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Expiration effects in The Netherlands
An inquiry applied to four derivative stocks
http://homepage.uvt.nl/~s900285/BachelorThesis.pdf
word count: 8874 (total, including e.g. appendix)
this thesis contains 35 pages (total, including e.g. appendix)
Tilburg University, The Netherlands
Bachelor Thesis Finance
Tilburg, June 30, 2006
Author: E. Cardon (900285)
Supervisor: Drs. J. Cui
Second reader: Prof. dr. F.C.J.M. de Jong
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Plagiarism statement
I understand that I must use research conventions to cite and clearly mark other people's ideas andwords within my paper. I understand that plagiarism is an act of intellectual dishonesty. I understand it
is academically unethical and unacceptable to do any of the following acts:
To submit an essay written in whole or in part by another student as if it were my own. To download an essay from the internet, then quote or paraphrase from it, in whole or in part,
without acknowledging the original source. To restate a clever phrase verbatim from another writer without acknowledging the source. To paraphrase part of another writer's work without acknowledging the source. To reproduce the substance of another writer's argument without acknowledging the source. To take work originally done for one instructor's assignment and re-submit it to another
teacher. To cheat on tests or quizzes through the use of crib sheets, hidden notes, viewing another
student's paper, revealing the answers on my own paper to another student through verbal or
textual communication, sign language, or other means of storing and communicating
information--including electronic devices, recording devices, cellular telephones, headsets,
and portable computers. To copy another student's homework and submit the work as if it were the product of my own
labor.
I understand that the consequences for committing any of the previous acts of academic dishonesty can
include a failing grade for the assignment or quiz, failure in the class as a whole, and even expulsion
from the university. I will not plagiarize or cheat.
Name: E. Cardon
Date: June 30, 2006
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Table of contents
1. Introduction................................................................................................................5
2. Expiration effects in international setting ..................................................................8
2.1 Introduction..........................................................................................................82.2 General reasons for abnormal behavior on expiration days.................................8
2.3 Trading volume effects on the underlying asset ..................................................9
2.4 Influence on stocks returns on expiration days..................................................11
2.5 Influence on stocks returns around expiration days...........................................12
2.6 Influence on volatility ........................................................................................13
2.7 Market conditions that might intensify or diminish expiration effects..............15
2.8 Conclusions........................................................................................................16
3. Market information, hypotheses, methodology and data.........................................19
3.1 Introduction........................................................................................................19
3.2 Option expiration characteristics in The Netherlands........................................19
3.3 Data collection ...................................................................................................213.4 Test methods ......................................................................................................22
3.5 Option volume in different expiration months...................................................23
3.6 Abnormal trading volume ..................................................................................25
3.7 Abnormal volatility............................................................................................26
3.8 Abnormal return on expiration day....................................................................27
3.9 Conclusions........................................................................................................28
4. Conclusions..............................................................................................................29
5. Limitations and future research ...............................................................................30
Appendix......................................................................................................................31
References....................................................................................................................34
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Abstract
This study examines expiration effects in The Netherlands based on four derivative
stocks for the period 1992-2003. Furthermore, the effects of long term options are
examined, given their expected higher option volume in October. The effects are
examined applying a Wilcoxon matched pair test for the overall test of expiration tests,
while the Mann-Whitney test is applied to test for abnormalities in October, when the
long term options expire. The results show, consistent with previous work, that
although the volume of the underlyings tends to be higher on expiration days, no price
distortions are found. For the long term options, the volume of the underlying is
slightly higher, but not significant. No price distortions were found.
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1. Introduction
The influence of options, futures and other derivatives has been extensively discussed
by both the academic world and regulators worldwide, as expiration days might be an
exploitable and pliant source for trading parties. As their use spreads internationally,
for example futures continue to be criticized for causing aberrations in the market
(Stoll and Whaley, 1997). Despite for example concern expressed by the Securities
and Exchange Commission (SEC), until now only mixed evidence has emerged that
derivatives could have an impact on underlying assets; some argue that volatility and
return effects can be found (e.g. Chamberlain et al, 1989), while others find the
opposite (e.g. Vipul, 2005). However, most agree that underlying volume is higher(e.g. Stoll and Whaley, 1987, Pope and Yadav, 1992, Chamberlain et al, 1989 and
Vipul, 2005). Much of the concern seems to be directed to small markets such as the
Oslo Stock Exchange (OSE) (Swidler et al, 1994) and the Spanish stock exchange
(Corredor et al, 2001), which are expected to be influenced more easily, due to the
low trading volumes. Also the so called triple watching hour (which is the last trading
day on the third Friday of the quarterly month when index futures, index options and
equity options expire simultaneously) has gained much attention from regulators.
