The second task of this study is to analyze the
relationship between a forward contract and a specific
portfolio of American options called "The American Options
Portfolio." It is easy to show that a portfolio of European
options, called "The European Options Portfolio" is exactly
equivalent to a forward contract on the underlying asset.
However, when extended to American options the relationship
between a similar options portfolio and a forward contract is
less definite because of the possibility that American
options may be exercised early.
To determine the importance of early exercise, it is
necessary to empirically analyze relevant data regarding
early exercises of American options. However, .'ire have been
no studies, to my knowledge, which have undertaken this task.
I carry out an empirical study to ascertain the importance of
early exercise. This study consists of determining the
absolute magnitude of early exercises as well as testing some
theoretical conjectures regarding early exercise of American
options. The detailed analysis and relevant empirical results
are presented in Chapter III.
The third task of the dissertation has its roots in the
analysis done in Chapter III. It is argued that "The American
Options Portfolio" closely approximates a forward contract.
Therefore, the relationship between forward and futures
contracts can be extended to cover "The American Options
Portfolio" and futures contracts. This argument is used to
infer forward prices from options and compare them to
corresponding futures prices. It is expected that the results
obtained in this section will be consistent with those
reported in Chapter II. This task is described in detail in
Chapter IV.
Chapter V summarizes and concludes the discussion by
bringing together all the results that bind together options,
forward contracts and futures contracts.
CHAPTER II
FORWARD CONTRACTS AND FUTURES CONTRACTS
Introduction
Forward contracts and futures contracts are quite similar
in the sense that both involve buying or selling an asset at
a future date for a fixed nominal price determined at the
time the contract is written. However, there are some
important economic and institutional differences between the
two contracts. The economic difference is that futures
contracts are settled daily whereas forward contracts are
settled only at maturity. The institutional differences vary
from contract to contract but, in general, forward contracts
are less standardized, have poor secondary markets, and are
not guaranteed by the exchange.
Forward Contract
A forward contract is an agreement to buy (or sell) the
underlying asset at time s at a price called the forward
price, g(t), determined at time t. The investment required at
time t is zero and there are no payoffs from the contract
until the maturity date s.
Futures Contract
A futures contract is an agreement to buy (or sell) the
underlying asset at time s, at a futures price f(t) fixed at
time t. Like a forward contract, the investment required for
a futures contract at time t is zero.1 However, unlike a
forward contract, the futures contract is markedtomarket at
the end of every day. The holder of a long position (the
buyer) can withdraw profits at the end of the day, and, in
case of a loss, must pay the difference to the seller. Since
the profits from a futures contract are realized as they are
earned, at maturity the price paid by the buyer to the seller
is the spot price prevailing at that time.
By definition both forward price and futures price at
maturity equal the spot price, i.e., g(s)=f(s)=S(s).
Previous Research
The relationship between forward and futures contracts has
been studied extensively. Black (1976) was one of the first
researchers to distinguish between the two contracts by
explicitly taking into account the daily resettlement feature
1The margin required for forward and futures contracts
is not considered an investment since the investor is assumed
to be able to borrow at the riskfree rate. He can buy
treasury securities with borrowed funds and deposit them as
margin. As a result, the net cost of establishing a position
is zero. Moreover, the treatment of margin is different for
futures contract as compared to other assets. This fact is
explained by Dusak (1973) as follows.
Unlike other capital assets such as common stocks where
the margin is transferred from buyer to seller, the margin
on a futures contract is kept in escrow by the broker. Not
only does the seller not receive tir capital transfer from
the buyer but he actually has to .cposit an equivalent
amount of his own funds in the broker's escrow account.
. The margin . is . merely a good faith deposit
to guarantee performance by the parties to the contract.
(page 1391)
of futures contracts. Since then, other researchers have also
focused their attention on this distinction and derived some
important analytical results. Margrabe (1978) and Jarrow and
Oldfield (1981) have derived a common result that if interest
rates are nonstochastic, forward prices should be equal to
futures prices. The intuition behind this result is quite
simple, as explained below.
As a result of daily resettlement of futures contracts,
the investor benefiting from the futures price movement on
any given day receives the cash proceeds from the investor
holding the opposite position and has the opportunity to
invest those proceeds at the prevailing interest rate. The
investor holding the opposite position must come up with the
requisite cash, presumably by borrowing at the prevailing
interest rate. For both investors the future interest rate is
an important variable in determining the net benefit due to
daily resettlement. Therefore, if there is no uncertainty
regarding interest rate that will prevail at each point of
time until maturity (i.e., if interest rates are non
stochastic), forward prices must be equal to futures prices.
(See Jarrow and Oldfield (1981) for a lucid proof of this
proposition).
Cox, Ingersoll and Ross (CIR) (1981) further investigate
the effect of stochastic interest rates on the magnitude of
the difference between forward prices and futures prices. CIR
derive an arbitrage proof to show that the difference between
the two prices depends upon the relationship between futures
prices and shortterm interest rates. If the two variables
have positive covariance, then forward prices must be lower
than futures prices. The opposite is true if futures prices
and shortterm interest rates have negative covariance. The
magnitude of the difference depends on the magnitude of the
covariance between futures price and shortterm interest
rates and the time to maturity of the contracts.
The intution behind the CIR result is explained well by
Klemkosky and Lasser (1985) as follows.
when the futures price falls, if there is a negative
correlation between the futures price and shortterm
interest rates, the buyer of the contract must borrow for
payment to the seller at a higher interest rate than
existed when the contract was issued. When the futures
price rises, the buyer will be able to invest the
resettlement, but at a lower rate. The seller, on the
other hand, will be able to invest when rates rise and
must borrow when rates fall. (page 610)
CIR (1981) show that if forward prices and futures prices
do not behave in this fashion, an arbitrage profit can be
obtained by undertaking the following strategy: Buy a forward
contract, sell B(j) futures contracts in each period j,
liquidate them in the next period and invest the (possibly
negative) proceeds into riskfree bonds. This arbitrage
Strictly speaking, it is the local covariance between
percentage changes in futures prices and percentage changes
in bond prices that should be either always negative or
always positive for this proposition to be meaningful (Cox,
Ingersoll, and Ross (1981), page 326). See French (1983) for
a discussion of this assumption for empircial testing (page
330, footnote 21).
process prescribes the following relationship between futures
prices and forward prices (CIR, Proposition 6, page 326).
s1
f(t)g(t) = PVt E [f(j+l)f(j)][B(j)/B(j+l)l]/B(t) (1)
j=t
where PV is the present value operator.
In a continuoustime framework this equation reduces to
the following equation.
s
g(t)f(t) = PVt[jf(u)cov(f'(u),B'(u))dul/B(t) (2)
t
where cov(f'(u),B'(u)) is defined as the local
covariance of the percentage change in the
futures price, f'(u), and the percentage
change in bond price, B'(u).
This result implies that if the local covariance between
futures prices and bond prices is positive for every time
from t to s, forward prices will be greater than futures
prices. Conversely, for negative covariance futures prices
will be greater than forward prices. Note that this equation
does allow for the possibility that forward prices and
futures prices may be equal even when interest rates are
stochastic. This is possible if the local covariance between
bond prices and futures prices is zero for each period until
maturity.
A number of studies have empirically investigated the
difference between futures prices and forward prices. Some of
them have tested whether the observed differences are in line
with the prediction of the CIR model.
Cornell and Reinganum (1981) find that there is no
significant difference between forward prices and futures
prices on foreign currencies. Since they find that the
covariance between shortterm interest rates and currency
futures prices is negligible, their findings are consistent
with the CIR model. Cornell and Reinganum also find that
Tbills show greater difference between forward prices and
futures prices even though the covariance between shortterm
interest rates and Tbill futures prices is negligible. This
difference is apparently inconsistent with the CIR model.
Cornell and Reinganum suggest that the inconsistency may be
caused by factors other than markingtomarket. They offer
tax treatment of Tbills and problems associated with
shorting Tbills as primary candidates for explaining the
discrepancy.
French (1983) compares forward prices and futures prices
on two commoditiessilver and copperand finds significant
differences between them. He finds some support for the CIR
model in explaining the differences between the two prices.
Park and Chen (1985) find that there are no significant
differences between forward and futures prices on foreign
currencies, but such differences are significant for
contracts based on physical commodities. They find strong
support for the CIR model.
A New Approach to Comparina
Forward and Futures Prices
The empirical studies described above have one or both of
the following drawbacks.
(1) The data on forward contracts are not only difficult to
obtain, they are often of poor quality too. This problem
is evident in the studies by French (1983) and Park and
Chen (1985).
French compares forward and futures prices which are
observed in different countries and at different times
and are denominated in different currencies. Park and
Chen have problems in getting a large number of
observations because forward contracts and futures
contracts trade under different conventions. Forward
contracts are issued with standard maturity periods,
i.e., on every day, a onemonth, a threemonth, a
sixmonth, and other such contracts are available. On the
other hand, futures contracts are traded on the basis of
standard maturity dates. Therefore, a threemonth futures
contract is available only on the day it is initiated or
when a longer maturity contract has exactly three months
left to maturity. For this reason it is difficult to
obtain enough observations for which forward contracts
and futures contracts have the same time to maturity.
(2) The second drawback of these studies is their inability
to take into account qualitative factors such as higher
liquidity, greater degree of standardization, and
guaranteed performance of futures contracts. These
studies use forward prices observed in the market,
compare them to futures prices and attribute the
difference to markingtomarket. Since forward contracts
differ from futures contracts along other qualitative
dimensions too, it is not clear how the differences
observed can be attributed solely to markingtomarket.
In order to isolate the markingtomarket effect forward
prices, that are free from these extraneous factors, are
needed. It is quite obvious that one cannot hope to observe
such "perfect" forward prices in the market. However, they
can be determined quite accurately by a simple, yet powerful,
arbitrage model.
This wellknown arbitrage model of forward prices,
sometimes known as the costofcarry model, simply says that
the forward price of an asset must equal its spot price plus
the net costs associated with buying the asset today and
holding it until maturity of the contract. If such is not the
case then arbitrage will take place.
For a financial asset which provides no intermediate cash
flows, the cost associated with holding the asset is simply
the interest cost (the opportunity cost of money). Thus, the
forward price for such an asset is given by the following
model.
glT) (T) (3)
T (T)
where g is the forward price
S is the spot price of the asset
B is the price of a discount bond which
pays $1 at maturity, s.
If this price does not prevail, an arbitrage profit is
available. For example, if g(T)>S(T)/B(T), an investor can
buy the asset in the spot market by borrowing the money at
the riskfree rate, and short a forward contract on the same
asset. This strategy costs nothing and gives a positive
payoff of [g(T)S(T)/B(T)J at maturity. If g(T)
then the strategy is reversed to make a riskless profit.
This model can be easily adjusted for assets that provide
intermediate cash flows (e.g., dividends on common stock).
