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
5
.
1
. The 6alue of the early exercise premium In Table 2, the average early exercise premium is presented with respect to
moneyness and time to expiration of the options. Panel A displays the results for the PCP measure, whereas the results for the theoretical BAW measure are
presented in Panel B. The table also reports the average market price for the options in each subgroup
10
. If the entire sample is analysed, the mean value of the PCP premium is SEK 1.52, which equals 9.33 of the average put option price.
The average premium is significantly different from zero at P-values B 0.001 in all subgroups. The theoretical early exercise premium is on average SEK 1.06, or
6.65 of the average BAW option price, for the entire sample. The average theoretical premium is significantly different from zero in all subgroups, but
appears to be lower than the corresponding PCP premium. Except for two subgroups, the deep and very deep in-the-money options with more than 150 days
left to expiration, the differences are also statistically significant. The average option price is also lower in the theoretical panel for the majority of the subgroups.
This could imply that some of the mispricing of the theoretical model might be explained by incorrect valuation of the early exercise premium.
The early exercise premium in out-of-the-money option prices reflects the possi- bility that the options may become in-the-money, and should, at least for deep
out-of-the-money options, not be very high. However, the average PCP premium appears to be rather high compared to the BAW premium even in the group with
deep out-of-the-money options with a month or less left to expiration. Perhaps the Kamara and Miller 1995 liquidity risk premium could explain this finding, given
10
In Panel B, the average BAW option price is reported.
the fact that deep out-of-the-money puts exhibit relatively low liquidity. The average BAW premium, on the other hand, is almost zero for these options.
The average premium that is found in this study is somewhat higher than the premium found in Blomeyer and Johnson 1988 and Jorion and Stoughton 1989.
Jorion and Stoughton 1989, who use option prices from two different markets to compute the premium, argue that their results could be affected by the fact that the
prices from the two markets are not synchronised. In some cases they even find the American option price to be below the corresponding European price. The results
in this study could also be influenced by measurement problems, and the reported value of the empirical early exercise premium might be an over or underestimation
of the actual value. The problems with non-synchronous data are hopefully minimised due to the screening procedure and the choice of data. Blomeyer and
Johnson 1988 use transactions data free from the non-synchronous trading problem. However, assessing the early exercise premium is not their issue, and the
premium implicit in their results is found using a completely different method than in other studies.
Although higher than in some of the previous studies, the early exercise premium found in this study is not unreasonably high. It is for instance very close to the
premium found in Zivney 1991, even though the premium should be lower for
Table 2 Average early exercise premium displayed with respect to moneyness and time to expiration
a
Time days left to expiration T−t 31–90
91–150 151–182
Total Moneyness XS
7–30 Panel A
:
de6iations from put-call parity 0.29 00.58
B 0.90
0.34 00.86 0.47 01.36
0.58 01.69 0.42 01.17
0.38 01.41 0.90–0.98
0.46 03.12 0.63 04.45
1.04 05.76 0.61 03.79
0.81 08.12 1.34 09.97
0.85 08.90 0.98–1.02
0.56 07.35 0.41 04.60
1.02–1.10 0.87 14.48
0.55 13.94 1.45 15.84
1.95 16.02 1.25 15.21
1.06 27.18 1.75 26.22
1.10–1.20 2.83 26.25
2.96 27.06 2.20 26.45
4.71 40.31 4.53 53.12
4.34 47.23 2.83 53.68
3.71 30.54 \
1.20 Total
1.50 18.85 1.65 14.90
1.78 13.87 0.84 15.98
1.52 16.30 Panel B
:
the BAW model B
0.90 0.01 00.20
0.02 00.54 0.07 01.16
0.12 01.47 0.05 00.91
0.27 04.45 0.44 05.75
0.02 01.17 0.10 02.94
0.90–0.98 0.20 03.69
0.06 04.59 0.27 07.33
0.98–1.02 0.57 08.97
0.80 10.05 0.46 08.16
0.99 15.13 1.02–1.10
0.34 13.68 1.47 15.95
1.20 15.81 0.70 14.40
1.10–1.20 1.58 25.81
0.77 26.52 2.39 25.79
1.92 25.99 2.86 26.73
1.10 51.49 \
1.20 2.97 45.46
2.82 50.98 3.62 30.21
3.68 38.88 1.24 14.64
1.34 13.77 1.06 15.93
Total 0.37 15.44
0.99 18.30
a
The table presents the average early exercise put premium in SEK, according to deviations from PCP Panel A, and according to the BAW model Panel B, for subgroups classified on the basis of
moneyness and time to expiration. The average option price in SEK for each subgroup is reported in parenthesis actually observed in Panel A and according to the BAW model in Panel B. The figures are
based on data from the first part of the sample July 1–December 31, 1995.
