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JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
Estimating Security Returns Variance:
A Review of Techniques for Classroom Discussion
John L. Teall1
Abstract
The typical financial management or investments textbook
offers variance as a security risk measure, though usually
omitting significant discussion concerning drawbacks to
standard historical variance estimators and failing to discuss
various alternatives to them. This paper reviews alternative
variance estimation procedures, the most commonly used of
non-relative risk measures. This review paper is intended to
draw together variance estimation procedures from a number
of sources for undergraduate or M.B.A. finance instructors
intending to provide a more complete applications-oriented
classroom discussion of risk measurement. Methodologies
discussed here include traditional sample variance estimates,
extreme value estimators, Black-Scholes implied volatility and
AutoRegressive techniques.
Introduction
In recent years, increased attention has been focused on forecasting security risk, as its measurement
and computation is problematic. Generally speaking, the risk of an investment might be defined as the
uncertainty associated with its returns or cash flows. Analysts typically use absolute risk measures such as
variance and relative risk measures such as beta to quantify security risk. Unfortunately, while defining
variance and standard deviation, many investments texts offer only scant discussion of “real-world”
computation and measurement problems. This paper reviews methodologies used for estimating variances
and relative advantages of with each. We intend to provide a discussion that financial educators can apply
to classroom presentations.
Although return variance is a quite simple mathematical construct with many desirable characteristics,
its estimation is hampered by lack of ideal data. Suppose that we wish to estimate the risk or variance
associated with a security's returns over the next year. Consider the following discrete expression for exante variance σ F that considers all potential return outcomes Ri and associated probabilities Pi:
2
n
(1)
σ F2 = ∑ ( Ri − E[ Ri ]) 2 Pi
i =1
While this expression for variance is, by definition correct, its computation requires that we delineate all
potential returns for the security (which might range from minus infinity to positive infinity). This is
actually practical when the list of potential returns is small or when we can ascertain a specific return
generating process. However, associating probabilities with these returns is a greater problem. For example,
what is the probability that the return for a given stock will range between five and six percent? In many
instances, we will be forced to either make probability assignments of a somewhat subjective nature or
define a joint return and probability generating process for the security. Availability of historical price data
typically makes risk estimation based historical variances more practical.
1
Professor of Finance, Pace University, One Pace Plaza, New York, NY 10038; email: [email protected]
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
Historical Volatility Indicators
Because it is frequently difficult to estimate the inputs necessary to estimate security ex-ante variance,
analysts often use the volatility of historical returns as a surrogate for ex-ante risk:
σ
(2)
2
H
n
=∑
t =1
(R
t
− Rt
n −1
)
2
where Rt represents the return realized during period t in this n time period framework. Table I presents
sample daily historical price data for a stock whose returns are given in the third column. The traditional
sample daily variance estimator for this stock based on these returns equals .003172; the monthly variance
(assuming 30 trading days per month; with weekends there are normally 20-21 days), if we were able
assume that returns follow a Brownian motion process is .095.2 Use of the traditional sample estimator to
forecast variance requires the assumption that stock return variances are constant over time, or more
specifically, that historical return variance is an appropriate indicator of future return uncertainty. While
this can often be a reasonable assumption, firm risk conditions can change and it is well documented that
market volatility does fluctuate over time (See for example Officer (1971)). In addition, note that the
sample variance estimator rather than the population estimator is proposed in Equation (2). This difference
becomes more significant with smaller samples. Smaller samples intensify the need for a reliable mean.
Table 1: Traditional Sample Estimators and Other Preliminary Computations
(1)
(2)
(4)
(5)
(6)
Time Pricet
Returnt Returnt-1
Errort
2
Error t
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
N.A.
