on pre
dicts stock price movements. Such an assumption does not allow for the occasional
wild days that many stocks, some futures, and the relatively rare index undergo. The
normal distribution pretty much says that a stock can't rise or fall by more than 3 stan
dard deviations. In fact, according to math, the probability of something that behaves
according to the normal distribution ( the "classic" bell curve is a normal distribution)
moving three standard deviations is 0.0013 (or just a little more than one tenth of one
percent). So, if there are 2,500 optionable stocks, say, then one would expect maybe
3 of them to move three standard deviations on any given day.
However, in real market trading, there are routinely moves in stocks of more
than 3 standard deviations - some as much as 5 standard deviations or more.
Statistically (if the lognormal distribution were correct), one would only expect to see
moves of that size maybe once in his lifetime, yet there are five or ten each day!
Specifically, the normal distribution's probability of something being able to move 8
standard deviations is 0.000000000000000629. This number is so small that one
would expect to see only one such occurrence in the known life of the universe, if
prices were truly distributed via the normal distribution. If the normal distribution
were the correct format for stock prices, such a small probability would indicate that
one would not see an 8-standard deviation move until billions of trading days had
passed. However, one can find several such moves on nearly any trading day, and it
is not necessary to use some low-priced, oddball stock that dropped from 1 to . 75 or
some such nonsense as an example.
If those numbers don't convince you that stocks aren't lognormally distributed,
perhaps the following study will. Table 38-1 lists moves that occurred on Monday,
April 5, 1999 - a day that was somewhat volatile (the Dow was up 174 points).
There were many other substantial moves that day. In the list in Table 38-1, all
three that fell in price were on earnings warnings, and the two that rose were just
swept up in the that day's version of the Internet mania. All in all, 58 stocks had
moves of greater than four standard deviations on that day! It was not any sort of spe
cial day, although it did fall during earnings warnings season and in the midst of the
Internet mania. But the fact that the market is somewhat volatile shouldn't justify all
of these huge moves, if one still adheres to the belief that lognormal is the correct
distribution.
Chapter 38: The Distribution ol Stock Prices 785
TABLE 38-1.
Stock Last Sale Change Standard Deviations
Aspect Devt (ASDV) 8 -14.38 -31.2
Axent (ANT) 8 -12 -11.2
Ameritrade (AMTD) 91.63 +29 .13 + 8.6
CheckPoint (CHKP) 28.75 -10.75 - 8.4
Sabre Gp. (TSG) 55 + 8.50 + 8.0
So, just to make sure that wasn't a bad sample, a low-volatility period was cho
sen to sample. Since the inception of the CBOE's Volatility Index ($VIX) trading, its
lowest market volatility readings were in January and July of 1993. The lowest single
day was July 25, 1993. On that day, twelve stocks had moves of more than four stan
dard deviations. They included some big names, like Adaptec (ADPT), Bethlehem
Steel (BS), U.S. Steel (X), Chiquita Brands (CQB), and Novell (NOVL).
The only way to tell how many standard deviations a stock has moved is to use
its historical volatility - say, the 20-day historical volatility, for example - in the meas
urement. Thus, a 4-point move for a nonvolatile stock like Bethlehem Steel in 1993
pales in comparison to Ameritrade's 29-point gain in 1999 (Table 38-1), but both
were large moves in terms of standard deviations, determined by using each stock's
historical volatility.
As another test, prices from October 8, 1998 - the day the market bottomed
after a severe and rather swift decline brought on by Russian debt problems (it was
a very volatile day after several volatile weeks of trading) - were tested to see how
many stocks had moves of four standard deviations or more. There were 33, but that
seemingly low number reflects the fact that many stocks' 20-day historical volatilities
were already well inflated by October 8, 1998. On that day, the Utility Index ($UTY)
fell over 14 points, which was about 5.5 standard deviations. American Power
Conversion (APCC) was up over six points that day, to 36.88 - a gain of over five stan
dard deviations.
Perhaps you might think that these one-day moves overstate things, that over a
more prolonged period of time, the lognormal distribution fits better. A study was
constructed to measure the volatility over a slightly longer time period. The results
of the study not only confirmed our suspicions, but actually were somewhat startling
in quantifying just how volatile certain individual issues can be.
The first example comprised the 30-trading day period between October 22,
1999 and December 7, 1999. There is nothing magical about these dates; that just
happened to be the most recent 30-day period for which data was available when the
study was conducted.
786 Part VI: Measuring and Trading VolatiRty
This particular period started out as a rather normal one in the market - in fact,
prices had perhaps been a little less volatile than normal leading into that period. To
support that statement, it should be noted that on October 22, the CBOE's Volatility
Index ($VIX) stood at about 23 - a relatively middle-of-the-range level. So it wasn't
as if this was an extremely volatile period.
The example is simple enough. The performance of 2,900 optionable stocks was
measured to see if, beginning on October 22, 1999, any of them experienced moves of
greater than three standard deviations at any time during the 30-day period. The stan
dard deviation was based on the 20-day historical volatility of each individual stock.
Obviously, a stock would have to make a much greater move to exceed the


ple is simple enough. The performance of 2,900 optionable stocks was
measured to see if, beginning on October 22, 1999, any of them experienced moves of
greater than three standard deviations at any time during the 30-day period. The stan
dard deviation was based on the 20-day historical volatility of each individual stock.
Obviously, a stock would have to make a much greater move to exceed the 3-standard
deviation limit if it waited until the end of the 30-day period to do it, as opposed to mak
ing a big move on the first day. So, before reading on, take a guess: How many of the
stocks do you think exceeded three standard deviation moves at some point during the
30 days? Remember that the lognormal distribution would predict virtually no moves
of that size. The answer is in the next paragraph.
There were more stocks that had large upside moves than there were that had
large downside moves over the period in question. That isn't too surprising, since the
market moved up during that time. The final total showed this: Of the 2,900 stocks,
nearly 650 experienced moves of 3 standard deviations or more during the life of the
study, including 65 that moved more than six standard deviations. If the lognormal
distribution were correct, the two lines in Table 38-2 would be filled with zeroes. This
clearly shows stocks during this period didn't conform to the "normal" expectations.
The study results are shown in Table 38-2. (Note that "cr" is the greek letter sigma,
which mathematicians traditionally use to denote standard deviations, so 3cr means
three standard deviations.)
TABLE 38·2
Stock price movements.
Total Stocks: 2,888
Upside Moves:
Downside Moves:
3cr
309
69
4cr
116
29
Sa
44
15
Dates: I 0/22/99-12/7 /99
>6cr
47
19
Total
516
132
Total number of stocks moving >=3cr: 648 [22% of the stocks studied!
The largest move was registered by a stock that jumped from a price of 5 to nearly
12 in about six trading days. One of the bigger downside movers was a stock that fell
from about 20 to 8 in a matter of a couple of weeks, with most of the damage occur
ring in a two-day time period.
Chapter 38: The Distribution of Stock Prices 187
Lest you think that this example was biased by the fact that it was taken during
a strong run in the NASDAQ market, here's another example, conducted with a dif
ferent set of data- using stock prices between June 1 and July 18, 1999 (also 30 trad
ing days in length). At that time, there were fewer large moves; about 250 stocks out
of 2,500 or so had moves of more than three standard deviations. However, that's still
one out of ten - way more than you've been led to expect if you believe in the nor
mal distribution. The results are shown in Table 38-3.
TABLE 38-3.
More stock price movements.
Total Stocks: 2,447 Dates: 6/1 /99-7 /18/99
Upside Moves:
Downside Moves:
3cr
104
54
4cr
28
19
Scr
13
7
>6cr
12
14
Total number of stocks moving >=3cr: 251 ( 10% of the stocks studied)
Total
157
94
Finally, one more example was conducted, using the least volatile period that
we had in our database - July of 1993. Those results are in Table 38-4.
TABLE 38-4.
Stock price movements during a nonvolatile period.
Total Stocks: 588 Dates: 7 /1 /93-8/17 /93
3cr 4cr Scr >6cr Total
Upside Moves: 14 5 1 1 21
Downside Moves: 28 5 3 4 40
Total number of stocks moving >=3cr: 61 ( 10% of the stocks studied)
At first glance, it appears that the number of large stock moves diminished dra
matically during this less volatile period in the market - until you realize that it still
represents 10% of the stocks in the study. There were just a lot fewer stocks with list
ed options in 1993 than there were in 1999, so the database is smaller (it tracks only
stocks with listed options). Once again, this means that there is a far greater chance
for large standard deviations moves - about one in ten - than the nearly zero percent
chance that the lognormal distribution would indicate.
VOLATILITY BUYER'S RULE!
The point of the previous discussion is that stocks move a lot farther than you might
expect. Moreover, when they make these moves, it tends to be with rapidity, gener-
788 Part VI: Measuring and Trading VolatiDty
ally including gap moves. There are not always gap moves, though, over a study of
this length. Sometimes, there will be a more gradual transition. Consider the fact that
one of the stocks in the study moved 5.8 sigma in the 30 days. There weren't any huge
gaps during that time, but anyone who was short calls while the stock made its run
surely didn't think it was a gradual advance.
So, what does this information mean to the average option trader? For one,
you should certainly think twice about selling stock options in a potentially volatile
market ( or any market, for that matter, since these large moves are not by any means
limited to the volatile market periods). This statement encompasses naked option
selling, but also includes many forms of option selling, because of the possibilities of
large moves by the underlying stocks.
For example, covered call writing is considered to be "conservative." However,
when the stock has the potential to make these big moves, it will either cause one to
give up large upside profits or to suffer large downside losses. ( Covered call writing
has limited profit potential and relatively large downside risk, as does its equivalent
strategy, naked put selling.) When these large stock moves occur on the upside, a cov
ered writer is often disappointed that he gave up too much of the upside profit poten
tial. Conversely, if the stock drops quickly, and one is assigned on his naked put, he
often no longer has much appetite for acquiring the stock ( even though he said he
"wouldn't mind" doing so when he sold the puts to begin with).
Even spreading has problems along these lines. For example, a vertical spread
limits profits so that one can't participate in these relati

