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closing options and futures prices between 2nd
January and 31st December, 2001. Resultsobtained demonstrate that the inclusion of
transaction costs in the model considerably reduces
Abstract
This paper investigates the put-call parity (PCP)relation using options on futures on the Standard
and Poors 500 (S&P 500) Index using daily
259D e r i v a t i v e s U s e , T r a d i n g & R e g u l a t i o n V o l u m e N i n e N u m b e r T h r e e 2 0 0 3
Tests of the put-call parity relation using options
on futures on the S&P 500 Index
Urbi Garay*, Mara Celina Ordonez and Maximiliano Gonzalez
*Instituto de Estudios Superiores de Administracion (IESA), Av. IESA, Edif. IESA,
San Bernardino, Caracas, 1010, Venezuela. Tel. 58 212 555 4242;
Fax. 58 212 555 4446; E-mail: urbi.garay@iesa.edu.ve
Received (in revised form): 6th May, 2003
Urbi Garay is Assistant Professor of Finance at the Instituto de Estudios Superiores de Administracion (IESA)
in Caracas, and is associated with the Center for International Studies and Derivatives Markets (CISDM) at the
University of Massachusetts, Amherst. He holds an MA in International Economics from Yale University and a
PhD in Finance from the University of Massachusetts, Amherst.
Mara Celina Ordonez is Accounting and Financial Reporting Services IT Manager at Procter & Gamble Latin
American headquarters in Caracas. She joined P&G in 1996, and holds a Masters degree in Finance from IESA
in Caracas.
Maximiliano Gonzalez is Assistant Professor of Finance at IESA in Caracas. He holds an MBA from IESA, and
a Masters degree in Management and a PhD in Finance from Tulane University.
Practical applications
This paper investigates the well-known put-call parity (PCP) relation using options on
futures on the Standard and Poors 500 (S&P 500) Index. The authors verify that when
transaction costs commission costs and bid-ask spreads on options and on futures are
included in the model, arbitrage opportunities are translated into the possibility of a gain
well below $1,000 for an option contract on futures on the S&P 500. This amount does
not represent an economically significant value, especially if it is noted that other factors
such as taxes have not been considered in this paper. These results offer support to the
efficient market theory.
Derivatives Use,ng & Regulation,ol. 9 No. 3, 2003,
pp. 259280 Henry Stewart
Publications,
1357-0927
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the number of times that a violation of the PCP
relation occurs at the same time that it diminishes
the magnitude of the distortion. Similarly, the
PCP relation applies more accurately to those
options that are the nearest to being at-the-money.When deep out-of-the-money or deep
in-the-money options were used in the tests the
number of violations increased. This may be the
result of the low liquidity levels of these contracts.
Finally, the authors verify in this study that when
transaction costs commission costs and bid-ask
spreads on options and on futures are included
in the model, arbitrage opportunities are translated
in the possibility of a gain well below $1,000 for
an option contract on futures on the S&P 500.
This amount does not represent an economically
significant value, especially if it is considered that
other factors such as taxes have not been considered
in this paper. These results offer support to the
efficient market theory.
INTRODUCTION
Numerous empirical studies about the put-call
parity (PCP) relation have been conducted in
the American, European and Australian stock
and stock index options markets. Some of
these studies have concluded that distortions
in the price of options are caused by
transaction costs. Other research has found
that violations to the PCP relation can be
attributed to incorrect sampling and to the use
of non-synchronous data. Finally, it has been
verified that some markets presentinefficiencies that cannot be explained by
transaction costs, and that these inefficiencies
could be translated into arbitrage
opportunities.
This paper investigates the PCP relation
using options on futures, more specifically,
options on futures on the Standard and Poors500 (S&P 500) Index. These contracts are
traded on the Chicago Mercantile Exchange
(CME). The authors inquire whether any
pattern of violation of the PCP relation is
present and to what extent transaction costs
and liquidity could reasonably explain any
possible distortions.
Following this, there is a section presenting
the different PCP relations that have been
proposed in the literature, and a review of the
empirical literature. The next section presents
the data and methodology followed to test
the PCP relation in the study; the final
section presents the results obtained and
concludes the paper.
PUT-CALL PARITY RELATIONS
The basic put-call parity relation
(Stoll)
The PCP relation is a theoretical derivation
initially outlined by Stoll.1 Stoll established
a relation between the prices of a put and a
call option written on the same underlying
asset, having the same exercise price and
the same expiration date. Stoll makes the
following assumptions in his derivation:
there are no transaction costs; options arealways exercised on their date of expiration
and never before this date; underlying assets
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stocks that do not pay dividends or that
they are protected against the payment of
dividends, it is certain that an American call
option should not be exercised before its
expiration date and, therefore, an Americancall should have the same value as an
European call option. On the other hand,
the case is not the same for put options
because if, at some moment, the underlying
asset reaches a value near zero, the put
option would have to be exercised, since
this would be a situation in which there
would exist a large possible gain. With this
reasoning, Merton proposes the following
inequality that must be fulfilled for
American options:
SKCPSKerT (3)
The PCP relation and the effect of
dividend payments (Klemkosky and
Resnick, and Cox and Rubinstein)
Based on the study of the effect of
dividend payments made by Klemkosky
and Resnick,3,4,5 Cox and Rubinstein6
present two propositions The first
proposition presents the effects of
dividends on the PCP relation for
European options whose underlying stocks
pay well-known dividends (D) and that
are not protected against them. This
relation appears next:
CDKerT P S (4)
The second proposition shows the effects of
dividends on the PCP relation for
do not pay dividends during the life of the
option contract or, in their defect, options
are protected against dividend payments;
there are no arbitrage opportunities,
borrowers and lenders can become indebtedthemselves or lend money at the risk-free
rate of interest.