0.00417
0.00415
-0.00413
0.06224
0.06250
-0.08824
0.03226
-0.04688
0.00820
0.00407
0.00405
-0.00403
0.00405
0.00403
-0.02811
0.09091
-0.09091
0.17083
-0.06050
-0.02652
0.00389
0.00388
-0.00772
0.00389
0.06202
0.06204
0.05842
-0.10714
-0.01455
-0.00738
-0.00189
-0.01018
0.05261
0.08157
-0.06905
-0.01375
-0.04077
-0.01992
-0.00023
-0.00204
-0.01013
-0.00554
-0.00206
-0.03421
0.07091
-0.05944
0.12367
0.00554
-0.06052
-0.01542
-0.00229
-0.01389
-0.00729
0.05585
0.08102
0.07741
-0.08972
-0.06873
-0.02151
0.000004
0.000104
0.002768
0.006654
0.004768
0.000189
0.001662
0.000397
0.000000
0.000004
0.000103
0.000031
0.000004
0.001171
0.005028
0.003533
0.015295
0.000031
0.003663
0.000238
0.000005
0.000193
0.000053
0.003120
0.006564
0.005992
0.008050
0.004723
0.000463
30.000
30.125
30.250
30.125
32.000
34.000
31.000
32.000
30.500
30.750
30.875
31.000
30.875
31.000
31.125
30.250
33.000
30.000
35.125
33.000
32.125
32.250
32.375
32.125
32.250
34.250
36.375
38.500
34.375
33.875
33.625
σ H2
2
(3)
0.00417
0.00415
-0.00413
0.06224
0.06250
-0.08824
0.03226
-0.04688
0.00820
0.00407
0.00405
-0.00403
0.00405
0.00403
-0.02811
0.09091
-0.09091
0.17083
-0.06050
-0.02652
0.00389
0.00388
-0.00772
0.00389
0.06202
0.06204
0.05842
-0.10714
-0.01455
= .095154
Violation of this assumption in our illustration is discussed later.
(7)
2
Error t −1
0.000004
0.000104
0.002768
0.006654
0.004768
0.000189
0.001662
0.000397
0.000000
0.000004
0.000103
0.000031
0.000004
0.001171
0.005028
0.003533
0.015295
0.000031
0.003663
0.000238
0.000005
0.000193
0.000053
0.003120
0.006564
0.005992
0.008050
0.004723
(8)
2
Error t − 2
0.000004
0.000104
0.002768
0.006654
0.004768
0.000189
0.001662
0.000397
0.000000
0.000004
0.000103
0.000031
0.000004
0.001171
0.005028
0.003533
0.015295
0.000031
0.003663
0.000238
0.000005
0.000193
0.000053
0.003120
0.006564
0.005992
0.008050
(9)
Φt
0.001861
0.002352
0.003493
0.003813
0.002679
0.002169
0.002193
0.001909
0.001842
0.001861
0.001864
0.001848
0.002051
0.002934
0.003318
0.005160
0.004427
0.002499
0.002502
0.001883
0.001877
0.001884
0.002406
0.003536
0.004015
0.004285
0.004040
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
Using Equation (2) to estimate security variance requires that the analyst choose a sample series of
prices (and dividends, if relevant) at n regular intervals from which to compute returns. Two problems arise
in this process:
1.
2.
Which prices should be selected and at what intervals?
How many prices should be selected?
First, since each of the major stock markets tend to close at regular times on a daily basis, closing prices
will usually reflect reasonably comparable intervals, whether selected on a daily, weekly, monthly, annual
or other basis. Those prices closer to the date of computation will probably better reflect security risk (e.g.,
Price volatility of a security thirty years ago hardly seems relevant today). On the other hand, longer-term
returns such as those computed on a monthly or annual basis will more closely follow a normal distribution
than returns computed on a daily or shorter-term basis. This is a highly desirable quality, since many of the
statistical estimation procedures used by analysts assume normal (or lognormal) distribution of inputs; the
characteristics of most non-normal distributions are not well known.