eBooks\Lawrence G. McMillan - Options as a Strategic Investment_ Fifth Edition-Prentice Hall Press (2012)\Lawrence G. McMillan - Options as a Strategic Investment_ Fifth Edition-Prentice Hall Press (2012).pdf


the upside profit poten
tial. Conversely, if the stock drops quickly, and one is assigned on his naked put, he
often no longer has much appetite for acquiring the stock ( even though he said he
"wouldn't mind" doing so when he sold the puts to begin with).
Even spreading has problems along these lines. For example, a vertical spread
limits profits so that one can't participate in these relatively frequent large stock
moves when they occur.
What can an option seller do? First, he must carefully analyze his position and
allow for much larger stock movements than one would expect under the lognormal
distribution. Also, he must be careful to sell options only when they are expensive in
terms of implied volatility, so that any decrease in implied will work in his favor.
Probably most judicious, though, is that an option seller should really concentrate on
indices (or perhaps certain futures contracts), because they are statistically much less
volatile than stocks. Hard as it is to believe, futures are less volatile than stocks
(although the leverage available in futures can make them a riskier investment overall).
Two 30-day studies, similar to those conducted on stocks, were run on option
able indices, covering the same time periods: 10/22/99 to 12/7/99 for one study and
7/1/93 to 8/17/93 for the other. The results are shown in Tables 38-5 and 38-6. This
may be a somewhat distorted picture, though, because many of these indices overlap
(there are four Internet indices, for example). The largest mover was the Morgan
Stanley High-Tech Index (5 standard deviations), but it should also be noted that
something that is considered fairly tame, such as the Russell 2000 ($RUT), also had
a 3-standard deviation move in one study. The first study showed that 37% of the
Chapter 38: The Distribution of Stock Prices
TABLE 38-5.
Index price movements.
Total Indices: 135
Upside Moves:
Downside Moves:
TABLE 38-6.
3cr
32
None
4cr
15
Scr
3
Index price movements, least volatile period.
Total Indices: 66
Upside Moves:
Downside Moves:
3cr
l
3
4cr
l
0
Scr
0
0
789
Dates: 10/22/99-12/7/99
>6cr
0
Total
50
Dates: 7/1/93-8/17/93
>6cr
0
0
Total
2
3
Total number of indices moving >=3cr: 5 (8% of the indices studied)
indices made oversized moves - probably a bias because of the strong Internet stock
market during that time period. The low-volatility period showed a more reasonable,
but still somewhat eye-opening, 8% making moves of greater than three standard
deviations. So, even selling index options isn't as safe as it's cracked up to be, when
they can make moves of this size, defying the "normal" probabilities.
Since that period in 1999 was rather volatile, and all on the upside, the same
study was conducted, once again using the least volatile period of July 1993.
In Table 38-6, the numbers are lower than they are for stocks, but still much
greater than one might expect according to the lognormal distribution.
These examples of stock price movement are interesting, but are not rigorous
ly complete enough to be able to substantiate the broad conclusion that stock prices
don't behave lognormally. Thus, a more complete study was conducted. The follow
ing section presents the results of this research.
THE DISTRIBUTION OF STOCK PRICES
The earlier examples pointed out that, at least in those specific instances, stock price
movements don't conform to the lognormal distribution, which is the distribution
used in many mathematical models that are intended to describe the behavior of
stock and option prices. This isn't new information to mathematicians; papers dating
back to the mid-1960s have pointed out that the lognormal distribution is flawed.
However, it isn't a terrible description of the way that stock prices behave, so many
applications have continued to use the lognormal distribution.
790 Part VI: Measuring and Trading Volatility
Since 1987, the huge volatility that stocks have exhibited - especially on certain
explosive down days such as the Crash of '87 or the mini-crash of April 14, 2000 - has
alerted more people to the fact that something is probably amiss in their usual assump
tions about the way that stocks move. The lognonnal distribution "says" that a stock
really can't move farther than three standard deviations (whether it's in a day, a week,
or a year). Actual stock price movements make a mockery of these assumptions, as
stocks routinely move 4, 5, or even 10 standard deviations in a day (not all stocks, mind
you, but some - many more than the lognormal distribution would allow for).
In order to further quantify these thoughts, computer programs were written to
analyze our database of stock prices, going back over six years. As it turns out, that is
a short period of time as far as the stock market is concerned. While it is certainly a
long enough time to provide meaningful analysis (there are over 2.5 million individ
ual stock "trading days" in the study), it is a biased period in that the market was ris
ing for most of that time.
THE "'BIG" PICTURE
The first part of the analysis shows that the total distribution of stock prices conforms
pretty much to what the expectations were for the study, and - not surprisingly - to
what others have written about the "real" distribution of stock prices. That is, there
is a much greater chance of a large standard deviation move than the lognormal dis
tribution would indicate. The high probabilities on the ends of the distribution are
called "fat tails" by most mathematicians and stock market practitioners alike. These
"tails" are what get option writers in trouble - and perhaps even leveraged stock own
ers - because margin buyers and naked writers figure that they will never occur. It is
not intuitively obvious to them and to many other stock market participants that stock
prices behave in this manner.
The graphs in Figure 38-1 show this total distribution. The top grap


ock market practitioners alike. These
"tails" are what get option writers in trouble - and perhaps even leveraged stock own
ers - because margin buyers and naked writers figure that they will never occur. It is
not intuitively obvious to them and to many other stock market participants that stock
prices behave in this manner.
The graphs in Figure 38-1 show this total distribution. The top graph is that of the
lognormal distribution and the actual distribution, using the data from September 1993
to April 2000 overlaid upon each other. The actual distribution was drawn using 30-
day moves (i.e., the number of standard deviations was computed by looking at the
stock price on a certain day, and then where it was 30 calendar days later). The x-axis
(bottom axis) shows the number of standard deviations moved. Note that the curves
have the shape of a normal distribution rather than a lognormal distribution, because
the x-axis denotes number of standard deviations moved rather than stock prices them
selves. For this reason, the term "normal" will be used in the remainder of this section;
it should be understood that it is the distribution of standard deviations that is "normal,"
while the distribution of the stock prices measured by those standard deviation moves
is "lognormal." The y-axis (left axis) shows the "count" - the number of times out of the
2.5 million data points computed that each point on the x-axis actually occurred (in the
Chapter 38: The Distribution of Stock Prices
FIGURE 38-1.
Stock price distribution is not ''normal."
Normal Stock Price Distribution
vs. Actual Stock Distribution (30-Day Moves)
0
£
C ::,
0 (.)
4000
3000
2000
1000
0
-1.0 0 +1.0
Sigmas
240 Includes All Moves 180
220 below-4.0 0 160
210 £ 140
200 C 120
0 180 ::,
0 100
£ 160 (.) 80 C 140 ::,
120 60 0
(.) 100 40
80 20
60 0
40 +3.0
20
0 -4.0
Sigmas
791
792 Part VI: Measuring and Trading Volatility
case of the "actual" distribution) or could be expected to occur (in the case of the "nor
mal" distribution). The notation on the y-axis shows the actual count divided by 10. So,
for example, the highest point (0 standard deviations moved) for the "normal" distri
bution shows that about 95,000 times out of 2.5 million, you could expect a stock to be
unchanged at the end of 30 calendar days.
At first glance, it appears that the two curves have almost identical shapes.
Upon closer inspection, however, it is clear that they do not, and in fact some rather
startling differences are evident.
Fat Tails
Figure 38-1 shows the fat tails quite clearly. Magnified views of the fat tails are pro
vided to show you the stark differences between the theoretical ("normal") distribu
tion and actual stock price movements. Consider the downside (the lower left circled
graph in Figure 38-1). First, note that both the "actual" and "normal" graphs lift up
at the end - the leftmost point. This is because the graph was terminated at -4.0 stan
dard deviations, and all moves that were greater than that were accumulated and
graphed as the leftmost data point. You can see that the "normal" distribution expects
fewer than 200 moves out of 2.5 million to be of -4.0 standard deviations or more
(yes, the "normal" distribution does allow for moves greater than 3 standard devia
tions; they just aren't very probable). On the other hand, actual stock prices - even
during the bull market that was occurring during the term of the data in this study -
fell more than -4.0 standard deviations nearly 2,500 times out of 2.5 million. Thus,
in reality, there was really more than 12 times the chance (2,500 vs. 200) that stocks
could suffer a severely dramatic fall, when comparing actual to theoretical distribu
tion. Also notice in that lower left circle that the actual distribution is greater than the
normal distribution all along the graph.
The upside fat tail shows much the same thing: Actual stock prices can rise far
ther than the normal distribution would indicate. At the extreme - moves of +4.0
standard deviations or more - there were about 2,000 such moves in actual stock
prices, compared with fewer than 100 expected by the normal distribution. Again, a
very large discrepancy: twenty-to-one.
Inflection Points
If the actual distribution is higher at both ends, it must be lower than the normal
distribution somewhere, because there are only a total of 2.5 million data points
plotted. It turns out in this case that the normal distribution is higher (i.e., is expect
ed to occur more often than it actually does) between -2.5 standard deviations and
+0.5 standard deviations. Those are the points where the two curves cross over each
other - the inflection points. Outside of that range, the actual distribution is more
frequent than it was expected to be.
Chapter 38: The Distribution of Stock Prices 793
It is probably the case that this data reflected an overly bullish period. That is,
actual stock prices rose farther than they were expected to, not necessarily at the tails,
but in the intermediate ranges, say between +0.5 and +1.5 standard deviations. This
does not change the results of the study as far as the tails go, but one may not always
be able to count on intermediate upside moves being more frequent than predicted.
SIDE BENEFITS OF THIS STUDY
In the course of doing these analyses, a lot of smaller distributions were calculated
along the way. One of these is the distribution on any individual trading day that was
involved in the study. Now, one must understand that one day's trading yields only
about 3,000 data points (there were about 3,000 stocks in the database), so the result
ing curve is not going to be as smooth as the ones shown in Figure 38-1.
Nevertheless, some days could be interesting. For example, consider the day of the
mini-crash, Friday, April 14, 2000. The Dow-Jones Industrials were down 617 that
day; the S&P 500 index was down 83 points; and the NASDAQ-100 was


