from the StSE. Dividends are paid only once a year and most dividend payouts occur around May. Hence, no dividend payments are made during the sample
period. Finally, daily rates of the Swedish 1-month Treasury bills are used as a proxy for the riskless interest rate.
The sample period is divided into an estimation period: July 1 – December 31, 1995 and an evaluation period: January 1 – February 1, 1996. Since prices of both
the put and the call in an option pair have to be used in the estimation period, only days for which non-zero quotes for both puts and calls exist are included in the
sample. To avoid possibly disturbing effects of the trading immediately before expiration, options with less than a week left to expiration are omitted. After a
screening procedure, observations that do not satisfy the American PCP boundary condition in Eq. 2 are excluded. In addition, observations are omitted if the
option value according to the BAW model is less than SEK 0.01, the lowest tick size allowed at OM. The final estimation sample consists of 11 594 put-call pairs.
Since only put options are included in the evaluation sample, the only restrictions for this sample is that the options should have at least a week left to expiration and
the BAW model prices should not be less than SEK 0.01. The final evaluation sample consists of 2742 observations.
Table 1 offers a summary of the option data divided into subgroups with respect to moneyness and time to expiration. Panel A displays the data for the estimation
period whereas Panel B contains data for the evaluation period. The observations are divided into six subgroups according to moneyness and four subgroups accord-
ing to the number of days left to expiration. Moneyness is defined as the ratio of the exercise price to the stock price, where the average of the closing bid and ask
quotes is used. In accordance with e.g. Jorion and Stoughton 1989, a put is defined to be at-the-money when its moneyness is between 0.98 and 1.02. For the
estimation period, around one third of the puts are out-of-the-money while over 50 of the observations belong to the in-the-money groups. The dispersion of the
options into different subgroups according to moneyness is roughly the same for the options in the evaluation period.
4. Methodology
Zivney 1991 uses SP 100 index option data since this is the most widely traded option in the world. The high trading frequency enhances the possibility of
simultaneous prices. Since the relationship between option prices and prices of the underlying is analysed, they should all be simultaneous. The results could otherwise
be misleading, and show an over or underestimated value of the premium. The use of Swedish equity option data, where trading is not as frequent as in the US, could
lead to difficulties with non-synchronised prices since it is improbable that the prices of put, call and underlying stock are exactly simultaneous. Unfortunately, no
intradaily time stamped record of option prices is available. In order to avoid too much time difference between the prices, the average of the daily closing bidask
quotes are used. In addition, only the most frequently traded options are analysed.
However, the bidask quotes are only valid for a certain number of options, and in addition, several transactions may be executed within the spread. It is also possible
that market markers do not price illiquid options as thoroughly as the more frequently traded ones. Nevertheless, the choice of bidask quotes instead of
transaction prices is likely to improve the analysis.
4
.
1
. Empirical and theoretical measures of the early exercise premium The early exercise premium according to the deviations from PCP is estimated
using the difference between the market value of the put, where the average of the closing bidask quotes is used, and the value of the put according to the PCP
relationship given in Eq. 1. The average of the closing bidask quotes of the call and the underlying stock respectively is used in this computation. To establish the
theoretical measure of the premium, the average of the same day’s closing bidask quotes of the option nearest-the-money is used to calculate the implied volatility for
each day. In the calculation of the implied volatility, the BAW model and the closing transaction stock price are used. For each day in the sample, volatility
Table 1 The number of observations displayed with respect to moneyness and time to expiration
a
Moneyness Time days left to expiration T−t
XS Total
7–30 31–90
91–150 151–182
Panel A
:
the number of obser6ations in the estimation period 736 06.35
147 01.27 1507 13.00
B 0.90
522 04.50 102 00.89
2327 20.07 395 03.41
818 07.06 0.90–0.98
777 06.70 337 02.91
1405 12.12 0.98–1.02
167 01.44 433 03.73
524 04.52 281 02.42
314 02.71 948 08.18
1.02–1.10 1071 09.24
499 04.30 2832 24.43
1.10–1.20 782 06.74
267 02.30 2041 17.60
223 01.92 769 06.63
\ 1.20
1482 12.78 470 04.05
728 06.28 160 01.38
124 01.07 4388 37.85
4190 36.14 1347 11.62
1669 14.39 Total
11594 100.00 Panel B
:
the number of obser6ations in the e6aluation period 70 02.55
106 03.87 B
0.90 123 04.49
4 00.15 303 11.05
33 01.20 631 23.01
0.90–0.98 157 05.73
169 06.16 272 09.92
351 12.80 32 01.17
152 05.54 0.98–1.02
93 03.39 74 02.70
60 02.19 667 24.33
272 09.92 1.02–1.10
158 05.76 177 06.46
210 07.66 152 05.54
10 00.35 126 04.60
1.10–1.20 498 18.16
0 00.00 157 05.73
292 10.65 76 02.77
59 02.15 \
1.20 Total
2742 100.00 742 27.06
773 28.19 1088 39.68
139 05.07
a
Panel A displays the number of observations that are included in the first part of the sample July 1–December 31, 1995, according to the moneyness and the time left to expiration of the put options.