Generally speaking, more prices or data points used in the computation process will increase the
statistical significance of variance estimates. However, this leads to a dilemma: more data points,
particularly pertaining to longer term returns will require prices from the more distant, but less relevant
past. On the other hand, shorter estimation intervals may result in non-normal return distributions as well as
autocorrelation issues. Hence, the analyst must balance the needs for a large sample to ensure statistical
significance, recent data for relevance and longer-term data for independence and normality of distribution.
These conflicting needs call for compromise. The convention that has developed over the years both in
academia and in the industry is based on computations of five years of monthly returns.
Nonetheless, numerous difficulties still remain with this estimation procedure. For example, as we
discussed above, variances are not necessarily stable over time. In our numerical illustration, highervolatility periods are clustered as are lower volatility periods, in a manner similar to actual return variances.
Second, returns themselves may not be independently distributed. In our numerical illustration, returns are
inversely correlated, leading to significant differences in our variance estimates. Both problems arise in our
numerical illustration data in Table I. In addition, non-trading may omit returns data for computations.
Extreme Value Estimators
Two difficulties associated with the traditional sample estimator procedure, time required for
computation and arbitrary selection of returns from which to compute volatilities may be dealt with by
using extreme value estimators. Extreme value estimators are based on high and low values (and sometimes
other parameters) realized by the security's price over a given period.
For example, consider the Parkinson Extreme Value Estimator (Parkinson (1980)). This estimating
procedure is based on the assumption that underlying stock returns are log-normally distributed without
drift. Given this distribution assumption, the underlying stock's realized high and low prices over a given
period provide information regarding the stock's variance. Thus, if we are willing to assume that the return
distribution is to be the same during the future period, Parkinson's estimate for the underlying stock return
variance is determined as follows:
(3)
  HI 
σ = .361 ⋅ ln

  LO 
2
2
p
where HI designates the stock's realized high price for the given period and LO designates the low price
over the same period.3
3
Accuracy of the Parkinson measure can be improved if the sample period can be subdivided into n equal sub-periods such that
variance is estimated as follows:
.361  n  HI t
⋅ ∑ ln
σ =
n  t =1  LOt
2
p



2
This is the more general form of the Parkinson Estimator. Each of the other extreme value estimators discussed in this paper can be
generalized in a similar manner. The constant, .361 is the normal density function constant, 1 /
2π .
32
33
JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
Figure I: Sample Prices and Range
P(t)
HI 38.5
LO
30
21.5
0
5
10
15
20
25
30
t
The Parkinson measure results in a variance estimate equal to .022465 for the stock whose historical
prices are listed Table 1. Clearly, this is likely to be a simple estimate to obtain when periodic high and low
prices for a stock are regularly published as they are for NYSE and many other stocks listed in the Wall
Street Journal. Furthermore, the efficiency of the Parkinson procedure is several times higher than the
traditional sample estimation procedure. To understand why this might be the case, consider Figure I.
Several sample prices from which periodic returns might be computed are plotted for a second security,
along with the high and low prices of the distribution. One might expect that high and low prices over the
life of a process will tell us more about the variance of the distribution than would open and close prices
alone. Using the extreme value estimator might be as simple as inserting into Equation (3) the 52-week
high and low prices from the Wall Street Journal. Note that the Parkinson estimate is significantly smaller
than the monthly historical estimate. This difference draws largely from the negative autocorrelation in the
returns series.
P(t)
Figure II: Sample Prices and the Normalized Range
HI 38.5
LO
30
21.5
0
5
10
15
t"
20
25
t'
30
t
34
JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
While the Parkinson extreme value estimator has the desirable characteristics of efficiency and
simplicity, it is based on the assumption of zero drift in security prices. This means that larger returns
(positive or negative) will reduce the effectiveness of the estimator and will tend to increase the computed
variance.