only
about 3,000 data points (there were about 3,000 stocks in the database), so the result
ing curve is not going to be as smooth as the ones shown in Figure 38-1.
Nevertheless, some days could be interesting. For example, consider the day of the
mini-crash, Friday, April 14, 2000. The Dow-Jones Industrials were down 617 that
day; the S&P 500 index was down 83 points; and the NASDAQ-100 was down 346.
Except for the Crash of 1987, these were the largest single-day declines in history.
The distribution graph is shown in Figure 38-2.
First of all, notice how heavily the distribution is skewed to the left; that agrees
with one's intuition that the distribution should be on the left when there is such a seri
ous down day as 4/14/2000. Also, notice that the leftmost data point- representing all
moves of -4.0 standard deviations and lower, shows that about 750 out of the 2,984
stocks had moves of that size! That is unbelievable, and it really points out just how
FIGURE 38-2.
Stock price distribution for 4/14/2000 - 2,984 Stocks in Study.
110
100
90
80
0 70 ,....
~ 60 C ::,
50 0
()
40
30
20
10
-4.0 -3.0 -2.0 -1.0 0.0 + 1.0 +2.0 _3.0 +4.0
Sigmas
794 Part VI: Measuring and Trading Volatility
FIGURE 38-3.
Stock price distribution, IBM, 7-year.
7
6
5
~ 4 :;:,
C: :::,
8 3
2
o----------------+---,.._,._""-¥-+-
-4.o -3.o -2.0 -1.0 o.o +1.0 +2.0 +3.o +4.o
Sigmas
dangerous naked puts and long stock on margin can be on days like this. No proba
bility calculator is going to give much likelihood to a day like this occurring, but it did
occur and it benefited those holding long puts greatly, while it seriously hurt others.
In addition to distributions for individual dates, distributions for individual stocks
were created for the time period in question. The graph for IBM, using data from the
same study as above (September 1993 to April 2000) is shown in Figure 38-3. In the
next graph, Figure 38-4, a longer price history of IBM is used to draw the distribution:
1987 to 2000. Both graphs depict 30-day movements in IBM.
FIGURE 38-4.
Actual stock price distribution, IBM, 13-year.
13
12
11
10
9
o a ,-
E 7
g 6
(.) 5
4
3
2
o -4.0 -3.0 -2.0 -1.0 0.0 + 1.0 +2.0 +3.0 +4.0
Sigmas
Chapter 38: The Distribution of Stock Prices 795
Figure 38-3 perhaps shows even more starkly how the bull market has affected
things over the last six-plus years. There are over 1,600 data points for IBM (i.e., daily
readings) in Figure 38-3, yet the whole distribution is skewed to the right. It appar
ently was able to move up quite easily throughout this time period. In fact, the worst
move that occurred was one move of -2.5 standard deviations, while there were
about ten moves of +4.0 standard deviations or more.
For a longer-term look at how IBM behaves, consider the longer-term distribu
tion of IBM prices, going back to March 1987, as shown in Figure 38-4.
From Figure 38-4, it's clear that this longer-term distribution conforms more
closely to the normal distribution in that it has a sort of symmetrical look, as opposed
to Figure 38-3, which is clearly biased to the right (upside).
These two graphs have implications for the big picture study shown in Figure
38-1. The database used for this study had data for most stocks only going back to
1993 (IBM is one of the exceptions); but if the broad study of all stocks were run
using data all the way back to 1987, it is certain that the "actual" price distribution
would be more evenly centered, as opposed to its justification to the right (upside).
That's because there would be more bearish periods in the longer study (1987, 1989,
and 1990 all had some rather nasty periods). Still, this doesn't detract from the basic
premise that stocks can move farther than the normal distribution would indicate.
WHAT THIS MEANS FOR OPTION TRADERS
The most obvious thing that an option trader can learn from these distributions and
studies is that buying options is probably a lot more feasible than conventional wisdom
would have you believe. The old thinking that selling an option is "best" because it
wastes away every day is false. In reality, when you have sold an option, you are exposed
to adverse price movements and adverse movements in implied volatility all during the
life of the option. The likelihood of those occurring is great, and they generally have
more influence on the price of the aption in the short run than does time decay.
You might ask, "But doesn't all the volatility in 1999 and 2000 just distort the
figures, making the big moves more likely than they ever were, and possibly ever will
be again?" The answer to that is a resounding, "Nol" The reason is that the current
20-day historical volatility was used on each day of the study in order to determine
how many standard deviations each stock moved. So, in 1999 and 2000, that histori
cal volatility was a high number and it therefore means that the stock would have had
to move a very long way to move four standard deviations. In 1993, however, when
the market was in the doldrums, historical volatility was low, and so a much smaller
794 Part VI: Measuring and Trading Volatility
FIGURE 38-3.
Stock price distribution, IBM, 7-year.
7
6
5
0 4
:g
::,
0 3 ()
2
0f-<--,.Jil,,J:.~------+---+---+----+--____;_,=!!:¥+-
-4.0 -3.0 -2.0 -1.0 0.0 + 1.0 +2.0 +3.0 +4.0
Sigmas
dangerous naked puts and long stock on margin can be on days like this. No proba
bility calculator is going to give much likelihood to a day like this occurring, but it did
occur and it benefited those holding long puts greatly, while it seriously hurt others.
In addition to distributions for individual dates, distributions for individual stocks
were created for the time period in question. The graph for IBM, using data from the
same study as above (September 1993 to April 2000) is shown in Figure 38-3. In the
next graph, Figure 38-4, a longer price history of IBM is used to draw the d


those holding long puts greatly, while it seriously hurt others.
In addition to distributions for individual dates, distributions for individual stocks
were created for the time period in question. The graph for IBM, using data from the
same study as above (September 1993 to April 2000) is shown in Figure 38-3. In the
next graph, Figure 38-4, a longer price history of IBM is used to draw the distribution:
1987 to 2000. Both graphs depict 30-day movements in IBM.
FIGURE 38-4.
Actual stock price distribution, IBM, 13-year.
13
12
11
10
9
0 8 ,...
E 7
5 6
() 5
4
3
2
1
o -4.0 -3.0 -2.0 -1.0 0.0 + 1.0 +2.0 +3.0 +4.0
Sigmas
Chapter 38: The Distribution of Stock Prices 195
Figure 38-3 perhaps shows even more starkly how the bull market has affected
things over the last six-plus years. There are over 1,600 data points for IBM (i.e., daily
readings) in Figure 38-3, yet the whole distribution is skewed to the right. It appar
ently was able to move up quite easily throughout this time period. In fact, the worst
move that occurred was one move of -2.5 standard deviations, while there were
about ten moves of +4.0 standard deviations or more.
For a longer-term look at how IBM behaves, consider the longer-term distribu
tion of IBM prices, going back to March 1987, as shown in Figure 38-4.
From Figure 38-4, it's clear that this longer-term distribution conforms more
closely to the normal distribution in that it has a sort of symmetrical look, as opposed
to Figure 38-3, which is clearly biased to the right (upside).
These two graphs have implications for the big picture study shown in Figure
38-1. The database used for this study had data for most stocks only going back to
1993 (IBM is one of the exceptions); but if the broad study of all stocks were run
using data all the way back to 1987, it is certain that the "actual" price distribution
would be more evenly centered, as opposed to its justification to the right (upside).
That's because there would be more bearish periods in the longer study (1987, 1989,
and 1990 all had some rather nasty periods). Still, this doesn't detract from the basic
premise that stocks can move farther than the normal distribution would indicate.
WHAT THIS MEANS FOR OPTION TRADERS
The most obvious thing that an option trader can learn from these distributions and
studies is that buying options is probably a lot more feasible than conventional wisdom
would have you believe. The old thinking that selling an option is "best" because it
wastes away every day is false. In reality, when you have sold an option, you are exposed
to adverse price movements and adverse movements in implied volatility all during the
life of the option. The likelihood of those occurring is great, and they generally have
more influence on the price of the option in the short run than does time decay.
You might ask, "But doesn't all the volatility in 1999 and 2000 just distort the
figures, making the big moves more likely than they ever were, and possibly ever will
be again?" The answer to that is a resounding, "Nol" The reason is that the current
20-day historical volatility was used on each day of the study in order to determine
how many standard deviations each stock moved. So, in 1999 and 2000, that histori
cal volatility was a high number and it therefore means that the stock would have had
to move a very long way to move four standard deviations. In 1993, however, when
the market was in the doldrums, historical volatility was low, and so a much smaller
796 Part VI: Measuring and Trading Volatility
move was needed to register a 4-standard deviation move. To see a specific example
of how this works in actual practice, look carefully at the chart of IBM in Figure 38-
4, the one that encompasses the crash of '87. Don't you think it's a little strange that
the chart doesn't show any moves of greater than minus 4.0 standard deviations? The
reason is that IBM's historical volatility had already increased so much in the days
preceding the crash day itself, that when IBM fell on the day of the crash, its move
was less than minus 4.0 standard deviations. (Actually, its one-day move was greater
than -4 standard deviations, but the 30-day move - which is what the graphs in Figure
38-3 and 38-4 depict - was not.)
STOCK PRICE DISTRIBUTION SUMMARY
One can say with a great deal of certainty that stocks do not conform to the normal
distribution. Actually, the normal distribution is a decent approximation of stock
price movement rrwst of the time, but it's these "outlying" results that can hurt any
one using it as a basis for a nonvolatility strategy.
Scientists working on chaos theo:ry have been trying to get a better handle on
this. An article in Scientific American magazine ("A Fractal Walk Down Wall Street,"
Februa:ry 1999 issue) met some criticism from followers of Elliot Wave theo:ry, in that
they claim the article's author is purporting to have "invented" things that R. N.
Elliott discovered years ago. I don't know about that, but I do know that the article
addresses these same points in more detail. In the article, the author points out that
chaos theo:ry was applied to the prediction of earthquakes. Essentially, it concluded
that earthquakes can't be predicted. Is this therefore a useless analysis? No, says the
author. It means that humans should concentrate on building stronger buildings that
can withstand the earthquakes, for no one can predict when they may occur. Relating
this to the option market, this means that one should concentrate on building strate
gies that can withstand the chaotic movements that occasionally occur, since chaotic
stock price behavior can't be predicted either.
It is important that option traders, above all people, understand the risks of
making too conservative an estimate of stock price movement. These risks are espe
cially great for the writer of an option (and that includes covered writers and spre