Under these assumptions, Stoll
demonstrates that buying a put (P) is the
same as buying a call option (C) on the
same stock, with the same expiration date
(T) and strike price (K) combined with a
short position in the underlying asset (S)
plus the loan of an amount of money
equivalent to the value of the exercise price
discounted at the risk free rate of interest
(r):
CSPKerT. (1)
Rearranging terms, Stoll obtains the PCP
relation:
CKert PS (2)
According to Stoll, since investors should
not exercise options prior to their
expiration dates, the relation outlined in (2)
should hold for both American and
European options.
The modified PCP relation for
American options (Merton)
Merton2 makes a comment on the work ofStoll in which he argues that, assuming that
both put and call options are written on
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American options whose underlying stocks
pay well-known dividends and that are not
protected. This relation is:
SDKCP SKerT (5)
If dividends are uncertain, equations (4) and
(5) are modified as presented in equations
(6) and (7), respectively:
(4) > CD+KerT P+SC
DKerT (6)
(5) > SD+KC
PSKerT (7)
Where D+ represents the maximum
expected dividends and D represents the
minimum expected dividends.
The PCP relation and the effect of
transaction costs
The previous derivations of the PCP
relation are based on the assumption that
transaction costs do not exist. Cusack7
studies the following relation for bothEuropean and American options whose
underlying stocks do not pay dividends:
SKTCuCPS
KerTTCo (8)
Where TCu represents those transaction
costs that are attributable to an arbitrage
strategy resulting from a call option that is
undervalued and TCo represents thetransaction costs that arise when an
arbitrage strategy is followed as a result of a
call option that is overvalued. Both TCu
and TCo should include: broker
commissions, stock-market commissions and
taxes. Bid-ask spreads represent another
component of transaction costs that must beconsidered.
REVIEW OF THE EMPIRICAL
LITERATURE
This section presents a brief review of the
empirical literature on the PCP relation in
the American, European and Australian
markets. But first, the effects of the use of
non-synchronous data in tests of the PCP
relation are considered.
Effects of the use of non-synchronous
data in previous empirical studies of
the PCP relation
A number of previous empirical studies on
the PCP relation suggest that there exist
real opportunities for arbitrage in markets as
a result of apparent mispricings of options.
On some occasions the mispricing can be
attributed to transaction costs since, in most
cases, these have not been included. In
other cases, however, the problem of
non-synchronicity in transactions between
options and the price of underlying assets
can explain what appears to be a violation
of the PCP relation. In this regard, and
depending on the liquidity of the options
and underlying assets that are used in theempirical study, a suitable form of sampling
ought to be selected so that the effect of
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(Chicago Board Options Exchange
(CBOE), Chicago Board of Trade (CBOT),
etc), finding possible inefficiencies in the
formal market for options in the USA).
The following three important factors,which may be the cause of the possible
inefficiencies found, were present in these
early studies:
The data used were not intra-daily data
and, in some cases, not even daily
closing data. Most of the samples taken
used weekly or monthly closing prices,
thus increasing the probability of errors
caused by non-synchronicity in data;
Transaction costs were not taken into
account. All studies were made on
American options, and in many cases it
was not possible to isolate the effect of
the value of the early exercise of
options;
Since over-the-counter (OTC)
transactions take place directly between
financial institutions and corporations,
and not through a formal market, the
registration of these transactions is not
very precise.
Kamara and Miller12 made an empirical
study of the PCP relation on European
options on the S&P 500, which is
negotiated on the CBOE. The authors
eliminated the problem posited by the
value of the early exercise of an Americanoption since they were using European
type options contracts. Kamara and Miller
non-synchronous trading can be
ameliorated.
For liquid options and stock markets,
Brown and Easton8 propose that the
following recommendations be taken intoaccount in the sampling process where the
only information available is the closing
price of options and stocks:
To use the closing stock price if the
spread is within the bid-ask spread,
otherwise the sample must be discarded.
To consider solely put and call options
that fulfill the following characteristics:
a) They have a volume of transactions
different from zero on the sampled
day;
b) That the date and hour of market
closing is the same one that was
taken for the stock; and
c) That the closing price of put and call
options is within the closing bid-ask
spread.
Previous studies of the PCP relation in
the USA
The PCP relation has been extensively
tested in the US markets. The first studies
were made by Stoll,1 who established
support for the PCP theory, and Gould and
Galai.9 These two later authors found that
the PCP relation held depending on the
magnitude of assumed transaction costs.