Garman and Klass (1980) refine the methodology of Parkinson by including the security's opening (O)
and closing (C) prices for the given period in their estimate, but still assume zero drift:
(4)


2
= .511 ln
σ GK
2
2
 C  HI
HI
LO 
LO 
LO
HI 
 C
− ln
+ ln
⋅ ln
 − .383 ln  T
 − .019 ln  ln
 − 2 ln
O
O 
O 
O
O 
 O
 O O
he Garman and Klass measure results in a variance estimate equal to .026275 for the stock in Table 1.4
Garman and Klass have estimated variations of their estimator efficiency to be as high as 8.4 times that of
the traditional sample estimator, though it is still biased when the security has non-zero drift. Rogers and
Satchell (1991) assume that stock prices follow a lognormal random walk with drift and propose a drift
adjusted extreme value estimator for variance:
2
= ln
σ RS
(5)
HI
O
C
LO  LO
C
 HI
ln O − ln O  + ln O ln O − ln O 
The Rogers and Satchell measure results in a variance estimate equal to .033774 for the stock in Table 1.
Ball and Torous (1984) propose a maximum likelihood estimation procedure based on extreme values.
Kunitomo (1992) extends the Parkinson procedure to Brownian Motion processes with drift by constructing
a Brownian Bridge (in a sense, neutralizing the drift by normalizing the minimum and maximum prices as
distances from the drift line):
(6)
σ
2
K

6  
t′
t ′′
 



= 2 ⋅  ln max  Pt − (C − O )  − ln min  Pt − (C − O )  
T
T
π  
 




2
where t' represents the time period associated with the maximum price Pt, t" represents the time associated
with the maximum price Pt, and T represents the time associated with the closing price C, given that the
time associated with the open price equals 0. The Kunitomo measure results in a variance estimate equal to
.016448 for the stock in Table 1. Figure II provides a graphical depiction of the Kunitomo risk measure.
Kunitomo estimates his procedure to be as much as ten times as efficient as the traditional sample
estimation procedure. Shu and Zhang (2004) test extreme value estimators and find biases resulting from
drift consistent with the descriptions given above.
When one converts the extreme value variances to standard deviation estimates, we find that the
estimate differences, in terms of the original units of measure, are relatively small. The reader is referred to
Shu and Zhang (2004) for various statistical properties of the different extreme value estimators.
Implied Volatilities
Analytical Procedures
A problem shared by both the traditional sample estimating procedures and the extreme value
estimators is that they require the assumption of stable variance estimates over time; more specifically, that
historical variances equal future variances. A third procedure first suggested by Latane and Rendleman
(1976) is based on market prices of options that may be used to imply variance estimates. For example, the
Black-Scholes Option Pricing Model and its extensions provide an excellent means to estimate underlying
stock variances if call prices are known. Essentially, this procedure determines market estimates for
underlying stock variance based on known market prices for options on the underlying securities. Consider
our stock example from Table 1 on day 30 where the stock is currently trading for $33.625. Suppose that a
one month (t =1) call on this stock with a striking price equal to $30 is currently trading for c0 = $4.50:
t=1
rf = .005
c0 = $4.50 X = $30
S0 = $33.625
4
Readers should refer to the Garman and Klass paper for a discussion of the models numerical constants.
JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
where rf equals the monthly riskless return rate and X is the option striking price. If investors use the BlackScholes Options Pricing Model to evaluate calls, the following must hold:
4.50 = 33.625 ⋅ N (d1 ) − 30e − rT ⋅ N (d 2 )
(
)
 33.625 
2
ln
 + .005 + .5σ ⋅ 1
30 
d2 = 
σ 1
(7)
d 2 = d1 − σ 1
We find that this system of equations holds when σ = .027459. Thus, the market prices this call as though
it expects that the variance of anticipated returns for the underlying stock is .027459.
Unfortunately, the system of equations required to obtain an implied variance has no closed form
solution. That is, we will be unable to solve explicitly for variance; we must search or substitute for a
2
solution. One can substitute values for σ until he finds one that solves the system. One may save a
2
significant amount of time by using one of several well-known numerical search procedures such as the
Method of Bisection or the Newton-Raphson Method.
