asuring 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
the pertinent historical volatilities. For example, one might require that implied
volatility is less than 80% of each of the 10-, 20-, 50-, and 100-day historical volatili
ty calculations. In addition, the current percentile of implied volatility should be
noted so that one has some relative basis for determining if all of the volatilities, his
torical and implied, are very high or very low. One would not want to buy options if
they were all in a very high percentile, nor sell them if they were all in a very low per
centile.
Often, a volatility chart showing both the implied and certain historical volatili
ties will be a useful aid in making these decisions. One can not only quickly tell if the
options are in a high or low percentile, but he may also be able to see what happened
at similar times in the past when implied and historical volatility deviated substan
tially.
Finally, one needs some measure to ensure that, if convergence between
implied and historical volatility does occur, he will be able to make money. So, for
example, if one is buying a straddle, he might require that if implied rises to meet his
torical (say, the lowest of the historicals) in a month, he will actually make money.
One could use a different time frame, but be careful not to make it something unrea
sonable. For example, if implied volatility is currently 40% and historical is 60%, it is
highly unlikely that implied would rise to 60% in a day or two. Using this criterion
also ensures that the absolute difference between implied and historical volatility is
wide enough to allow for profits to be made. That is, if implied is 10% and historical
is 13%, that's a difference of 30% in the two - ostensibly a "wide" divergence
between implied and historical. However, if implied rises to meet historical, it will
mean only an absolute increase of 3 percentage points in implied volatility - proba
bly not enough to produce a profit, after costs, if any length of time passes.
If all of these criteria are satisfied, then one has successfully found "mispriced"
options using the implied versus historical method, and he can proceed to the next
step in the volatility analysis: using the probability calculator.
READING THE VOLATILITY CHART
Another technique that traders use in order to determine if options are mispriced is
to actually try to analyze the chart of volatility - typically implied volatility, but it
could be historical. This might seem to be a subjective approach, except that it is real-
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
the pertinent historical volatilities. For example, one might require that implied
volatility is less than 80% of each of the 10-, 20-, 50-, and 100-day historical volatili
ty calculations. In addition, the current percentile of implied volatility should be
noted so that one has some relative basis for determining if all of the volatilities, his
torical and implied, are very high or very low. One would not want to buy options if
they were all in a very high percentile, nor sell them if they were all in a very low per
centile.
Often, a volatility chart showing both the implied and certain historical volatili
ties will be a useful aid in making these decisions. One can not only quickly tell if the
options are in a high or low percentile, but he may also be able to see what happened
at similar times in the past when implied and historical volatility deviated substan
tially.
Finally, one needs some measure to ensure that, if convergence between
implied and historical volatility does occur, he will be able to make money. So, for
example, if one is buying a straddle, he might require that if implied rises to meet his
torical (say, the lowest of the historicals) in a month, he will actually make money.
One could use a different time frame, but be careful not to make it something unrea
sonable. For example, if implied volatility is currently 40% and historical is 60%, it is
highly unlikely that implied would rise to 60% in a day or two. Using this criterion
also ensures that the absolute difference between implied and historical volatility is
wide enough to allow for profits to be made. That is, if implied is 10% and historical
is 13%, that's a difference of 30% in the two - ostensibly a "wide" divergence
between implied and historical. However, if implied rises to meet historical, it will
mean only an absolute increase of 3 percentage points in implied volatility - proba
bly not enough to produce a profit, after costs, if any length of time passes.
If all of these criteria are satisfied, then one has successfully found "mispriced"
options using the implied versus historical method, and he can proceed to the next
step in the volatility analysis: using the probability calculator.
READING THE VOLATILITY CHART
Another technique that traders use in order to determine if options are mispriced is
to actually try to analyze the chart of volatility - typically implied volatility, but it
could be historical. This might seem to be a subjective approach, except that it is real-
818 Part VI: Measuring and Trading Volatility
quickly revert back to where they were. But for a position trader, the problems cited
a


other technique that traders use in order to determine if options are mispriced is
to actually try to analyze the chart of volatility - typically implied volatility, but it
could be historical. This might seem to be a subjective approach, except that it is real-
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
the pertinent historical volatilities. For example, one might require that implied
volatility is less than 80% of each of the 10-, 20-, 50-, and 100-day historical volatili
ty calculations. In addition, the current percentile of implied volatility should be
noted so that one has some relative basis for determining if all of the volatilities, his
torical and implied, are very high or very low. One would not want to buy options if
they were all in a very high percentile, nor sell them if they were all in a very low per
centile.
Often, a volatility chart showing both the implied and certain historical volatili
ties will be a useful aid in making these decisions. One can not only quickly tell if the
options are in a high or low percentile, but he may also be able to see what happened
at similar times in the past when implied and historical volatility deviated substan
tially.
Finally, one needs some measure to ensure that, if convergence between
implied and historical volatility does occur, he will be able to make money. So, for
example, if one is buying a straddle, he might require that if implied rises to meet his
torical (say, the lowest of the historicals) in a month, he will actually make money.
One could use a different time frame, but be careful not to make it something unrea
sonable. For example, if implied volatility is currently 40% and historical is 60%, it is
highly unlikely that implied would rise to 60% in a day or two. Using this criterion
also ensures that the absolute difference between implied and historical volatility is
wide enough to allow for profits to be made. That is, if implied is 10% and historical
is 13%, that's a difference of 30% in the two - ostensibly a "wide" divergence
between implied and historical. However, if implied rises to meet historical, it will
mean only an absolute increase of 3 percentage points in implied volatility - proba
bly not enough to produce a profit, after costs, if any length of time passes.
If all of these criteria are satisfied, then one has successfully found "mispriced"
options using the implied versus historical method, and he can proceed to the next
step in the volatility analysis: using the probability calculator.
READING THE VOLATILITY CHART
Another technique that traders use in order to determine if options are mispriced is
to actually try to analyze the chart of volatility - typically implied volatility, but it
could be historical. This might seem to be a subjective approach, except that it is real-
Chapter 39: Volatility Trading Techniques 819
ly not much different from the GARCH approach, which is considered to be highly
advanced. When one views the volatility chart, he is not looking for chart patterns like
technical analysts might do with stock charts: support, resistance, head-and-shoul
ders, flags, pennants, and so on. Rather, he is merely looking for the trend of volatil
ity to change.
This is a valid approach in the use of many indicators, particularly sentiment
indicators, that can go to extreme levels. By waiting for the trend to change, the user
is not subjecting himself to buying into the midst of a downtrend in volatility, nor sell
ing into the midst of a steep uptrend in volatility.
Example: Suppose a volatility trader has determined that the current level of implied
volatility for XYZ stock is in the 1st percentile of all past readings. Thus, the options
are as cheap as they've ever been. Perhaps, though, the overall market is experienc
ing a very dull period, or XYZ itself has been in a prolonged, tight trading range -
either of which might cause implied volatility to decline steadily and substantially.
Having found these cheap options, he wants to buy volatility. However, he has no
guarantee that implied volatility won't continue to decline, even though it is already
as cheap as it's ever been.
If he follows the technique of waiting for a reversal in the trend of implied
volatility, then he would keep an eye on XYZ's implied volatility daily until it had at
least a modest increase, something to indicate that option buyers have become more
interested in XYZ's options. The chart in Figure 39-1 shows how this situation might
look.