on building strate
gies that can withstand the chaotic movements that occasionally occur, since chaotic
stock price behavior can't be predicted either.
It is important that option traders, above all people, understand the risks of
making too conservative an estimate of stock price movement. These risks are espe
cially great for the writer of an option (and that includes covered writers and spread
ers, who may be giving away too much upside by writing a call against long stock or
long calls). By quantifying past stock price movements, as has been done in this chap
ter, my aim is to convince you that "conventional" assumptions are not good enough
for your analyses. This doesn't mean that it's okay to buy overpriced options just
because stocks can make large moves with a greater frequency than most option
Chapter 38: The Distribution of Stock Prices 797
models predict; but it certainly means that the buyer of underpriced options stands
to benefit in a couple of ways. Conversely, an option seller must certainly concentrate
his efforts where options are expensive, and even then should be acutely aware that
he may experience larger-than-expected stock price movements while the option
position is in place.
So what does this mean for option strategies? On the surface, it means that if
one uses the normal (or lognormal) distribution for estimating the probability of a
strategy's success, he may get a big move in the stock that he didn't originally view as
possible. If one were long straddles, that's great. However, if he is short naked
options, then there could be a nasty surprise in store. That's one reason why extreme
caution should be used regarding selling naked options on stocks; they can make
moves of this sort too often. At least with indices, such moves are far less frequent,
although the Dow drop of over 550 points in October 1997 was a move of seven stan
dard deviations, and the crash of '87 was about a 16-standard deviation move - which
Professor Mark Rubenstein of the University of California at Berkeley says was some
thing that should occur about once in ten times the life of our current universe! That's
according to lognormal distribution, of course, which we know understates things
somewhat, but it's still a big number under any distribution.
There are two approaches that one can take, then, regarding option strategies.
One is to invent another method for estimating stock price distributions. Suffice it to
say that that is not an easy task, or someone would have made it well-known already.
There have been many attempts, including some in which a large history of stock
price movements is observed and then a distribution is fitted to them. The problem
with accounting for these occasional large price moves is that it is perhaps an even
more grievous error to overestimate the probabilities of such moves than to underes
timate them.
The second approach is to continue to use the normal distribution, because it's
fast and accessible in a lot of places. Then, either rely on option buying strategies
( straddles, for example) where implied volatility appears to be low - knowing that you
have a chance at better results than the statistics might indicate - or adjust your cal
culations mentally for these large potential movements if you are using option selling
strategies.
THE PRICING OF OPTIONS
The extreme movements of the fat tail distribution should be figured into the pricing
of an option, but they really are not, at least not by most models. The Black-Scholes
798 Part VI: Measuring and Trading Volatility
model, for example, uses a lognormal distribution. Personally, this author believes
that the Black-Scholes model is an excellent tool for analyzing options and option
strategies, but one must understand that it may not be affording enough probability
to large moves by the underlying.
Does this mean that most options are underpriced, since traders and market
makers are using the Black-Scholes model (or similar models) to price them?
Without getting too technical, the answer is that yes, some options - particularly out
of-the-money options - are probably underpriced. However, one must understand
that it is still a relatively rare occurrence to experience one of these big moves - ifs
just not as rare as the lognormal distribution would indicate. So, an out-of-the-money
option might be slightly underpriced, but often not enough to make any real differ
ence.
In fact, futures options in grains, gold, oil, and other markets that often experi
ence large and sudden rallies display a distinct volatility skew. That is, out-of-the-money
call options trade at significantly higher implied volatilities than do at-the-money
options. Ironically, there is far less chance of one of these hyper-standard-deviation
moves occurring in commodities than there is in stocks, at least if history is a guide. So,
the fact that some out-of-the-money futures options are expensive is probably an incor
rect overadjustment for the possibility of large moves.
THE PROBABILITY OF STOCK PRICE MOVEMENT
The distribution information introduced in this chapter can be incorporated into
somewhat rigorous methods of determining probabilities. That is, one can attempt to
assess the chances of a stock, futures contract, or index moving by a given distance,
and those chances can incorporate the fat tails or other non-lognormal behavior of
prices.
The software that calculates such probabilities is typically named a "probability
calculator." There are many such software programs available in the marketplace.
They range from free calculators to completely overpriced ones selling for more than
$1,000. In reality, high-level probability calculation software can be created by some
one with a good understanding of statistics, or a program can be purchased for a
rather nominal fee - perhaps $100 or so.
Before getting into these various methods of probability estimation, it should b


e in the marketplace.
They range from free calculators to completely overpriced ones selling for more than
$1,000. In reality, high-level probability calculation software can be created by some
one with a good understanding of statistics, or a program can be purchased for a
rather nominal fee - perhaps $100 or so.
Before getting into these various methods of probability estimation, it should be
noted that all of them require the trader to input a volatility estimate. There are only
a few other inputs, usually the stock price, target price(s), and length of time of the
study. The volatility one inputs is, of course, an estimate of future volatility - some-
Chapter 38: The Distribution of Stock Prices 799
thing that cannot be predicted with certainty. Nevertheless, any probability calcula
tor requires this input. So, one must understand that the results one obtains from any
of these probability calculators is an estimate of what might happen. It should not be
relied on as "gospel."
Additionally, probability calculators make a second assumption: that the volatil
ity one inputs will remain constant over the entire length of the study. We know this
is incorrect, for volatility can change daily. However, there really isn't a good way of
estimating how volatility might change in the course of the study, so we are pretty
much forced to live with this incorrect assumption as well.
There is no certain way to mitigate these volatility "problems" as far as the prob
ability calculator is concerned, but one helpful technique is to bias the volatility pro
jection against your objectives. That is, be overly conservative in your volatility pro
jections. If things tum out to be better than you estimated, fine. However, at least
you won't be overstating things initially. An example may help to demonstrate this
technique.
Example: Suppose that a trader is considering buying a straddle on XYZ. The five
month straddle is selling for a price of 8, with the stock currently trading near 40. A
probability calculator will help him to determine the chances that XYZ can rise to 48
or fall to 32 (the break-even points) prior to the options' expiration. However, the
probability calculator's answer will depend heavily on the volatility estimate that the
trader plugs into the probability calculator. Suppose that the following information is
know about the historical volatility of XYZ:
l 0-day historical volatility:
20-day historical volatility:
50-day historical volatility:
l 00-day historical volatility
22%
20%
28%
33%
Which volatility should the trader use? Should he choose the 100-day historical
volatility since this is a five-month straddle, which encompasses just about 100 trad
ing days until expiration? Should he use the 20-day historical volatility, since that is
the "generally accepted" measure that most traders refer to? Should he calculate a
historical volatility based exactly on the number of days until expiration and use that?
To be most conservative, none of those answers is right, at least not for the right
reasons. Since one is buying options in this strategy, he should use the lowest of the
above historical volatility measures as his volatility estimate. By doing so, he is taking
a conservative approach. If the straddle buy looks good under this conservative
assumption, then he can feel fairly certain that he has not overstated the possibilities
800 Part VI: Measuring and Trading Volatility
of success. If it turns out that volatility is higher during the life of the position, that
will be an added benefit to this position consisting of long options. So, in this exam
ple, he should use the 20-day historical volatility because it is the lowest of the four
choices that he has.
Similarly, if one is considering the sale of options or is taking a position with a
negative vega ( one that will be harmed if volatility increases), then he should use the
highest historical volatility when making his probability projections. By so doing, he
is again being conservative. If the strategy in question still looks good, even under an
assumption of high volatility, then he can figure that he won't be unpleasantly sur
prised by a higher volatility during the life of the position.
There have been times when a 100-day lookback period was not sufficient for
determining historical volatility. That is, the underlying has been performing in an
erratic or unusual manner for over 100 days. In reality, its true nature is not described
by its movements over the past 100 days. Some might say that 100 days is not enough
time to determine the historical volatility in any case, although most of the time the
four volatility measures shown above will be a sufficient guide for volatility.
When a longer lookback period is required, there is another method that can be
used: Go back in a historical database of prices for the underlying and compute the
20-day, 50-day, and l 00-day historic volatilities for all the time periods in the data
base, or at least during a fairly large segment of the past prices. Then use the medi
an of those calculations for your volatility estimates.
Example: XYZ has been behaving erratically for several months, due to overall mar
ket volatility being high as well as to a series of chaotic news events that have been
affecting XYZ. A trader wants to trade XYZ's options, but needs a good estimate of
the "true" volatility potential of XYZ, for he thinks that the news events are out of the
way now. At the current time, the historical volatility readings are:
20-day historical: 130%
50-day historical l 00%
100-day historical 80%
However, when the trader looks farther back in XYZ's trading history, he sees
that it is not normally this volatile. Since he suspects that XYZ's recent trading histo
ry is not typical of its true long-term performance, what volatility should he use in
either an option model or a probability calculator?
Rather than just using the