Consider again the above example pertaining to a one-month call currently trading for $4.50 while its
underlying stock current trades for $33.625. We wish to solve the above Black-Scholes system of equations
for σ through use of the Method of Bisection. This is equivalent to solving for the root of:
f( σ *) = 0 = 33.625N(d1) - 30e-.005N(d2) - 4.50
(8)
based on equations above for d1 and d2. There exists no closed form solution for σ . Thus, we will use the
Method of Bisection to search for a solution. We will first arbitrarily select two endpoints of a segment, b1
= .1 and a1 = .3 such that f(b1) < 0 and f(a1) > 0. Next, we will bisect this segment to obtain next
our σ estimate, which will be half way between .1 and .3. Since these endpoints result in f( σ ) with
opposite signs and we have bisected our segment, our first iteration will use σ 1 = .5(.1+.3) = .2. We find
that this estimate for sigma results in a value of 0.34858 for f( σ ). Since this f( σ ) is positive, we know
that σ * is in the segment b2 = .1 and a2 = .2. We repeat the iteration process, continuing to bisect
appropriate segments, finding after 19 iterations that σ *=.165708 and that implied variance is .027459.
Table 2 provides step-by-step details on the process of iteration.
Although the Method of Bisection will reliably generate solutions for most implied volatility problems,
its chief drawback is that it frequently converges to a solution rather slowly. An alternative, and usually
more efficient method for finding a root to Equation (8) is the Newton-Raphson Method, which is based on
a first order Taylor Series Expansion. By the Newton-Raphson Method we first select an initial trial
solution σ 0. Based on a first order Taylor approximation, we would determine our next trial solution by
solving for σ 1 as follows:
(9)
0 = f( σ 0) + ( σ 1 -
σ
0)f`(
σ
0)
Consider again the above example where we wish to estimate the volatility implied by a six-month call
option. We now solve for implied standard deviation using the Newton-Raphson method, with an
arbitrarily selected initial trial solution of σ 0=.3. We need the derivative of the Black-Scholes model with
respect to the underlying stock return standard deviation:
( )
− d12
2
(10)
∂C
e
=S t⋅
>0
f ' (σ ) =
∂σ
2π
35
36
JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
Table 2: Using the Bisection Method to Estimate Implied Volatility
INITIAL EQUATION:
SN(d1)-Xe-rtN(d2)
a1 =
0.3
b1=
0.1
rf =
0.005
S0=
33.625
σ
C0= 4.5
T=1
X =30
n
d1( σ n)
d2( σ n)
c( σ n)
f( σ n)
f(a1)=
1.46317
0.3
0.546908
0.246908
0.7077791
0.5975103
5.9631666
1.4631666
f(b1)=
-0.5444
0.1
1.240725
1.140725
0.8926462
0.8730077
3.9556220
-0.5443780
d1( σ n)
d2( σ n)
N[d1( σ n)]
N[d2( σ n)]
n
σ
N[d1( σ n)] N[d2( σ n)]
C( σ n)
f( σ n)
an
bn
1
0.30000
0.10000
0.200000
0.695362
0.4953623
0.7565859
0.6898278
4.8485838
0.3485838
2
0.20000
0.10000
0.150000
0.868816
0.7188164
0.8075263
0.763873
4.3511742
-0.1488258
3
0.20000
0.15000
0.175000
0.767914
0.5929140
0.7787309
0.7233807
4.5916430
0.0916430
4
0.17500
0.15000
0.162500
0.814004
0.6515036
0.7921786
0.7426393
4.4689437
-0.0310563
5
0.17500
0.16250
0.168750
0.789990
0.6212396
0.7852331
0.7327791
4.5297351
0.0297351
6
0.16875
0.16250
0.165625
0.801741
0.6361155
0.7886485
0.7376495
4.4991931
-0.0008069
7
0.16875
0.16563
0.167188
0.795803
0.6286153
0.7869267
0.7351996
4.5144284
0.0144284
8
0.16719
0.16563
0.166406
0.798756
0.6323497
0.7877841
0.7364209
4.5068017
0.0068017
9
0.16641
0.16563
0.166016
0.800244
0.6342286
0.7882154
0.7370343
4.5029951
0.0029951
10
0.16602
0.16563
0.165820
0.800991
0.6351711
0.7884318
0.7373416
4.5010935
0.0010935
11
0.16582
0.16563
0.165723
0.801366
0.6356431
0.7885401
0.7374955
4.5001432
0.0001432
12
0.16572
0.16563
0.165674
0.801553
0.6358792
0.7885943
0.7375725
4.4996681
-0.0003319
13
0.16572
0.16567
0.165698
0.801459
0.6357611
0.7885672
0.737534
4.4999056