There are a number of items marked on the chart, so it will be described in
detail. There are two graphs in Figure 39-1: The top line is the implied volatility
graph, while on the bottom is the stock price chart. The implied volatility chart shows
that, near the first ofJune, it made new all-time lows near 28% (i.e., it was in the 0th
percentile of implied volatility). Hence, one might have bought volatility at that
point. However, it is obvious that implied volatility was in a steep downtrend at that
time, so the volatility trader who reads the charts might have decided to wait for a
pop in volatility before buying. This turned out to be a judicious decision, because
the stock went nowhere for nearly another month and a half, all the while volatility
was dropping. At the right of the chart, implied volatility has dropped to nearly 20%.
The solid lines on the two graphs indicate the data that is known about the
implied volatility and price history of XYZ. The dotted lines indicate a scenario that
might unfold. If implied volatility were to jump ( and the stock price migh

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


nowhere for nearly another month and a half, all the while volatility
was dropping. At the right of the chart, implied volatility has dropped to nearly 20%.
The solid lines on the two graphs indicate the data that is known about the
implied volatility and price history of XYZ. The dotted lines indicate a scenario that
might unfold. If implied volatility were to jump ( and the stock price might jump, too),
then one might think that the trend of implied volatility was no longer down, and he
would then buy volatility.
820 Part VI: Measuring and Trading VolatHity
The reason that this approach has merit is that one never knows how low volatil
ity can go, and more important, how high it can get. It was mentioned that the same
sort of approach works well for other sentiment indicators, the put-call ratio, in par
ticular. During the bull market of the 1990s, the equity-only put-call ratio generally
ranged between about 30 and 55. Thus, some traders became accustomed to buying
the market when the put-call ratio reached numbers exceeding 50 (high put-call
ratio numbers are bullish predictors for the market in general). However, when the
bull market ended, or at least faltered, the put-call ratios zoomed to heights near 70
or 75. Thus, those using a static approach (that is, "Buy at 50 or higher") were buried
as they bought too early and had to suffer while the put-call ratios went to new all
time highs. A trend reversal approach would have saved them. It is a more dynamic
procedure, and thus one would have let the put-call ratio continue to rise until it
peaked. Then the market could have been bought.
This is exactly what reading the volatility chart is about. Rather than relying on
past data to indicate where the absolute maxima and minima of movements might
occur, one rather notes that the volatility data is at extreme levels ( 1st percentile or
100th percentile) and then watches it until it reverses direction. This is especially
useful for options sellers, because it avoids stepping into the vortex of massive option
FIGURE 39-1.
Chart of the trend of implied volatility.
XYZ
,J'
········································ · ························ ....................... ··················· 50. 0
... · ····················· 40. 0
Implied Volatility I/vi A r
···-····························· .. •·•······•·········-·············································-··vw•···················)·•······ 30. 0
All-Time Volatility Low -
, .... , ..... , .. ··················,·······~················
t :t·•····•····· ...•. ':·LJJSL5~?.
--;---··········•·:
o, b ::r F f1 h j ::r
34.000
32.000
30. 000
28. 000
26.000
24.000
22.000
20. 000
Chapter 39: VolatiDty Trading Techniques 821
buying, where the buyers perhaps have inside information about some forthcominf
corporate event such as a takeover. True, the options might be very expensive ( 10ot
percentile), but there is a reason they are, and those with the inside information know
the reason, whereas the typical volatility trader might not. However, if the volatility
trader merely waits for a downturn in implied volatility readings before selling these
options, he will most likely avoid the majority of trouble because the options will
probably not lose implied volatility until news comes out or until the buyers give up
(perhaps figuring that the takeover rumor has died).
Volatility buyers don't face the same problems with early entry that volatility
sellers do, but still it makes sense to wait for the trend of volatility to increase (as in
Figure 39-1) before trying to guess the bottom in volatility. Just as it is usually fool
hardy to buy a stock that is in a severe downtrend, so it may be, too, with buying
volatility.
A less useful approach would be to apply the same techniques to historical
volatility charts, for such charts say nothing about option prices. See the next section
for expansion on these thoughts.
COMPARING HISTORICAL VERSUS HISTORICAL
The above paragraphs summarize the three major ways that traders attempt to find
options that are out of line. Sometimes, another method is mentioned: comparing
current levels of historical volatility with past levels of the same measure, historical
volatility. This method will be described, but it is generally an inferior method
because such a comparison doesn't tell us anything about the option prices. It would
do little good, for example, to find that current historical volatility is in a very low per
centile of historical volatilities, only to learn later that the options are expensive and
that perhaps implied volatility is even higher than historical volatility. One would nor
mally not want to buy options in that case, so the initial analysis of comparing histor
ical to historical is a wasted effort.
Comparing current levels of historical volatility with past measures of historical
volatility is sort of a backward-looking approach, since historical volatility involves
strictly the use of past stock prices. There is no consideration of implied volatility in
this approach. Moreover, this method makes the tacit assumption that a stock's
volatility characteristics do not change, that it will revert to some sort of "normal" past
price behavior in terms of volatility. In reality, this is not true at all. Nearly every stock
can be shown to have considerable changes in its historical volatility patterns over
time.
822 Part VI: Measuring and Trading Volatility
Consider the historical volatilities of one of the wilder stocks of the tech stock
boom, Rambus (RMBS). Historical volatilities had ranged between 50% and ll0%,
from the listing of RMBS stock, through February 2000. At that time, the stock aver
aged a price of about $20 per share.
Things changed mightily when RMBS stock began to rise at a tremendous rate
in February 2000. At that time, the stock blasted to ll5, pulled back to 35, made a
new high near 135, and then collapsed to a price n


(RMBS). Historical volatilities had ranged between 50% and ll0%,
from the listing of RMBS stock, through February 2000. At that time, the stock aver
aged a price of about $20 per share.
Things changed mightily when RMBS stock began to rise at a tremendous rate
in February 2000. At that time, the stock blasted to ll5, pulled back to 35, made a
new high near 135, and then collapsed to a price near 20. Hence, the stock itself had
completed a wild round-trip over the two-year period. See Figures 39-2 and 39-3 for
the stock chart and the historical volatility chart of RMBS over the time period in
question.
As this happened, historical volatility skyrocketed. After February 2000, and
well into 2001, historical volatility was well above 120%. Thus it is clear that the
behavior patterns of Rambus changed greatly after February 2000. However, if one
had been comparing historical volatilities at any time after that, he would have erro
neously concluded that RMBS was about to slow down, that the historic volatilities
were too high in comparison with where they'd been in the past. If this had led one
to sell volatility on RMBS, it could have been a very expensive mistake.
While RMBS may be an extreme example, it is certainly not alone. Many other
stocks experienced similar changes in behavior. In this author's opinion, such behav
ior debunks the usefulness of comparing historical volatility with past measures of
historical volatility as a valid way of selecting volatility trades.
FIGURE 39-2.
Historical volatilities of RMBS.
RMBS 19.000 17.250 18.875 20010410
............ ······· 60.0
· ·················· 50.0
...... ······· 40. 0
39 M i·h··-:; s ·~ s a ~--f:i--·5--r;;····~··h s·s- A s a N □ s--r·;; r
Chapter 39: Volatility Trading Techniques 823
FIGURE 39-3.
