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An implicit assumption underlying the above procedure is that the
SDHIST is the best available estimate of the future standard deviation of stock
returns. If equilibriull~ market prices reflect a more accurate estimate of future
standard deviations ol \I!‘< I, IC‘~LII.I~tYh e above method of option evaluation
would not be useful in the calculation of ‘fair’ values for stock options. The
difference between the observed and the calculated option price would reflect
the difference between the estimated SDHIST and the aggregate market measure
of expected future standard deviation of returns and not, as some option
traders believe, a profit opportunity. The use of historic price data to construct
trading rules for options would be ineffective (except for those individuals who
received the commissions generated by such an unproductive procedure). If
option prices contain information which can be used to calculate better indicators
of future stock return variances than the estimates obtained from
historic stock price data, that information may be of substantial value to stock
or option traders. The procedure for obtaining estimates of future standard
deviations of stock returns would be simplified. One would need only to
examine the current option price instead of processing numerous historic stock
price data. The current option price when entered into the evaluation equation
would permit the calculation of the standard deviation inferred from the market
price. In this study that value obtained by use of the current option price is
called the ‘implied standard deviation’ (ISD).
3.1. Weighted implied standard deviations
During the period encompassed by this study there was an average of 6.3
option prices recorded per stock for each observation date. Each of those
options had a different ISD. The ISDs must be combined in order to produce a
single estimate of future standard deviation of returns for each stock. LatanC
and Rendleman combined ISDs to produce WISD values. Although they
presented a biased weighting function, their reasoning for not accepting an
arithmetic average of the ISDs is sound. There are considerable differences in
the sensitivities of option prices to changes in expected stock return variances.
It is reasonable that the more sensitive the option price is to changes in standard
deviation the more accurately the ISD will estimate the value of v.
LatanC and Rendleman had intended to weight the ISDs by the partial
derivatives of the Black-Scholes model with respect to each implied standard
deviation. That is equivalent to weighting 1SDs according to the sensitivity of
the dollar price change for the options relative to the incremental change in the
implied standard deviations. A rational investor measures returns as the ratio
of the dollar price change to the size of the investment. The reasoning of Latant
and Rendleman emphasizes the total dollar return without regard to the size
of the investment3 In order to be consistent with a rational measure of returns
the price elasticity of options with respect to their ISDs must be considered. One
must be concerned with the percentage change in the price of an option with
respect to the percentage change in its ISD.
The equation used to obtain the weighted implied standard deviation of the
options on one stock for each observation date is
2 ,,,,a~$
WIsD=‘=’ j i,
N aWj Vj c j=l avj W,
(3)
where
N = the number of options recorded on a particular stock for the
observation date,
WISD = the weighted implied standard deviation for a particular stock on the
observation date,
ISDj = the implied standard deviation of option j for the stock,
aw. 0. II= the price elasticity of option j with respect to its implied standard
aa, Wl deviation (v).
In an efficient market prices will fully reflect all available information.
Therefore, estimated variances calculated from option prices should reflect not
only the informational content of stock price history but also any other available
information. Thus one may suspect that the WISD values reflect future standard
deviations more accurately than do the historic sample standard deviations.
3.2. The hypothesis
A test of the following hypothesis will aid in the determination of the predictive
characteristics of stock option prices :
Standard deviations inferred from option prices have been better predictors
of standard deviations of future stock returns than standard deviations obtained
from historic stock returns.
The following graph and linear regression equations aid the explanation of
test procedures for the hypothesis.
‘A one-dollar price change on a one-dollar stock is considered equivalent to the same price
change on a ISty-dollar stock.