index options than for equity options due to the lower probability of early exercise. The premium in Zivney 1991, which is found for one of the most widely traded
options in the world, is around 10 for put options. Even though most studies find roughly similar values of the early exercise premium, the exception is the results of
de Roon and Veld 1996, where the average premium is over 50 of the put option price. No other study finds a premium nearly as high as that, but the results could
perhaps be explained by their rather small sample size.
The lower average premium for the theoretical measure compared to the PCP measure is to some extent in contrast to the results of McMurray and Yadav
1996. They find that a theoretical measure of the premium, computed using the binomial model, yields higher early exercise premiums for in-the-money puts than
both the PCP measure and the direct measure. To examine if our results depend of the choice of theoretical model, the computations of the early exercise premium are
repeated with the binomial model prices. However, the results not reported, but available upon request show that the average premiums for each subgroup are
almost identical to those reported for the BAW measure.
In Table 2, the average premium increases with both moneyness and time to expiration for both measures. Using a two-way ANOVA, the hypothesis of equality
between subgroups when it comes to the average early exercise premium is tested. A dummy variable regression with interaction terms shows that the difference in
average premium between the subgroups is statistically significant at a P-value B 0.001, for both the PCP and the BAW measure
11
. The regression results are presented in Table 3, with and without dummies for the
moneyness of the options
12
. The results for the PCP measure of the premium, which are displayed in Panel A, show that the early exercise premium as expected
increases with moneyness and time to expiration. The coefficients are positive and highly statistically significant for all moneyness groups. The effect of the interest
rate and volatility however, depends on the moneyness of the option. For at-the- money and out-of-the-money options, all coefficients are of the expected signs. For
the in-the-money options, the coefficients for interest rate and volatility are both negative. They are also both highly statistically significant. The negative coefficient
for volatility could possibly be explained with the increased value of the insurance element of the option associated with a higher volatility. This will in turn make
early exercise less attractive. The negative interest rate relationship however, is hard to explain. The signs of the coefficients in the regression without dummies in Table
3 are the same as those for the in-the-money options. This is not surprising, since the average moneyness for all options is 1.05. The goodness of fit, as measured by
regression R
2
, is significantly higher for the dummy variable regression than for the regression without dummies
13
.
11
The results are not reported but available upon request.
12
The condition number, computed from the matrix including the explanatory variables, equals 1.39. Hence, according to the guidelines in Belsley et al. 1980, there is no problem with multicollinearity
between the explanatory variables. Correction for heteroscedasticity in the residuals is made according to White 1980.