Stock chart of RMBS.
19. 000 17. 250 18. 875 20010410
30.000
22. 000
14. 000
' ' : :
! : : : ; 1 : : 1 r : ! : : : : ; : : : : : i l : l : 6. ooo
99 M il M :i J ii s b N b J F M il M :i J il s b N b J F h 1
What this method may be best used for is to complement the other methods
described previously, in order to give the volatility trader some perspective on how
volatile he can expect the underlying instrument to be; but it obviously has to be
taken only as a general guideline.
CHECK THE FUNDAMENTALS
Once these mispriced options have been found, it is always imperative to check the
news to see if there is some fundamental reason behind it. For example, if the options
are extremely cheap and one then checks the news stories and finds that the under
lying stock has been the beneficiary of an all-cash tender offer, he would not buy
those options. The stock is not going to go anywhere, and in fact will disappear if the
deal goes through as planned.
Similarly, if the options appear to be very expensive, and one checks the news
and finds that the underlying has a product up for review before a governmental
agency (FDA, for example), then the options should not be sold because the stock
may be about to undergo a large gap move based on the outcome of FDA hearings.
There could be any number of similar corporate events that would make the options
very expensive. The seller of volatility should not try to intercede when such events
or rumors are occurring.
824 Part VI: Measuring and Trading Volatllity
However, if there is no news that would seem to explain why the options are so
cheap or so expensive, then the volatility trader can continue on to the rest of his
analyses.
SELECTING THE STRATEGY TO USE
In general, when one wants to trade volatility, a simple approach is best, especially if
one is buying volatility. If there is a volatility skew involved, then there may be other
strategies that are superior, and they are discussed in the latter part of this chapter.
However, when one is interested in the straight trading of volatility because he thinks
implied volatility is out of line, then only a few strategies apply.
If volatility is too low, then either a straddle or a strangle should be purchased.
One would normally choose a straddle if the underlying instrument is currently trad
ing near an available striking price. However, if the underlying is currently trading
between two ~riking prices, then a strangle might be the better choice. In either case,
a position trader would want to buy a straddle with several months of life remaining,
in order to improve his chances of making a profit. There is no "best" time length to
use, so one should use a probability calculator to aid in that decision. The use of prob
ability calculators will be discussed shortly.
Example: XYZ is trading at 39.60 and a volatility trader has determined that he wants
to buy volatility. With this information, he should attempt to buy a straddle with a
striking of 40 for both the put and the call.
Suppose that the current date is in December, and the available expiration
months for XYZ are January, February, April, July, and October, plus LEAPS for
January of the next year. Then he would analyze each straddle (January 40, February
40, April 40, etc.) to see which is the best one to buy. It generally seems to work out
that the midrange straddles have the best probabilities of success, given the way that
option prices are usually structured. Of course, the actual prices of each straddle
would be considered when using the probability calculator. In this case, then, the July
40 or October 40 straddle would probably be the best choices from a statistical view
point for a position trader.
If XYZ had been trading at a price of 37.50, say, then the trader would proba
bly want to consider buying a strangle: buying a call with a striking price of 40 and a
put with a striking price of 35. From the viewpoint of buying strangles, it does not
make sense to separate the strikes by more than one striking price unit - 5 points for
stock options, for example. This just makes the position more neutral to begin with.
Chap


ice of 37.50, say, then the trader would proba
bly want to consider buying a strangle: buying a call with a striking price of 40 and a
put with a striking price of 35. From the viewpoint of buying strangles, it does not
make sense to separate the strikes by more than one striking price unit - 5 points for
stock options, for example. This just makes the position more neutral to begin with.
Chapter 39: Volatility Trading Techniques 825
Speaking of neutrality, one can also use the deltas of the options in question to
alter the ratio of puts to calls, making the position initially as neutral as possible. This
is the suggested approach, since the volatility buyer does not care whether the stock
goes up or down. He is merely interested in stock movement and/or an increase in
implied volatility.
Example: XYZ is again trading at 39.60, and the trader wants a neutral position. He
should use the deltas of the options to construct a neutral position. Consider the
October 40 straddle, for example. Assume the volatility used for the probability cal
culations is 40%. Under those conditions (and the ones assumed in the previous
example), the October 40 call has a delta of 0.60 and the October 40 put has a delta
of -0.40. Thus a ratio of buying 2 calls and 3 puts is a neutral ratio. If the call is sell
ing for 6 and the put is selling for 5, then the break-even points for a 2-by-3 position
would be 53.5 on the upside and 31 on the downside. This information is summarized
as follows:
Delta of October 40 call:
Delta of October 40 put:
+0.60
-0.40
Delta-neutral ratio: buy 2 calls and 3 puts
Price of October 40 coll:
Price of October 40 put:
Net cost of 2-by-3 position: 27 points
Break-even points:
6.00
5.00
Upside = 40 + 27 /2 = 53.50
Downside = 40 - 27 /3 = 31 .00
So, the probability calculations would endeavor to determine what the chances are of
the stock ever trading at either 53.50 or 31.00 at any time prior to expiration. In fact,
since there are straddles available in several expiration months, the strategist would
want to analyze each of them in a similar fashion. Table 39-1 shows how his choices
might look. If one were considering buying a strangle, similar calculations could be
made using the deltas of the put and the call, where the call strike is higher than the
put strike.
Another simple strategy that can be used when volatility is low is the calendar
spread, because it has a positive vega. That is, it can be expected to expand if implied
volatility increases. For most traders, though, the limited profit nature of the calen-
826 Part VI: Measuring and Trading Volatility
TABLE 39-1
January February April July October January LEAP
Call price 1.25 2.25 3.50 5.00 6.00 7.15
Put price 1.50 2.35 3.35 4.35 5.00 5.55
Call delta 0.48 0.52 0.55 0.58 0.60 0.62
Put delta -0.52 -0.48 -0.45 -0.42 -0.40 -0.38
Neutral ~ 1-to-1 ~l-to-1 ~ 1-to-1 ~2-to-3 2-to-3 ~2-to-3
Debit 2.75 4.60 6.85 23.05 27.00 30.95
Upside break-even 42.75 44.60 46.85 51.57 53.50 55.47
Downside break-even 37.25 35.40 33.15 32.30 31.00 29.68
dar spread is too much of a burden"° either psychologically or in terms of commis
sions, and so this strategy is only modestly used by volatility traders. Some traders will
use the calendar spread if they don't see immediate prospects for an increase in
implied volatility. They perhaps buy a call calendar slightly out-of-the-money and also
buy a put calendar with slightly out-of-the-money puts. Then, if not much happens
over the short term, the options that were sold expire worthless, and the remaining
long straddle or strangle is even more attractive than ever. Of course, this strategy has
its drawback in that a quick move by the underlying may result in a loss, something
that would not have happened had a simple straddle or strangle been purchased.
SELLING VOLATILITY
If one were selling volatility (i.e., volatility is too high), his choices are more complex.
Virtually anyone who has ever sold volatility has had a bad experience or two with
either exploding stock prices or exploding volatility. Some of the concerns regarding
the sale of volatility will be discussed at length later in this chapter. For now, the sim
pler strategies will be considered, in keeping with the discussion involving the cre
ation of a volatility trading position.