Later, Klemkosky and Resnick,3,4,5 Evnineand Rudd,10 and Chance11 tested the PCP
relation in the American formal markets
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found that the number of PCP violations
was much smaller than that found in
previous studies in which only American
options were used. Additionally, the authors
concluded that the pattern of violations ofthe PCP relation can be associated with a
premium value that results from liquidity
risk, that is, the risk that an investor incurs
when he or she is trying to engage in
arbitrage transactions and one of the
transactions cannot be completed at the
correct price. Finally, Kamara and Miller
found that the frequency and the number
of violations is related to moneyness
(options farther from being at-the-money
present a greater number of violations to
the PCP relation than those that are closer
to being at-the-money).
Previous studies of the PCP relation in
Europe
Nisbet13 makes an empirical study based on
negotiated American options traded on the
London Traded Options Market (LTOM),
using intra-daily data and including
transaction costs and dividend payments.
Nisbet finds that when the only transaction
cost considered is the bid-ask spread a
significant number of violations of the PCP
relation are present. In a second model, in
which he includes the costs of commission
and the effect of dividends in addition to
the bid-ask spread, he finds that the volume
and frequency of the violations in relationto the PCP are reduced to the point where
the possibilities of potential arbitrage gains
are relatively low. In a recent paper,
Capelle-Blancard and Chaudhury14 find
support for the PCP relation in France and
present a review of the literature on the
PCP relation in different Europeancountries.
Previous studies of the PCP relation in
Australia
The main studies conducted in Australia are
those of Loudon,15 Gray,16 Taylor,17
Easton,18 Brown and Easton8 and Cusack.7
Loudon and Taylor each undertake
empirical tests of the PCP relation in the
Australian Options Market (AOM) using
the same model and the same source of
information. Nevertheless, they reach
diametrically opposite conclusions. Brown
and Easton made a study attempting to
reconcile the results obtained by Loudon
and Taylor.
The authors next present the model
employed by Loudon and Taylor, and later
by Brown and Easton, as well as a
comparative table of the results obtained in
each study (see Table 1).
CSKe_rTPC SKVp(D)
(9)
Brown and Easton obtain results that are
similar to those of Loudon and conclude
that the main reason why Taylors study
produced different results lay in theproblem of using non-synchronous data,
since 60 per cent of their samples were
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Finally, Cusack7 makes an empirical study
on the AOM for American options,
including transaction costs (but excluding
the bid-ask spread), but without including
the effect of dividends and using intra-daily
data for time intervals between five and 15
minutes verifying that their results are
consistent with those obtained by Loudon,
Brown and Easton and Gray. These results
are consistent with the existence of
inefficiencies in the Australian market, even
when transaction costs are included in the
analysis, and the use of intra-daily data
versus the use of closing daily data not
making a difference in the results obtained.
Data and general methodology
There follows an outline of the data and
general methodology used to test the PCP
relation for options on futures on the S&P
500 Index:
For the accomplishment of the study the
information on closing prices on the
invalid. Taylor only used monthly closing
data and included closing data for days for
which the volume of put or call
transactions was zero. Additionally, Brown
and Easton found some computational
errors in the procedure employed to
calculate the put-call values.
The studies of Loudon and Brown and
Easton demonstrate the existence of
apparent inefficiencies in the AOM. In
most cases, inefficiencies come from an
underestimation of the price of puts (lower
boundary). For this reason, apparent
arbitrage opportunities were present. Gray16
makes a study on the AOM using a model
that includes transaction costs, the value of
early exercise of option contracts and the
effects of dividends, using closing prices for
options and stocks that have been traded
during the day. Gray finds significant
violations of the PCP relation, even when
he includes commission costs. The
frequency and volume of violations isreduced, however, when transaction costs
include the bid-ask spread.
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Table 1: Comparison of empirical studies of the PCP relation in Australia: Loudon,15
Taylor,17 Brown & Easton8
Loudon15 Taylor17 Brown and Easton8
Non-violations (%)
Lower boundary violations (%)
Upper boundary violations (%)
60
38.5
1.5
83.8
0
16.2
70.8
26.3
3
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following derivatives traded on the
CME is considered: Futures contracts on
the S&P 500 Index, American put
option contracts on futures on the S&P
500 Index (the size of the contract is250), American call option contracts on
futures on the S&P 500 Index (the size
of the contract is 250).
The sample period starts on 2nd January,
2001 and ends on 31st December, 2001.
The information chosen corresponds
only to closing daily prices of those
option contracts for which at least one
transaction occurred during the day. This
procedure was followed as a way to
eliminate part of the problem of
working with non-synchronous data.
Contracts that show a price of zero on a
certain date are eliminated from the
sample.
Only transactions made in the regular
trading time were taken from the CME
(regular transactions or R). All
after-hours transactions were eliminated
(electronic transactions or E). These
transactions can be made through the
GLOBEX system in night schedules
(since trades on futures on the S&P 500
Index are closed in the regular period,
the use of these after-hours transactions
could have introduced distortions in the
study because of the problem of
non-synchronicity in trades).
Tests of the PCP relation wereundertaken for every working day of the
stock market, using closing put, calls,
and futures prices of contracts that have
the same maturity date and the same
exercise price. This procedure diminishes
part of the problem of using
non-synchronous data.Yields on Treasury Bills are used as the
risk-free rate of interest. The rates were
taken according to the following criteria:
T-bill to three months for contracts
that possess from zero to three months
to maturity; T-bill to six months for
contracts that mature between three and
six months from the day the sample was
taken; T-bill to one year for
contracts maturing between six months
and one year.