In Panel B, the regression results for the theoretical measure of the early exercise premium are displayed. Overall, the results are similar to those for the PCP
measure. Also the theoretical premium increases with moneyness and time to expiration, and as in Panel A, the coefficients for interest rate and volatility are
negative for the in-the-money options. However, the volatility coefficient is here significantly negative for all levels of moneyness, whereas the interest rate coeffi-
Table 3 Regression results for the early exercise premium
a
Regression coefficients XS
T−t r
s R
2
Constant Panel A
:
the PCP measure −
65.638 −
3.3201 0.3611
− 1.2477
Without dummies 8.4672
2.4372 0.0001
0.0001 0.0513
0.0001 0.0001
8.3064 1.1390
2.1037 XSB0.98
− 2.7709
1.3788 With dummies
0.0001 0.0343
0.0001 0.0001
0.0001 2.3692
27.980 2.0140
0.985XS −
7.6614 4.8113
B 1.02
0.0051 0.0001
0.0042 0.0001
0.0001 XS]1.02
3.2348 −
134.25 −
9.7243 0.4270
1.9144 12.331
0.0001 0.0001
0.0896 0.0001
0.0001 Panel B
:
the BAW measure −
42.903 −
5.2788 0.4923
− 1.6499
7.0116 2.3575
Without dummies 0.0001
0.0001 0.0001
0.0001 0.0001
0.4841 −
0.0687 With dummies
XSB0.98 −
1.4550 1.4997
0.8401 0.5819
0.0092 0.0001
0.0001 0.0001
6.2785 0.985XS
1.8026 10.720
− 1.2325
− 6.9433
B 1.02
0.0017 0.0001
0.0001 0.0001
0.0001 −
74.286 −
10.894 0.5812
3.5454 0.2553
8.8986 XS]1.02
0.0001 0.0001
0.0001 0.7236
0.0001
a
Panel A shows the results of regressions of the early exercise premium according to the deviations from PCP on the pricing parameters; moneyness XS, time to expiration T−t, interest rate r and
volatility s. Panel B shows the corresponding results of regressions of the early exercise premium according to the BAW model. Each set of results is presented for the entire sample regression without
dummies and for different subgroups according to moneyness regression with dummies respectively. The figures in parentheses are p-values associated with each regression coefficient. Correction for
heteroscedasticity in the residuals is made according to White 1980.
13
The null hypothesis, that the inclusion of the dummy variables in the regression does not improve the goodness of fit, is rejected at any significance level. It is also possible, at any significance level, to
reject each null hypothesis of equality between the coefficients for the three moneyness levels for each independent variable. The test results are available upon request.
cient for out-of-the-money options is not significantly different from zero. No plausible explanation can be offered for these findings. Since there has to be a
comparatively high volatility for a deep out-of-the-money option to become enough in-the-money to even be considered for early exercise, the negative coefficient for
these options is unexpected. The BAW premium is not affected by possible problems with non-synchronised prices, except perhaps for the calculation of
implied volatility. However, since the implied volatility for the option nearest-the- money is used to value all options, this should not affect the results for the entire
sample. As in Panel A, the goodness of fit is significantly higher for the dummy variable regression than for the regression without dummies. Furthermore, the
goodness of fit is higher for the theoretical measure than for the PCP measure
14
. Although the expected correlations are found in most of the previous studies,
also the results of Jorion and Stoughton 1989 show a negative volatility coefficient for the put options. The coefficients of their other variables are of the expected sign,
but no coefficient is significantly different from zero. The authors propose that the negative volatility coefficient probably could be explained by a low variation in the
independent variable, for the part of the sample with the highest premium. This is probably not the explanation for the results of this study, since the variation in
volatility and interest rate is approximately the same in each moneyness group.
5
.
2
. E6aluation of the modified control 6ariate approach The value of the early exercise premium and its dependence on the pricing
parameters is established in the estimation period. The results are then used in the out-of-sample evaluation period to value American puts according to the modified
control variate approach. Table 4 displays the number of put option values within the market bidask spread according to the PCP control variate approach Panel A
and the BAW control variate approach Panel B. For comparison, the number of put values that are within the bidask spread using the BAW and the Black – Sc-
holes model are also displayed Panel C and D, respectively.
The deep out-of-the-money options show similar results, regardless of the model used, but the number of PCP control variate values within the bidask spread is
slightly higher than for the other models. For some of the timemoneyness subgroups the PCP control variate technique does not work very well. Even the
Black – Scholes model works better in many cases. However, for the very deep-in- the-money options, Panel A of Table 4 shows a higher number of values within the
bidask spread than any of the others. The results in Panel C show that the BAW model works best for at-the-money options. This is not surprising, since the implied
volatility is found using these options. However, the BAW model does not, in relative terms, work very well for very deep-in-the-money options, and a possible
explanation is that the theoretical models underestimate the early exercise premium for these options. The BAW control variate values, presented in Panel B, works
14
See footnote 13.