Simplistically, a volatility seller would generally have a choice between one of
two strategies (although there is a more complicated strategy that can be introduced
as well). The simplest strategy is just to sell both an out-of-the-money put and an out
of-the-money call. The striking prices chosen should be far enough away from the
current underlying price so that the probabilities of the position getting in trouble
(i.e., the probabilities that the underlying actually trades at the striking prices of the
naked options during the life of the position) are quite small. Just as the option buyer
Chapter 39: Volatility Trading Techniques 827
above outlined several expiration months, then computed the break-even prices, so
should a volatility seller. Generally he will probably want to sell short-term options,
but all expiration months should be considered, at least initially. Also, he may want to
try different strike prices in order to get a balance between a low probability of the
stock reaching the striking price of the naked options and taking in enough premium
to make the trade worthwhile. To this author, the sale of naked options at small frac
tional prices does not appear attractive.
Of course, merely selling such a put and a call means the options are naked, and
that strategy is not suitable for all traders. The next best choice then, I suppose, is a
credit spread. The problem with a credit spread is that one is both selling expensive
options and also buying expensive options as protection. The ramifications of volatil
ity changes on the credit spread strategy were detailed two chapters p


y selling such a put and a call means the options are naked, and
that strategy is not suitable for all traders. The next best choice then, I suppose, is a
credit spread. The problem with a credit spread is that one is both selling expensive
options and also buying expensive options as protection. The ramifications of volatil
ity changes on the credit spread strategy were detailed two chapters previously, so
they won't be recounted here, except to say that if volatility decreases, the profits to
be realized by a credit spreader are quite small (perhaps not even enough to over
come the commission expense of removing the position), whereas a naked option
seller would benefit to a greater and more obvious extent.
The choice between naked writing and credit spreading should be made based
largely on the philosophy and psychological makeup of the trader himself. If one feels
uncomfortable with naked options, or if he doesn't have the ability to watch the mar
ket pretty much all the time (or have someone watch it for him), or ifhe doesn't have
the financial wherewithal to margin the positions and carry them until the stock hits
the break-even point, then naked writing is not for that trader.
Another factor that might affect the choice of strategy for the option seller is
what type of underlying instrument is being considered. Index options are by far the
best choices for naked option selling. Futures are next, and stocks are last. This is
because of the ways those various instruments behave; stocks have by far the great
est capability of making huge gap moves that are the bane of naked option selling. So,
if one has found expensive stock or futures options, that might lend more credence
to the credit spread strategy.
There is one other strategy that can be employed, upon occasion, when options
are expensive. It is called the volatility backspread, but its discussion will be deferred
until later in the chapter.
USING A PROBABILITY CALCULATOR
No matter which method is used to find options that are out of line, and no matter
which strategy is preferred by the trader, it is still necessary to use a probability cal
culator to get a meaningful idea of whether or not the underlying has the ability to
828 Part VI: Measuring and Trading Volatility
make the move to profitability (or not make the move into loss territory, if you're sell
ing options). This is where historical volatility plays a big part, for it is the input into
the probability calculator. In fact, no probability calculator will give reasonable pre
dictions without a good estimate of volatility. Please refer to the previous chapter for
a more in-depth discussion of probability calculators and stock price distributions.
The use of probability analysis also mitigates some of the problems inherent in
the method of selection that compares implied and historical volatilities. If the prob
abilities are good for success, then we might not care so much whether the options
are currently in a low percentile of implied volatility or not (although we still would
not want to buy volatility when the options were in a high percentile of implied
volatility and we would not want to sell options that are in a low percentile).
In using the probability calculator, one first selects a strategy (straddle buying,
for example, if options are cheap) and then calculates the break-even points as
demonstrated in the previous section. Then the probability calculator is used to
determine what the chances are of the underlying instrument ever trading at one or
the other of those break-even prices at any time during the life of the option position.
It was shown in the previous chapter that a Monte Carlo simulation using the fat tail
distribution is best used for this process.
An attractive volatility buying situation should have probabilities in excess of
80% of the underlying ever exceeding the break-even point, while an attractive
volatility selling situation should have probabilities of less than 25% of ever trading
at prices that would cause losses. The volatility seller can, of course, heavily influence
those probabilities by choosing options that are well out-of-the-money. As noted
above, the volatility seller should, in fact, calculate the probabilities on several dif
ferent striking prices, striving to find a balance between high probability of success
and the ability to take in enough premium to make the risk worthwhile.
Example: The OEX Index is trading at 650. Suppose that one has determined that
volatilities are too high and wants to analyze the sale of some naked options.
Furthermore, suppose that the choices have been narrowed down to selling the
September options, which expire in about five weeks. The main choices under con
sideration are those in Table 39-2. The option prices in this example, being index
options, reflect a volatility skew (to be discussed later) to make the example realistic.
The two right-hand columns should be rejected because the probabilities of
the stock hitting one or the other of the striking prices prior to expiration are too
high well in excess of the 25% guideline mentioned earlier. That leaves the
September 500-800 strangle or the September 550-750 strangle to consider. The
probabilities are best for the farthest out-of-the-money options (September 500-
800 strangle), but the options are selling at such small prices that they will not pro-
Chapter 39: Volatility Trading Techniques 829
TABLE 39-2
September September September September
800 call 750 call 730 call 700 call
September September September September
Naked sale: 500 put 550 put 570 put 600 put
Call price 0.20 1.50 3.50 8.80
Put price 0.40 2.00 3.70 8.50
Probability of call strike 4% 17% 30% 44%
Probability of put strike 1% 11% 20% 40%
vide much of a return even if they expire worthless. Remember that one is
required to establish the position with margin of at least 10% of the index price
for naked index


er September September September
Naked sale: 500 put 550 put 570 put 600 put
Call price 0.20 1.50 3.50 8.80
Put price 0.40 2.00 3.70 8.50
Probability of call strike 4% 17% 30% 44%
Probability of put strike 1% 11% 20% 40%
vide much of a return even if they expire worthless. Remember that one is
required to establish the position with margin of at least 10% of the index price
for naked index options, which would be $6,500 in this case. In fact, it has been
recommended that one margin the position at the striking price itself (15% of 800,
or $12,000 in this case). So, taking in only $60, less commissions, for the sale of
the September 500-800 strangle doesn't seem to provide enough of a reward.
Thus, the best choice seems to be the September 550-750 strangle. One would be
making about $320 after commissions if the options expired worthless, and the
recommended margin would be 15% of 750 (the higher strike), or $11,250 - a
return of about 2.8% for one month. One cannot annualize these returns, for he
has no idea if the same option pricing structure will exist in five weeks, when these
options expire.
Other probabilities can be calculated as well. For example, suppose one has
decided to buy a straddle. He might want to know what the odds are not only of
breaking even, but also of making at least a certain percentage return- say 20%. One
could also calculate the probability of the stock moving 20% past the break-even
points. That distance - 20% - is a reasonable figure to use because one would most
likely be taking some partial profits or adjusting his position if the stock did indeed
move that far.