Contracts possessing the following
expiration dates were considered: March,
June, August and December, 2001; and
March, June and September, 2002.
The date of expiration of each contract
was the third Friday of the expiration
month.
The initial contract sample, including
traded volume and prices different from
zero, is constituted by: 16,367 call options
(number of days number of contracts)
on futures on the S&P 500 Index; 22,252
put options (number of days number
of contracts) on futures on the S&P 500
Index; and 1,770 futures contracts
(number of days number of contracts)
on the S&P 500 Index.
For every working day, put and calloption contracts with the same exercise
price and the same expiration date to the
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In most cases, commission costs depend
on the volume of the transaction. For
this study, consider the minimum costs
necessary to trade a single contract (ie
size of 250). The commission costs weretaken from an article published in
CNNMoney.19 This article presents a
schedule of the commissions charged by
the most important discount brokers in
the market (see Table 2).
The bid-ask spread for futures contracts
traded on the S&P 500 Index was
estimated, sampling five months of data
from the CME (April 2001August
2001). For each date the magnitude of
the closing bid-ask spread was recorded
in relation to the closing price of the
futures contract. Following this
procedure, the authors determined that
the average bid-ask spread was 0.09 per
cent of the futures price, with a standard
deviation of 0.04 per cent. This value
was employed as the transaction cost
associated with the bid-ask spread of any
futures transaction.
The data supplied by the CME do not
contain any information pertaining to
the bid-ask spread associated with put
and call options on futures on the S&P
500 Index (http://www.cme.com). The
authors estimated this bid-ask spread
using information supplied by Yahoo
Finance (http://www.yahoofinance.com).
A two-week sample was taken in whichthe bid-ask spread of those options
closer to being at-the-money was
futures contract are grouped. This yields a
total of 2,773 samples ready to be tested
for the PCP relation.
The equation that is used to prove the
model, without taking into accounttransaction costs, is (model 1):
F0erTKPCF0Ke
rTP
Where: F0 is the price of the futures
contract on the S&P 500 Index; K is
the exercise price; P is the price of the
American put option on futures contract
on the S&P 500 Index; C is the price
of the American call option on futures
contract on S&P 500 Index; r is the
interest rate (T-bills); T is the time
remaining until the option expires.
The equation that is used to prove the
model, taking into account transaction
costs (model 2), is:
F0erTKPBid_AskTCu
CF0KerTPBid_AskTCo
Where: TCu represents the commission
costs incurred in an arbitrage transaction
when a call option is undervalued; buy
a call, sell a put, short a futures contract,
and invest K at the rate of interest of
T-bills. Tco represents the commission
costs incurred in an arbitrage transaction
when a call option is overvalued; sell a
call, buy a put, buy a futures contract,and borrow K at the rate of interest of
T-bills.
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recorded for each day. Although this
approach yields only a rough estimation
of bid-ask spreads, it constitutes the best
approach the authors could have
followed, given the lack of information.
For call options the average bid-ask
spread was 5.40 per cent of the price of
the option, with a standard deviation of
2.31 per cent. For put options the
average bid-ask spread was 4.99 per cent
of the option price, with a standard
deviation of 1.39 per cent. These values
were considered to be the bid-ask
spread-associated transaction costs
applicable to transactions on call and put
options on futures on the S&P 500
Index. For each trio of put, call and futures
prices the test of the PCP relation was
carried out for the four different groups
of studies that are presented below.
Study 1
This study was done using the whole
sample, with and without considering
transaction costs. The authors considered:
The number of times that the PCP
relation was violated by the upper
boundary.
The number of times that the PCP
relation was violated by the lower
boundary, and the number of times that
the relation was satisfied. Two values
were eliminated from the sample because
they were introducing a distortion factorto the sample, thus reducing the final
sample size to 245 days.
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Table 2: Commission costs charged by main brokers (CNNMoney)
Options Stocks
Minimum ($) Per contract ($) Minimum ($) Per contract ($)
Charles Schwab
E-Trade
TD Waterhouse
Fidelity
Ameritrade
Brown & Co (Chase)
DLJ Direct
Scottrade
CyberBroker
Arithmetic mean
35
29
28.13
27
29
25
35
20
19.95
27.56
0
0
0
0
0
0
1.75
1.60
0
0
29.95
14.95
12
25
8
5
20
7
14.95
15.21
0
0
0
0
0
0
0
0
0
0
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EMPIRICAL RESULTS
This section presents the results obtained
from each of the four studies.
Study 1: Determination of the numberof violations of the PCP relation for
the whole sample
In this study the authors calculated from the
data (put prices, futures price contracts,
interest rates, exercise prices, expiration
dates), the rank within which the theoretical
price of a call option should have oscillated.
The observed closing price of a call was then
compared with the obtained theoretical rank.
When the observed price surpassed the
upper boundary of the rank, it was classified
as a violation of the upper boundary. When
the violation occurred below the lower
boundary, it was classified as a violation of
the lower boundary. When the call price
observed was within the rank it was classified
as a non-violation of the PCP relation. This
study was first made without including
transaction costs (model 1) and was later
repeated adding transaction costs (model 2).