3The assumption that investors can borrow and lend at
the riskfree rate is not inordinately restrictive. The
following quote from Cox and Rubinstein (1985) attests to
that.
The main reasons private borrowing rates exceed lending
rates are transaction costs and differences in default
risk. Transactions cost per dollar decline rapidly as the
scale increases, so they are of secondary importance in a
large operation. And if the arbitrage operation in which
we are using these funds is indeed riskless, it should be
possible to collateralize the loan so that the lender will
bear no possibility of default. (page 40)
Moreover, this assumption does not require that all investors
be able to borrow and lend at the riskfree rate. So long as
there are a few previleged investors who can do so, the
equilibrium forward prices will prevail.
Such cash inflows can be considered as negative carrying
costs and the model can be rewritten as follows.
g(T) S(T) D(T) (4)
g(T) = (T) D(T)
B(T)
where D(T) is the known future value of all
dividends to be paid from T to s
This study proposes that in order to isolate the marking
tomarket effect, forward prices, to be used for comparison
with futures prices, be based upon the arbitarge model
described above. This will eliminate the extraneous factors
and also resolve the problem of mismatch of maturities of
forward and futures contracts.
4Most companies do not like their dividends to fluctuate
dramatically. Hence, dividends on individual companies are,
generally, quite predictable. The predictability of dividends
is substantially higher for stock indices because
fluctuations in the dividends of individual companies are
smoothed out when they are aggregated in an index. Therefore,
in my opinion, the assumption that dividends on a stock index
are known, is reasonable.
Wu (1984) does not agree with this assumption. He argues
that index futures prices are affected by uncertainty of
dividends. He reports that when this uncertainty is taken
into account the theoretical futures prices, given by the
costofcarry model, decrease significantly. Since this model
is used in this study to determine forward prices of stock
indices, the forward prices used in this study would be
upwardly biased according to Wu. A major result of this study
is that futures prices inordinately exceed forward prices
leading to quasiarbitrage opportunities. This argument will
become even stronger if Wu's contention is correct.
Selection of an Appropriate Asset
To get the most reliable results from the approach
described above, one must be careful in the selection of
contracts for empirical testing. Some of the criteria for
selection are as follows.
(1) The underlying asset should be such that the cost of
buying the asset and carrying it over a period can be
determined fairly accurately. This criterion is essential
for the equilibrium forward price, given by equation (4),
to be measured accurately. On this criterion, all
nonfinancial assets are ruled out since it is difficult
to estimate their carrying costs precisely. For financial
assets the carrying cost can reasonably be assumed to be
the opportunity cost of money measured by the interest
rate which is easily observable.
(2) The asset selected should have sufficient liquidity in
futures trading so that a large sample can be obtained
for reliable statistical testing. A number of financial
assets satisfy this criterion as can be seen from Table
1. The most liquid futures contracts are TBond futures
but they are not given to easy testing of the markingto
market effect, primarily because of the delivery option
associated with them. Stock index futures offer a good
alternative since there is no delivery option associated
with them.
(3) The choice of a particular index futures contract is
dictated by the ease with which the index can be
duplicated in the spot market. This criterion is
essential for the arbitrage process, that determines the
equilibrium forward price, to be successful. In these
days of "program trading" it is quite easy to duplicate
even a large index such as the S&P 500, but it is even
easier to duplicate the Major Market Index (MMI) which
consists of only 20 bluechip stocks.
On the basis of these three criteria, the Major Market
Index contracts are chosen for this study.
Data
The data for this part of the study are provided by the
Chicago Board of Trade. Included in the data are daily
observations on Major Market Index futures contracts for
different maturities. The sample period is from January 2,
1985, to December 31, 1985. Usually three or four different
maturities are available every day. The futures price used
for analysis is the settlement price established at the end
of trading every day. The closing values of the Major Market
Index are also provided by the Chicago Board of Trade. A
trtal of 945 observations are available for testing.
In order to determine the equilibrium forward price,
dividends on the stocks comprising the MMI as well as prices
of the discount bonds which mature on the same day as the
futures contracts, are required. The dividends are obtained
from Moody's Dividend Record and their dollar value is
adjusted to the index.5 The prices of the discount bonds are
proxied by the prices of TBills which mature one day before
the maturity of the futures contracts. The mismatch of one
day in the maturity of Tbills and the futures contract is
negligible and should not be of any consequence in
statistical testing. The Tbill prices are calculated from
the yields published daily in The Wall Street Journal.
The maturities of the futures contracts in the sample
range from one day to 186 days but maturities greater than
120 days are observed only infrequently.
bMajor Market Index (MMI) is an equallyweighted (or
priceweighted) index designed to emulate the DowJones
Industrial Average. Changes in the index correspond to
changes in the sum of the prices of one share each of the
MMI's 20 stocks. The prices of 20 stocks are added and the
sum is divided by a standard divisor. Periodically, this
divisor is adjusted to reflect changes in the capitalization
of the 20 companies. During 1985, the divisor value was
changed twiceon May 20, 1985 and on December 31, 1985. The
second change is of no significance to this study since none
of the observations extend beyond the date of the change. The
first change, made to reflect a stock dividend by Eastman
Kodak, is relevant. On May 20, 1985, the value of the divisor
was changed from 4.49699 to 4.41560. The adjustment to the
dividends takes this fact into account. For example, a
dividend of 65 cents by Procter and Gamble, on January 14,
1985, is divided by 4.49699, but an identical dividend by the
same company on July 15, 1985, is divided by 4.41560.
usually, for the last two days before maturity of the
future contract, yield on a Tbill which expires one day
before the maturity of the futures contract, is not
available. In this case, the nearest Tbill, which in the
sample used in this study is always a Tbill expiring on the
thrusday after the maturity of the futures contract, is used.
The bias in bond prices caused by this approximation is
miniscule because the time to maturity is extremely small.
An interesting aspect of the data is related to the
procedure for establishing the settlement price. The concept
of settlement price is of special significance for futures
contracts and it is necessitated by the markingtomarket
feature of these contracts.7 The task of establishing the
settlement price is easy if trading in the contracts is heavy
and some trades take place near the end of the trading
session. In such a case the average price of the trades in
the last few seconds (usually 2030 seconds) is used as the
settlement price. If, however, the trading in the contracts
is thin, the exchange establishes a settlement price which
may or may not reflect the true closing price.
A technical detail may be of interest to some readers.
The daily resettlement feature also applies to contracts
other than futures contracts, but it gets more prominence for
futures contracts. Any position in any security that requires
a margin, e.g., a short position in options, is generally
markedtomarket. An option seller must deposit additional
margin if the balance in the margin account is depleted as a
result of the losses. By the same token, he can withdraw the
excess balance in the margin account that may result because
of profits on the position.
For most conventional options traded on the Chicago Board
Options Exchange (CBOE), margin requirements are a function
of the price of the underlying asset with an adjustment for
the fact tht the option may be in or outofthemoney. For
some recently introduced options (e.g., option on S&P 500
futures, traded on the Chicago Mercantile Exchange), margin
is a function of the option price itself. For such options,
the exchange establishes a settlement price for the option.
A hidden reason behind establishment of settlement price
is to encourage "spread trading." If there is merit in this
argument it is conceivable that CBOE may attempt to change
its margin rules and start establishing settlement price even
for conventional options.
The data used in this study explicitly distinguish between
observations for which there is sufficient activity in the
contracts at the end of the day, from those observations for
which not enough trades take place at the close of the day.
One of the subtasks of the study is to find out if there is
a difference between observations for which the settlement
price reflects the true closing price and those for which the
settlement price is established, somewhat artificially, by
the exchange. The empirical results based on these data are
presented in the next section.
Empirical Results
Direction and Magnitude of the Difference
between Futures and Forward Prices
The first task is to examine the extent of the difference
between futures prices and forward prices. Since equation (2)
implies that the difference, if it exists, is likely to be
related to the time to maturity of the contracts, the sample
The data obtained from the Chicago Board of trade come
in a format that allows two closing prices to be published.
These two prices describe the closing range. However, two
prices are not available if trading at the close of the
trading session is thin. In that case, only one closing price
is published, and there is no way to determine the time at
which the trade took place. Whenever two closing prices are
published, it is reasonable to assume that the settlement
price reflects the true closing price. Such observations are
included in the first subset of observations. Whenever only
one price is available, it is indeterminate whether the
settlement price would also have been the closing price, if
some trades had taken place near the end of the trading
session. Such observations are included in the second subset
of observations.
is divided into five categories on the basis of this
criterion. The variable X is defined to denote the difference
between futures prices and forward prices, i.e., X=fg. Some
relevant statistics for X are presented in Table 2.
Table 2
Difference between Futures Prices
and Forward Prices (in Cents)
Days to 014 1529 3059 6089 >90 Total
Maturity
Number of
observations 128 126 253 219 219 945
X 12.1 20.5 47.1 81.0 160.1 72.9
** ** ** ** ** **
tstatistic 3.45 4.19 10.5 15.4 24.9 24.6
**
significant at 1% level
It is evident from this table that there is a significant
difference between futures prices and forward prices. Futures
prices are found to be in excess of forward prices by 73
cents on the average. The difference is highly significant as
can be judged from the tstatstic of 24.64. As expected, the
difference increases with an increase in time to maturity of
the contracts. The average difference for contracts with less
than 15 days to maturity is 12 cents but for contracts with
more than 90 days to maturity, it is as high as 160 cents.
Test of Settlement Price Effect
Having observed significant difference between futures
prices and forward prices, the next step is to check whether
the difference is real, or due to the fact that some of the
settlement prices used in testing may not reflect true
closing prices. To carry out this test, the total sample is
divided into two subsets.
The first set contains 405 observations for which there
are at least two trades in the closing seconds and,
therefore, it is almost certain that the settlement price is
also the true closing price. This set of observations is
loosely designated as the "Set of Liquid Contracts." The
second set contains 540 observations for which there are
either no trades or only one trade near the end of the
trading session and, therefore, the settlement price set by
the exchange may not reflect the true closing price. This set
of observations is loosely designated as the "Set of Illiquid
Contracts." The results obtained from the analysis are
presented in Table 3.
Table 3
Difference between Futures Prices
and Forward Prices (in cents)
Days to Maturity
014
1529
3059
6089
>90
Total
Liquid Contracts Illiquid Contracts
106
15.9
**
4.39
t =2.25
100
27.7
**
5.09
t
117
63.2
**
8.87
t
53
101.7
**
9.52
t
29
192.9
**
22.19
t
405
56.4
14.33
14.33
3 **
= 3.17
S.35
 3.35
= 2.23
= 2.34
= 5.03
significant 5% level
significant at 1% leave
significant at 1% level
22
6.2
0.69
26
7.0
**
0.74
136
33.4
**
6.21
166
74.4
**
12.46
190
155.2
**
22.19
540
85.3
20.40
20.40
Two types of tstatistics are presented in Table 3. First,
there is one tstatistic each for liquid contracts and
illiquid contracts corresponding to the null hypothesis that
the mean value of X is zero for each of the two sets of
observations. Second, there is a joint tstatistic which
corresponds to the null hypothesis that the mean values of X
for liquid contracts and illiquid contracts are equal.
