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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: [email protected]

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

260 Ga r a y, Or done z a n d G o n z ale z

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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).


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.


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.


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.


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


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


Table 4: Study 2A Violations of the PCP relation for at-the-money contracts


(model 1)

Including transaction costs

(model 2)


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.


Table 5: Study 2B Violations of the PCP relation for the most liquid option contracts


(model 1)


(model 2)


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


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%


contracts including transaction costs


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.



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


Figure 4: Observed versus theoretical percentage price deviation including all contracts

possessing the largest volume and ignoring transaction costs


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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possessing the largest volume and including transaction costs



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


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

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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

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and the Relationship between Put and Call Prices,

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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,

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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

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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

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Apparent Option Mispricing in Tests of Put-Call

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No. 1, pp. 4760.

19 CNNMoney (2002) A Web Buying Guide,March, www.cnnfn.com.

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The relationship between options, moneyness

and liquidity: Evidence from options on futures

on S&P 500 Index, Derivatives Use, Trading &

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