Additionally, the authors computed the
theoretical arbitrage gain an investor could
have obtained in each one of the cases
should he/she have detected the violation of
the PCP relation (see Table 3).
Analysis of study 1
Of the total of samples that include
in-the-money, out-of-the-money andat-the-money options, it can be observed
that when transaction costs are not included
The average gain that could have been
earned by an investor who took
advantage of the arbitrage opportunities
arising as a result of overvaluations or
undervaluations in the prices of options.
Study 2
Study 1 was repeated for the most liquid
options contracts in each day, with and
without transaction costs.
Study 3
For all option contracts nearest to being
at-the-money in each day, with and
without transaction costs, the authors:
Computed the percentage deviation of
the theoretical price in relation to the
maximum/minimum price determined
by the PCP relation.
Computed a descriptive statistic of the
percentage deviation of the theoretical
price between 2nd January, 2001 and
31st December, 2001. Two values were
eliminated from the sample because they
were introducing a factor of distortion
to the sample, thus reducing the final
sample size to 245 days.
Drew a graph of the daily deviation of
the theoretical price from 2nd January,
2001 to 31st December, 2001.
Study 4
Study 3 was repeated for the most liquidoptions contracts in each day, with and
without transaction costs.
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the PCP relation holds in only 43.56 per
cent of the cases (see Table 3). The greater
number of violations of the relation occur
by the lower boundary (38.11 per cent).
These violations can be attributed to the
fact that transaction costs are being ignored
in model 1 and to the low liquidity levels
of some of the instruments considered in
the sample (this may contribute to the
distortion since the data being used is
non-synchronous). This problem can be
important especially for those options
contracts that are deep in-the-money and
deep out-of-the-money.
When the study was repeated using
model 2, that is, including transaction costs,
the number of non-violations increases to
75.78 per cent, compared to 43.56 per cent
in model 1. Most of the violations still
occur in the lower boundary, althoughdiminishing to 16.17 per cent. It can be
observed that the inclusion of transaction
costs in the model causes the PCP relation
to be fulfilled a larger number of times.
Nevertheless, according to this study,
arbitrage opportunities are still present in 28
per cent of the cases.
When calculating the gains that an
investor could have potentially obtained
when engaging in arbitrage transactions, it
can be observed that, in the case in which
transaction costs are not included in the
model, the average gain is $1,591 for each
contract of 250, whereas in the case when
transaction costs are included, the gain
diminishes to $974.
Study 2: Determination of the number
of violations of the PCP relation for
options that are the nearest to being
at-the-money and for options that
present greater liquidityThis study is similar to study 1, with the
difference that, in this case, in the first part
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Table 3: Study 1 Violations of the PCP relation for the whole sample
Excluding transaction costs
(model 1)
Including transaction costs
(model 2)
Sample size % Sample size %
Violation upper boundary
Violation lower boundary
Non-violations
Total
Gain (Arbitrage)
508
1,056
1,207
2,771
$1,591
18.33
38.11
43.56
223
448
2,100
2,771
$974
8.05
16.17
75.78
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can be observed that, when transaction
costs are not included, the relation holds in
51.42 per cent of the cases (see Table 5).
This represents a 9 per cent increase in the
fulfillment of the PCP relation compared
to study 1. Both results are an indication
that the liquidity factor influences the
results of the study. The greater number of
violations of the PCP relation happens by
the lower boundary in all of the cases.
This result is consistent with the results
obtained in study 1.
The violations can still be attributed to
the transaction costs that are ignored in
model 1. When transaction costs are
included in the study, according to model
2, it can be observed that, for the subgroup
of options that are the nearest to being
at-the-money as well as for the case of the
most liquid option contracts, the number oftimes that the PCP relation is satisfied
increases to 87 per cent (options
only those options that are the nearest to
being at-the-money every day are used (see
Table 4) and, in the second part, the most
liquid options every day (largest volume)
are employed (see Table 5). This is done
with the purpose of isolating the impact
that including in the sample instruments
that could be less liquid can have, and that
could, therefore, cause distortions because
of the use of non-synchronous data.20
Analysis of study 2
Of the total of samples that include only
options that are the closest to being
at-the-money it can be observed that,
when transaction costs are considered, the
PCP relation is fulfilled in 65 per cent of
cases (see Table 4). This represents an
increase of 22 points in relation to the
previous result in which all the optionswere included. Also, for options with the
greatest volume of transactions every day it
271Ga r a y , Or done z a n d G o n z alez
Table 4: Study 2A Violations of the PCP relation for at-the-money contracts
Excluding transaction costs
(model 1)
Including transaction costs
(model 2)
Sample size % Sample size %
Upper boundary violation
Lower boundary violation
Non-violations
Total
Gain (Arbitrage)
37
49
161
247
$862
14.98
19.84
65.18
13
15
219
247
$460
5.26
6.07
88.66
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at-the-money) and 86 per cent (options
with the largest traded volume) respectively.
These numbers represent a substantial
increase in relation to model 1 (in which
transaction costs were not considered) and
in relation to the same scenario in study 1,
in which all the operations were included
(both liquid and illiquid).