During these three hours, at the end of the expiration day, most effects are assumed to
occur (e.g. Bollen and Whaley, 1999).
As expiration days are followed attentively by many parties and their importance
evident, given the changes on expiration procedures implemented in for example the
S&P 500 some years ago, the main purpose of this inquiry is to make a contribution to
existing documentation by examining four derivative Dutch stocks. On top of this, the
stocks are also examined on a possible intensifying circumstance on the underlying
stocks. As for example Vipul (2005) finds evidence on more impact of relatively high
derivative stocks, this element is also taken into account as long term options might
be such a factor, given the expected higher option volume on expiration days. A
fundamental factor here is that the increase in volatility is the result of additional cost
of liquidity (Stoll and Whaley, 1997). As the option volume is higher for long term
options over normal (short term) options, this fundamental theorem might be
applicable.
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The four stocks (Akzo Nobel, Royal Dutch, Philips and Unilever) were selected as the
long term options were first contingent on them and so a longer data period available,
which make the test results more reliable. The Dutch market is thereby characterized
for its uniqueness as it was the first worldwide with the introduction of such long term
options. Due to there long term view, investors are able to take positions for a longer
period and as such, the importance of the expiration day should increase as more
arbitrageurs are able to influence the market. Despite their success, to my best
knowledge no one has conducted research on the expiration effects of long term
options on for example volatility, returns, volume and expiration days of the
underlying stock. De Roon et al (1998) were one of the first who studied the
efficiency of the market for Dutch long-term call options by testing any deviations
from pricing formulas. They found no serious price inefficiencies in the market.
This inquiry is especially directed to market regulators, investors and traders. If there
is indeed an imperfect market, traders and investors can benefit from this by for
example taking speculative strategies based on any possible abnormalities in the
market. This could be a reason for regulators to revise existing expiration procedures.
At first, an overview of existing literature will be given in chapter two. Chapter three
deals with an overview of the Dutch stock market and subsequently the expiration
effects will be will be tested in it. In appendix 2 an overview of options pricing is
available. These chapters should be able to answer the main research question of this
thesis:
To what extent do expiration effects exist in The Netherlands compared to other
countries?
The major findings of this inquiry indicate that on expiration days no special effects
occur on the four derivative stocks in The Netherlands. Only the volume of the
underlying shares tends to be higher than normal, but volatility and return remain
unaffected. For the high derivatives stocks, no effects are observed as well. Despite a
slightly higher volume, volatility and return than normal expiration days, they remain
insignificant.
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Like any other study, this thesis has its limitations, which create opportunities for
future research. Briefly, only four stocks were studied because of the limited time for
the Bachelor thesis, high frequency data was not available and therefore suffers from
precise measurement techniques, different countries should be examined as well and
other factors might influence October expiration days.
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2. Expiration effects in international setting
2.1 Introduction
The main purpose of this chapter is to summarize the yielded results of previous
studies in different markets. With respect to the influences corresponding to this
inquiry, different markets are examined and by doing so, for example different
settlement methods and different market structures. These will be considered in this
chapter as well and on their possible relevant implications will be elaborated.
Furthermore, it is interesting to find out the rationale beyond the authors conclusions.
Respectively conclusions with respect to trading volume, returns and volatility on
underlying stocks will be summarized at the end of this chapter.
As there are many studies about expiration effects available, most of them take into
account the effects of options and other derivatives on the underlying asset in terms of
abnormal volume, abnormal volatility, abnormal return and reversals. Abnormal is in
this context what is different from the corresponding benchmark on non-expiration
days. Many authors use a different benchmark for what is normal on an expiration day.