At first glance there seems to be a sharp difference
between the two sets of observations. The difference between
futures prices and forward prices is consistently higher for
liquid contracts as compared to illiquid contracts. Care
should be taken in interpreting the average difference for
the total sample reported in the last row. Even though
illiquid contracts seem to have a higher mean difference,
just the opposite is true. The deceptive result in the last
row is a result of a sampling bias as illiquid contracts are
usually the contracts with long maturities, for which it is
natural to observe a greater difference between the two
prices.
The preceding results are somewhat puzzling at first. It
seems that the settlement price set by the exchange is
usually a downward biased estimate of the true closing price.
The term "settlement price effect" is used to refer to this
apparently systematic discrepancy between liquid cotnracts
and illiquid contracts. It seems odd that such a systematic
difference should exist between futures prices set by the
exchange and those observed in a liquid market. After a
25
thorough analysis, a partial explanation of this phenomenon
is uncovered. The apparent "settlement price effect" is
partly due to the period to which the observations belong. It
so happens that there were more illiquid contracts during the
latter part of 1985. During the same period, the difference
between futures prices and forward prices declined
significantly. This fact is evident from the results
presented in Table 4.
Table 4
Difference between Futures Prices
and Forward Prices (in Cents)
Quarter Liquid Contracts Illiquid Contracts Total
N = 137 N = 111 N = 248
Quarter 1 X = 91.5 X = 157.4 X = 120.9
** ** **
t = 13.55 t = 17.47 t = 20.63
N = 119 N = 130 N = 249
Quarter 2 X = 64.6 X = 130.5 X = 99.0
** X* **
t = 9.47 t = 18.97 t = 18.78
N = 85 N = 165 N = 250
Quarter 3 X = 39.5 X = 72.8 X = 61.5
** ** **
t = 5.47 t = 12.52 t = 13.31
N = 64 N = 134 N = 198
Quarter 4 X = 11.6 X = 3.1 X = 5.9
t = 1.84 t = 0.61 t = 1.46
N = 405 N = 540 N = 945
Total X = 56.4 X = 85.3 X = 72.9
t = 14.33 t = 20.40 t = 24.64
**
significant at 1% level
The results presented in Table 4 indicate that the
apparent "settlement price effect" implied by the results
presented in Table 3 may be illusory. The real cause of the
difference may, in fact, be the sampling feature that more
liquid contracts and fewer illiquid contracts are observed
during the time period when the differences between futures
prices and forward prices are higher.
It is possible to isolate the "settlement price effect,"
if it does exist, by eliminating the timeoftheyear effect.
However, an attempt to do so is thwarted by the small size of
the data at hand.
Test of Daily Resettlement Effect
Having observed significant differences between forward
and futures prices, the obvious question is "Are these
differences in line with our expectations?" To answer this
question, I go back to the CIR model which attmepts to
predict the difference between the two prices. The CIR
equation is reproduced below.
s1
f(t)g(t) = PV E [f(j+l)f(j)][B(j)/B(j+l)l]/B(t)
j=t
where PV is the present value operator.
The first noteworthy feature of the equation is the
present value operator in front of the parentheses. The lack
of a specific arithmetic expression for the present value
operator indicates that the model is not set in an
equilibrium pricing framework and, therefore, one does not
know the discount rate which would adjust for the risk of the
payoff given by the expression within the parentheses.
The second important feature of this equation is that it
is formulated at time t in terms of future values of
variables f (futures prices) and B (bond prices) which are
unknown exante (at time t).
These two characteristics, at first, seem to render
testing of the model infeasible. However, the following
discussion shows that one can still make useful comparisons
between the predicted difference between forward and futures
prices and that actually observed in the market.
As a first pass, assume that the correct discount rate, to
be used for the payoff in question, is the riskfree rate.
Also assume, for the moment, that the realized values of the
two variables, f and B are exactly in line with exante
expectations. Under these two assumptions, the predicted
value of the difference between futures and forward price is
calculated using expost data in equation (1). This variable
is denoted by X. Table 5 presents some relevant statistics
for X.
Table 5
Predicted Difference between Futures Prices
and Forward Prices (in Cents)
Days to 014 1529 3059 6089 >90 Total
Maturity
Number of
observations 116 125 252 218 219 930
Mean(X) 0.03 0.04 0.13 0.15 0.42 0.18
tstatistic 1.87 1.84 5.71 4.18 8.65 10.80
**
significant at 1% level
A comparison of Table 2 and Table 5 reveals startling
differences between predicted values of X and those actually
9
observed in the market. Not only is the direction of the
difference between forward and futures prices exactly the
opposite of that predicted by the model, the difference in
magnitude also seems extremely large. For example, for
contracts with less than 15 days to maturity, the CIR model
predicts that forward prices should exceed futures prices by
threehundredths of one cent. Instead, it is found that
futures prices on an average are higher than forward prices
by 12.1 cents. For contracts with more than 90 days to
maturity, futures prices exceed forward prices by 160 cents
on the average whereas the average difference predicted by
the CIR model is less than onehalf of one cent.
The reader may wonder if the simplifying assumptions used
to calculate the predicted values of X may have caused the
discrepancy between observed and predicted values of X. The
following arguments will show that these two assumptions are
unlikely to explain the magnitude of the discrepancy.
First, consider the assumption that the riskfree interest
rate can be used for discounting the payoff inside the
present value operator in equation (1). we know that this
The total number of observations, 93'. reported in
Table 5, is smaller than the total number of observations,
945, reported in Table 2. This happens because in calculating
the value of predicted X using equation 2, the first
observation for every contract with a different maturity is
eliminated since it nas no lagged value.
payoff is not riskfree and, therefore, should be discounted
at a rate higher than the riskfree rate. But a higher
discount rate will have no effect on the direction of the
difference predicted by the CIR model. It would only affect
the magnitude of the predicted difference, and considering
the miniscule differences predicted by the model (e.g.,
onehalf of one cent for contracts with greater than 90 days
to maturity), this effect would also pale in comparison with
the observed differences between the two prices. Hence, it
seems that the assumption of using the riskfree rate as the
discount rate has virtually no effect on the discrepancy
between observed X and predicted X.
The second assumption is somewhat more crucial. It says
that the futures prices and bond prices observed in the
market are exactly in line with the exante expectations of
market participants. It is possible that this this assumption
may introduce some bias, but there is no reason to believe
that the bias will be systematic. So long as measurement
errors caused by this assumption are random, statistical
results presented in Table 5 should still be useful. More
importantly, an even stronger statement, without utilizing
this assumption, is made in the next section to show that the
CIR model does not correctly predict the difference between
futures and forward prices.
QuasiArbitrage Opportunities
in the Futures Market
Since futures prices are found to be higher than forward
prices by an amount greater than that predicted by the CIR
model, and since the CIR model is based on a nonarbitrage
condition, it would seem that a profit opportunity exists
which can be exploited by taking opposite positions in spot
market and futures market. However, the problem with this
strategy is that it is not riskfree because futures prices
and bond prices in future are unknown. An investor shorting a
futures contract can lose on the futures position if futures
prices go up, and this loss can be accentuated if interest
rates also go up. At the same time, there will likely be a
profit on the long spot position since spot prices usually
move in tandem with futures prices. But the uncertainty
regarding the future values of futures prices, spot prices,
and bond prices implies that it is impossible to design a
"perfect" (riskless) arbitrage strategy. Nevertheless, it is
shown in this study that the market does offer opportunities
to design trading strategies which require zero investment
and still yield positive payoffs even under some of the most
pessimistic scenarios. In order to distinguish such
opportunities from "perfect" arbitrage opportunities, I use
the term "quasiarbitrage" opportunities. A "quasiarbitrage"
opportunity is one which requiFr.s zero investment but yields
a positive payoff even under some of the most pessimistic
scenarios.
Given the empirical result that futures prices seem
excessively higher than forward prices, the correct strategy
for the arbitrageur is to short a futures contract, buy the
20 stocks underlying the MMI using borrowed money and close
out the position at maturity. Table 6 describes the payoff
from this strategy.
Table 6
Payoff from the QuasiArbitrage Strategy
Time t t+l . s
Short one
futures 0 f(t)f(t+l) . f(sl)f(s)
Buy Spot
Asset S(t) 0 .. S(s)+D
Borrow S(t) f(t+l)f(t) S(tl s2[E fj+l)f()
B(t) B(j+l)
sl f(j l)f(j)
0 0 . g(s)g(t) Z B(j+l)
j=t
This strategy requires zero investment in every period
until maturity. The loss on the futures position to the
sl
investor is E [(f(j+l)f(j))/B(j+1)] and the profit on the
j=t
spot position is given by g(s)g(T). The strategy is
profitable if
s1
g(s)g(T) > E [(f(j+l)f(j))/B(j+l)] (5)
j=t
Before testing inequality (5) in order to examine the
possible existence of quasiarbitrage opportunities, the
definition of a pessimistic scenario is needed. Instead of
one, three different definitions based on varying degrees of
pessimism are provided and the existence of quasiarbitrage
opportunities is examined under all three definitions. Since
the arbitrageur shorts a futures contract, he stands to lose
if futures prices go up and also if interest rates go up.1
With this in mind the following three definitions of a
pessimistic scenario are given.
(1) Futures prices and interest
rate such that in 180 days they
levels.
(2) Futures prices and interest
rate such that in 180 days they
levels.
(3) Futures prices and interest
rate such that in 180 days they
levels.
Inequality (5) is tested for
rates go up every day at a
are 1.5 times their current
rates go up every day at a
are twice their current
rates go up every day at a
are 2.5 times their current
all three definitions using
daily data on Major Market Index futures. Out of a total of
It can be easily shown that the other possibility
where futures prices instead go down in every period is an
obviously profitable one. To see this simply modify Table 6
for this new scenario. It would be seen that the strategy
yields a nonnegative payoff in each period, and the cash
outflow at maturity is less than the sum of the cash inflows
during intermediate periods.
945 observations a substantial proportion satisfy this
inequality indicating that they could have been exploited by
using the strategy outlined in Table 6. The number of such
quasiarbitrage opportunities and their average profit is
given in Table 7.
Table 7
Number of QuasiArbitrage Opportunities in
Spot and Futures Markets for MMI during 1985
Definition 1 Definition 2 Definition 3
Number of
Opportunities 430 301 249
Average Profit
Per Index 11
Unit (Cents) 55.22 49.34 45.56
This analysis shows that there are opportunities for
profit which are technically not riskfree but yield a profit
even under some severely pessimistic scenerios. The next step
is to check if these opportunities are profitable after
transactions costs are taken into account.