istorical: 130%
50-day historical l 00%
100-day historical 80%
However, when the trader looks farther back in XYZ's trading history, he sees
that it is not normally this volatile. Since he suspects that XYZ's recent trading histo
ry is not typical of its true long-term performance, what volatility should he use in
either an option model or a probability calculator?
Rather than just using the maximum or minimum of the above three numbers
(depending on whether one is buying or selling options), the trader decides to look
Chapter 38: The Distribution of Stock Prices 801
back over the last 1,000 trading days for XYZ. A 100-day historical volatility can be
computed, using 100 consecutive trading days of data, for 901 of those days (begin
ning with the 100th day and continuing through the l,000th day, which is presumably
the current trading day). Admittedly, these are not completely unique time periods;
there would only be ten non-overlapping (independent) consecutive 100-day periods
in 1,000 days of data. However, let's assume that the 901 periods are used. One can
then arrive at a distribution of 100-day historical volatilities. Suppose it looks some
thing like this:
Percentile 100-Day Historical
oth 34%
10th 37%
20th 43%
30th 45%
40th 46%
50th 48%
60th 51%
70th 58%
aoth 67%
90th 75%
1 ooth 81%
In other words, the 901 historical volatilities (100 days in each) are sorted and then
the percentiles are determined. The above table is just a snapshot of where the per
centiles lie. The range of those 901 volatilities is from 34% on the low side to 81 % on
the high side. Notice also that there is a very flat grouping from about the 20th per
centile to the 60th percentile: The 100-day historical volatility was between 43% and
51 % over that entire range. The median of the above figures is 48% - the 100-day
volatility at the 50th percentile.
Referring to the early part of this example, the current 100-day historical is
80%, a very high reading in comparison to what the measures were over the past
1,000 days, and certainly much higher than the median of 48%.
One could perform similar analyses on the 1,000 days of historical data to deter
mine where the 10-day, 20-day, and 50-day historical volatilities were over that time.
Those, too, could be sorted and arranged in percentile format, using the 50% per
centile (median) as a good estimate of volatility. After such computations, the trader
might then have this information:
802 Part VI: Measuring and Trading Volatility
Using 1,000 days of data:
Median 100-day historical volatility: 48%
Median 50-day historical volatility: 49%
Median 20-day historical volatility: 52%
Median 10-day historical volatility: 49%
If these were all the data that one had, then he would probably use a volatility esti
mate of 48% or so in his option models or probability calculators. Of course, this is
starkly different from the current levels of historical volatility (shown at the begin
ning of this example). So, one must be careful in assessing whether he expects the
stock to perform more in line with its longer-term (1,000 trading days) characteristics
or if there is some reason to think that the stock's behavior patterns have changed and
the higher, more recent volatilities should be used.
The pertinent volatilities to consider, then, in a strategy analysis are the medi
ans as well as the current figures. If the trader were going to be buying options in his
strategy, should he use the minimum of the volatilities shown, 48%? Probably.
However, if he's a seller of options, should he use the maximum, 130%? That might
be a little too much of a penalty, but at least he would feel safe that if his volatility
selling position had a positive expected return with that high a volatility projection,
then it must truly be an attractive position.
In an analysis like that shown in this example, there is nothing magical about
using 1,000 trading days. Perhaps something like 600 trading days would be better.
The idea is to use enough trading days to bring in some historic data to counterbal
ance the recent, erratic behavior of the stock.
Among other things, this example also shows that volatilities are unstable, no
matter how much work and mathematics one puts into calculating them. Therefore,
they are at best a fragile estimate of what might happen in the future. Still, it's the
best guess that one can make. The trader should realize, though, that when volatili
ties are this disparate when comparing recent and more distant activity, the results of
any mathematical projections using those volatilities should not be relied upon too
heavily. Those results will be just as tenuous as the volatility projections themselves.
Of course, in any case, the actual volatility that occurs while the position is in place
may be even more unfavorable than the one the trader used in his initial analysis. There
is nothing that one can do about that. But if you choose what appears to be a somewhat
unfavorable volatility, and the position still looks good under those assumptions, then it
is likely that the trader will be pleasantly surprised more often than not - that actual
volatility during the life of the position will tend to be more in his favor than not.
Chapter 38: The Distribution of Stock Prices 803
In a recent chapter, the various methods of trying to predict volatility were out
lined, using either historical volatility, implied volatility, a moving average of either of
those, or even GARCH volatility. None of these will predict with certainty what is
going to happen in the future. Hence, the prediction of volatility is necessarily vague
at best.
In addition to the vagaries of estimating volatility, the probability calculators will
return an answer that represents the probability of something happening "in the long
run." That is, if the same scenario were to arise many, many times, the answer is rel
evant to how many times the s


ty what is
going to happen in the future. Hence, the prediction of volatility is necessarily vague
at best.
In addition to the vagaries of estimating volatility, the probability calculators will
return an answer that represents the probability of something happening "in the long
run." That is, if the same scenario were to arise many, many times, the answer is rel
evant to how many times the stock would move to the indicated target price. This is
small solace if one happens to be caught in the vortex of the Crash of '87, for exam
ple. So, just remember that these probability calculators are tools that can help you
in assessing the relative risks of similar positions ( evaluating various naked option
sales, say), but the resulting stock movement in any one case can be quite different
from what any probability calculator describes as the chances of that move actually
happening.
THE ENDPOINT CALCULATION
The following paragraphs describe how the various probability calculation mecha
nisms work. The simplest and most straightforward probability calculation has
already been presented in Chapter 28 on mathematical applications. It was included
in the section on "expected returns" in that chapter. The formula is presented here
again, for completeness.
The formula gives the probability of a stock, which is currently at price p, being
below some other price, q, at the end of the time period. The lognormal distribution
is assumed.
Probability of stock being below price q at end of time period, t
where
N = cumulative normal distribution
p = current price of the stock
q = price in question
ln = natural logarithm for the time period in question
804 Part VI: Measuring and Trading Volatility
If one is interested in computing the probability of the stock being above the
given price, the formula is
P (above) = 1 - P (below)
In the above formula, Vt = v✓t where t is time to expiration in years and v is
annual volatility, as usual.
This formula is quite elementary for predictive purposes, and it is used by
many traders. This calculator can be found for free at the Web site www.option
strategist.com. Its main problem is that it gives the probability of the stock being
above or below the target price at the end of the time period, t. That's not a totally
realistic way of approaching probability analysis. Most option traders are very con
cerned with what happens to their positions during the life of the option, not just at
expiration.
Example: suppose a trader is a seller of naked put options. He sells $OEX October
550 puts naked, with $OEX currently trading at 600. He would not normally just walk
away from this position until October expiration, because of the large risk involved
with the sale of a naked option. There are essentially three scenarios that can occur:
1. $OEX might never fall to 550 by expiration. In this case, he would have a
very comfortable trade that was never in jeopardy, and the options would
expire worthless.
2. $OEX might fall below 550 and remain there until expiration. In this case,
he would surely have a loss unless $OEX were just a tiny bit below 550.
3. $OEX might fall below 550 at some time between when the trade was estab
lished and when expiration occurred, but then subsequently rally back above
550 by the time expiration arrived.
An experienced option trader would almost certainly adjust if scenario 3 arose,
in order to prevent large losses from occurring. He might roll his naked puts down
and out to another strike, or he might just close them out altogether. However, it is
unlikely that he would do nothing.
The simple probability calculator formula shown above does not take into
account the trader's third scenario. Since it is only concerned with where the stock is
at expiration of the options, only scenarios 1 and 2 apply to it. Hence the usage of this
simple calculator is not really descriptive of what might happen to a trade during its
lifetime.
Let's assign some numbers to the above trade, so that you might see the differ
ence. Suppose that the volatility estimate is 25%, there are 30 days until expiration,
and the prices are as stated in the previous example: $OEX is at 600, and the strik-
Chapter 38: The Distribution of Stock Prices 805
ing price of the naked put being sold is 550. The resulting probabilities might be
something like this:
Scenario Actual Probability of Occurrence
1 . $OEX never falls below 550
2. $OEX falls below 550 and remains there
3. $OEX falls below 550 but rallies later
67%
19%
14%
The probabilities stated above are the "real" probabilities of the three various
scenarios occurring. However, if one were using the simple probability calculator
presented above, he would only have the following information:
Probability of $OEX being above 550 at expiration: 81 %
Probability of $OEX being below 550 at expiration: 19%
So, with the simple calculator, it looks like there's an 81 % chance of a worry-free
trade. Just sit back and relax and let the option expire worthless. However, in real
life - as shown by the previous set of probabilities, there's only a 67% chance of a
worry-free trade. The difference - the other 14% - is the probability of the third
scenario occurring ($OEX falls below 550, but rallies back above it by expiration).
The simple probability calculator doesn't account for that scenario at all.
Hence, most serious traders don't use the simple model. Does that mean it's not
useful at all? No, it is certainly viable as a comparative tool; for example, to compare
the chances of the $OEX put expiring worthless versus those of another put sale
being considered, perhaps something in a stock option. However, better analyses can
be undertaken.
Before leaving the scenario of the simple probability calculator, one more point
should be made. It has been mentioned earlier in this book that the delta of an option
is actually a fairly good estimate of the probability of the