For example, Vipul (2005) uses the average of two preceding Friday trading days and
the two Fridays after the expiration day, while Alkebck and Hagelin (2004) use a
regression model to predict the normal changes in the market. On this will be
elaborated in this chapter as the methods will be discussed. The reversal is defined as
the release of stock returns after the expiration day as they are possibly depressed on
and prior to the expiration day.
2.2 General reasons for abnormal behavior on expiration days
The possible explanations for the concern about abnormal behavior on and around
expiration days may be the result of respectively, the unwinding of arbitrageurs,
speculative strategies by traders and market manipulation (e.g. Stoll and Whaley,
1987 and Jarrow, 1994). Two forms of manipulation may occur: namely action-based
and trade-based manipulation. The former occurs as a result of revealing information
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(which can be false information) and the latter occurs by trading in the market with
for example large volumes on an underlying stock (e.g. Allen and Gale, 1992). Trade-
based information is especially important to this inquiry and documented by many
authors. Especially during the last trading hour prior to the expiration, some effects
should be found, but they remain controversial. The argument here is that during the
last trading hour large block transactions occur (Stoll and Whaley, 1997). The option
market can also be a good exploitable market for informed traders as a result of low
transaction costs, lesser capital outlay, a higher leverage and a limited downward
potential (Bhuyan and Chaudhury, 2005). Finally, they provide a fast and inexpensive
means of changing stock market exposures, both domestically and internationally
(Bollen and Whaley, 1999).
2.3 Trading volume effects on the underlying asset
It seems clear that price, volatility and volume can be possibly distorted near
expiration days as a result of the unwinding of arbitrageurs. This claim seems to be
certainly true with respect to large volumes in the index (Stoll and Whaley, 1986,
1987 and 1991). The effects not only seem to occur on the expiration day itself but
also a day or some days prior to the expiration days. These results were for example
yielded in The United Kingdom (Pope and Yadav, 1992).
For testing volume effects, several methods are applied. Lien and Yang (2005)
examine the Australian stock market and use trading volume and relative trading
volume of a stock to measure the trading activity. They define the trading volume of a
given day as the dollar value of all trades that occur during that day. The relative
trading volume is defined as the ratio of the dollar trading volume in the last half-hour
on a given day to the total dollar trading volume on that day. So, relative trading
volume =
ii
ofTradesno
iofSharesnoeicePerShar .Pr
.
1
=
Subsequently, the Mann-Whitney rank test is applied to test for abnormal volume.
They find small effects of individual stock futures and options expiration-days on
trading volume.
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This t-test method in connection with the trading volume formula discussed above is
used by Stoll and Whaley (1997), who also examine the Australian Stock Exchange
by examining expiration-day effects of SPI futures. By using high-frequency data,
they find significant results (higher volume) on eight out of fourteen expirations. The
remaining six show higher trading at the close, but the differences remain
insignificant.
The method used by Alkebck and Hagelin (2004) is quite different. They estimated
trading volume as the sum of open-to-close trading for each stock in the index in
Sweden (SEK index). A time trend was used to calculate the predicted volume on an
expiration day. So, volume= tttD +++2
92921 . The dummy is used to correct
for a trend started by the removal of transaction tax, which increased trading volume.
A time-independent trading volume proxy was calculated by dividing the trading
volume for a given observation by its predicted trading volume,
)/( tVolumeVolume . Subsequently, they used the pooled t-test and the Wilcoxon
test to test for any abnormalities. They find evidence for significant higher volumes
on expiration days of index futures and options compared with the respective
benchmark.
Vipul (2005) estimates the benchmark, i.e. what should be normal on an expiration
day in India, by taking the averages of the 14,7,7,14 ++ EEEE trading volume,
where E denotes the expiration day. Subsequently, the nonparametric Wilcoxon
squared rank test is applied to test for potential higher volumes on the expiration. The
author finds indeed evidence for such a higher volume on futures and options
expiration-days.
In the small Spanish market, an increase in the trading volume is found as well
(Corredor et al, 2001). The increase in trading volume is consistent when future
expirations are studied. Karolyi (1996) examined futures contract expirations on the
Nikkei 225 and finds also abnormal trading volumes. In the very large USA market,
the volume exist as well (Stoll and Whaley, 1987).