The results of this study, compared with
the previous investigations of Loudon15 and
Cusack7 in Australia, are different in the
sense that the number of violations of the
upper boundary of the PCP relation is
greater than the number of violations of the
lower boundary in these studies. This can
have several causes, such as: the costs of
transactions in the Australian market differ
from the costs of transactions at the CME,
partly because the Australian market is
substantially less liquid than the CME. Thestudies of Loudon15 and Cusack7 were
made on options on stocks, whereas the
present study is made on options of futures
on the S&P 500 Index. These instruments
have different liquidity patterns and
transaction costs.
When calculating the gains that an
investor could have obtained when
engaging in arbitrage transactions it can be
observed that, in the case of options that
are the nearest to being at-the-money (and
excluding transaction costs from the
model), the average gain is $862 for each
contract of 250. In the case where
transaction costs are included, the gain
diminishes to $460. In the case of more
liquid options these values are $1,485 (with
transaction costs) and $653 (without
transaction costs). In all cases it can be
considered that these amounts are not
economically significant, especially because
other factors exist (taxes, for example) thatcould reduce further the margin of gain
with zero investment.
272 Ga r a y, Or done z a n d G o n z ale z
Table 5: Study 2B Violations of the PCP relation for the most liquid option contracts
Excluding transaction costs
(model 1)
Excluding transaction costs
(model 2)
Sample size % Sample size %
Upper boundary violation
Lower boundary violation
Non-violations
Total
Gain (Arbitrage)
20
90
127
247
$1,485
12.15
36.44
51.42
10
25
212
247
$653
4.05
10.12
85.83
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average and the median deviation of the
theoretical price with respect to the real
price. This is done with the purpose of
identifying the percentage deviation that
better represents the sample. Figures 1, 2and 3 illustrate the findings.
Analysis of study 3
For contracts that are closest to being
at-the-money it can be observed that, on
average, when transaction costs are not
included, the percentage deviation is
around 7 per cent with a median of 4.5
per cent (see Figure 1). For the majority
of the samples (192 of 245) the deviation
is in the interval between 0 per cent and
7.14 per cent, with a midpoint of 3.57
per cent. Similarly, when transaction costs
are included, the average percentage
deviation is 3.0 per cent with a median
of 1.96 per cent (see Figure 2). For 225
of the 245 samples the deviation is in
the interval between 0 per cent and 3.6
per cent with a midpoint of 1.8 per
cent. In Figure 3, it can be observed
that only 15 samples of 245 seem to
turn aside from the average pattern. This
deviation could be a consequence of the
use of non-synchronous data, or it could
be due to some specific events in the
stock market that generated information
asymmetries.
Through this study it can be
corroborated that the introduction oftransaction costs into the PCP relation
diminishes the number of violations and the
In order to measure the percentage
economic significance of the violations, the
percentage of deviation for each one of the
samples of options that are the nearest to
being at-the-money are calculated in thefollowing study.
Study 3: Magnitude of violations of
the PCP relation for options that are
the nearest to being at-the-money
In this study, the price obtained in the
upper boundary (PUL) and the price
obtained in the lower boundary (PLL) are
compared (in percentage terms) with the
observed value of the closing price of the
call (PT). This comparison is made using
the following procedure:
Deviation PT PLU PT PLU
PT
This procedure can be used to determine
the magnitude by which the price of those
options contracts that are the nearest to
being at-the-money deviate from the
theoretical price range with the purpose of
determining if the price deviation is
economically significant.
Each day the authors analysed the option
contract that was the nearest to being
at-the-money, resulting in a total of 245samples. For these 245 samples, statistical
calculations were made to determine the
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percentage deviation by an importantmagnitude, as well as the percentage
magnitude of the violation with relation to
the price. The 3 per cent deviation of theobserved price with respect to the
theoretical price of the option for the cases
274 Ga r a y, Or done z a n d G o n z ale z
Figure 1: Observed versus theoretical percentage price deviation for all at-the-money
contracts ignoring transaction costs
Notes: mean: 7.27%; standard deviation: 5.80%; median: 4.56%; median deviation: 4.26%
Figure 2: Observed versus theoretical percentage price deviation for all at-the-money
contracts including transaction costs
Notes: mean: 3.02%; standard deviation: 2.24%; median: 1.96%; median deviation: 1.35%
0
50
100
150
200
250
3.57 10.71 17.86 25.00 32.14 39.29 46.43 53.57 60.71
Class rank
Percentage
devia
tion
0
50
100
150
200
250
1.80 5.39 8 .99 12.59 16.18 19.78 23.38 26.97 30.57
Class rank
P
ercentage
deviation
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Analysis of study 4
For contracts with the largest liquidity,
when transaction costs are not included it
can be observed that the average
percentage deviation is around 22 per
cent, with a median of 10 per cent (see
Figure 4). For the majority of the
samples (179 of 245) the deviation is in
the interval of 0 per cent and 13.76 per
cent, with a midpoint of 6.88 per cent.
Similarly, when transaction costs are
included, the average percentage deviation
is 10.21 per cent with a median of 7.68
per cent (see Figure 5). For 226 of the
245 samples the deviation is in theinterval between 0 per cent and 14.17
per cent, with a midpoint of 7.08 per
in which violations occur in the model
(approximately 11 per cent) can be
attributed to the use of non-synchronous
data in the study, the fact that taxes were
not included, the variability of the costs of
transaction depending on the contract and
the commissions charged by brokers.