f the $OEX put expiring worthless versus those of another put sale
being considered, perhaps something in a stock option. However, better analyses can
be undertaken.
Before leaving the scenario of the simple probability calculator, one more point
should be made. It has been mentioned earlier in this book that the delta of an option
is actually a fairly good estimate of the probability of the option being in-the-money
at its expiration date. Thus, the delta and the simple endpoint probability calculator
shown above attempt to convey the same information to a trader. In reality, because
of the fact that implied volatility might be different for various strikes (a volatility
skew), especially in index options, the delta of the option might not agree exactly with
the probability calculator. Even so, the delta is a quick and dirty way of estimating the
probability of the stock being above the strike price (in the case of call options) or
below the strike price (in the case of put options) at expiration.
THE nEVER" CALCULATOR
Having seen the frailties of the endpoint calculator, the next step is to try to design a
calculator that can estimate the probability of the stock ever hitting the target price(s)
806 Part VI: Measuring and 1iading Vo/atillty
at any time during the life of the probability study, usually the life of an option. It
turns out that there are a couple of ways to approach this problem. One is with a
Monte Carlo analysis, whereby one lets a computer run a large number of random
ly-generated scenarios (say, 100,000 or so) and counts the number of times the tar
get price is hit. A Monte Carlo analysis is a completely valid way of estimating the
probability of an event, but it is a somewhat complicated approach.
In reality, there is a way to create a single formula that can estimate the "ever"
probability, although it is not any easy task either. In the following discussion, I am
borrowing liberally from correspondence with Dr. Stewart Mayhew, Professor of
Mathematics at the University of Georgia. For proprietary reasons, the exact formu
la is not given here, but the following description should be sufficient for a mathe
matics or statistics major to encode it. If one is not interested in implementing the
actual formula, the calculation can be obtained through programs sold by McMillan
Analysis Corp. at www.optionstrategist.com.
This discussion is quite technical, so readers not interested in the description of
the mathematics can skip the next paragraph and instead move ahead to the next sec
tion on Monte Carlo studies.
These are the steps necessary in determining the formula for the "ever" proba
bility of a stock hitting an upside target at any time during its life. First, make the
assumption that stock prices behave randomly, and perform at the risk-free rate, r.
Mathematicians call random behavior "Brownian motion." There are a number of
formulae available in statistics books regarding Brownian motion. If one is to esti
mate the probability of reaching a maximum (upside target) point, what is needed is
the known formula for the cumulative density function (CDP) for a running maxi
mum of a Brownian motion. In that formula, it is necessary to use the lognormal
function to describe the upside target. Thus, instead of using the actual target price
in the CDF formula, one substitutes ln( qlp ), where q is the target price and p is the
current stock price.
The "ever" probability calculator provides much more useful information to a
trader of options. Not only does a naked option seller have a much more realistic esti
mate of the probability that he's going to have to make an adjustment during the life
of an option, but the option buyer can find the information useful as well. For exam
ple, if one is buying an option at a price of 10, say, then he could use the "ever" prob
ability calculator to estimate the chances of the stock trading 10 points above the
striking price at any time during the life of the option. That is, what are the chances
that the option is going to at least break even? The option buyer can, cf course, deter
mine other things too, such as the probability that the option doubles in price ( or
reaches some other return on investment, such as he might deem appropriate for his
analysis).
Chapter 38: The Distribution of Stock Prices 807
THE MONTE CARLO PROBABILITY CALCULATOR
Up to this point, the calculators we have discussed are subject to the limitations
described earlier - mainly, that they rely heavily on one's volatility estimate, that they
assume the volatility will remain constant over time, and that they assume a lognor
mal distribution. The early part of this chapter was spent explaining that the lognor
mal distribution is not the real distribution that stock prices adhere to. So, what we'd
like to see in a probability calculator is one that could adjust for various volatility sce
narios as time passed and one in which the assumed distribution of stock prices was
not lognormal.
When one starts to make these sorts of assumptions, I do not believe there is a
single formula that can be derived for the probability calculations. Rather, what is
known as a Monte Carlo simulation must be undertaken. Essentially, one "tells" the
computer what he is trying to simulate. It could be any number of things in real life,
perhaps the rocket engine components in a NASA space shuttle, or the operation of
an internal combustion engine, or the movement of a stock. As long as the process
can be described, it can be simulated by a computer. Then, the computer can run a
large number of those simulations to determine the answers to such things as "What
is the failure rate of the NASA engine components," or "How long can the internal
combustion engine go without an oil change," or "What is the probability of the stock
trading at a certain target price?" The Monte Carlo simulation technique can be
thought of as letting the co


puter. Then, the computer can run a
large number of those simulations to determine the answers to such things as "What
is the failure rate of the NASA engine components," or "How long can the internal
combustion engine go without an oil change," or "What is the probability of the stock
trading at a certain target price?" The Monte Carlo simulation technique can be
thought of as letting the computer run through the simulation a lot of times and
counting how many times a certain outcome occurs. If the number of trials (simula
tions) is large enough and the model is good enough, then the resulting count divid
ed by the number of trials undertaken is a good probability estimate of the said event
occurring. The reason one runs a lot of trials is that over a large number of trials, the
frequency with which an event occurs will approximate the actual probability of its
occurrence for a single trial - the single trial being your trade, for example.
The next three paragraphs describe the general process necessary for con
structing a stock probability calculator using a Monte Carlo simulation. Again, this is
fairly technical, so if the reader is not interested in the background behind the math
ematics, then skip ahead three paragraphs. In the case of a stock probability calcula
tor, the Monte Carlo simulation can be undertaken as follows.
We know what the distribution of stock prices looks like. The fat tails can be
built into the distribution if one wants to simulate real life. See Figure 38-1 for both
the lognormal distribution and the actual distribution. It's a simple matter to tell the
computer this information. For example, recall that 2.5 million points went into mak
ing up Figure 38-1. In the actual distribution in Figure 38-1, about 92,000 (or 3.68%)
808 Part VI: Measuring and Trading Volatility
of them resulted in the stock being unchanged. Also, only about 2,500 or them, or
1110th of one percent, resulted in a move of-4.0 standard deviations or more. Those
percentages, along with all of the others, would be built into the computer, so that
the total distribution accounts for 100% of all possible stock movements.
Then, we tell the computer to allow a stock to move randomly in accordance
with whatever volatility the user has input. So, there would be a fairly large proba
bility that it wouldn't move very far on a given day, and a very small probability that
it would move three or more standard deviations. Of course, with the fat tail distri
bution, there would be a larger probability of a movement of three or more standard
deviations than there would be with the regular lognormal distribution. The Monte
Carlo simulation progresses through the given number of trading days, moving the
stock cumulatively as time passes. If the stock hits the break-even price, that partic
ular simulation can be terminated and the next one begun. At the end of all the tri
als (100,000 perhaps), the number in which the upside target was touched is divided
by the total number of trials to give the probability estimate.
Is it really worth all this extra trouble to evaluate these more complicated prob
ability distributions? It seems so. Consider the following example:
Example: Suppose that a trader is considering selling naked puts on XYZ stock,
which is currently trading at a price of 80. He wants to sell the November 60 puts,
which expire in two months. Although XYZ is a fairly volatile stock, he feels that he
wouldn't mind owning it if it were put to him. However, he would like to see the puts
expire worthless. Suppose the following information is available to him via the vari
ous probability calculators:
Simple "end point" probability of XYZ < 60 at expiration: 10%
Probability that XYZ ever trades < 60 (using the lognormal distribution) 20%
Probability that XYZ ever trades < 60 (using the fat tail distribution): 22%
If the chances of the put never needing attention were truly only 10%, this trader
would probably sell the puts naked and feel quite comfortable that he had a trade
that he wouldn't have to worry too much about later on. However, if the true proba
bility that the put will need attention is 22%, then he might not take the trade. Many
naked option sellers try to sell options that have only probabilities of 15% or less of
potentially becoming troublesome.
Hence, the choice of which probability calculation he uses can make a differ
ence in whether or not a trade is established.
Other strategies lend themselves quite well to probability analysis as well.
Credit spreaders - sellers of out-of-the-money put spreads - usually attempt to quan
tify the probability of having to take defensive action. Any action to adjust or remove
Chapter 38: The Distribution of Stock Prices 809
a deeply out-of-the-money put credit spread usually destroys most or all of its prof
itability, so an accurate initial assessment of the probabilities of having to make such
an adjustment is important.
Option buyers, too, would benefit from the use of a more accurate probability
estimate. This is especially true for neutral strategies, such as straddle or strangle
buying, when the trader is interested in the chances of the stock being able to move
far enough to hit one or the other of the straddle's break-even points at some time
during the life of the straddle.
The Monte Carlo probability calculation can be expanded to include other sorts
of distributions. In the world of statistics, there are many distributions that define ran
dom patterns. The lognormal distribution is but one of them (although it is the one
that most closely follows stock prices movements in general). Also, there is a school of
thought that says that each stock's individual price distribution patterns should be ana
lyzed when looking at strategies on that stock, as opposed to using a general stock
price distribution accumulated over the entire market. There is much debate about
that, because an individual st