-0.0000909
14
0.16572
0.1657
0.165710
0.801413
0.6357021 0.7885536
Implied standard deviation is .165708; Implied variance is .027459
0.7375148
4.5000244
0.0000243
n
This derivative represents the sensitivity of the call's value with respect to the standard deviation of
underlying asset returns. Traders often refer to this sensitivity as vega or lambda. We see from Table 3 that
this standard deviation results in a value of f( σ 0) = 1.4631666. Plugging into Equation (10) .3 for σ 0, we
find that f'( σ 0) = 11.55016. Thus, our second trial value for σ is determined by: σ 1 = .3 (1.4631666/11.55016) = .17333. This process continues until we converge to a solution of approximately
.165708, with variance equal to approximately .027459. Notice that the rate of convergence is much faster
by using the Newton-Raphson Method than by using the Method of Bisection.
A second difficulty that arises with use of implied variances results from the fact that there will
typically be more than one option trading on the same stock. Each option's market price will imply its own
underlying stock variance, and these variances are likely to differ. How might we use this conflicting
information to generate the most reliable variance estimate? Each of our implied variance estimates is
likely to provide some information, yet has the potential for having measured σ with error. We might be
able to preserve much of the information from each of our estimates and eliminate some of our estimating
error if we use for our own implied volatility a value based on an average of all of our estimates. The
following suggests two means of averaging the implied standard deviation estimates:
1.
2.
Simple average: In this case, the final standard deviation estimate is simply the mean of
the standard deviations implied by the market prices of the various calls.
Average based on price sensitivities to σ : Calls that are more sensitive to σ as indicated
by ∂c 0 / ∂σ are more likely to imply a correct standard deviation estimate. Suppose we
have n calls on a given stock, each of which imply an underlying stock standard deviation
σ j. Each of the call prices will have a sensitivity to its implied underlying stock standard
deviation ∂c0 / ∂σ j . The sensitivities can be summed, and a weighted average standard
deviation estimate for the underlying stock can be computed from the following
weighting scheme:
JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
Table 3: The Newton-Raphson Method and Implied Volatilities
Initial Equation: SN(d1)-Xe-rtN(d2)
σ0 =
0.3
rf =
C0=
0.005
T=
1
4.5
X =
30
n
σn
1
2
3
4
5
6
0.30000
0.17333
0.16579
0.16571
0.16571
0.16571
S0=
f'(σn)
11.55106
9.945080
9.732214
9.729797
9.729797
9.729797
33.625
d1(σn)
0.5469082
0.7736327
0.8011121
0.8014221
0.8014222
0.8014222
d2(σn)
0.246908
0.600302
0.635323
0.635714
0.635714
0.635714
N(d1)
0.7077791
0.7804261
0.7884667
0.7885564
0.7885564
0.7885564
N (d2)
0.5975103
0.7258476
0.7373913
0.7375187
0.7375187
0.7375187
f(σn)
1.4631666
0.0750033
0.0007869
0.0000008
0
0
Implied standard deviation is .16571; Implied variance is .027459
∂c 0 j
(11)
wi =
n
∂σ j
∂c0 j
∑ ∂σ
j =1
j
where wi represents the weight for the implied standard deviation estimate for call option i. Thus,
the final standard deviation estimate for a given stock k based on all of the implied standard
deviations from each of the call prices is:
n
(12)
σ K = ∑ wiσ i
i =1
Simple Closed Form Solution Procedures
The implied volatilities described above have the desirable characteristic of having an ex-ante
orientation. However, their use is somewhat complicated by the sometimes time-consuming methodology
of iterating for solutions. Spreadsheet file and even computer program solutions packages can sometimes
be cumbersome. This section provides two methodologies for obtaining simple closed form solutions for
implied underlying asset volatilities.