11The face value of a 1985 MMI futures contract is
calculated by multiplying the quoted price by 100. Thus, each
contract consists of 100 index units. The profit given in
Table 7 is cents per index unit. To get profit per contract
shorted, these figures should be multiplied by 100.
Incidentally, a similar contract with 250 index units (called
maxii') has become popular recently and the volume in the 100
unit contract has declined sharply.
Transactions Costs and QuasiArbitrage
The first thing to note in this regard is that any
arbitrage opportunities in the market are likely to be
exploited by market professionals. These traders obviously
have extremely low transaction costs since they pay no
brokerage fee for trading on their own account. The major
costs paid by them are mostly fixed in nature, e.g., the cost
of obtaining a seat on the exchange. It is assumed here that
these fixed cost are allocated to brokerage business for
customers. The only relevant costs, then, are the marginal
(or variable) costs of undertaking the arbitrage operation.
I know of only one truly variable cost for traders trading
on their own accountthe cost of clearing the trade through
the clearing facility. For the type of arbitrage operation
described above, clearing charges will have to be incurred in
both futures and spot markets.
The clearing charge per MMI futures contract is 10 cents
per side. This implies that per index unit, the cost is
onetenth of one cent. For spot assets, the 20 stocks that
make up the MMI, clearing charges are levied by the National
Securities Clearing Corporation. Two types of relevant costs
are Trade Comparison Fees and Trade Clearance Fees.
Trade Comparison Fees represents the fees to enter trade
data. Currently, for each side of each stock trade submitted,
the fees is 3.3 cents per 100 shares, with a minimum fee of
6.6 cents and a maximum fee of $1.65.
Trade Clearance Fees represents fees for netting, issuance
of instructions to receive or deliver and effecting book
entry deliveries. Currently, there are seven types of
clearing fees out of which only the following two are
applicable to a simple trade.
1. Receipts from CNS (Continuous Net Settlement) to satisfy a
long valued position45 cents per issue received.
2. Deliveries to CNS in the night processing cycle to cover a
short valued position45 cents per delivery.
Either one of these two types of fees is paid by a
brokerage firm one time every day for each stock that it
bought or sold during the day, whether for its customers or
for its own account. For an arbitrage operation of the kind
described in the previous section, the marginal Trade
Clearance Fees is zero if the firm trades these stocks for
its customers. Since MMI consists of 20 bluechip stocks it
is quite likely that there will a large number of customer
trades in these stocks. Thus, it is reasonable to assume that
Trade Clearance Fees of the kind mentioned above is zero.
Nevertheless, the following calculations are made on the
conservative assumption that customers do not trade in the
stocks on the day that the arbitradeurs wants to undertake
the arbitrage operation, and therefore, trade clearance fees
is allocated to the arbitrage operation.
Assuming a hypothetical arbitrage operation of $1 million
face value, some estimates are made for transactions costs.
It is worth mentioning that most arbitrage operations used in
practice are of much bigger size, thus reducing the per unit
cost even more.
July 1, 1985, is arbitrarily chosen as the day for which
transactions costs are calculated. The average price of a
share for 20 MMI companies on that day is $57.29375. These
stocks are to be bought in equal proportion so that the total
price is approximately $1 million. The number of shares thus
calculated (rounded to nearest 100 to reduce transactions
costs), is 900 per stock. The total investment is $1.03
million. The transactions costs are calculated as follows.
Trade Comparison Fees = (3.3 cents x 900/100) x 20
= $5.94
Trade Clearance Fees = (45 cents x 20)
= $9.00
Total Fees = $14.94
Oneside fees of $14.94 is simply multiplied by 2 to get
the twoway transactions costs of $29.88. To find out
transactions costs per index unit, note that a face value of
$1,031,287.50 implies 3940.12 index units (base on spot index
value of 261.74 at the close of July 1, 1985). Thus,
transactions costs per index unit are 2988/3940.12 = 0.76
cents. The roundtrip clearing cost for the futures contract
is 0.2 cents per index unit, thus giving the total roundtrip
transactions costs of 0.96 cents per index unit for the whole
arbitrage operation.
Comparing these transactions costs with the discrepancy
between observed and predicted differences, given in Table 7,
it is easy to see that even after taking transactions costs
into account, substantial pretax profit can still be made.
As regards taxes, there is no need to distinguish between
ordinary gains and capital gains for professional traders.
The only effect of taxes, then, is that they reduce, but do
not eliminate, the profit from quasiarbitrage.
It is worth noting that the strategy outlined in Table 6
involves buying the underlying stocks which is easier than
shorting the stocks. Therefore, none of the objections
concerning shorting of stocks apply to this strategy.
Out of curiosity, I decide to check whether the transaction
costs for ordinary individuals are large enough to eliminate
their participation in quasiarbitrage activities. To get
some idea of transactions costs for such individuals,
commission schedules for stocks and futures contracts are
obtained from two discount brokers.12
For the futures contract, a quote of $27 per contract, per
side for minimum volume and minimum frequency, is obtained.
For larger volume and frequent trades, the cost declines to
$12 per contract, per side. For a $1 million face value
arbitrage program, the lower rate is applicable.
1The commission schedules used in this study are not
claimed to be representative of the market. Moreover,
commission rates do vary frequently. Therefore, any results
presented in this study should be interpreted accordingly.
Unfortunately, the quoted commission on futures contracts
is for the newer and more popular MMI contract (called
"Maxi") which contains 250 index units. The 1985 Major Market
Index contracts consisted of only 100 units. The transactions
costs for the old contracts are almost certainly lower than
those specified above. By using these higher transactions
costs, this study is being more conservative in its approach
to finding quasiarbitrage opportunities.
The commission schedule on stocks is used to calculate the
transaction costs for buying 900 shares of each of the 20
stocks. For all stocks, except A T & T, the commission is
5112 plus 0.10 percent of the dollar amount. For A T & T, the
commission is $61 plus 0.31 percent of the dollar amount. The
total oneway commission thus calculated is $1250.12. The
number of index units was calculated earlier to be 3940.12.
Hence, per index unit cost for stocks is 31.73 cents per
side. Adding to this the per side cost of 12 cents on the
futures contract, the total oneway cost is 43.73 cents. The
roundtrip cost, therefore, is about 87 cents.
Comparing this cost with the average differences mentioned
in Table 7, it can be seen that it is much more difficult for
ordinary individuals to undertake the quasiarbitrage
operation. It is true that there will be some observations
for which the discrepancy will be more than the transactions
cost of 87 cents. But the profitability of such opportunities
is, at best, substantially reduced and at worst, almost
completely eliminated.
Time Trend in QuasiArbitrage Opportunities
Given that quasiarbitrage opportunities existed in the
market it seems reasonable to expect that they must have been
utilized by discerning investors. It is also likely that
actions of such investors would cause these opportunities to
disappear over time. Since the quasiarbitrage opportunities
exist because of futures prices being inordinately higher
than forward prices, the conjecture is that the difference
between the two prices declined over time. To test this
conjecture the sample is divided into four quarterly subsets.
Table 8 presents the results of the test carried out to test
this conjecture.
Table 8
Difference between futures prices
and forward prices (in cents)
Quarter Qtr.l Qtr.2 Qtr.3 Qtr.4 Total
Days to Mat.
N
014 X
t
N
1529 X
t
N
3059 X
t
N
6089 X
t
N
>90 IX
t
N
Total X
t
33
26.9
**
4.48
30
62.7
**
6.55
64
105.7
**
13.46
54
120.8
**
9.82
67
208.2
**
22.19
248
120.9
20.63
29
16.2
2.35
32
23.0
**
2.79
64
61.1
**
9.41
61
118.4
**
20.40
63
195.5
**
23.33
249
99.0
**
18.78
33
10.1
1.63
32
3.7
0.42
65
39.9
**
6.32
62
60.3
**
8.81
58
148.1
17.90
17.90
33
4.3
0.55
32
4.6
0.56
60
22.3
**
3.28
42
6.0
0.63
31
7.1
0.57
250 198
61.5 5.9
13.21 1.46
128
12.1
X*
3.45
126
20.5
**
4.19
253
47.1
**
10.5
219
81.0
**
15.40
219
160.1
**
24.98
945
72.9
**
24.64
significant at 5 % level
Esrnificant at 1 % level
The results presented in Table 8 lend support to the
conjecture that quasiarbitrage opportunities in MMI
contracts disappeared over time. This is reflected in the
narrowing difference between futures and forward prices. The
decline from the third to the fourth quarter is dramatic. One
possible explanation for this behavior is that "program
trading," which became extremely popular during 1985, caused
arbitrage opportunities to decline sharply over time. The
following quote from a recent news article by Zaslow (1986)
supports this hypothesis.
Arbitrage is one technique often employed by inter
market traders, who frequently swap baskets of large
capitalization stocks for offsetting stockindex futures
to take advantage of price discrepencies. . .
Diminishing opportunities in the fouryear old S&P 500
futures contract are driving some arbitragers to seek new
trading frontiers. . With stock prices at their
current lofty levels, professional traders find the MMI a
cheaper vehicle for arbitrage (emphasis added). "To
replicate the S&P index by buying underlying securities
you need 40 to 50 stocks" . "You could replicate the
entire MMI, all 20 stocks, and still get away cheaper."
Conclusion
Significant differences are found between futures and
forward prices for MMI contracts during 1985. The differences
are significantly higher than those predicted by the CIR
model. This discrepancy between observed and predicted
differences implies that quasiarbitrage opportunities
existed in MMI spot and futures markets during 1985. The
empirical analysis confirms the existence of such
quasiarbitrage opportuities. However, it is also found that
these opportunities have since disappeared, presumably
because of the actions of the arbitrageurs.
CHAPTER III
OPTIONS AND FORWARD CONTRACTSTHE ISSUE
OF EARLY EXERCISE
It has been shown by researchers (for example, Moriarty,
Phillips, and Tosini (1981) and Cox and Rubinstein (1985))
that a specific portfolio of European options can be created
such that its payoff is identical to the payoff from a
forward contract on the underlying asset. However, the same
is not quite true if instead the portfolio consists of
American options because American options may be exercised
before expiration. In this chapter the issue of early
exercise and its effect on the relationship between "The
American Options Portfolio" and a forward contract are
discussed. The empirical analysis is done using daily data
regarding actual exercises of MMI options. As a byproduct of
the main analysis, some wellknown theoretical conjectures
regarding early exercises of options are also tested.
Forward Contracts and
The European Options Portfolio
At time t, "The European Options Portfolio"3 is created
by taking a long position in a European call option and a
3Moriarty, Phillips, and Tcini (MPT), 1981, first
described this portfolio as being equivalent to a futures
contract under certain restrictive assumptions. Cox and
Rubinstein (1985) realizing the nature of one of the
restrictions imposed by MPT, correctly describe the portfolio
as being equivalent to a forward contract.
short position in a European put option with the same
maturity and identical exercise price. To finance this
portfolio the investor borrows k(t) so that the investment
required at time t is zero (recall that k(t)sc(t)p(t)). The
assumption here is that investors can borrow and lend at the
riskfree rate.