USING STOCK PRICE HISTORY
All of the work done so far - determining which options are expensive, selecting a
strategy, and calculating the probabilities of success - has been somewhat theoretical
in that we haven't done any "back testing" with regard to the volatility of the under
lying instruments. At this point, one should look at past prices to see if the stock has
been able to make large moves (whether or not such a move is desired).
830 Part VI: Measuring and Trading Volatility
Example: A trader is considering the purchase of the XYZ October 40 straddle for
11 points, with the stock at 39.60. The options are cheap and the probabilities of suc
cess appear to be good, according to the probability calculator. The question that now
needs to be asked and answered is this: "In the past, has this stock been able to move
11 points in 10 months (the time remaining in the straddle's life)?" Or, more impor
tantly, since 11 divided by 39.60 is about 28%, "Has this stock been able to make
moves of 28% over 10 months, in the past?" The answers to these questions can be
readily obtained if stock price history data is available. One could even look at a chart
of the stock and attempt to answer the questions himself without the aid of a com
puter, but computer analysis of the price history is more rigorous and is therefore
encouraged.
The answers can be expressed in the form of probabilities, much as the results
of the probability calculator are.
Suppose one determines that the stock has been able to move 11 points in 10
months 77% of the time in the past. That's okay, but not great. However, when one
looks at the price chart ofXYZ, he sees that it traded at much lower prices - near $10
a share - for a long time before rising to its current levels. It would be very hard to
expect a $10 stock to move 11 points in 10 months. That's why the second figure, the
one involving the 28% move, is the more significant one. In this case, one might find
that XYZ has been able to move 28% in 10 months over 90% of the time in the past.
Now one has what appears to be a decent-looking straddle buy.
This analysis of past prices can be done in a more sophisticated manner. Rather than
just asking whether or not the stock has moved the required distance in the past, one
might want to see just how the stock's movements "look." That is, there are a couple
of scenarios under which the past movements might look attractive, but upon closer
examination, one would not be so sanguine.
For example, what ifXYZ had repeatedly moved 28%, but never much more in
most of the IO-month periods that comprise its stock history? Then, one would be
less inclined to want to own this straddle.
Another scenario of past movements might be that XYZ had made moves that
one could not reasonably expect to be repeated. Perhaps there was a huge gap down
on an earnings shortfall, or if it was an Internet stock around the tum of the millen
nium, it had a huge move upward, followed by a huge move downward. That would
be another nonrepeating type of move, because absent the Internet mania, the stock
might have been a rather range-bound item both prior to and after the one huge,
round-trip move.
Chapter 39: Volatility Trading Techniques 831
FIGURE 39-4.
Histogram of XYZ movements. {Testing 28% move in ten months.)
15%
X
xx
XXX
X XXX
XX XXX
5% xxxxxxx xxxxxxx
X XXXXXXXX
XXX XXXXXXXX
xxxxxxxxxxxxxxxx
-3 -2 -1 0
Movement
1
X X
X X X X
XXX XX X XXXX
xxxxxxxxxxxxxxxxxxx
2 3
These problems could be addressed by merely looking at the chart, but the
naked eye can be deceiving in many cases. Rather, a more rigorous approach would
be to construct a histogram of these past stock movements and analyze the histogram.
Figure 39-4 shows such a histogram. The x-axis shows the magnitude of each
10-month move that is in the database of XYZ stock prices. A move to 'T' would
mean that it moved the 28% and no further over the 10-month period. A move to
"-2" indicates that it fell 56% (twice the required distance) during the 10-month
period. The y-axis (left-hand scale) shows the percentage of times that the move
occurred. The sample histogram shown in Figure 39-4 is actually a very favorable
one. Notice that the stock was always able to move at least 28%. Furthermore, it
FIGURE 39-5.
Example o


no further over the 10-month period. A move to
"-2" indicates that it fell 56% (twice the required distance) during the 10-month
period. The y-axis (left-hand scale) shows the percentage of times that the move
occurred. The sample histogram shown in Figure 39-4 is actually a very favorable
one. Notice that the stock was always able to move at least 28%. Furthermore, it
FIGURE 39-5.
Example of poor movement.
15%
10%
5%
-3 -2
X
xx xx xxxx
X XXXX XXXX
X
X X
xx xxxx,"lrx~x~x~x--...----=.:.;:r·
0
X
X
X XXX
2 3
832 Part VI: Measuring and Trading Volatility
moved two or three times that far with great frequency. Finally, there is a continu
ity to the points on the histogram: There are some y-axis data points at almost all
points on the x-axis (between the minimum and maximum x-axis points). That is
good, because it shows that there has not been a clustering of movements by XYZ
that might have dominated past activity.
As for what is not a "good" histogram, we would not be so enamored of a his
togram that showed a huge cluster of points near and between the "-1" and 'T' points
on the X-axis. We want the stock to have shown an ability to move farther than just the
break-even distance, if possible. As an example, see Figure 39-5, which shows a stock
whose movements rarely exceed the "-1" or "+l" points, and even when they do, they
don't exceed it by much. Most of these would be losing trades because, even though
the stock might have moved the required percentage, that was its maximum move
during the 10-month period, and there is no way that a trader would know to take
profits exactly at that time. The straddles described by the histogram in Figure 39-5
should not be bought, regardless of what the previous analyses might have shown.
Nor would it be desirable for the histogram to show a large number of move
ments above the "+3" level on the histogram, with virtually nothing below that. Such
a histogram would most likely be reflective of the spiky, Internet-type stock activity
that was referred to earlier as being unreasonable to expect that it might repeat itself.
In a general sense, one doesn't want to see too many open spaces on the histogram's
X-axis; continuity is desired.
If the histogram is a favorable one, then the volatility analysis is complete. One
would have found mispriced options, with a good theoretical probability of profit,
whose past stock movements verify that such movements are feasible in the future.
ANOTHER APPROACH?
After having considered the descriptions of all of these analyses, one other approach
comes to mind: Use the past movements exclusively and ignore the other analyses
altogether. This sounds somewhat radical, but it is certainly a valid approach. It's
more like giving some rigor to the person who "knows" IBM can move 18 points and
who therefore wants to buy the straddle. If the histogram (study of past movements)
tells us that IBM does, indeed, have a good chance of moving 18 points, what do we
really care about the relationship of implied and historical volatility, or about the cur
rent percentiles of either type of volatility, or what a theoretical probability calcula
tor might say? In some sense, this is like comparing implied volatility (the price of the
straddle) with historical volatility (the history of stock price movements) in a strictly
practical sense, not using statistics.
Chapter 39: Volatility Trading Techniques 833
In reality, one would have to be mindful of not buying overly expensive options ( or
selling overly cheap ones), because implied volatility cannot be ignored. However, the
price of the straddle itself, which is what determines the x-axis on the histogram, does
reflect option prices, and therefore implied volatility, in a nontechnical sense. This author
suspects that a list of volatility trading candidates generated only by using past movements
would be a rather long list. Therefore, as a practical matter, it may not be useful.
MORE THOUGHTS ON SELLING VOLATILITY
Earlier, it was promised that another, more complex volatility selling strategy would
be discussed. An option strategist is often faced with a difficult choice when it comes
to selling (overpriced) options in a neutral manner - in other words, "selling volatili
ty." Many traders don't like to sell naked options, especially naked equity options, yet
many forms of spreads designed to limit risk seem to force the strategist into a direc
tional (bullish or bearish) strategy that he doesn't really want. This section addresses
the more daunting prospect of trying to sell volatility with protection in the equity
and futures option markets.