SDHIST,,, WISDi,, SDFUT,,, i= 1,2 ,..., 23,
.___----_.__---___-_-.
t-20 I t+20 t = 1,2 ,..., 23,
SDFUTi.1 = ah+BhSDHISTi,,, (4)
SDFUTi.1 = Uo+BoWISDi,r, (5)
SDFUT,,, = U,+B,,SDHISTi,,+B,,WISDi,l, (6)
where http://www.1daixie.com/dxlxslw/
WISDI,, = the weighted implied standard deviation of returns for stock
i at time t,
SDH1STi.t = the sample standard deviation of returns for stock i from time
r-20 to time t,
SDFUTi,, = the sample standard deviation of returns for stock i from time t
to time f +20,
*h, Bh = coefficients of the simple regression model for SDFUT,,, on
SDHISTi,, ,
a,, 4 = coefficients of the simple regression model for SDFUT,,, on
WISDi,r,
ac, Bch, B,, = coefficients of the multiple regression model for SDFUT,,, on
SDHISTi,, and WISDi,, .
The SDHISTs and the WISDs are compared to the SDFUTs to determine
which predictor was superior during the test period. The period examined
began two months after the start of listed option trading and lasts for 22 months
(23 observations). The above method allows one to compare the predictive
characteristics of option prices over time.
The combination of SDHISTs and WISDs are also compared to the SDFUTs
to determine whether each predictor provides unique information or contains
only information already captured by the other. The values of SDHIST, WISD
and SDFUT are available on request.
3.3. Data
There are 23 monthly observation periods (t) beginning June, 1973 and ending
April, 1975. Data are recorded for the last trading day of each month for each
option on stocks whose options were traded on the Chicago Board Options
Exchange (CBOE) as of June 29, 1973. Data recorded on each data are:
(1) the discounts on the U.S. Treasury bills which mature closest to the
standard option exercise dates,
(2) the closing price of each underlying stock,
(3) the number of days to the standard exercise dates,
(4) the standard exercise prices,
(5) values unique to each option:
(a) the closing price,
(b) the dividend that the underlying stockholder is entitled to receive as a
result of his ownership up to the exercise date,
(6) the monthly price ratios of each underlying stock for each twenty-month
period preceding and following each observation date.4
For (l), (2), (4) and (5a) above the data are recorded directly from the Wall
Street Journal. For number (3) the standard exercise month is recorded and the
number of days are calculated. The U.S. Treasury bill discount rate is converted
to an equivalent continuous interest rate. The values for (5b) are taken from
Moody’s Handbook of Common Stocks published by Moody’s Investors Service,
Inc. 5 The dividends are converted to an equivalent continuous rate. For number
(6) above the stock return data are taken from the computer tapes available
from the Center for Research in Security Prices at the University of Chicago.
The logarithms of the monthly price ratios are computed and the sample
standard deviations of the logarithms of monthly ratios are calculated for the
twenty-month period preceeding and following each observation date. From the
data in (1) through (5) above an implied standard deviation (ISD) for each
option can be calculated by the use of an iterative search process.6 The weighted
average of the ISDs for each stock on each observation date is calculated by the
use of eq. (3).
3.4. Obseroations on the processing of data
While calculating ISDs to obtain the WISD values for the regression models,
several items of interest were discovered. The use of the dividend adjusted mode1
increased the value of the ISD calculated for each option on which the dividend
adjustment was relevant. Most option data which previously had produced no
solution for an ISD by insertion into the Black-Scholes model now enabled
one to produce meaningful LSD values. For options on stocks with high yields
and low WISD values the effect was most pronounced. The ISDs which differed
most from their associated WISD values were discovered on options which
violated a CBOE rule. The Chicago Board Options Exchange generally prohibits
establishing a new position either as a buyer or seller in options selling below
$0.50 whose underlying stock price is more than $5.00 below the option’s
4The monthly price ratio is the price at the end of the month divided by the price at the end
of the previous month. Ratios have been corrected for stock distributions other than dividends.
51t is assumed that the market’s forecast of dividends is perfect during the life of each
option. Analysis of final data indicates that such an assumption is not crucial to this study.
‘It is impossible to solve the evaluation equation directly for the 1SD. By use of the partial
derivative of the option price with respect to its implied standard deviation the value of the
ISD is altered until the calculated option price is within 50.001 of the observed option price.