Study 4: Magnitude of violations of
the PCP relation for options
possessing the largest volume of
trades
This study is similar to study 3, with the
difference that it is based on options that
exhibit the largest liquidity every day(largest traded volume). Figures 4, 5 and 6
show the results obtained.
275Ga r a y , Or done z a n d G o n z alez
Figure 3: Observed versus theoretical percentage price deviation for all at-the-money
contracts with and without transaction costs
0
10
20
30
40
50
60
70
02/01/01
02/02/01
02/03/01
02/04/01
02/05/01
02/06/01
02/07/01
02/08/01
02/09/01
02/10/01
02/11/01
02/12/01
Date
Percentage
deviation
% without transaction costs
% with transaction costs
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cent. In Figure 6, it can be observed
how the introduction of transaction costs
diminishes the number of violations.
Through this study it can be once again
corroborated that the introduction of
transaction costs into the PCP model
substantially diminishes both the number
of times that the relation is violated and
the percentage magnitude of the violation
in relation to the price. Nevertheless, this
study shows values that are higher than
those shown in study 3, which leads the
authors to think that the best way to
diminish the noisy effect of using
non-synchronous data in the samples is to
consider options that are the nearest to
being at-the-money instead of those that
present the greatest traded volume oneach day. Apparently, a greater relation of
synchrony (or simultaneity between
purchases and sales of puts, calls and
futures contracts) between the samples of
options is present when we analyse
at-the-money contracts.
SUMMARY
Results obtained in the determination of
the pattern of violations of the PCP
relation using options on futures on the
S&P 500 Index demonstrate that the
inclusion of transaction costs in the model
considerably reduces both the number of
times that a violation occurs and the
magnitude of the distortion. Similarly, the
authors have verified in this study that
when transaction costs are included in the
model, arbitrage opportunities are translatedin the possibility of a gain well below
$1,000 for an option contract on futures on
276 Ga r a y, Or done z a n d G o n z ale z
Figure 4: Observed versus theoretical percentage price deviation including all contracts
possessing the largest volume and ignoring transaction costs
Notes: mean: 21.93%; standard deviation: 22.19%; median: 9.42%; median deviation: 16.22%
0
30
60
90
120
150
180
210
6.88 20.64 34.40 48.16 61.92 75.68 89.44 103.20 116.96
Class rank
Percentage
deviation
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277Ga r a y , Or done z a n d G o n z alez
Figure 5: Observed versus theoretical percentage price deviation including all contracts
possessing the largest volume and including transaction costs
Notes: mean: 10.21%; standard deviation: 5.76%; median: 7.68%; median deviation: 3.63%
Figure 6: Observed versus theoretical percentage price deviation including all contracts
possessing the greatest volume with and without transaction costs
0
50
100
150
200
250
7.08 21.25 34.42 49.59 63.76 77.93 92.10 106.26 120.43
Class rank
Percentage
deviation
0
20
40
60
80
100
120
140
160
180
02/01/01
02/02/01
02/03/01
02/04/01
02/05/01
02/06/01
02/08/01
02/07/01
02/09/01
02/10/01
02/11/01
02/12/01
Date
Percentage
de
viation
% without transaction costs% with transaction costs
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the S&P 500. This amount does not
represent an economically significant value,
especially if it is noted that other factors,
such as taxes, have not been considered in
this model. These results offer support tothe efficient market theory and can be
appraised through Tables 6 and 7.
The average deviation of the observed
price of an American option call compared
to the calculated theoretical interval
oscillates between 0 per cent and 7 per
cent when Mertons PCP relation for
American options excluding transaction
costs is used. Considering only those
options contracts that are the nearest to
being at-the-money, and including
transaction costs in the model, it can be
observed that the average deviation
diminishes to 3 per cent. This 3 per cent
can be caused by estimation errors in the
bid-ask spread, estimation errors in
commission costs, taxes, and the fact that
options on futures on the S&P 500 Index
are American.
It could be stated that the PCP relation
applies more accurately for options that are
the nearest to being at-the-money. When
deep out-of-the-money or deep
in-the-money options were used in the
tests the number of violations increased.
This may be the result of the low liquidity
levels of these contracts. The authors also
verified that options with the largest
volumes of trades introduced a percentageof violations in the study (in absolute and
percentage amount of deviation) slightly
higher than those found in the study in
which options were the nearest to being
at-the-money. This finding demonstrates
that there is more synchrony in samples
where options are nearer to beingat-the-money than in the case where
options presented the largest volume.
The pattern of violations of the PCP
relation is opposed to that found by
Cusack7 and Loudon14 for the case of the
Australian market, in the sense that the
largest percentage of violations occurred by
the lower boundary (upper in the case of
the Australian studies, which are based on
determining the rank of prices of put
options). The reasons for this can be
diverse. For example, differences between
transaction costs that apply in the Australian
and the American markets, or the fact that
the studies in Australia were made on stock
options, whereas the authors study was
based on options on futures. Nevertheless,
in both studies the introduction of
transaction costs in the model considerably
diminishes the number of non-violations.