The quandary in trying to sell volatility is in trying to find a neutral strategy that
allows one to benefit from the sale of expensive options without paying too much for
a hedge - the offsetting purchase of equally expensive options. The simple strategy
that most traders first attempt is the credit spread. Theoretically, if implied volatility
were to fall during the time the credit spread position is in place, a profit might be
realized. However, after commissions on four different options in and possibly out
(assuming one sold both out-of-the-money put and call spreads), there probably
wouldn't be any real profit left if the position were closed out early. In sum, there is
nothing really wrong with the credit spread strategy, but it just doesn't seem like any
thing to get too excited about. What other strategy can be used that has limited risk
and would benefit from a decline in implied volatility? The highest-priced options are
the longer-term ones. If implied volatility is high, then if one can sell options such as
these and hedge them, that might be a good strategy.
The simplest strategy that has the desired traits is selling a calendar spread
that is, sell a longer-term option and hedge it by buying a short-term option at the
same strike. True, both are expensive (and


d volatility? The highest-priced options are
the longer-term ones. If implied volatility is high, then if one can sell options such as
these and hedge them, that might be a good strategy.
The simplest strategy that has the desired traits is selling a calendar spread
that is, sell a longer-term option and hedge it by buying a short-term option at the
same strike. True, both are expensive (and the near-term option might even have a
slightly higher implied volatility than the longer-term one). But the longer-term one
trades with a far greater absolute price, so if both become cheaper, the longer-term
one can decline quite a bit farther in points than the near-term one. That is, the vega
of the longer-term option is greater than the vega of the shorter-term one. When one
sells a calendar spread, it is called a reverse calendar spread. The strategy was
834 Part VI: Measuring and Trading Volatility
described in the chapter on reverse spreads. The reader might want to review that
chapter, not only for the description of the strategy, but also for the description of the
margin problems inherent in reverse spreads on stocks and indices.
One of the problems that most traders have with the reverse calendar spread is
that it doesn't produce very large profits. The spread can theoretically shrink to zero
after it is sold, but in reality it will not do so, for the longer-term option will retain
some amount of time value premium even if it is very deeply in- or out-of-the-money.
Hence the spread ·will never really shrink to zero.
Yet, there is another approach that can often provide larger profit potential and
still retain the potential to make money if implied volatility decreases. In some sense
it is a modification of the reverse calendar spread strategy that can create a poten
tially even more desirable position. The strategy, known as a volatility backspread,
involves selling a long-term at-the-money option (just as in the reverse calendar
spread) and then buying a greater number of near-er term out-of-the-money options.
The position is generally constructed to be delta-neutral and it has a negative vega,
meaning that it will profit if implied volatility decreases.
Example: XYZ is trading at 115 in early June. Its options are very expensive. A trad
er would like to construct a volatility backspread using the following two options:
Call Option
July 130 call:
October 120 call:
Price
2.50
13
Delta
0.26
0.53
Vega
0.10
0.27
A delta-neutral position would be to buy 2 of the July 130 calls and sell one of
the October 120 calls. This would bring in a credit of 8 points. Also, it would have a
small negative position vega, since tvvo times the vega of the July calls minus one
times the vega of the October call is -0.07. That is, for each one percentage point
drop in implied volatility of XYZ options in general, this position would make $7 -
not a large amount, but it is a small position.
The profitability of the position is shown in Figure 39-6. This strategy has lim
ited risk because it does not involve naked options. In fact, if XYZ were to rally by a
good distance, one could make large profits because of the extra long call.
Meanwhile, on the downside, if XYZ falls heavily, all the options would lose most of
their value and one would have a profit approaching the amount of the initial credit
received. Furthermore, a decrease in implied volatility produces a small profit as
well, although time decay may not be in the trader's favor, depending on exactly
which short-term options were bought. The biggest risk is that XYZ is exactly at 130
at July expiration, so any strategist employing this strategy should plan to close it out
Chapter 39: Volatility Trading Techniques
FIGURE 39-6.
Volatility backspread neutral position.
Underlying Price
835
in advance of the near-term expiration. It should not be allowed to deteriorate to the
point of maximum loss.
Modifications to the strategy can be considered. One is to sell even longer-term
options and of course hedge them with the purchase of the near-term options. The
longer-term the option is, the bigger its vega will be, so a decrease in implied volatil
ity will cause the heftier-priced long-term option to decline more in price. This mod
ification is somewhat tempered, though, by the fact that when options get really
expensive, there is often a tendency for the near-term options to be skewed. That is,
the near-term options will be trading with a much higher implied volatility than will
the longer-term options. This is especially true for LEAPS options. For that reason,
one should make sure that he is not entering into a situation in which the shorter
term options could lose volatility while the longer-term ones more or less retain the
same implied volatility, as LEAPS options often do. This concept of differing volatil
ity between near- and long-term options was discussed in more detail in Chapter 36
on the basics of volatility trading. As a sort of general rule, if one finds that the longer
term option has a much lower implied volatility than the one you were going to buy,
this strategy is not recommended. As a corollary, then, it is unlikely that this strate
gy will work well with LEAPS options.
One other thing that you should analyze when looking for this type of trade is
whether it might be better to use the puts than the calls. For one thing, you can estab
lish a position in which the heavy profitability is on the downside (as opposed to the
upside, as in the XYZ example above). Then, of course, having considered that, it
might actually behoove one to establish both the call spread and the put spread. If
836 Part VI: Measuring and Trading Volatility
you do both, though, you create a "good news, bad news" situation. The good news
is that the maximum risk is reduced; for example, if XYZ goes exactly to 130 (the
worst point for the call spread), the companion put spread's credit would redu


ing considered that, it
might actually behoove one to establish both the call spread and the put spread. If
836 Part VI: Measuring and Trading Volatility
you do both, though, you create a "good news, bad news" situation. The good news
is that the maximum risk is reduced; for example, if XYZ goes exactly to 130 (the
worst point for the call spread), the companion put spread's credit would reduce that
risk a little. However, the bad news is that there is a much wider range over which
there is not profit, since there are two spots where losses are more or less maximized
(at the strike price of the long calls and again at the strike price of the long puts).
Margin will be discussed only briefly, since it was addressed in the chapter on
reverse spreads. For both index and stock options, this strategy is considered to have
naked options - a preposterous assumption, since one can see from the profit graph
that the position is fully hedged until the near-term options expire. This raises the
capital requirement for nonmember traders. The margin anomaly is not a problem
with futures options, however. For those options, one need only margin the differ
ence in the strikes, less any credit received, because that is the true risk of the posi
tion. In summary, the volatility trader who wants to sell volatility in equity and futures
options markets needs to be hedged, because gaps are prevalent and potentially very
costly. This strategy creates a more neutral, less price-dependent way to benefit if
implied volatility decreases, especially when compared with simple credit spreads.