Convergence is rapid, sometimes requiring only three iterations. See Kohler and Manaster
exercise price. Such options are deleted from this study.’ Large discrepancies
between ISDs and their associated WlSDs were also found on options whose
stocks were selling substantially above the option exercise price. Those options
are not very sensitive to changes in expected standard deviation. The weighting
function effectively eliminates their impact on the WISD values calculated.
Thus it is possible to base this study on an improved evaluation formula and a
better weighting function for WlSDs which do not require the exclusion of as
much option data as was necessary in the research of Trippi and that of LatanC
and Rendleman.
3.5. Results
Tables I and 2 present estimates of the regression parameters for eqs. (4) and
(5). Subscripts have been deleted to simplify presentation of data. Values are
Table 1
Results for the regression (4),
SDFUT = a+ B (SDHIST).
Month a (SE) B (SE) t= R2
2
3
4
5
6
I
8
9
IO
11
:5
14
15
16
17
18
19
20
21
22
23
0.205 (0.07)
0.232 (0.06)
0.224 (0.06j
0.226 (0.06)
0.235 (0.07)
0.284 (0.06j
0.277 (0.06)
0.269 (0.06)
0.288 (0.06)
0.277 (0.06)
0.271 (0.06)
0.268 (O.OSj
0.264 (0.05)
0.238 (0.06)
0.236 (0.06)
0.181 (0.04)
0.177 (0.06)
0.174 (0.05)
0.185 (O.OSj
0.158 (0.04)
0.127 (0.04)
0.123 (0.04)
0.145 (0.04)
0.776 (0.25) 3.00b 0.30
0.622 (0.20) 3.08“ 0.31
0.653 (0.20) 3.23b 0.34
0.662 (0.22) 2.98b 0.30
0.640 (0.24) 2.70b 0.26
0.348 (0.19) 1.84 0.14
0.357 (0.19) 1.92 0.15
0.377 (0.18) 2.08 0.17
0.326 (0.18) 1.80 0.13
0.368 (0.18) 2.04 0.17
0.386 (0.18) 2.14 0.18
0.402 (0.16) 2.47 0.23
0.413 (0.16) 2.51b 0.23
0.443 (0.16) 2.71b 0.26
0.430 (0.16) 2.63b 0.25
0.440 (0.12) 3.69b 0.39
0.352 (0.14) 2.51b 0.23
0.353 (0.14) 2.59b 0.24
0.311 (0.13) 2.30 0.20
0.306 (0.09) 3.45b 0.36
0.363 (0.09) 3.90b 0.42
0.374 (0.10) 3.70b 0.39
0.291 (0.11) 2.74b 0.26
‘All t values are significant at the 0.05 level (one-tail test).
bSignificance at the 0.01 level (one-tail test).
‘Also deleted are options priced below their intrinsic value (i.e., options with price less than
X-C). These option prices are probably not reflecting the actual closing price of the underlying
stock for if they were the difference between the observed price and the intrinsic value would be
arbitraged away. Also, options with a remaining life of less than twenty-four days are not
observed due to the selection of end of month observations
recorded for each model for every month. Data were observed for twenty-three
stocks on each date. The values of SDHIST, WISD and SDFUT were calculated
for each stock. The SDHIST and the WISD values were compared to the
SDFUT values to determine which of the two was the better predictor of the
SDFUT values.
The parameters estimated in table 1 indicate that historic standard deviations
(SDHISTS) explained approximately 26 percent (the average value of R2 is
0.26) of the future standard deviations (SDFUTs) of stock returns. From table 2
the corresponding value of R2 for the weighted implied standard deviations
(WISDs) obtained from option prices is 0.32. The increase is 23 percent and the
trend in the increase is interesting. During the first several months of the study
trading of listed options was a relatively new experience for investors. Option
trading on the Chicago Board Options Exchange began in April 1973. Until
February 1974 the data indicate little difference between the predictive characteristics
of the WISD and the SDHIST values. Beginning in March 1974, less than
a year after the start of listed option trading, the option implied standard
deviations showed a sudden increase in predictive ability. They then begin to
value of R’ for the remainder of the study increases to 0.39 as compared to
0.21 in the prior period. There is no reliable way to determine if the increase is
statistically significant. The regression using SDHIST does not indicate any
trend in predictive ability over time. The evidence suggests that the predictive
abilities of option implied standard deviations were continuing to improve
during the entire period under study. That evidence suggests that the option
market became more efficient as traders gained more experience. The t values
in the tables relate to the testing of the hypothesis that the true values of B are
greater than zero. Negative values of B may be ruled out on theoretical grounds.