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As most authors agree on the increased volume on expiration days, this increase in
volume of the underlying shares tends to start building up slowly a day prior to
expiration. A possible reason for this is that arbitrageurs unwind their positions prior
to the contract expiration (Bollen and Whaley, 1999), but their effects depend on the
microstructure of the market (Vipul, 2005), the test method and the use of high-
frequency data. Most authors give as a primary reason for higher volumes the
squaring-up by arbitrageurs. This even continues the day after the expiration. The
only reason for this is that the selling activity of investors stimulated by an increase in
the prices after they were depressed for two days. This is shown in for example India
(Vipul, 2005).
2.4 Influence on stocks returns on expiration days
The influence on stock returns is a very important issue as they might be an
exploitable source for trading parties if they exist in the markets on expiration days.
The different findings in several markets in connection with the method used will
therefore be discussed next.
The major difference in the methods is that concerning the use of log transformations.
Log transformations are often made in finance to make the data more normally
distributed (e.g. Chatfield, 1996), as the assumption of normality in financial markets
is often violated (Vipul, 2005).
Lien and Yang (2005) use high frequency data to investigate the option-expiration day
effects by comparing the mean returns of the first hour, the last hour and the whole of
trading hours on the expiration days with the benchmark groups on non-expiration
days in the Australian market. They computed the stock return as
)log()log( 1= ttt ppR . tp denotes the stock price at the end at the end of a five
minute interval. Subsequently, a Mann-Whitney non-parametric test is applied. The
authors find significant effects on returns.
Alkebck and Hagelin (2004) compare the relevant means and medians on the
expiration days to non-expiration-days (comparison group) by respectively applying
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the t-test and the W (Wilcoxon)-test as they also use them for other tests such as are
for abnormal volume. They found no serious price distortions in the market.
Vipul (2005) applies again the Wilcoxon test, for the same reasons as indicated in the
previous section. He compares the returns on the expiration day with its respective
benchmark to test for any price distortions. The expiration return is for each period
subtracted from its benchmark. He finds no specific pattern for the expiration-day
returns. He argues that this is the result of a downward pressure on prices prior to the
expiration day which causes the return to remain normal on the expiration days.
In summary, with respect to the influence on returns on expiration days, literature
does not agree. Some authors argue that no significant price return movements on
expiration days can be found (e.g. Vipul, 2005), while others find significant results
(e.g. Ni et al, 2004). One reason for the different findings tends to be the use of high
frequency data, which is likely to make a contribution in detecting abnormalities in
the market. Other reasons for the opacity tend to be the market characteristics and the
test methods used. On the latter will be elaborated later in this chapter.
2.5 Influence on stocks returns around expiration days
As it is empirically still unclear what the effect is of expiration days, one could also
consider stock movements one day prior to an expiration day and one day after. Not
all authors test expiration effects on the expiration day itself. Bollen and Whaley
(2005) and Stoll and Whaley (1997) for example only tested reversals on the
underlying stock. This reversal can be measured in different ways; Lien and Yang
(2005) for example, measure the price reversal as follows:
=
+
+
1
1
t
t
R
RREV if
>
n ) Tis approximately
distributed with mean (Keller and Warrack, 2003)
4
)1()(
+=
nnTE and the standard deviation
24
)12)(1( ++=
nnnT
The standardized test statistic is calculated as
T
TETz
)(=
3.5 Option volume in different expiration months
To test whether option expiration days in October cause more option volume than
January, April and July, which is a necessity to infer that long term option might
cause stronger effects on expiration days, it is necessary to compute the different
option volumes per month and compare them with their respective benchmark. For
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this test, a different method is used as by Vipul. Option volumes are much easier to
influence than stock volumes, if for example one trader sells one large block of
options, this could disturb the total picture. Therefore, option volume on expiration
days is taken by the average of the week in which the expiration day takes place. As a
benchmark for non-expiration days, the average of the option volume of the rest of the
month is taken. Table 5 depicts the ratio for each stock each month, where the ratio is
defined as actual option volume / benchmark option volume. The following
hypotheses are advanced:
=0H The distribution of the ratio of the option volume in October is not different
from other months in October.