SUMMARY: TRADING THE
VOLATILITY PREDICTION
Attempting to establish trades when implied volatility is out of line is a theoretically
attractive strategy. The process outlined above consisted of a few steps, employing
both statistical and theoretical analysis. In any case, though, probability calculators
must "say" that a volatility trade has good probabilities of success. It's merely a mat
ter of what criteria we apply to limit our choices before we run the probability analy
sis. So, it might be more useful to view volatility trading analysis in this light:
Step I: Use a selection criterion to limit the myriad of volatility trading choices. Any
of these could be used as the first criterion, but not all of them at once:
a. Require implied volatility to be at an extreme percentile.
b. Require historical and implied volatility to have a large discrepancy
between them.
c. Interpret the chart of implied volatility to see if it has reversed trend.
Step 2: Use a probability calculator to project whether the strategy can be expected
to be a success.
Step 3: Using past price histories, determine whether the underlying has been able
to create profitable trades in the past. (For example, if one is considering
Chapter 39: Volatility Trading Techniques 837
buying a straddle, ask the question, "Has this stock been able to move far
enough, with great enough frequency, to make this straddle purchase prof
itable?") Use histograms to ensure that the past distribution of stock prices
is smooth, so that an aberrant, nonrepeatable move is not overly influenc
ing the results.
Each criterion from Step 1 would produce a different list of viable volatility
trading candidates on any given day. If a particular candidate were to appear on more
than one of the lists, it might be the best situation of all.
TRADING THE VOLATILITY SKEW
In the early part of this chapter, it was mentioned that there are two ways in which
volatility predictions could be "wrong." The first was that implied volatility was out of
line. The second is that individual options on the same underlying instrument have
significantly different implied volatilities. This is called a volatility skew, and presents
trading opportunities in its own right.
DIFFERING IMPLIED VOLATILITIES ON THE SAME UNDERLYING SECURITY
The implied volatility of an option is the volatility that one would have to use as input
to the Black-Scholes model in order for the result of the model to be equal to the
current market price of the option. Each option will thus have its own implied volatil
ity. Generally, they will be fairly close to each other in value, although not exactly the
same. However, in some cases, there will be large enough discrepancies between the
individual implied volatilities to warrant the strategist's attention. It is this latter con
dition of large discrepancies that will be addressed in this section.
Example: XYZ is trading at 45. The following option prices exist, along with their
implied volatilities:
Actual Implied
Option Price Volatility
January 45 call 2.75 41%
January 50 call 1.25 47%
January 55 call 0.63 53%
February 45 call 3.50 38%
February 50 call 4.00 45%
838 Part VI: Measuring and Trading Volatility
Note that the implied volatilities of the individual options range from a low of
38% to a high of 53%. This is a rather large discrepancy for options on the same
underlying security, but it is useful for exemplary purposes.
A neutral strategy could be established by buying options with lower implied
volatilities and simultaneously selling ones with higher volatilities, such as buy 10
February 45 calls and sell 20 January 50 calls. Examples of neutral spreads will be
expanded upon in the next chapter, when more exact measures for determining how
many calls to buy and sell are discussed.
Before jumping into such a position, the strategist should ask himself if there is
a valid reason why the different options have such different implied volatilities. As a
generalization, it might be fair to say that out-of-the-money options have slightly
higher implieds than at-the-money ones, and that longer-term options have lower
implieds than short-term ones. But there are many instances in which such is not the
case, so one must be careful not to overgeneralize.
Speculators often desire the lowest dollar-cost option available


nt implied volatilities. As a
generalization, it might be fair to say that out-of-the-money options have slightly
higher implieds than at-the-money ones, and that longer-term options have lower
implieds than short-term ones. But there are many instances in which such is not the
case, so one must be careful not to overgeneralize.
Speculators often desire the lowest dollar-cost option available. Thus, in a
takeover rumor situation, they would buy the out-of-the-moneys as opposed to the
higher-priced at- or in-the-moneys. If the out-of-the-moneys are extremely expen
sive because of a takeover rumor, then the strategist must be careful, because the
neutral strategy concept may lead him into selling naked calls. This is not to say
he should avoid the situation altogether; he may be able to structure a position
with enough upside room to protect himself, or he may believe the rumors are
false.
Returning to the general topic of differing implied volatilities on the same
underlying stock, the strategist might ask how he is to determine if the discrepancies
between the individual options are significantly large to warrant attention. A mathe
matical approach is presented at the end of the next chapter in a section on advanced
mathematical concepts. Suffice it to say that there is a way that the differences in the
various implieds can be reduced to a single number - a sort of "standard deviation of
the implieds" that is easy for the strategist to use. A list of these numbers can be con
structed, comparing which stocks or futures might be candidates for this type of neu
tral spreading. On a given day, the list is usually quite short - perhaps 20 stocks and
10 futures contracts will qualify.
The concept of the implied volatilities of various options on the same underly
ing stock remaining out of line with each other is one that needs more discussion. In
the following section, the idea of semipermanent distortion between the volatilities
of different striking prices is discussed.
Chapter 39: Volatility Trading Techniques
VOLATILITY SKEWING
839
After the stock market crashed in 1987, index options experienced what has since
proven to be a permanent distortion: Out-of-the-money puts have remained more
expensive than out-of-the-money calls. Furthermore, out-of-the-money puts are
more expensive than at-the-money puts; out-of-the-money calls are cheaper than at
the-money calls. This distorted effect is due to several factors, but it is so deep-seat
ed that it has remained through all kinds of up and down markets since then. Other
markets, particularly futures markets, have also experienced a long-lasting distortion
between the implied volatilities at various strikes.
The proper name given to this phenomenon is volatility skewing: the long-last
ing effect whereby options at different striking prices trade with differing implied
volatilities. It should be noted that the calls and puts at the same strike must trade
for the same implied volatility; otherwise, conversion or reversal arbitrage would
eliminate the difference. However, there is no true arbitrage between different strik
ing prices. Hence, arbitrage cannot eliminate volatility skewing.
Example: Volatility skewing exists in OEX index options. Assume the average volatil
ity of OEX and its options is 16%. With volatility skewing present, the implied volatil
ities at the various striking prices might look like this:
OEX: 580
Implied Volatility
Strike of Both Puts and Calls
550 22%
560 19%
570 17%
580 16%
590 15%
600 14%
610 13%
In this form of volatility skewing, the out-of-the-money puts are the most
expensive options; the out-of-the-money calls are the cheapest. This pattern of
implied volatilities is called a reverse volatility skew or, alternatively, a negative
volatility skew.
840 Part VI: Measuring and Trading Volatility
The causes of this effect stem from the stock market's penchant to crash occa
sionally. Investors who want protection buy index puts; they don't sell index futures
as much as they used to because of the failure of the portfolio insurance strategy dur
ing the 1987 crash. In addition, margin requirements for selling naked index puts
have increased, especially for market-makers, who are the main suppliers of naked
puts. Consequently, demand for index puts is high and supply is low. Therefore, out
of-the-money index puts are overly expensive.
This does not entirely explain why index calls are so cheap. Part of the reason
for that is that institutional traders can help finance the cost of those expensive index
puts by selling some out-of-the-money index calls. Such sales would essentially be
covered calls if the institution owned stocks, which it most certainly would. This strat
egy is called a collar.