The payoff from this portfolio, if held until maturity, is
given in Table 9.
Table 9
Payoff from The European Options Portfolio
Initial Cash Flow at Expiration
Cash Flow S(s)>E S(s)=E S(s)
Buy a call c(t) S(s)E 0 0
Sell a put +p(t) U 0 S(s)E
Borw k) k(t) k(t) k(t)
Borrow k(t) Bt Bt Bt
B(t) B(t) B(t)
0 S(s)E k(t) k(t) S(s)E k(t)
B(t) B(t) B(t)
Intuitively, it is easy to see why this portfolio is like
a forward contract on the underlying asset. If it is held
until maturity, the holder of the portfolio ends up buying
the underlying asset either by exercising the call option, if
(S(s)E)'C, or by being forced to buy by the put holder, if
(S(s)E)<0. In either case, he owns the underlying asset by
paying the exercise price E and repaying the loan whose value
at time s is k(t)/B(t). Thus, "The European Options
Portfolio" is like a forward contract with the implied
forward price, g given by the following equation.
k(t) (6)
g (t) = E +(t
B(t)
Forward Contracts and
The American Options Portfolio
If the portfolio of a long call and a short put is created
with American options it is not sufficient to consider the
payoff only at maturity because an American option gives its
holder the right of early exercise. This causes a distortion
in the possible equivalence of a forward contract and "The
American Options Portfolio" because if either of the options
comprising the portfolio is exercised before expiration,
there is an intermediate payoff from the portfolio but no
such payoff is forthcoming from the forward contract.
Note that once the equivalence is distorted due to early
exercise it cannot, theoretically, be restored by simply
buying or selling an identical option because the exercised
option is exercised presumably because it is "optimal" for
everyone to do so. Under this scenario the open interest in
the option should go to zero.
It is quite obvious that the exact equivalence of a
forward contract and "The European Options Portfolio" is of
little practircl interest since almost all traded options are
American. The more interesting issue is that of the
relationship between "The American Options Portfolio" and a
forward contract. In order to make any meaningful statement
about this relationship it is imperative that the issue of
early exercise be analyzed in greater depth. This task is
described in the next section.
Early Exercise of American Options
Previous Research
The theoretical implications of the possibility of early
exercise have been discussed by a number of researchers. The
most important contribution in this area is by Merton (1973)
who developed a number of propositions regarding early
exercise of options. Since then, other researchers (e.g., Cox
and Rubinstein (1985), Geske and Shastri (1985), Evnine and
Rudd (1985)) have elaborated on and supplemented his work.
The major results known regarding early exercise of
options are summarized below. For ease of understanding,
these propositions are presented assuming the option is
written on a share of common stock.
(1) A call option on a stock that pays no dividend before the
expiration of the option, should not be exercised before
expiration. A call option on a stock that does pay some
intermediate dividends may be exericsed early but the
only times when it may be optimal to do so are when the
stock is about to go exdividend.
(2) A put option on a stock which pays no dividend may be
exercised early. Theoretically, it may be optimal to
exercise a put at any time before expiration, but the
more likely points of time are immediately after the
stock goes exdividend.
(3) If it is optimal to exercise a call then it is never
optimal to leave unexercised an otherwise identical call
that has either a lower exercise price or a shorter time
to expiration.
(4) If it is optimal to exercise a put then it is never
optimal to leave unexercised an otherwise identical put
that has either a higher exercise price or a shorter time
to expiration.
(5) It is not optimal to exercise an option if a better
price can be obtained in the secondary market. By the
same token, if it is optimal to exercise an option, its
price in the secondary market should be exactly equal to
its exercise value.
(6) Assuming that future dividends and interest rates are
known, if the present value of all future dividends is
less than the present value of the interest that can be
earned on the exercise price, the call should never be
exercised before expiration. Thus, higher dividends
increase the probability of early exercise of call
options.
(7) Assuming that future dividends and interest rates are
known, if the present value of all future dividends is
greater than the present value of interest that can be
earned on the exercise price, a put option should not be
exercised early. This proposition implies that higher
dividends deter early exercise of put options.
(8) From Propositions (3) and (4) it follows that an option
with a longer time to expiration is less likely to be
exercised than an otherwise identical option with a
shorter time to expiration. Also, an option with smaller
exercise value is less likely to be exercised than an
otherwise identical option with higher exercise value.
(9) The last proposition has special relevance for index
options. The dividend on an index is far more continuous
than the dividend on an individual stock. Combining this
knowledge with Proposition (7) and Proposition (7) one
can see that dividends act as a weaker stimulant for
early exercise of index call options than they do for
individual stock options. Analogously, dividends act as a
weaker deterrent to the early exercise of index put
options than they do for individual stock options.
Even though the analytical results listed above have been
known for quite some time, I know of no empirical research to
validate these conjectures. The lack of data may partially
explain the lack of research in this area. In order to carry
out the second task outlined in the beginning, it is
necessary to carry out an empirical study of early exercises
of index optionsspecifically, MMI options.
Data
The data regarding early exercise of MMI options are
provided by the Options Clearing Corporation (OCC). These
data contain a listing of calls and puts exercised every day
during 1985. The exercises are categorized according to
exercise price and expiration month. These data are
supplemented by other data regarding volume and open interest
from The Wall Street Journal and Stock Option Guide
respectively.
The data regarding open interest have two deficiencies
which need to be pointed out. First, they are available for
only 32 of the most popular MMI options contracts. This
implies that a small number of contracts are left out.
However, the contracts that are so missed are those for which
open interest is low. The remaining sample is still of
sufficient size to yield good statistical results. The second
deficiency is that only weekly observations are available.
This compels one to estimate daily open interest by
interpolating between two adjacent weekly observations and
making an adjustment for the number of options exercised. It
is possible that this estimation may create some bias but it
is not likely to be systematic. Moreover, a look at open
interest data shows that they usually do not fluctuate
dramatically over short periods and, therefore, the magnitude
of the bias caused by interpolation is likely to be small.
Open interest for each series of calls and puts for each
day is estimated by using the following equation.
EOI(t) = EOI(tl) + DAILY(w) EX(t) (7)
where EOI(t) = Estimated Open Interest for day t
EOI(t1) 5 Estimated Open Interest for day ti
EX(t) 5 Number of Options Exercised on day t
DAILY(w) 5 Daily factor defined as
(OI(W) OI(wl) + TOTEX(w))/5
OI(w) S Actual Open interest at the end of
the current week
OI(wl) = Actual Open Interest at the end of
TOTEX(w) ]
the previous week
Total Number of Options Exercised
during the current week.
The last piece of information needed is the number of
"Opening Purchases." This term refers to purchases which give
the investor a new position where previously he had none. In
contrast to "Opening Purchases," the term "Closing Purchases"
refers to purchases which are entered into in order to close
out an existing position. The following equation is used to
determine the number of "Opening Purchases."
OP = (VOL + EX + EXP)/2 (8)
where OP C Number of Opening Purchases
VOL 5 Total Volume (i.e., Total Number of
Options Traded)
EX S Number of Options Exercised
EXP S Number of Options that Expired without
being Exercised
The number of options that expired without being exercised
is estimated by aggregating the open interest in options that
are outofthemoney at expiration. The implicit assumption
here is that all options that are inthemoney are either
exercised or closedout in secondary market. This assumption
is reasonable since every rational investor will likely
14
exercise an inthemoney option at maturity. Since at any
time, data regarding only 32 contracts is available, some
observations for variable EXP (Number of Options that Expired
without being Exercised) in equation (8) are missed. However,
it is not expected to introduce a serious bias since the
number of contracts so missed is likely to be small.
Moreover, the effect of a small error in estimating EXP is
miniscule since the major driving force in equation (8) is
Total Volume (VOL).
Empirical Results
The data are analyzed with the objective of determining
the importance or nonimportance of early exercise of MMI
options. This is achieved by two different means. First, some
summary statistics regarding the magnitude of early exercise
are presented. Second, a test of some of the propositions
14
14Investors for whom transactions costs of exercising
are not close to zero may decide to let an option expire
without being exercised if transactions costs are more than
the exercise value of the option.
regarding early exercise is conducted in order to understand
the motivation behind early exercises by investors.
Summary statistics
During the year 1985 a total of 11.17 million MMI options
contracts were traded. Of these, 6.44 million were call
options and 4.73 million were put options. During the same
period a total of 231,272 calls and 115,631 puts were
actually exercised. The rest were either closedout by taking
an opposite position or allowed to expire worthless.
To measure the percentage of options that were actually
exercised, one should not calculate the number exercised as a
percentage of total number of options traded. In order to
make a meaningful statement about the magnitude of the
options exercised, the number of exercised options should be
expressed as a percentage of total "Opening Purchases" for
the following reason. An "Opening Purchase" can either be
closedout by taking an opposite position or the option can
be exercised or the option can be allowed to expire if it is
worthless. Therefore, to find out how many of the originating
contracts are indeed exercised it is necessary to take the
number of contracts exercised as a percentage of "Opening
Purchases."
Using equation (8), it is found that there are a total of
3.70 million "Opening Purchases" for call options and 2.P)
million "Opening Purchases" for put options, thus giving a
total of 6.50 million "Opening Purchases" for MMI options
during 1985.
Using these numbers in conjunction with the early exercise
data it is found that 6.25 percent of MMI calls and 4.13
percent of MMI puts were actually exercised during 1985.
However, for this study the important variable is not the
percentage of options that were exercised, but the percentage
of options that were exercised early. The analysis shows that
out of a total of 231,272 calls that were exercised, 61.8
percent were exercised on maturity date itself. Out of a
total of 115,631 puts exercised, 69.2 percent were exercised
on maturity date. This implies that the number of calls and
puts that were exercised early is 88,395 and 35,595
respectively. As a percentage of "Opening Purchases," it
implies that a mere 2.39 percent of calls and 1.27 percent of
puts were exercised early. For both types of options put
together 1.91 percent of all opening contracts were exercised
early. A summary of these statistics is provided in Table 10.
Table 10
Summary Statistics for
Major Market Index Options (1985)
(1) (2) (3) (4) (5) (6) (7)
Total Opening Number (3) as Early (5) as (5) as
Volume Purchases Exer % of Exer % of % of
(millions) (millions) cised (2) cises (3) (2)
Calls 6.44 3.70 231,272 6.25 88,395 38.2 2.39
Puts 4.73 2.80 115,631 4.13 35,595 30.8 1.27
Total 11.17 6.50 346,903 5.34 124,190 35.8 1.91
From these statistics it is clear that a very small
percentage of options are acutally exercised. Not only is the
proportion of options exercised small in absolute terms, it
is even smaller as compared to marketwide statistics
presented in Table 11.