=aH The distribution of the ratio of the option volume in October is different from
other months.
Table 5: Option volume averages for each month (ratio with benchmark) 1992-2003
Stock January April July October
Akzo 1.70 1.47 1.53 1.65
Philips 1.49 1.62 1.78 1.73
Royal Dutch 1.60 1.69 1.43 1.43
Unilever 1.85 1.77 1.64 2.07
Average 1.66 1.64 1.60 1.72
The average ratio tends to be slightly higher on October expiration days. Table 6
depicts the test results.
Table 6: Statistical results abnormal option volume tests
Jan-Oct April-Oct July-Oct Overall option
volume effect
Mann-Whitney
Rank test
1054.50 1092.50 1039.50
Z -0.204 -0.091 -0.492 -10.910
Significance
(2-tailed)*
0.839 0.928 0.623 0.000*
*significant p-value
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3.6 Abnormal trading volume
As there is no evidence that October expiration days cause more option volume than
the other expiration months, then this should also not be noticeable in the trading
volume of their underlying. Table 7 depicts the ratio for each stock each month, where
the ratio is defined as actual trading volume / benchmark option volume. The
following hypotheses are advanced:
=0H The distribution of the ratio of the trading volume of the underlying asset is not
different from other months in October
=aH The distribution of the ratio of the trading volume of the underlying asset in
October is different from other months
Table 7: stock volume averages for each month (ratio with benchmark) 1992-2003
Stock January April July October
Akzo 2.05 2.19 2.61 2.66
Philips 1.99 1.74 2.10 1.76
Royal Dutch 1.30 1.78 1.45 1.72
Unilever 1.99 1.74 2.10 1.76
Average 1.83 1.86 2.07 1.98
The October average is again slightly higher than the other months, except July. Table
8 depicts the statistical results.
Table 8: Statistical results abnormal stock volume tests
Jan-Oct April-Oct July-Oct Overall option
volume effect
(wilcoxon
matched pair
test)
Mann-Whitney
Rank test
850.50 904.500 826.500
Z -0.161 -0.718 -0.773 -10.954
Significance
(2-tailed)
0.872 0.473 0.439 0.000*
*significant p-value
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but there is not enough evidence to refute oH . The overall effect turns out to be
significant.
3.7 Abnormal volatility
As mentioned earlier, a positive correlation between trading volume and volatility has
been documented in the financial markets (e.g. Daigler and Wiley, 1999). The
previous section showed that abnormal volume of October expirations was on average
slightly high, but not statistically shown. Abnormal volatility is measured by taking
the procedure from Vipul (2005):
2/)( DayLowDayHigh
DayLowDayHigh
+
Consequently, each number is compared with its benchmark, which is measured by
taking the average of the 14,7,7,14 ++ EEEE days, where E denotes the
expiration day. As the numbers are calculated as a ratio yet, the difference is
calculated with the benchmark. (Volatility-benchmark volatility)
The following hypotheses are advanced:
=0H The distribution of the difference of the volatility of the underlying asset in
October is not different from other months.
=aH The distribution of the difference of the volatility of the underlying asset in
October is different from other expiration months.
Table 9: volatility averages for each month (ratio with benchmark) 1992-2003
Stock January April July October
Akzo 0.004 0.002 0.016 -0.002Philips 0.003 0.002 0.014 0.001
Royal Dutch 0.004 0.000 0.003 0.004
Unilever -0.004 -0.001 -0.005 -0.003
Average 0.00175 0.00075 0.007 0
Table 9 shows quite unexpected results, the October expiration is on average less
volatile compared with its benchmark in the other months. Table 10 depicts the
statistical results.
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Table 10: Statistical abnormal stock volatility tests
Jan-Oct April-Oct July-Oct Overall option
volume effect
(wilcoxon
matched pair
test)Mann-Whitney
Rank test
850.50 904.500 826.500
Z -0.087 -0.173 -0.673 -0.221
Significance
(2-tailed)
0.931 0.863 0.501 0.825
There is not enough evidence to refute 0H
3.8 Abnormal return on expiration day
As discussed in chapter 2, some studies conclude that expiration days cause a
downward pressure on stock prices. The following hypotheses are advanced:
=0H The distribution of difference of the return with its benchmark of the
underlying asset in October is not different from other months.