Table 4 The number of estimated option values within the market bidask spread, displayed with respect to
moneyness and time to expiration
a
Time days left to expiration T−t 31–90
91–150 151–182
Total 7–30
Moneyness XS
Panel A
:
option 6aluation using the PCP control 6ariate technique 60 85.71
81 76.41 97 78.86
0 00.00 238 78.55
B 0.90
136 50.00 7 21.21
114 67.46 360 57.05
103 65.61 0.90–0.98
77 50.66 14 43.75
0.98–1.02 218 62.11
56 75.68 71 76.34
162 59.56 39 65.00
94 53.11 407 61.02
1.02–1.10 112 70.89
127 60.48 1 10.00
1.10–1.20 301 60.44
82 65.08 91 59.87
34 57.63 0 00.00
47 61.84 213 72.95
132 84.08 \
1.20 Total
633 58.18 545 73.45
61 43.88 1,737 63.35
498 64.42 Panel B
:
option 6alution using the BAW control 6ariate technique 51 72.86
79 74.53 104 84.55
0 00.00 234 77.22
B 0.90
212 77.94 16 48.48
128 75.74 460 72.90
0.90–0.98 104 66.24
72 97.30 90 96.77
131 86.18 25 78.13
318 90.60 0.98–1.02
196 72.06 39 65.00
148 83.62 518 77.66
135 85.44 1.02–1.10
122 58.10 2 20.00
1.10–1.20 310 62.25
88 69.84 98 64.47
35 59.32 0 00.00
57 75.00 201 68.84
\ 1.20
109 69.43 Total
600 77.62 800 73.53
82 58.99 2,041 74.43
559 75.34 Panel C
:
option 6aluation using the BAW model 102 82.93
B 0.90
0 00.00 51 72.86
233 76.90 80 75.47
217 79.78 0.90–0.98
16 48.48 102 64.97
458 72.58 123 72.78
140 92.11 23 71.88
90 96.77 326 92.88
0.98–1.02 73 98.65
253 93.01 49 81.67
1.02–1.10 622 93.25
150 94.94 170 96.04
185 88.10 6 60.00
113 74.34 402 80.72
1.10–1.20 98 77.78
26 34.21 40 25.48
41 69.49 0 00.00
107 36.64 \
1.20 514 69.27
602 77.88 938 86.21
94 67.63 2,148 78.34
Total Panel D
:
option 6aluation using the Black–Scholes model 97 78.86
2 50.00 B
0.90 227 74.92
51 72.86 77 72.64
215 79.04 19 57.58
118 69.82 450 71.32
98 62.42 0.90–0.98
88 94.62 72 97.30
123 80.92 30 93.75
313 89.17 0.98–1.02
123 69.49 134 84.81
155 56.99 23 38.33
435 65.22 1.02–1.10
47 22.38 6 60.00
40 26.32 159 31.93
1.10–1.20 66 52.38
\ 1.20
7 09.21 3 05.08
0 00.00 25 08.56
15 09.55 640 58.82
80 57.55 453 58.60
1,609 58.68 Total
436 58.76
a
Panel A and B show the number of control variate option values, according to PCP and BAW, respectively, that are within the market bidask spread. Panel C and D show the number of option
values, according to the BAW model and the Black–Scholes model, respectively, that are within the market bidask spread. The observations are displayed with respect to moneyness and time to expiration.
In parentheses, the numbers of values within the spread are expressed as percentages of the total number of observations during the evaluation period January 1–February 1, 1996.
better for these options. Overall, the number of BAW control variate values within the bidask spread is rather close to the BAW model values.