In parentheses the numbers are expressed as percentages of the total number of observations. Panel B contains the corresponding information for the observations that are included in the second part of the
sample January 1–February 1, 1996.
estimates are made for the two different expiration days traded at that time. These estimates are then used to value both the American and the European options.
Finally, the theoretical early exercise premium is calculated as the difference between the value given the BAW model and the value given the Black – Scholes
model.
4
.
2
. Dependence on option pricing parameters Jorion and Stoughton 1990 issue proofs regarding the sensitivity of the early
exercise premium to the pricing parameters. Their analysis focuses on foreign currency options but the proofs can be modified to hold also for equity options.
The probability of early exercise is higher the more deep in-the-money the option is. When put options are concerned, the value of the early exercise premium is
expected to increase as the ratio of the exercise price to the stock price increases. The time to expiration is also expected to have a positive effect on the premium,
since the holder of a longer American option has all the possibilities the holder of a shorter option has, plus the additional possibilities coming from the extra time to
expiration. Also the interest rate and the volatility of the underlying stock could affect the size of the premium. An increase in the interest rate leads to a reduction
in the present value of the exercise amount of the option. Hence, the possibility to exercise early becomes more attractive, and the value of the early exercise premium
is expected to increase with increases in the interest rate. The effect of the volatility is intuitively not as clear. A higher volatility would increase the likelihood that the
stock price reaches a level low enough to trigger early exercise, but this exercise boundary level will simultaneously also be affected. The actual outcome depends on
which effect dominates. In accordance with the proofs of Jorion and Stoughton 1990, the early exercise premium is hypothesised to be an increasing function of
the volatility of the underlying security.
To investigate these hypotheses, the average early exercise premium is computed for all the subgroups according to moneyness and time to expiration mentioned
above. It is then tested whether they are statistically different from each other. Furthermore, each estimate of the early exercise premium is regressed on the
explanatory variables moneyness XS, the riskless interest rate r, the time left to expiration measured in years T − t and the implied volatility s. Furthermore,
dummy variables are used in the regression to take into account if the option is out, at or in-the-money, respectively.
4
.
3
. The modified control 6ariate approach The original control variate technique is suitable for valuing an American option
when the corresponding European option value is known. Boyle 1977 discusses how to use the control variate technique with Monte Carlo simulation, while Hull
and White 1988 show that it can be used in connection with the binomial model. The theoretical model is used to value both the American and the European option.
A theoretical early exercise premium is then computed as the difference between the
two model values. After that, the value of the American option can be established by adding the theoretical premium to the known value of the European option.
In the modified version of the control variate approach, the value of the early exercise premium, which is found through empirical analysis, is added to a
theoretical value of the European option. In this study, the theoretical European option value is obtained according to the Black and Scholes 1973 formula, while
the early exercise premium is determined using the regression coefficients for both the PCP and the BAW measure of the premium. The early exercise premium is
determined by multiplying the appropriate coefficient with the value of the corre- sponding pricing parameter. Each estimate of the premium during the 1-month
evaluation period is then added to the corresponding Black – Scholes value. The accuracy of the resulting American option values is examined by comparing them
to market quotes and the values according to the BAW model and the Black – Sc- holes model, respectively.
5. Empirical results