Finally, it was observed that the market
reacts to distort the theoretical prices of call
when certain events follow one another. For
instance, it was verified that for all
at-the-money contracts there appears to be a
strong distortion of prices between the
months of March and May. These distortions
cannot be explained by transaction costs or
by the presence of non-synchronous tradessince the pattern was consistent for all
contracts. These distortions could have been
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Put and Call Option Prices: Comment, Journal of
Finance, Vol. 28, No. 1, pp. 183184.
3 Klemkosky, R. and Resnick, B. (1979) Put-Call
Pariry and Market Efficiency, Journal of Finance,
Vol. 34, No. 5, pp. 11411155.
4 Klemkosky, R. and Resnick, B. (1980) An Ex
Ante Analysis of Put-Call Parity, Journal of
Financial Economics, Vol. 8, pp. 363378.
5 Klemkosky, R. and Resnick, B. (1992) A Note
on the No Premature Exercise Condition ofDividend Payout Unprotected American Call
Options: A Clarification, Journal of Banking and
Finance, Vol. 16, No. 2, pp. 373379.
caused by information asymmetries that
translate in arbitrage opportunities seen
during certain periods of time and could be
economically significant.
References
1 Stoll, H. (1969) The Relationship Between Putand Call Option Prices, Journal of Finance, Vol.
24, No. 5, pp. 801824.
2 Merton, R. (1973) The Relationship Between
279Ga r a y , Or done z a n d G o n z alez
Table 6: Summary number of violations of the PCP relation
Without transaction costs (TCs)
(model 1)
With transaction costs (TCs)
(model 2)
Wholesample
Nearest to beingat-the-money
Mostliquid
Wholesample
Nearest to beingat-the-money
Mostliquid
Upper boundary
violation (%)
Lower boundary
violation (%)
Non-violations (%)
Gain (Arbitrage)
18.33
38.11
43.56
14.98
19.84
65.18
12.15
36.44
51.42
8.05
16.17
75.78
5.26
6.07
88.66
4.05
10.12
85.83
$1,591 $862 $1,485 $974 $460 $653
Table 7: Summary magnitude of the PCP violation
At-the-money Most liquid
Without TC With TC Without TC With TC
Mean (%)
Standard Dev (%)
Median (%)
Median Dev (%)
7.27
5.80
4.56
4.26
3.02
2.24
1.96
1.35
21.93
22.19
9.42
16.22
10.21
5.76
7.68
3.63
8/13/2019 pcp_spx
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6 Cox, J. C. and Rubinstein, M. (1985) Options
Markets, Prentice Hall, Englewood Cliffs, NJ.
7 Cusack, A. (1997) Are there consistent abnormal
profits arising from mispricing of options in the
Australian Market, University of Melbourne,
Working Paper.
8 Brown, S. A. and Easton, S. A. (1992) EmpiricalEvidence on Put-Call Parity in Australia: A
Reconciliation and Further Evidence, Australian
Journal of Management, Vol. 17, No. 1, pp. 1120.
9 Gould, J. P. and Galai, D. (1974) Transaction Costs
and the Relationship between Put and Call Prices,
Journal of Financial Economics, Vol. 1, No. 2, pp.
105129.
10 Evnine, J. and Rudd, A. (1985) Index Options:
The Early Evidence, Journal of Finance, Vol. 40,
No. 3, pp. 743756.
11 Chance, D. (1987) Parity Tests of Index
Options, Advances in Futures and Options Research,
Vol 2, pp. 4764.12 Kamara, A. and Miller, T. (1995) Daily and
Intradaily Tests of European Put-Call Parity,
Journal of Financial and Quantitative Analysis, Vol.
30, No. 4, pp. 519539.
13 Nisbet, M. (1992) Put-Call Parity Theory and
an Empirical Test of Efficiency of the London
Traded Option Market, Journal of Banking and
Finance, Vol. 16, No. 2, pp. 381403.
14 Capelle-Blancard, G. and Chaudhury, M. (2001)
Efficiency Tests of the French Index (CAC 40)
Options Market, unpublished manuscript, Paris,
France.
15 Loudon, G. F. (1988) Put-Call Parity: Evidencefrom the Big Australian, Australian Journal of
Management, Vol. 13, No. 1, pp. 5367.
16 Gray, S. F. (1989) Put-Call Parity: an Extension
of Boundary Conditions, Australian Journal of
Management, Vol. 14, No. 2, pp. 151169.
17 Taylor, S. L. (1990) Put-Call Parity: Evidence
from the Australian Options Market, Australian
Journal of Management, Vol. 15, pp. 203216.
18 Easton, S. A. (1994) Non-simultaneity and
Apparent Option Mispricing in Tests of Put-Call
Parity, Australian Journal of Management, Vol. 19,
No. 1, pp. 4760.
19 CNNMoney (2002) A Web Buying Guide,March, www.cnnfn.com.
20 Garay, U., Justiniano, R. and Lopez, M. (2003)
The relationship between options, moneyness
and liquidity: Evidence from options on futures
on S&P 500 Index, Derivatives Use, Trading &
Regulation, Vol. 8, No. 4, pp. 305323.
280 Ga r a y, Or done z a n d G o n z ale z
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