Brenner and Subrahmanyam (1988) provide a simple formula to estimate an implied standard deviation
(or variance) from an option whose striking price equals the current market price of the underlying asset.
As the market price differs more from the option striking price, the estimation accuracy of this formula will
worsen:
(13)
σ
2
BS
2πC o2
=
tS o2
This problem is apparent with our Brenner-Subrahmanyam estimate of .1125; our striking price differed
from the market price by more than 10%. The accuracy of formula (13) when the option strike and
underlying asset market prices are unequal can be improved with use of a quadratic procedure proposed by
Corrado and Miller (1996). Their formula makes use of a second order approximation of the cumulative
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
normal density function and an approximation for ln(S0/X) as 2(S0-X)/(S0+X). Several additional
simplifications and approximations lead to the following formula for σ :
(14)
2
σ CM
=
2π
t (S o + X )
2
2

(S o − X )2
So − X
So − X 


⋅ co −
+  co −
 −
2
2 
π







2
While the accuracy of this formula is improved when the market price of the underlying asset is close to the
striking price of the option, it is not nearly as sensitive to differences between these prices as is the
Brenner-Subrahmanyam formula. The Corrado-Miller estimate was .0274, much closer to our original
Black-Scholes estimate.
Autoregressive Techniques
While the implied volatility techniques discussed above may be suited where anticipated variances may
differ from historical variances, they do assume that anticipated variances are constant over the lives of the
options. Furthermore, they rely option prices, requiring an appropriately active options market. Techniques
discussed in this section do not require assumptions of variance stability nor do they require market prices
of options; instead, these methods make use of historical relationships among variances and forecast errors
in deriving volatility estimates. For example, Cohen et al. (1982) and others have found significant
autocorrelation in security returns and variances, leading to some degree of predictability. To estimate
variances conditional on patterns of disturbances, Engle (1982) developed the AutoRegressive Conditional
Heteroskedasticity model (ARCH).
Table 1 provides sample calculations of variance forecasts based on the ARCH methodology. The first
step in constructing the ARCH forecast of volatility is to perform an OLS regression of security returns
(column 3) against one period lagged returns (column 4): rt = A0 + B1rt-1 + ε t. The fifth column consists of
ε t, the forecast error or disturbance for time t, which is squared for the sixth column. Notice in Column 6
that small squared disturbances seem to be clustered together as do the larger squared disturbance terms.
This might suggest that variances are unstable over time. Next, one decides how many periods of
disturbances will be relevant (n is assumed to equal 2 in Table 1, hence we will work with an ARCH (2)
model) to obtain the variance estimate. Columns seven and eight represent one and two period squared
disturbance terms. An OLS regression of squared disturbances for time t (in the sixth column in Table 1) is
run against n squared disturbances ε for s = (t-i) from i = 1 to n. The ARCH (2) model in Table 1 predicts
a daily variance in time t equal to .004040 or a 30-day variance of .1212, based on relationships the n = 2
prior forecast errors ε t. More generally, the ARCH (n) model forecasts a variance as follows:5
(15)
ARCH ( n ) : σ t2 = a 0 +
n
∑bε
i =1
i
2
T −i
Time varying variance forecasts do tend to be consistent with market observations. The ARCH model
has been extended by Bollerslev (1986) into the Generalized Auto-Regressive Conditional
Heteroskedasticity model (GARCH), which makes use of n lagged squared disturbances and lagged
conditional variances:
(16)
n
m
i =1
j =1
GARCH (n, m) : σ t2 = a 0 + ∑ bi ε T2−i + ∑ γ j σ t2− j
The ARCH (2) model in Table 1 can be extended to create the GARCH (2,1) model by performing the
regression ε t = 0 + b1 ε t-1 + b2 ε t-2 + b3 σ t-1 + µt. This particular GARCH (2,1) model results an a variance
estimate equal to .003311, based on the following estimates (standard errors in parentheses):
ε t = 0.005713 + 0.25481 ε t-1 + 0.571437 ε t-2 - 1.91491 σ
t-1 + µt
(0.0035) (0.207)
(0.331)
(1.271)
R Squared = 0.16493; No. of Observations = 26; Degrees of Freedom = 22
5
The analyst may wish to impose constraints on 0 and each bi and on the GARCH coefficients to ensure that variance estimates are
non-negative.