Table 11
Comparison of Early Exercises
for MMI Options Vs. All Others (1985)
Percent of Opening Purchases
MMI Options All Options
Closing Early Exercises Expired Closing Total Expired
Sales Exer at Sales Exer
cises Maturity cises
Calls 74.01 2.39 3.86 19.74 58.4 12.5 31.2
Puts 68.89 1.27 2.86 26.98 63.6 13.0 27.3
Note: The figures for "All Options" do not add up to 100
percent for technical reasons. See CBOE brochure entitled
Market Statistics, for details.
(Source: Chicago Board Options Exchange and Options Clearing
Corporation)
The statistics given in Table 11 show that the magnitude of
exercise for MMI options is substantially lower than the
magnitude for all options put together. For all options on
all exchanges, the percentage of "Opening Purchases" settled
by exercising during 1985 is 12.3 percent for call options
and 7.5 percent for put options. The comparable figures for
MMI options are 6.25 percent and 4.13 percent respectively.
These results are not all that surprising considering the
cash settlement feature of index options. Since exercising an
index option does not involve physical transfer of the
underlying stocks it eliminates investors who otherwise may
have wanted to exercise the options in order to obtain the
underlying stocks. For call options, there is an additional
reason for fewer exercises. The dividend on an index is much
more continuous than the dividend on an individual stock and,
therefore, the nonexercise condition (given by Proposition
6) is satisfied more easily for call index options.
Even though summary statistics strongly support the view
that early exercises are negligible, the data are further
examined to test some of the early exercise propositions. Of
particular interest is the role of dividends in the decision
to exercise since most propositions have it as an important
variable.
Test of dividend effect
The first task is to test the dividend effect on exercises
as hypothesized by Proposition (1) and Proposition (2). This
test should shed light on investors' motivations behind
exercising options before expiration.
To carry out this test a distinction is made between
"relevant days" and "other days." The term "relevant days"
simply means the days for which there is a greater likeli.'od
of early exercise than on other days, other things being
equal. From Proposition (1), it can be seen that a "relevant
day" for call options is the day just before the stock goes
exdividend. From Proposition (2). a "relevant day" for put
options is the exdividend day itself. If investors are
indeed swayed by the dividend factor in making the decision
to exercise, one should observe a greater percentage of
exercises taking place, on an average, on "relevant days"
than on "other days."
During 1985, dividends were paid by one or more Major
Market Index companies on 59 different days. This implies
that there were 59 "relevant days" for calls as well as for
puts. For both types of options, the data are divided into
two groups. The first group consists of observations from
"relevant days," and the second group consists of all other
observations. These data are used to test the null hypothesis
that the mean value of percentage of options outstanding
exercised on any day is equal for both the groups. The
results for this hypothesis are presented in Table 12.
Table 12
Test of Dividend Effect on Option Exercises
% of Calls % of Calls % of Puts % of Puts
outstanding outstanding outstanding outstanding
exercised on exercised on exercised on exercised on
relevant days other days relevant days other days
N = 781 N = 2541 N = 744 N = 2470
a = 0.99 u = 0.72 u = 0.41 u = 0 55
t = 1.04 t = 0.85
The tstatistics presented in Table 12 show that the null
hypothesis cannot be rejected even at 10 percent level of
significance, i.e., it cannot be claimed with a high level of
confidence that more exercises take place on "relevant days."
Moreover, the results show that for put options the direction
of the difference between the two groups is just the opposite
of that suggested by theory. A smaller percentage of
outstanding puts are exercised on "relevant days" than on
"other days" but the difference is statistically
insignificant. For call options the difference is in line
with that predicted by Proposition (1). A larger percentage
of outstanding calls are exercised on "relevant days" than on
"other days," but the difference between the two groups is
not statistically significant.
These results seem to imply that not only the magnitude of
early exercises for index options is negligible, it is also
not systematically related to dividends as predicted by
Propositions (1) and (2). This does not, by any means, imply
that Propositions (1) and (2) are incorrect. It simply means
that for the special case of index options they do not hold
as strongly as they are expected to, and probably do, for
individual stock options.
Choice between closingout in secondary market
and exercisingRole of transactions costs
Propositions (3) and (4) alongwith some other propositions
are tested in a regression analysis framework in the next
section. Proposition (5) which is related to the choice
between exercising and closing out an option in the secondary
market is interesting. It is analyzed in detail next.
The data are examined to see if the price that could have
been realized in the secondary market, at the time the option
was exercised, is greater than the exercise value (intrinsic
value) of the option. In order for the test to be reliable,
the time at which the price of the option is observed should
coincide with the time the option is exercised. A special
feature of index options ensures that the optimal time to
make a decision regarding exercising the option is after 4:00
P.M. (Eastern Time) each day. The feature which causes such a
phenomenon is cash settlement. When an index option is
exercised, there is no delivery of underlying stocks. The
holder of the option simply gets the difference between
exercise price and the spot index value at the close of the
day on which the exercise notice is tendered to the Options
Clearing Corporation. It is, therefore, in the interest of
the option holder to wait until the stock market has closed
and the closing index value has been determined. Since the
stock market closes at 4:00 P.M. (Eastern Time) the optimal
time to make the exercise decision is after 4:00 P.M.
The next step is to check if there are any trades between
4:00 P.M. and 4:10 P.M. for the options that were exercised
during 1985. Since this study focuses exclusively on early
exercises, the data for 88,395 calls and 35,595 puts that
were exercised early, are examined further. It is found .hat
for 40,862 calls and 15,590 puts a better price in the
secondary market was foregone by the investors.
At first glance, such behavior on the part of the
investors seems to be irrational. However, there is one
important variable that has not been considered yet. So far
transactions costs have been ignored on the basis that for
professional traders, they are minimal. However, when data on
actual exercises is analyzed, it is inappropriate to make
such an assumption since a number of these exercises are
possibly undertaken by ordinary individuals, who are not
immume from transactions costs.
The most common practice regarding transactions costs for
exercising an index option is to charge the customer assuming
the option had been closed out in the secondary market. For
such cases there is no difference between transactions costs
for exercising and those for closing out. 5 Therefore, the
regular commission is irrelevant for comparing closingout
and exercising. However, there is another type of transaction
cost that is still relevant. The bidask spread of the dealer
is relevant when considering closingout in the market. Cox
and Rubinstein (1985) state that the oneway spread borne by
an option investor is less than 1/16 of one dollar. With this
in mind, I calculate the number of options for which the
price available in the secondary market was higher than
1All four brokerage houses (2 discount brokers anJ2 2
full service brokers) contacted, follow the practice of
charging a customer for exercising an option by assuming that
the option had been closedout in secondary market. The
possibility that some brokers may not follow this practice,
cannot be denied.
exercise value to cover a bidask spread of $1/16. A second
scenario envisaged is that of a higher bidask spread of
$1/8. These calculations are presented in Table 13. The
numbers in the second column entitled "Number Analyzable"
refer to the number of options for which a trade between 4:00
P.M. and 4:10 P.M. could be traced. The term "Prem." refers
to the premium over exercise value of the option.
Table 13
Choice between Exercising and ClosingOut
(1) (2) (3) (4) (5) (6) (7) (8)
4 of Number 4 for (3) as # for (5) as # for (7) as
Early Analy which % of which % of which % of
Exer zable Prem. (2) Prem. (2) Prem. (2)
cises >0 >1/16 >1/8
Calls 88395 64949 40862 62.9 16502 25.4 13506 20.8
Puts 35595 29070 14768 50.8 7562 26.0 3843 13.2
From the results presented above it can be seen that when
bidask spread is taken into account, the number of options
that apparently should not have been exercised declines
sharply. Nevertheless, there are still a few option exercises
that cannot be explained even by a bidask spread of $1/8.
An interesting phenomenon is discovered for some
exercises. There are instances where options are being
exercised at maturity even though their exercise value is
extremely low, and apparently not enough to cover the
transaction costs. Two cases stand out in this regard. On
March 16, 1985, a total of 2,678 March 250 calls were
exercised at a time when their exercise value was a mere
eight cents. Similarly, on May 18, 1985, a total of 10,827
May 255 puts were exercised when their exercise value was
eight cents. Since the transaction costs of exercising are
usually greater than eight cents (see Chapter IV for details
regarding transactions costs) it seems surprising that these
options should have been exercised. Three possible
explanations exist.
(1) These exercises were undertaken by brokers themselves
and, therefore, transactions costs were limited to the
nominal sum charged by the OCC.
(2) It is possible that some brokerage houses do not follow
the practice of charging their customers for exercises as
if they had been closedout.
(3) These exercises are irrational.
The first two explanations seem more likely, but it is
impossible to prove one way or the other since no information
is available as to who indeed exercised these options.
Test of other early exercise propositions
Propositions (6), (7), and (8) are tested next in a
regression analysis framework. Since Proposition (8) _, based
on Propositions (3) and (4), the following test also applies
to the latter two. These propositions offer the following
conjectures.
(1) The greater the exercise value (intrinsic value) of an
option, greater the likelihood of its early exercise.
12) The greater the present value of future dividends minus
the present value of interest on exercise price, greater
the likelihood of early exercises for call options. For
put options, the opposite holds true.
(3) The greater the time to maturity of an option, lower the
likelihood of its early exercise.
Based on these three conjectures the following regression
model is formulated.
Y = a + 1 2X2 + 3X3 + e
where Y s Percentage of Outstanding Options Exercised
(i.e., Y = EX(t)/EOI(t)*100)
X1, Exercise Value (Intrinsic Value) of Option
(for calls: X1 = SE, for puts: X1 = ES)
X2 = Difference between PV(Dividend) and PV(Interest)
(for calls:X2 a PV(Dividend)PV(Interest on E)
for puts: X2 2 PV(Interest on E)PV(Dividend)
X3 3 Time to Maturity of the Contract
On the basis of the analytical results listed earlier, a
priori, the coefficients are expected to have the following
signs.
Bi>0, P2>0, B3<0
The results from this regression model are presented in
Table 14.
Table 14
Results from the Regression
Y = a + )3X1 + 2X2 + 3X3 + e
Y N Intercept X1 X2 X3 R2
Calls 3322 2.004 0.099 0.326 0.023 5.5 %
** ** **
(11.06) (9.99) (1.61) (3.72)
Puts 3214 1.849 0.095 0.024 0.021 6.1 %
** ** **
(13.10) (12.55) (0.16) (5.16)
Total 6536 1.753 0.098 0.158 0.025 5.8 %
** ** **
(17.22) (16.29) (2.50) (9.597)
significant at 5 % level
**
significant at 1 % level
All three coefficients have their expected signs. The
difference, however, is that "Intrinsic Value" and "Time to
Maturity" are good explanatory variables, but the "Difference
between Present Value of Dividends and Present Value of
Interest on Exercise Price" is not as significant in
explaining the magnitude of early exercise. Once again, a
relative lack of dividend effect for index options is
demonstrated.
64
Conclusion
In this chapter a number of results regarding early
exercises of index options were presented. It was shown that
a small number of options are exercised early, and especially
for index options, a negligible number are exercised early.