The higher the t value the lower is the probability that the sample could have
been obtained from a distribution whose actual value of B was zero. The null
hypothesis, that B equals zero, is rejected at the 0.05 significance level in each
month for both regression models. The r values in table 2 show a tendency to
increase over time similar to the trend in the corresponding values of R2.
Table 3 lists the estimates of the parameters B, and B,, the corresponding t
values and R2 for the regression model SDFUT = u + B1(WISD) + B,(SDHIST).
The tabulated values support the results obtained from the analysis of the first
two regression models. The R2 values are substantially higher than corresponding
values in table 2 for only three months (July, August, and October
1973). After October 1973, the fifth month of twenty-three in this study, the
multiple regression model does not produce substantially better values of R2
than those obtained by use of the option implied standard deviations. The t
values do not indicate consistently high levels of confidence as in the previous
models. However, the t values for the coefficients of WISD are at a substantially
higher value than those of the coefficients of SDHIST. After February 1974
the t values associated with the coefficients of SDHlSTs deteriorate so markedly
relative to those of the coefficients of WISDs that the use of the SDHIST values
appears to add no information that is not already contained in the values of
WISD.
Table 4 presents the regression parameters estimated by pooling all the
observations into one ‘grand regression’ for each model. First the regression
parameters are presented for the ‘uncorrected’ model. Results are distorted by
possible violations of the basic assumptions of the classical normal linear
regression model. The regression parameters presented in the second half of
table 4 indicate the substantial improvement in the t values for each regression
coefficient obtained by correcting each model for heteroskedasticity (variance
of disturbances not consistent for all observations). The results are obtained by
normalizing the data (subtracting the mean from each monthly value of WED,
SDHIST and SDFUT and dividing the remainders by the standard error of
their associated monthly regression). Mutual correlation (standard deviation of
returns correlated between stocks) and autoregression (disturbances at one point
in time carrying over into another period) are ignored.* Data in table 4 support
the conclusions previously obtained from the estimates of the monthly regression
parameters. The t values are substantially higher for the coefficients of WISD
values than for those of the SDHIST values. The analysis of the transformed
multiple regression model results in an R2 of 0.66, an increase of only 0.03
over the value of R2 obtained using the WlSDs alone. That tends to confirm
the previous conclusion that the WISDS have been a better predictor of the
future standard deviation of stock returns. The informational content of the
SDHISTs might be improved if the sample standard deviations were calculated
by weighting the most recent observations more heavily or if daily, rather than
monthly, data were employed. Although the t statistic for the coefficient of
SDHIST is significant, any additional information contained does not appear to
adequately reward the extra effort required to include the SDHIST values in the
analysis.
OIt is also possible to correct for mutual correlation and autoregression by applying modified
Aitkin’s estimation formulas to the data after transformation as indicated by Kmenta
(1971,~~. 512-514). However one ofthe steps wouldinvolvecalculationofavariance-covariance
matrix of the order 506 x 506. The analysis of data up to this point has provided very satisfactory
results and this problem is left for future research.
3.6. Conclusion
The information contained in tables 1, 2, 3 and 4 provides substantial support
for the hypothesis for the period of this study after February, 1974. The change
in the predictive characteristics over time is also an important discovery. The
results indicate that during the first nine months covered by this study the WISDs
and the SDHISTs were both relatively poor indicators of the SDFUTs. However,
during the last fourteen months the WISDs were clearly the superior
predictors of SDFUTs.
The evidence presented in this section is strong. The conclusion is that the
WISDs have been substantially better predictors of SDFUTs than have the SDHISTs.