The values according to the different pricing approaches are also compared to the average of the daily closing bidask quotes of the options. In Table 5, the root
Table 5 RMSE for the difference between estimated option values and actual option prices, displayed with
respect to moneyness and time to expiration
a
Time days left to expiration T−t 31–90
Moneyness XS 91–150
7–30 151–182
Total Panel A
:
option 6aluation using the PCP control 6ariate technique 0.2995
1.5234 0.3478
0.3942 B
0.90 0.4016
0.90-0.98 0.5010
0.4364 0.7577
1.4788 0.6894
0.98-1.02 0.3862
0.6787 1.1954
0.7493 0.4590
0.9542 0.9001
0.8808 0.9014
1.02-1.10 0.9104
1.6613 1.10-1.20
1.5288 1.3107
1.8570 1.5271
\ 1.20
2.4059 1.7931
3.2445 –
2.3156 1.2303
1.2102 Total
1.1830 1.1755
1.1030 Panel B
:
option 6aluation using the BAW control 6ariate technique 0.3720
0.3841 1.0407
B 0.90
0.4101 0.4442
0.3634 0.4991
0.9647 0.90-0.98
0.4917 0.4497
0.3425 0.6005
0.3918 0.98-1.02
0.2474 0.0911
1.02-1.10 0.7638
0.7406 0.6427
1.0213 0.7396
1.10-1.20 1.6608
1.8689 1.6598
1.8979 1.2990
2.8229 3.9688
2.6782 –
\ 1.20
3.0188 1.2964
Total 1.2608
1.4083 1.0182
1.3014 Panel C
:
option 6aluation using the BAW model 0.4030
0.4161 1.0655
0.4423 0.3667
B 0.90
0.3667 0.5001
1.1269 0.90-0.98
0.5107 0.4509
0.98-1.02 0.2590
0.0807 0.3752
0.7976 0.3716
1.02-1.10 0.6625
0.7191 0.3985
0.6384 0.5835
1.3548 1.1537
1.2442 1.10-1.20
1.6004 1.2163
\ 1.20
3.2686 2.7594
2.9014 –
3.0704 0.8673
0.9464 Total
1.1869 1.2316
1.6383 Panel D
:
option 6aluation using the Black–Scholes model 0.3819
0.4557 0.7529
B 0.90
0.4341 0.4442
0.4550 0.90–0.98
0.4061 0.5072
0.6912 0.4812
0.4021 0.4353
0.98–1.02 0.1022
0.3282 0.5123
1.0484 1.1055
0.8375 1.8593
1.02–1.10 1.1257
3.1130 1.10–1.20
2.8515 1.7001
2.5354 2.6920
\ 1.20
4.0104 4.5484
5.7399 –
4.5485 2.0265
Total 2.0640
1.9458 1.4588
1.9805
a
Panel A and B show the RMSE for the difference between the control variate option value, according to PCP and BAW, respectively, and the average of the closing bid and ask option prices.
Panel C and D show the RMSE for the difference between the theoretical option value, according to the BAW model and the Black–Scholes model, respectively, and the average of the closing bid and ask
option prices. The observations are displayed with respect to moneyness and time to expiration.
mean square error RMSE for the differences between theoretical values and average market quotes is presented for each pricing model. Panel A presents the
RMSE for the PCP control variate approach whereas the results for the BAW control variate approach are reported in Panel B. The RMSE for values according
to the BAW and the Black – Scholes model are displayed in Panel C and D.
Overall, the PCP control variate approach produces option values that minimise RMSE compared to the other approaches. The RMSE for all options in the
evaluation sample is 1.18 for this approach, compared to 1.23 and 1.98 for the BAW and Black – Scholes model, respectively. Again, this result is mainly due to a
relatively better performance of the PCP control variate approach for deep in-the- money options with a short time left to expiration. As in Table 4, the results for the
deep out-of-the-money options are similar for the different models, with the PCP control variate approach being slightly superior. The RMSE results for the BAW
control variate model are rather close to the actual BAW model results. Surpris- ingly though, for the at-the-money options it outperforms the BAW model when it
comes to RMSE.
6. Concluding remarks