Given this information it seems plausible that even "The
American Options Portfolio" can closely approximate a forward
contract. The next chapter uses this conjecture further for
exploring the relationship between options and futures
contracts.
CHAPTER IV
FUTURES CONTRACTS AND
THE AMERICAN OPTIONS PORTFOLIO
Introduction
The year 1985 will be remembered in financial circles as
the year when words like "arbitrage" and arbitrageurss"
became commonly known. Much of the credit for this goes to
"program trading" which was increasingly used by arbitrageurs
to lock in nearly riskfree profits by taking advantage of the
discrepancy in prices between different markets. The Major
Market Index contracts were not immune from this phenomenon
as this quote from a news article by Zaslow (1986) confirms.
A growing number of big institutional traders are using
a Chicago Board of Trade futures contract and an American
Stock Exchange option on the Major Market Index . As
traders buy or sell underlying MMI stock or options and
offset their positions in futures, they are creating some
of the most unpredictable price moves yet in the booming
stock and stockindex markets. (emphasis added) (April 7,
1986, page 40)
The analysis presented in the preceding chapters provides
the tools to explore the relationship between MMI options and
MMI futures. From the analysis of early exercises of MMI
options presented in Chapter III, it seems that their effect
is minimal, and that "The American Options Portfolio" can b1
used to approximate a forward contract. In that case, the
relationship between forward and futures contracts described
and tested in Chapter II should also hold true for "The
American Options Portfolio" and a futures contract.
Recall that equation (6) provided the definition of
inferred forward prices using European options. This equation
is now rewritten for American options.
K(T) (9)
g = E + (T)
B(T)
Forward prices inferred using this equation are compared
to futures prices to determine the direction and magnitude of
the differences between them. Futher, the possibility of
existence of arbitrage opportunities is examined.
Data
To study the relationship between "The American Options
Portfolio" and a futures contract, daily observations for all
MMI calls and puts traded during 1985 are obtained from Daily
Market Publications. A major strength of the data is that
they are tradebytrade with time of trade stamped alongside
the price of the option. This feature of the data enables me
to minimize the nonsynchronicity problem so common in
studies using options data.
The nonsynchronicity problem arises when the prices of
two contracts being compared are observed at different times.
The following process is used to minimize the effect of this
problem. It is known that trading in the three markets, i.e.,
the stock market, the options market, and the futures market,
stops within a span of 15 minutes. The market for spot asset
is the New York Stock Exchange and it stops trading at 4:00
P.M. (Eastern Time). The MMI options, traded on the American
Stock Exchange, stop trading at 4:10 P.M. (Eastern Time) and
the futures contracts, traded on the Chicago Board of Trade,
stop trading at 4:15 P.M. (Eastern Time).
A data set containing the last trade of each call and put
for every day in 1985 is created, provided a trade took place
between 4:00 P.M. and 4:10 P.M. This means that the maximum
discrepancy between the time the option prices and the
futures prices are observed, is 15 minutes. Even though, in
my opinion, this is within acceptable limits, I doublecheck
by creating two subsets of dataone consisting of only those
options for which a trade took place exactly at 4:10 P.M.,
and another consisting of all other options. The results of
this exercise are presented in the next section.
Minimizing the nonsynchronicity problem has one drawback
in that the deep in and outofthemoney and longer maturity
options are represented only infrequently in the sample.
Empirical Results16
Direction and Magnitude of the Difference between
Futures Prices and Inferred Forward Prices
Similar to Chapter II, the first task is to examine the
direction and the magnitude of the difference between futures
16The empirical results presented in this section should
be interpreted carefully. An unexpected phenomenon, found in
later research, seems to weaken these results. See footnote
17 for more details regarding this phenomenon.
prices and inferred forward prices. The data are divided into
four categories on the basis of time to maturity. The
following table contains some statistics regarding the
difference between futures prices and inferred forward
prices. The difference (fg ) is denoted by variable Z.
Table 15
Difference between Futures Prices
and Inferred Forward Prices (in cents)
Days to 014 1529 3044 >45 Total
Maturity
Number of
observations 340 350 254 153 1097
Z 7.9 11.9 21.2 24.2 14.5
** ** ** ** **
tstatistic 5.55 6.29 8.77 6.40 13.37
Comparing the results presented in Table 15 to those in
Table 2, a few similarities as well as several important
differences are obvious. The two sets of results are similar
in that futures prices are found to be greater than forward
prices as well as inferred forward prices. In both cases, the
difference between the two prices increases with time to
maturity. Even though the direction of the results in both
tables is identical, the difference in magnitude is
unmistakable. The difference between futures prices and
inferred forward prices is substantially smaller than the
difference between futures prices and forward prices. To
examine the cause of this discrepancy the first possibility
examined is the effect of nonsynchronicity.
Test of NonSynchronicity Effect
To examine whether a significant difference is caused by
the small degree of nonsynchronicity left in the sample, the
sample is divided into two sets. The first set contains only
those option trades which took place at 4:10 P.M. Since the
futures market closes at 4:15 P.M., the nonsynchronicity for
such trades is 5 minutes. The second set contains the last
trade for each call and put series, provided the trade took
place between 4:00 P.M. and 4:09 P.M. The nonsynchronicity
for this set ranges from 6 minutes to 15 minutes. Table 16
presents the results of a test of the difference in the value
of Z for the two groups.
Difference
inferred
Table 16
between futures prices and
forward prices (in cents)
Days to Maturity 4:10 P.M. Trades 4:004:09 P.M. Trades
N = 186 N = 154
014 X = 8.1 X = 7.7
** **
t = 4.57 t = 3.32
t = 0.13
N = 188 N = 162
1529 X = 11.5 X = 12.4
** **
t = 4.57 t = 4.31
t = 0.24
N = 85 N = 169
3044 X = 19.3 X = 22.1
** **
t = 4.79 t = 7.34
t = 0.56
N = 32 N = 121
>45 X = 13.9 X = 26.9
**
t = 1.61 t = 6.43
t = 1.36
significant at 5% level
**
significant at 1% level
The results presented in the Table 16 indicate that the
two groups are are very similar. It is not surprising that
observations from 4:00 P.M. trades are, on an average,
similar to those observed at 4:10 P.M. because an important
determinant of option prices, the spot index, is fixed at
4:00 P.M. when the stock market stops trading. Since the two
groups are so similar, the following discussion does not
distinguish between the two sets.
Time Trend in the Difference between Futures
Prices and Inferred Forward Prices
Recall that in Chapter II it was argued that the
differences between futures and forward prices are large
enough to allow for the existence of quasiarbitrage
opportunities. If a similar argument is be applied to the
difference between futures prices and inferred forward
prices, it would seem that quasiarbitrage opportunities, if
they exist, may not be as abundant in options and futures
markets as they are in futures and spot markets because the
differences between futures and inferred forward prices are
relatively smaller.
Another argument advanced in Chapter II was that there is
a decline in the difference between forward and futures price
over time, presumably as a result of the actions of the
arbirageurs. A similar reasoning can be applied to inferred
forward prices and futures prices. It is expected that the
difference between futures prices and inferred forward prices
will decline over time but the decline may not be as sharp or
consistent since the difference is relatively small to begin
with. To test this conjecLre the sample is divided into four
quarters and the trend in Z is observed over time.
Table 17
Difference between Futures Prices
and Inferred Forward Prices (in Cents)
Days to Qtr.l Qtr.2 Qtr.3 Qtr.4 Total
Maturity
N = 86 70 82 102 340
014 Z = 7.4 15.3 10.4 1.3 7.9
t = 3.37 5.47 4.13 0.40 5.55
N = 83 81 81 105 350
1529 Z = 14.1 17.8 16.9 1.7 11.9
** ** ** **
t = 2.94 4.35 5.34 0.59 6.29
N = 53 47 66 88 254
3044 Z = 23.2 12.4 33.2 15.6 21.2
** ** ** ** **
t = 3.28 3.17 6.82 4.68 8.77
N = 14 30 55 54 153
>45 Z = 8.8 45.8 22.4 17.9 24.2
** ** ** **
t = 0.67 5.44 4.04 2.69 6.40
N = 236 228 284 349 1097
Total Z = 13.4 19.6 19.9 7.6 14.5
t = 5.18 8.66 9.96 4.03 13.37
**
significant at 1 % level
As expected the decline in the difference between futures
prices and inferred forward price is not as noticeable as the
decline in the difference between futures prices and forward
prices as reported in Table 8. This is consistent with the
conjecture that quasiarbitrage opportunities in options and
futures market, if they existed at all during 1985, were
probably not very large to begin with and, therefore, the
difference between futures price and inferred price does not
decline as dramatically.
QuasiArbitrage Opportunities
in Options Markets
An interesting aspect of equation (9) is that g the
inferred forward price, can be observed from any exercise
price so long as the corresponding call and put prices can be
observed. It can be easily shown that if two different
exercise prices yield two different g for European options,
arbitrage profit can be made as described below.
Without any loss of generality, assume that E1
prices inferred from these options are denoted by gl and g2
respectively. An arbitrage strategy is described in Table 18
A* **
for the case where gl>g2. In case g
simply be reversed.
Table 18
Arbitrage Strategy Using European Options
At Maturity
Initial
Cash Flow S(s)>E >E1 E2>S(s)>E1 E2>E >S(s)
Buy Call 2 c2 S(s)E2
Sell Put 2 p S(s)E, S(s)E2
Buy Put 1 p E1S(s)
Sell Call 1 c! E1S(s) E1S(s)
Lend B(E2E1) E2E1 E2E1 E2E1
e>0 0 0 0
This strategy gives a positive payoff at the beginning and
a zero payoff at maturity date. In an efficient market, such
opportunities should be eliminated instantly. Thus, for
European options, a nonarbitrage condition is that options
with two different exercise prices should yield the same g
In chapter III it was argued that the early exercise
feature of American options on stock indicies is sparingly
utilized. Hence, it seems that even for American options, a
change in the exercise price should have a negligible effect
on the inferred forward price.
To test this conjecture the data are divided into two
subsets. Every day for which inferred forward prices based on
two or more different exercise prices are available, is
selected for the test. Prices of options with the lowest
exercise price are denoted by CMIN and PMIN and those with
highest exercise price are denoted by CMAX and PMAX. Based on
these two sets of observations, two different values of g
are calculated using the following equations.
CMINPMIN
g MIN = EMIN M B
SCMAX PMAX
g'MAX = EMAX B
(10)
(11)
*
To test whether the mean value of g is equal to the
mean value of qMAX' a simple ttest is performed. A new
*
variable, W, is defined to denote the difference between gMIN
and gMAX. The relevant statistics for W are presented in
Table 19.
Table 19
between gMIN
and gMAX (in Cents)
Days to 014 1529 3044 >45 Total
Maturity
Number of
observations 116 114 82 38 350
W 14.4 21.7 21.2 16.9 18.6
** ** ** **
tstatistic 5.49 4.97 4.63 1.53 8.07
**
significant at 1 % level
The results presented in Table 19 are somewhat surprising.