=aH The distribution of the difference of the return with its benchmark of the
underlying asset in October is different from other expiration months.
Table 11: Return averages for each month (ratio with benchmark) 1992-2003
Stock January April July October
Akzo -0.0005 -0.001 0.0093 -0.005
Philips -0.0081 0.0092 -0.0189 0.0053
Royal Dutch -0.0023 0.0038 0.0031 0.0026
Unilever -0.0029 0.0055 0.0027 0.0113
Average 0.0035 -0.0044 -0.001 0.0036
In several studies a downward pressure was documented on the expiration day itself.
The expectation is therefore that this is also the case for the October series, as their
impact is slightly higher than the other expiration months. As can be seen from table
11, the benchmark return is higher in October, which suggests that returns are slightly
depressed. To test for significance, the Mann-Whitney test was performed.
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Table 12: Statistical abnormal stock return tests
Jan-Oct April-Oct July-Oct Overall option
volume effect
(wilcoxon
matched pair
test)Mann-Whitney
Rank test
982.000 1022.500 1019.000
Z -1.087 -0.786 -0.643 -0.019
Significance
(2-tailed)
0.277 0.432 0.520 0.985
The tests show again that on October expiration days, there is more pressure on stock
prices, but it is not statistically significant. There is also no evidence that the overall
effect of price pressure exists on the expiration day.
3.9 Conclusions
The major conclusion from this chapter is that an overall volume effect exists on the
four Dutch stocks. However, volatility and return effects remain insignificant. The
option volume of the long term options is also not shown. There tends to be a slightly
higher option and stock volume, although they remain insignificant.
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4. Conclusions
The growth in derivatives trading has gained substantial attention from regulators and
trading parties. As their use is spreading internationally, the major concern expressed
tends to be the increased voaltility which might be an indirect result of the higher
volume, as this inference is often made in the financial markets. The concern has
therefore been tested extensively in different markets. Only the abnormal volume
seems to be significant, which tends to be higher on expiration days. The reason for
this tends to be the unwinding of arbitrageurs. The growth in abnormal volume,
however, does not lead to a substantial increase in volatility in most markets. Also
returns tend to remain unaffected. When some authors find significant effects, it isstill difficult to benefit from the abnormalities as transactioncosts outweigh the
benefits and only large parties can profit from them. The test method used depend on
the choice of the researcher, and might influence the existence of an expiration effect.
Most popular are the t-test, the Wilcoxon test and the Mann Whitney test. Only the
first is a parametric test. As the assumption of normal distributions in the financial
markets tends to be violated, for this inquiry the Wilcoxon test and the Mann Whitney
test are applied. The market characterics tend to play a crucial role, for example the
existence of futures has its implications, as they can make the market more complete
and lessen the expiration effects. In The Netherlands, only a higher volume is
observed in the market. The volatility and return remain insignificant. The option
volume of the long term options tends to be slightly higher than normal options,
which gives no reason to infer that their influence exists. This indeed the case as no
abnormal behavior is found for the four Dutch Stocks. The findings are in line with
existing literature, which also only document a higher volume, but mixed findings for
volatility and return.
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5. Limitations and future research
Like any other inquiry, this thesis has its limitations, which create opportunities for
future research. Because of the lack of time, it was only possible to study fourindividual stocks. It might be interesting to test if more individual stocks included
change the total picture. Unfortunately, high frequency data was not available; it was
only possible to gather data from DataStream. Lien and Yang (2005) tested some
effects, when high-frequency data was available and observed more significant results.
Stoll and Whaley (1997) also tested with high frequency data the last thirty minutes
around the expiration day. The non-availability of high frequency data is therefore a
major drawback to this inquiry, although they seem to be almost captured by the
Vipuls procedure. As made clear in chapter two, in several markets different
expiration effects occur. It is therefore interesting to test several effects in different
markets. Finally, although is it is likely that long term options might be the major
influencer in the October months, other factors can play a role and should therefore be
a incentive for future research.