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
The ARCH and GARCH variance estimates obviously differ substantially from our other variance
estimates; they were much higher. However, these procedures are the only ones that accounted for
autocorrelation in returns data and clustering of higher- and lower-variance periods. As of the end of our
estimating period, the variance might have been characterized as high relative to other potential estimating
periods.
Summary
This paper has reviewed alternative risk or volatility estimation procedures in the financial literature.
Methodology discussed here included traditional sample variance estimates, extreme value estimators,
Black Scholes implied volatility estimators and AutoRegressive techniques. These methodologies are
compared and contrasted in Table 4. Numerous other variance estimator and correction techniques have
been proposed in the financial literature as well, including relative measures, measures based on
fundamental factors and those which correct for non-trading of securities.
Table 4: Comparison of Risk Measurement Methodologies
Measure
Best used when
Ex-Ante Measure Based
on Probabilities
(1) Ex-ante or future-oriented measure is needed such as when:
a. The asset's historical volatility does not properly indicate its future risk
b. The asset's risk characteristics have recently changed
c. The asset has no price or returns history
(2) All potential future return or cash flow outcomes can be specified
(3) Probabilities can be associated with each potential return or cash flow outcome
(4) Instead of (2) & (3), there is a specific return generating process with known parameters
Traditional Sample
Estimator
(1) Variances are expected to be constant between historical and future time periods
(2) There are an appropriate number of sampling intervals where:
a. More periods increase statistical significance
b. More periods increase reliance on older, less relevant historical data
(3) Appropriate interval lengths can be determined; longer periods approach normality
Parkinson Extreme Value
Estimator
(1) The computationally simplest measure based on a minimum of data is desired
(2) Asset returns are log-normally distributed without drift
(3) Historical volatility is a good indicator of future risk
(1) Same as for Parkinson, plus open and closing prices are conveniently available
Garman and Klass
Extreme Value Estimator
Rogers and Satchell
Extreme Value Estimator
Kunitomo Extreme Value
Estimator
Implied Volatility:
Analytical Procedures
(1) Same as for Garman and Klass but is superior when returns drift exists
(1) Same as for Rogers and Satchell measure but is computationally more cumbersome
(1) Option prices on asset are readily available
(2) Option pricing model assumptions hold in the relevant market
(3) Can be used when historical volatility does not indicate future risk
(4) User is able to use the appropriate analytical procedures
(5) The market can be assumed capable of assessing risk
(6) Method of Bisection does not require sensitivity computation; Newton-Ralphson is faster
Implied Volatility: Simple
Closed Form Procedures
(1) Same as for analytical procedures above except that mathematical sophistication and much
time is not needed if one is willing to accept an approximation
(2) Brenner and Subrahmanyam is easier and less accurate than Corrado and Miller
ARCH Method
(1) Variances are not constant; There is a linear pattern in variance changes over time
GARCH Method
(1) Same as for ARCH; One can ascertain a non-linear pattern in variance over time
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 3 • Number 2 • Winter 2004
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