The variable gMIN is higher than gMAX for all maturities.
Difference
This implies that as exercise price decreases, the inferred
forward price, g also increases.17
If early exercise is not an issue then an arbitrage as
shown in Table 18 should be feasible. Since it was argued in
the previous chapter that early exercise is minimal, but not
zero, a "perfect" arbitrage strategy seems infeasible. A
quasiarbitrage strategy may be to adopt the strategy
outlined in Table 18 and take positions with different
brokers. It has been shown empirically that early exercise
probability itself is small. Even if that small probability
were to become a reality, it is unlikely that it would happen
for investor's positions with all brokers. It is, however,
difficult to quantify the net effect of this strategy given
the subjective factors involved.
Transactions Costs and QuasiArbitrage
A casual check of transactions costs for floor traders
shows that they are minimal. The only kind of transactions
costs that these traders pay are clearing charges to the OCC
and any per contact charges levied by the options exchange.
At present, the OCC charges 7.5 cents per contract per side
for clearing the trade. Since the strategy outlined in Table
1This unexpected phenomenon reduces the credibility of
the results presei'ed in the previous section. It was shown
that futures prices, f,,exceeded inferred forward prices, g*.
Now that we find that g varies systematically with E, it is
not clear whether the difference between f and g* is real or
caused by the exercise prices that happened to be selected
for the results presented in Table 15 through Table 19.
18 involves taking a position in four different options, the
total oneway transaction costs are 30 cents for one contract
in each option. For each index unit it implies a total one
way transactions cost of 0.3 cents. As regards the charges by
the options exchange, traders must pay the exchange 6 cents
per contract per side for trading on their own account. This
implies an additional charge of 24 cents per side charge for
trading in four options simultaneously. The addition to the
per index unit cost is a negligible 0.24 cents.
From this discussion it is obvious that the transactions
costs for floor traders are negligible. Therefore, such costs
will not be a deterrent to exploitation of quasiarbitrage
opportunities, if such opportunities exist.
I also check to see if transactions costs for ordinary
individuals are low enough for them to consider jumping into
the fray to exploit the quasiarbitrage opportunities.
Ignoring early exercise, this strategy is feasible for
ordinary individuals if the transactions costs are less than
the differences reported in Table 19. The transactions costs
are obtained from two discount brokers and two fullservice
brokers. Since similar results are obtained from different
transactions costs data, only one of them is reported here.
Dollar Amount Commission
$ 0 2,000 $18 + 1.8 % of dollar anount
$ 2,001 $11,000 $38 + 0.8 % of dollar amount
$ 11,001 and over $98 + 0.25 % of dollar amount
These commission rates are applied to the data on options
to see if the arbitrage strategy described in Table 18 is
feasible for ordinary individuals. Three different scenerios
are envisaged on the basis that the arbitrage strategy is
undertaken by taking a position in 10 contracts, 100
contracts, and 1,000 contracts. Table 20 presents the summary
of oneway transactions costs for all three scenerios.8
Table 20
OneWay Transactions Costs Per Index Unit for Ordinary
Individuals using the Strategy Outlined in Table 18
(in cents)
Days to
Maturity 014 1529 3044 >44 Total
No. of Contracts
10 21.2 25.7 28.4 29.9 25.3
100 6.0 7.5 8.4 8.8 7.4
1000 3.3 4.3 5.0 5.4 4.2
Comparing the transaction costs given in Table 20 to the
*
magnitude of difference between gMIN and gMAX given in Table
18
18Only oneway transactions costs are presented in this
table since they are sufficient to make the point here. The
roundtrip costs cn be approximated by noting that out of
the four options in which a position is taken, two will
expire worthless. For the other two, closing trades will have
to be made and transactions costs will have to be incurred. A
rough estimate of roundtrip transaction costs, therefore, is
to multiply the numbers presented in Table 20 by 1.5.
19, it is easy to see that it is much more difficult for
ordinary individuals to undertake the operation, unless they
are willing to take on large positions. The transactions
costs for ordinary individuals are not negligible as they are
for floor traders.
Conclusion
It was shown in this chapter that for the sample used,
futures prices exceed forward prices inferred from option
prices. The difference between the two prices is a function
of time to maturity, but not a function of time of trade for
observations taken between 4:00 P.M. and 4:10 P.M. A weakness
of the results is that they are sensitive to the exercise
prices chosen for inferring forward prices. Such sensitivity
weakens the results presented in Table 15 through Table 18.
However, it proves to be the basis of the discovery of
potential quasiarbitrage opportunities in the options
market.
CHAPTER V
SUMMARY AND CONCLUSIONS
This dissertation examines the interrelationships among
options, forward contracts and futures contracts. The first
issue addressed is the relationship between forward contracts
and futures contracts. The controversy about the difference
between forward prices and futures prices is addressed from a
different perspective using stock index futures. It is shown
that during 1985 significant differences existed between
equilibrium forward prices and observed futures prices on
Major Market Index. The differences were too large to be
explained by the CIR model which takes into account the daily
resettlement feature of futures contracts. The magnitude of
the differences is large enough to allow for the existence of
quasiarbitrage opportunities between spot and futures
markets. Even after the transactions costs are taken into
account, profitable opportunities exist for professional
traders. For ordinary investors, transactions costs are
apparently high enough to perclude their participation in
arbitarge operations. Further, it is found that during 1985
such opportunities declined over time presumably because of
the actions of the aihitrageurs.
The second issue addressed is the relationship between a
forward contract and a specific portfolio of options. It is
wellknown that the two are equivalent if the portfolio
consists of European options. It is argued in this study that
even when the portfolio consists of American options, it may
closely approximate a forward contract. The essence of this
argument is that a miniscule proportion of options are
exercised early thus reducing the distinction between
European and American options. The argument is especially
strong for the special case of index options which experience
a much smaller incidence of early exercise. It is also shown
that for index options, some of the theoretical conjectures
regarding early exercises are not very useful in explaining
even the small incidence of early exercise. In particular,
dividends do not seem to influence early exercise of index
options in the manner suggested by theory for individual
stock options.
The third issue addressed is the relationship between
American options and futures contracts. It is argued that
since "The American Options Portfolio" closely approximates a
forward contract, it should be related to a futures contract
in the same way as a forward contract. In other words,
forward prices inferred from option prices should be related
to futures just like the eqlibrium forward prices would be. A
test of this conjecture cofirms that the direction of the
difference between inferred forward prices and futures prices
is the same as the difference between equilibrium forward
prices and futures prices, but their magnitudes differ
significantly. The difference between futures prices and
inferred forward prices are much smaller indicating that
fewer, if any, quasiarbitrage opportunities existed between
options and futures markets as compared to such opportunities
in spot and futures markets. In doing this analysis, it is
discovered that a different kind of quasiarbitrage
opportunity may have existed within the options market. This
discovery is a result of the observation that option prices
based on different exercise prices yield different inferred
forward prices. Lower exercise prices yield systematically
higher forward prices. If early exercise is not a major
problem, as argued in chapter III, then a quasiarbitrage
opportunity seems to exist. However, the gain from this
opportunity is not quantifiable.
BIBLIOGRAPHY
Black, F., 1976, The pricing of commodity contracts, Journal
of Financial Economics, 3, 167179.
Cornell, B. and M. R. Reinganum, 1981, Forward and futures
prices: evidence from the foreign exchange markets,
Journal of Finance, 36(5), 10351045.
Cox, J., J. Ingersoll and S. Ross, 1981, The relationship
between forward prices and futures prices, Journal of
Financial Economics, 9, 321346.
Cox, J. and M. Rubinstein, 1985, Options Markets (Englewood
Cliffs, New Jersey: PrenticeHall).
Daily Market Publications: Option Sales on the American Stock
Exchange, 1985, published by Francis Emory Fitch, Inc.,
New York, NY.
Dusak, K., 1973, Futures trading and investor returns: An
investigation of commodity market risk premiums, Journal
of Political Economy, NovemberDecember, 13861407.
Evnine, J. and A. Rudd, 1985, Index options: the early
evidence, Journal of Finance, 40(3), 743756.
French, K. R., 1983, A comparison of futures and forward
prices, Journal of Financial Economics, 12, 311342.
Geske, R. and K. Shastri, 1985, The early exercise of
American Puts, Journal of Banking and Finance, 9,
207219.
Jarrow, R. and G. Oldfield, 1981, Forward contracts and
futures contracts, Journal of Financial Economics, 9,
373382.
Klemkosky, R. C. and D. J. Lasser, 1985, An efficiency
analysis of the Tbond futures market, Journal of Futures
Markets, 5(4), 607620.
Margrabe, W., 1976, A theory of forward and futures prices,
Working paper (The Wharton School, University of
Pennsylvania, Philadelphia, PA).
Merton, R. C., 1973, Theory of rational option pricing, Bell
Journal of Economics, 4, 141183.
Moody's Dividend Record, 1986, published by Moody's Investors
Service, New York, NY, 56(1).
Moriarty, E., S. Phillips and P. Tosini, 1981, A comparison
of options and futures in the management of portfolio
risk, Financial Analysts Journal, JanuaryFebruary, 6167.
Park, H. Y. and A. H. Chen, 1985, Differences between futures
and forward prices: a further investigation of the
markingtomarket effects, Journal of Futures Markets,
5(1), 7788.
Stock Option Guide: Daily Graphs, 1985, published by William
O'Neil and Co., Inc., Los Angeles, CA, Volume X.
The Wall Street Journal, 1985, published by DowJones and
Company, Inc., New York, NY, Volumes CCVI and CCVII.
Wu, S., 1984, An equilibrium model of index futures pricing,
Ph.D. dissertation (University of Florida, Gainesville,
FL).
Zaslow, J., 1986, Blue chip stockindex contracts fuel
unusually volatile swings in prices, The Wall Street
Journal, April 7, 1986, Page 40.
BIOGRAPHICAL SKETCH
Gautam Dhingra was born on December 11, 1961, in New
Delhi, India. He earned a Bachelor of Commerce (Honors)
degree from University of Delhi in 1980 and was awarded the
MBA degree in 1982 by the same institution. In 1983, he
entered the University of Florida and was awarded a doctoral
degree in Finance in August 1986. Upon completion of the
degree he joined Hewitt Associates, Lincolnshire, Illinois,
as an Investment Analyst.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Phil sophy.
 G/ 
Robert C. Radcliffe, airman
Associate Professor of Finance
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
r ^}^
Stephen Cosslett
Associate Professor of Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Rog_ang T
Roger uang "
Associate Professor of Finance
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree o Doctor o Philosophy.
M.P.N fyanan
Assistant Professor of Finance
This dissertation was submitted to the Graduate Faculty of
the Department of Finance, Insurance, and Real Estate in the
College of Business Administration and to the Graduate
School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
August 1986 
Dean, Graduate School