lthough it is the one
that most closely follows stock prices movements in general). Also, there is a school of
thought that says that each stock's individual price distribution patterns should be ana
lyzed when looking at strategies on that stock, as opposed to using a general stock
price distribution accumulated over the entire market. There is much debate about
that, because an individual stock's trading pattern can change abruptly just consider
any of the Internet stocks in the late 1990s and early 2000s. Thus, a probability esti
mate based on a single stock's behavior, even if that behavior extends back several
years, might be too unreliable a statistic upon which to base a probability estimate.
In summary, then, one should use a probability calculator before taking an
option position, even an outright option buy. Perhaps straight stock traders should
use a probability calcutor as well. In doing so, though, one should be aware of the
limitations of the estimate: It is heavily biased by the volatility estimate that is input
and by the assumption of what distribution the underlying instrument will adhere to
during the life of the position. While neither of those limitations can be overcome
completely, one can mitigate the problems by using a conservative volatility estimate.
Also, he can look at the results of the probability calculation under several distribu
tions (perhaps lognormal, fat tail, and the distribution using only the past price
behavior of the underlying instrument in question) and see how they differ. In that
case, he would at least have a feeling for what could happen during the life of the
option position.
EXPECTED RETURN
The concept of expected return was described in the chapter on mathematical appli
cations. In short, expected return is a position's expected profit divided by its invest
ment ( or expected investment if the investment varies with stock price, as in a naked
option position or a futures position). The crucial component, though, is expected
profit.
810 Part VI: Measuring and Trading Volatility
Expected profit is computed by calculating the profitability of a position at a
certain stock price times the probability of the stock being at that price, and summing
that multiple over all possible stock prices. When the concept was first introduced,
the "probability of the stock being at that price" was given as what we now know is
the "endpoint" probability. In reality, a much better measure of the expected profit
of a position can be obtained by using one of the more advanced probability estima
tion models presented above.
In generalized expected return studies done using the fat tails Monte Carlo sim
ulation, certain general conclusions can be drawn about some strategies.
• A bull spread is an inferior strategy when the options are fairly priced, no matter
which distribution is assumed. This more or less agrees with observations that
have been made previously regarding the disappointments that traders often
encounter when using vertical spreads.
• While covered writing might seem superior to stock ownership under the log
normal distribution, the two are about equal under a fat tail distribution.
• Most startling, though, is the fact that option buying strategies fare much, much
better under a fat tail distribution than a lognormal one. This most clearly
demonstrates the "power" of the fat tail distribution: A limited-risk investment
with unlimited profit potential can be expected to perform very well if the fat tails
are allowed for.
Using the lognormal distribution more or less represents the conventional wisdom
regarding option strategies - the one that many brokers promote: "Don't buy options,
don't mess with spreads, either buy stocks or do covered call writes." The fat tail dis
tribution column stands much of that advice on its head. In real life (as demonstrat
ed by the fat tail distribution), strategies with limited profit potential and unlimited
or large risk potential are inferior strategies.
One should be aware that the phrase "expected return" is used in many quasi
sophisticated option analyses (and even in analyses not using options). Many
investors accept these "returns" on blind faith, figuring that if they're generated by a
computer, they must be correct. In reality, they may be not be representative, even
for comparisons.
SUMMARY
This chapter has demonstrated that probability analysis is an inexact science, because
markets behave in ways that are very difficult to describe mathematically. However,
probability analysis is also necessary for the option strategist; without it he would be
Chapter 38: The Distribution of Stock Prices 811
in the dark as to the likelihood of profitable outcomes for his strategy. Overall, in a
diversified set of positions, the option strategist should use the fat tail distribution in
a Monte Carlo simulation to estimate probabilities. However, if that is not available,
he can use the normal or lognormal distribution with the proviso that he understands
it is not "gospel." He should require ve:ry stringent criteria on any strategies that are
antivolatility strategies, such as naked option writing of stock options, for there is a
greater than normal chance of a large move by the underlying, especially if the
underlying is stock.
The sophisticated trader may want to view his probabilities in the light of more
than one proposed distribution of prices. Of course, this type of analysis ( using sev
eral distributions) puts the onus on the investor to choose the distribution that he
wants to use in order to analyze his investment. However, such an approach should
be extremely illustrative in that he can compare returns from different strategies and
have a reasonable expectation as to which ones might perform the best under differ
ent market conditions.
CHAPTER 39
Volatility Trading Techniques
The previous three chapters laid the foundation for volatility tra


ion that he
wants to use in order to analyze his investment. However, such an approach should
be extremely illustrative in that he can compare returns from different strategies and
have a reasonable expectation as to which ones might perform the best under differ
ent market conditions.
CHAPTER 39
Volatility Trading Techniques
The previous three chapters laid the foundation for volatility trading. In this chapter,
the actual application of the technique will be described. It should be understood
that volatility trading is both an art and a science. It's a science to the extent that one
must be rigorous about determining historical volatility or implied volatility, calculat
ing probabilities, and so forth. However, given the vagaries of those measurements
that were described in some detail in the previous chapters, volatility trading is also
something of an art. Just as two fundamental analysts with the same information
regarding earnings, sales projection, and so on might have two different opinions
about a stock's fortunes, so also can two volatility traders disagree about the potential
for movement in a stock.
However, volatility traders do agree on the approach. It is based on comparing
today's implied volatility with what one expects volatility to do in the future. As noted
previously, one's expectations for volatility might be based on volatility charts, pat
terns of historical volatility and implied volatility, or something as complicated as a
GAR CH forecasting model. None of them guarantees success. However, we do know
that volatility tends to trade in a range in the long run. Therefore, the approach that
traders agree upon is this: If implied volatility is "low," buy it. If it's "high," sell it with
caution. So simple: Buy low, sell high (not necessarily in that order). The theory
behind volatility trading is that it's easier to buy low and sell high (or at least to deter
mine what's "low" and "high") when one is speaking about volatility, than it is to do
the same thing when one is talking about stock prices.
Most of the time, implied volatility will not be significantly high or low on any
particular stock, futures contract, or index. Therefore, the volatility trader will have
little interest in most stocks on any given day. This is especially true of the big-cap
stocks, the ones whose options are most heavily traded. There are so many traders
812
Chapter 39: Volatility Trading Techniques 813
watching the situation for those stocks that they will rarely let volatility get to the
extremes that one would consider "too high" or "too low." Yet, with the large num
ber of optionable stocks, futures, and indices that exist, there are always some that
are out of line, and that's where the independent volatility trader will concentrate
his efforts.
Once a volatility extreme has been uncovered, there are different methods of
trading it. Some traders - market-makers and short-term traders - are just looking
for very fleeting trades, and expect volatility to fall back into line quickly after an
aberrant move. Others prefer more of a position traders' approach: attempting to
determine volatility extremes that are so far out of line with accepted norms that it
will probably take some time to move back into line. Obviously, the trader's own sit
uation will dictate, to a certain extent, which strategy he pursues. Things such as
commission rates, capital requirements, and risk tolerance will determine whether
one is more of a short-term trader or a position trader. The techniques to be
described in this chapter apply to both methods, although the emphasis will be on
position trading.
TWO WAYS VOLATILITY PREDICTIONS CAN BE WRONG
When traders determine the implied volatility of the options on any particular under
lying instrument, they may generally be correct in their predictions; that is, implied
volatility will actually be a fairly good estimate of forthcoming volatility. However,
when they're wrong, they can actually be wrong in two ways: either in the outright
prediction of volatility or in the path of their volatility predictions. Let's discuss both.
When they're wrong about the absolute level of volatility, that merely means that
implied volatility is either "too low" or "too high." In retrospect, one could only make
that assessment, of course, after having seen what actual volatility turned out to be
over the life of the option. The second way they could be wrong is by making the
implied volatility on some of the options on a particular underlying instrument much
cheaper or more expensive than other options on that same underlying instrument.
This is called a volatility skew and it is usually an incorrect prediction about the way
the underlying will perform during the life of the options.
The rest of this chapter will be divided into two main parts, then. The first part
will deal with volatility from the viewpoint of the absolute level of implied volatility
being "wrong" (which we'll call "trading the volatility prediction"), and the second
part will deal with trading the volatility skew.
814 Part VI: Measuring and Trading Volatility
TRADING THE VOLATILITY PREDICTION
The volatility trader must have some way of determining when implied volatility is
sufficiently out of line that it warrants a trade. Then he must decide what trade to
establish. Furthermore, as with any strategy- especially option strategies - follow-up
action is important too. We will not be introducing any new strategies, per se, in this
chapter. Those strategies have already been laid out in the previous chapters of this
book. However, we will briefly review important points about those strategies and
their follow-up actions where it is appropriate.
First, one must try to find situations in which implied volatility is out of line.
That is not the end of the analysis, though. After that, one needs to do some proba
bility work and needs to see how the und