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Appendix
1. Table of expiration days in The Netherlands 1992-2003
Year/Month January April July October1992 17 17 17 16
1993 15 16 16 15
1994 21 15 15 21
1995 20 21 21 20
1996 18 19 19 18
1997 17 18 18 17
1998 16 17 17 16
1999 15 16 16 15
2000 21 21 21 20
2001 19 20 20 192002 18 19 19 18
2003 17 18 18 17
2. Option pricing
The use of options has become quite popular and has therefore been extensively
studied. Many models for pricing options and other derivatives have been developed
and are still being developed. All models in the option pricing theory agree on four
assumptions. First, traders have symmetric information. Second, markets are complete.
Third, markets are frictionless and fourth, all investors are price takers (Jarrow, 1994).
Quite logical, these assumptions apply to long-term options as well. Until the
beginning of 1970, it was necessary to have knowledge of the expected return of the
underlying asset, which should filled in into different formulas, which limited their
practical use (Stulz, 2003). However, Black and Scholes (1972) were the first with an
easier applicable opting pricing model. It was no longer necessary to have more
knowledge of the expected return of the underlying asset. The model can be applied to
common stock as well as other financial assets.
The price for a European call option is as follows:
)()()(),,,( tTdKNTPdSNtTKSc t =
Where tTtT
KPSd t +
=
5.0)/ln(
and
N(d)=the cumulative standard normal distribution evaluated at d.
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Unfortunately, the model by Black and Scholes is only applicable to European options,
since the put-call parity does not hold for American options. The reason is
straightforward: American options can also be exercised prior to the expiration day
and a rather large loss of the underlying violates the put-call parity, which is defined
as:
SXPVCP += )(
In this inquiry the individual options contingent on the five stocks are of the American
type. Many models are available for American options, but none of them seems to
predict option prices precisely. However, if mispricing is large enough, arbitrage
opportunities may occur. Therefore, bounds exist on the call price of an American
option to avoid the possibility of arbitrage opportunities (Stulz, 2003).
KTPStTKSPtTKSCKStTKSP ttt )(),,,(),,,(),,,( ++
Black developed a formula for the pricing of an American call option on a stock that
pays one dividend before maturity:
{ }),',,(),,,,)'((),,,( ttKSctTKDtPScMaxtTKSC tBlack
=
Fortunately, all the models agree on the five important influencing variables that willbe discussed next. The model by Black and Scholes consists of five variables. These
are delta (the exposure of the option price with respect to the stock price), vega
(exposure of the call option with respect to the volatility), rho (the exposure of the
option price with respect to the interest rate), theta (the exposure of the option price
with respect to time maturity) and the impact of a change in the exercise price on the
call option price.
The delta is increasing when the option is more in the money. The vega is positive
correlated to the option price, since more volatility makes it more likely that the
underlying will reach the exercise price. An increase in the interest rate also has an
impact on the value of the option, since it lowers the present value of the exercise
price, which has a positive impact on the value. Theta is trivial, since the longer time
an option has to reach the exercise price, the higher the probability that this will occur.
An increase in the exercise price causes a decrease in the call option price. This is in
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contrast to an increase in the exercise price of the put option, which increases the
value of the put.
Quite logical, the largest impact can be found on time to maturity, which increases the
price of long-term options. Options with similar strike price but a longer maturity
have therefore always more value than a short term option. Nevertheless, future
planned dividend payment may violate this statement, as they decrease the share price
by the future planned dividend payments. A difficult estimation, however, is volatility.
It is easier to estimate the volatility of an option expiring in three months than an
option expiring in five years. This is consistent with results from literature, which
indeed indicate that some imperfections were not taken into account in the Black and
Scholes model. With respect to short term options, the models fits quite well, because
it is easier to estimate their volatility, while for long term options, there is still a lot of
uncertainty in the market. Bakshi et al, 2000 studied the distinctiveness of alternative
models based on long term options criteria. It seemed indeed that all the models
assuming stochastic volatility produce stock price option deltas that are drastically
different from those based on the Black and Scholes model. This could mean that it is
quite difficult to value long-term options and this might be a disadvantage of the
introduction of long term options.
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