ve already been laid out in the previous chapters of this
book. However, we will briefly review important points about those strategies and
their follow-up actions where it is appropriate.
First, one must try to find situations in which implied volatility is out of line.
That is not the end of the analysis, though. After that, one needs to do some proba
bility work and needs to see how the underlying has behaved in the past. Other fine
tuning measures are often useful, too. These will all be described in this chapter.
DETERMINING WHEN VOLATILITY IS OUT OF LINE
There is much disagreement among volatility traders regarding the best method to use
for determining if implied volatility is "out ofline." Most favor comparing implied with
historical volatility. However, it was shown two chapters ago that implied volatility is
not necessarily a good predictor of historical volatility. Yet this approach cannot be dis
carded; however it must be used judiciously. Another approach is to compare today's
implied volatility with where it has been in the past. This concept relies heavily on the
concept of the percentile of implied volatility. Finally, there is the approach of trying
to "read" the charts of implied and historical volatility. This is actually something akin
to what GARCH tries to do, but on a short-term horizon. So the approaches are:
1. Compare implied volatility to its own past levels (percentile approach).
2. Compare implied volatility to historical volatility.
3. Interpret the chart of volatility.
In addition, we will examine two lesser-used methods: comparing current levels of
historical volatility to past measures of historical volatility, and finally, using only a
probability calculator and trading the situation that has the best probabilities of
success.
THE PERCENTILE APPROACH
In this author's opinion, there is much merit in the percentile approach. When one says
that volatility tends to trade in a range, which is the basic premise behind volatility trad-
Chapter 39: Volatility Trading Techniques 815
ing, he is generally talking about implied volatility. Thus, it makes sense to know where
implied volatility is within the range of the past readings of implied volatility. If volatil
ity is low with respect to where it usually trades, then we can say the options are cheap.
Conversely, if it's high with respect to those past values, then we can say the options are
expensive. These conclusions do not draw on historical volatility.
The percentile of implied volatility is generally used to describe just where the
current implied volatility reading lies with respect to its past values. The "implied
volatility" reading that is being used in this case is the composite reading - the one
that takes into account all the options on an underlying instrument, weighting them
by their distance in- or out-of-the-money (at-the-money gets more weight) and also
weighting them by their trading volume. This technique has been referred to many
times and was first described in Chapter 28 on mathematical applications. That com
posite implied volatility reading can be stored in a database for each underlying
instrument every day. Such databases are available for purchase from firms that spe
cialize in option data. Also, snapshots of such data are available to members of
www.optionstrategist.com.
In general, most underlying instruments would have a composite implied
volatility reading somewhere near the 50th percentile on any given day. However, it
is not uncommon to see some underlyings with percentile readings near zero or 100%
on a given day. These are the ones that would interest a volatility trader. Those with
readings in the 10th percentile or less, say, would be considered "cheap"; those in the
90th percentile or higher would be considered expensive.
In reality, the percentile of implied volatility is going to be affected by what the
broad market is doing. For example, during a severe market slide, implied volatilities
will increase across the board. Then, one may find a large number of stocks whose
options are in the 90th percentile or higher. Conversely, there have been other times
when overall implied volatility has declined substantially: 1993, for example, and the
summer of 2001, for another. At those times, we often find a great number of stocks
whose options reside in the 10th percentile of implied volatility or lower. The point
is that the distribution of percentile readings is a dynamic thing, not something stat
ic like a lognormal distribution. Yes, perhaps over a long period of time and taking
into account a great number of cases, we might find that the percentiles of implied
volatility are normally distributed, but not on any given day.
The trader has some discretion over this percentile calculation. Foremost, he
must decide how many days of past history he wants to use in determining the per
centiles. There are about 255 trading days in a year. So, if he wanted a two-year his
tory, he would record the percentile of today's composite implied volatility with
respect to the 510 daily readings over the past two years. This author typically uses
816 Part VI: Measuring and Trading Volatility
600 days of implied volatility history for the purpose of determining percentiles, but
a case could be made for other lengths of time. The purpose is to use enough implied
volatility history to give one a good perspective. Then, a reading of the 10th per
centile or the 90th percentile will truly be significant and would therefore be a good
starting point in determining whether the options are cheap or expensive.
In addition to the actual percentile, the trader should also be aware of the width
of the implied volatility distribution. This was discussed in an earlier chapter, but
essentially the concept is this: If the first percentile is an implied volatility of 40% and
the 100th percentile is an implied volatility of 45%, then that entire range


determining whether the options are cheap or expensive.
In addition to the actual percentile, the trader should also be aware of the width
of the implied volatility distribution. This was discussed in an earlier chapter, but
essentially the concept is this: If the first percentile is an implied volatility of 40% and
the 100th percentile is an implied volatility of 45%, then that entire range is so nar
row as to be meaningless in terms of whether one could classify the options as cheap
or expensive.
The advantage of buying options in a low percentile of implied volatility is to
give oneself two ways to make money: one, via movement in the underlying (if a
straddle were owned, for example), and two, by an increase in implied volatility. That
is, if the options were to return to the 50th percentile of implied volatility, the volatil
ity trader who has bought "cheap" options should expect to make money from that
movement as well. That can only happen if the 50th percentile and the 10th per
centile are sufficiently far apart to allow for an increase in the price of the option to
be meaningful. Perhaps a good rule of thumb is this: If the option rises from the cur
rent (low) percentile reading to the 50th percentile in a month, will the increase in
implied volatility be equal to or greater than the time decay over that period?
Alternatively stated, with all other things being equal, will the option be trading at
the same or a greater price in a month, if implied volatility rises to the 50th percentile
at the end of that time? If so, then the width of the range of implied volatilities is
great enough to produce the desired results.
The attractiveness to this method for determining if implied volatility is out of
line is that the trader is "forced" to buy options that are cheap ( or to sell options that
are expensive), on a relative basis. Even though historical volatility has not been taken
into consideration, it will be later on when the probability calculators are brought to
bear. There is no guarantee, of course, that implied volatility will move toward the
50th percentile while the position is in place, but if it does, that will certainly be an
aid to the position.
In effect this method is measuring what the option trading public is "thinking"
about volatility and comparing it with what they've thought in the past. Since the pub
lic is wrong (about prices as well as volatility) at major turning points, it is valid to want
to be long volatility when "everyone else" has pushed it down to depressed levels. The
converse may not necessarily be true: that we would want to be short volatility when
everyone else has pushed it up to extremely high levels. The caveat in that case is that
Chapter 39: Volatility Trading Techniques 817
someone may have inside information that justifies expensive options. This is another
reason why selling volatility can be difficult: You may be dealing with far less infor
mation than those who are actually making the implied volatility high.
COMPARING IMPLIED AND HISTORICAL VOLATILITY
The most common way that traders determine which options are cheap or expensive
is by comparing the current composite implied volatility with various historical
volatility measures. However, just because this is the conventional wisdom does not
necessarily mean that this method is the preferred one for determining which options
are best for volatility trades. In this author's opinion, it is inferior to the percentile
method (comparing implied to past measures of implied), but it does have its merits.
The theory behind using this method is that it is a virtual certainty that implied and
historical volatility will eventually converge with each other. So, if one establishes
volatility trading positions when they are far apart, there is supposedly an advantage
there.
However, this argument has plenty of holes in it. First of all, there is no guar
antee that the two will converge in a timely manner, for example, before the options
in the trader's position can become profitable. Historical and implied volatility often
remain fairly far apart for weeks at a time.
Second, even if the convergence does occur, it doesn't necessarily mean one will
make money. As an example, consider the case in which implied volatility is 40% and
historical volatility is 60%. That's quite a difference, so you'd want to buy volatility.
Furthermore, suppose the two do converge. Does that mean you'll make money? No,
it does not. What if they converge and meet at 40%? Or, worse yet, at 30%? You'd
most certainly lose in those cases as the stock slowed down while your options lost
time value.
Another problem with this method is that implied volatility is not necessarily
low when it is bought, nor high when it is sold. Consider the example just cited. We
merely knew that implied volatility was 40% and that historical volatility was 60%. We
had no perspective on whether 40% was high, medium, or low. Thus, it is also nec
essary to see what the percentile of implied volatility is. If it turns out that 40% is a
relatively high reading for implied volatility, as determined by looking at where
implied volatility has been over the past couple of years, then we would probably not
want to buy volatility in this situation, even though implied and historical volatility
have a large discrepancy between them.
Many market-makers and floor traders use this approach. However, they are
often looking for an option that is briefly mispriced, figuring that volatilities will
818 Part VI: Measuring and Trading Volatility
quickly revert back to where they were. But for a position trader, the problems cited
above can be troublesome.
Having said that, if one looks to implement this method of trying to determine
when options are out of line, something along the following lines should be imple
mented. One should ensure that implied volatility is significantly different from all of
th