This distortion in volatilities is not in accordance with the probability distribu
tion of stock prices. These distorted implied volatilities define a different probability
curve for stock movement. They seem to say that there is more chance of the market
dropping than there is of it rising. This is not true; in fact, just the opposite is true.
Refer to the reasons for using lognormal distribution to define stock price move
ments. Consequently, there are opportunities to profit from volatility skewing, if one
is able to hold the position until expiration.
It was shown in previous examples that one would attempt to sell the options
with higher implied volatilities and buy ones with lower implieds as a hedge. Hence,
for OEX traders, three strategies seem relevant:
1. Buy a bear put spread in OEX.
Example: Buy 10 OEX June 560 puts
Sell IO OEX June 540 puts
2. Buy OEX puts and sell a larger number of out-of-the-money puts - a ratio write
of put options.
Example: Buy 10 OEX June 560 puts
Sell 20 OEX June 550 puts
3. Sell OEX calls and buy a larger number of out-of-the-money calls - a backspread
of call options.
Example: Buy 20 OEX June 590 calls
Sell IO OEX June 580


spread in OEX.
Example: Buy 10 OEX June 560 puts
Sell IO OEX June 540 puts
2. Buy OEX puts and sell a larger number of out-of-the-money puts - a ratio write
of put options.
Example: Buy 10 OEX June 560 puts
Sell 20 OEX June 550 puts
3. Sell OEX calls and buy a larger number of out-of-the-money calls - a backspread
of call options.
Example: Buy 20 OEX June 590 calls
Sell IO OEX June 580 calls
In all three cases, one is selling the higher implied volatility and buying options
with lower implied volatilities. The first strategy is a simple bear spread. While it will
Chapter 39: Volatility Trading Techniques 841
benefit from the fact that the options are skewed in terms of implied volatility, it is not
a neutral strategy. It requires that the underlying drop in price in order to become
profitable. There is nothing wrong with using a directional strategy like this, but the
strategist must be aware that the skew is unlikely to disappear ( until expiration) and
therefore the index price movement is going to be necessary for profitability.
The second strategy would be best suited for moderately bearish investors,
although a severe market decline might drive the index so low that the additional
short puts could cause severe losses. However, statistically this is an attractive strat
egy. At expiration, the volatility skewing must disappear; the markets will have moved
in line with their real probability distribution, not the false one being implied by the
skewed options. This makes for a potentially profitable situation for the strategist.
The backspread strategy would work best for bullish investors, although some
backspreads can be created for credits, so a little money could also be made if the
index fell. In theory, a strategist could implement both strategies simultaneously,
which would give him an edge over a wide range of index prices. Again, this does not
mean that he cannot lose money; it merely means that his strategy is statistically
superior because of the way the options are priced. That is, the odds are in his favor.
In reality, though, a neutral trader would choose either the ratio spread or the
backspread - not both. As a general rule of thumb, one would use the ratio spread
strategy if the current level of implied volatility were in a high percentile. The back
spread strategy would be used if implied volatility were in a low percentile current
ly. In that way, a movement of implied volatility back toward the 50th percentile
would also benefit the trade that is in place.
Another interesting thing happens in these strategies that may be to their ben
efit: The volatility skewing that is present propagates itself throughout the striking
prices as OEX moves around. It was shown in the previous section's example that one
should probably continue to project his profits using the distorted volatilities that
were present when he establishes a position. This is a conservative approach, but a
correct one. In the case of these OEX spreads, it may be a benefit.
Assuming that the skewing is present wherever OEX is trading means that the
at-the-money strike will have 16% as its implied volatility regardless of the absolute
price level; the skewing will then extend out from that strike. So, if OEX rises to 600,
then the 600 strike would have a volatility of 16%; or if it fell to 560, then the 560
puts and calls would have a volatility of 16%. Of course, 16% is just a representative
figure; the "average" volatility of OEX can change as well. For illustrative purposes,
it is convenient to assume that the at-the-money strike keeps a constant volatility.
Example: Initially, a trader establishes a call backspread in OEX options in order to
take advantage of the volatility skewing:
842
Initial situation: OEX: 580
Option
June 590 call
June 600 call
A neutral spread would be:
Buy 2 June 600 calls
Sell I June 590 call
Implied Volatility
15%
14%
since the deltas are in the ratio of 2-to-l.
Part VI: Measuring and Trading Volatility
Delta
0.40
0.20
Now, suppose that OEX rises to 600 at a later date, but well before expiration.
This is not a particularly attractive price for this position. Recall that, at expiration, a
backspread has its worst result at the striking price of the purchased options. Even
prior to expiration, one would not expect to have a profit with the index right at 600.
However, the statistical advantage that the strategist had to begin with might be
able to help him out. The present situation would probably look like this:
Option
June 590 call
June 600 call
Implied Volatility
17%
16%
The June 600 call is now the at-the-money call, since OEX has risen to 600. As
such, its implied volatility will be 16% ( or whatever the "average" volatility is for OEX
at that time - the assumption is made that it is still 16% ). The June 590 call has a
slightly higher volatility (17%) because volatility skewing is still present.
Thus, the options that are long in this spread have had their implied volatility
increase; that is a benefit. Of course, the options that are short had theirs increase as
well, but the overall spread should benefit for two reasons:
1. Twice as many options are owned as were sold.
2. The effect of increased volatility is greatest on the at-the-money option; the in
the-money will be affected to a lesser degree.
All index options exhibit this volatility skewing. Volatility skewing exists in other
markets as well. The other markets where volatility skewing is prevalent are usually
Chapter 39: Volatility Trading Techniques 843
futures option markets. In particular, gold, silver, sugar, soybeans, and coffee options
will from time to time display a form of volatility skewing that is the opposite of that
displayed by index options. In these futures markets, the cheapest options are out-of
the-money puts, while the most expensive options are out-of-the-money calls.
Example: January soybeans are tradin


ing Techniques 843
futures option markets. In particular, gold, silver, sugar, soybeans, and coffee options
will from time to time display a form of volatility skewing that is the opposite of that
displayed by index options. In these futures markets, the cheapest options are out-of
the-money puts, while the most expensive options are out-of-the-money calls.
Example: January soybeans are trading at 580 ($5.80 per bushel). The following
table of implied volatilities shows how volatility skewing that is present in the soybean
market is the opposite of that shown by the OEX market in the previous examples:
January beans: 580
Strike Implied Volatility
525 12%
550 13%
575 15%
600 17%
625 19%
650 21%
675 23%
Notice that the out-of-the-money calls are now the more expensive items, while
out-of-the-money puts are the cheapest. This pattern of implied volatilities is called
forward volatility skew or, alternatively, positive volatility skew.
The distribution of soybean prices implied by these volatilities is just as incor
rect as the OEX one was for the stock market. This soybean implied distribution is
too bullish. It implies that there is a much larger probability of the soybean market
rising 100 points than there is of it falling 50 points. That is incorrect, considering the
historical price movement of soybeans.
A strategist attempting to benefit from the forward ( or positive) volatility skew
in this market has essentially three strategies available. They are the opposite of the
three recommended for the $OEX, which had a reverse (or negative) volatility skew.
First would be a call bull spread, second would be a put backspread, and third would
be a call ratio spread. In all three cases, one would be buying options at the lower
striking price and selling options at the higher striking price. This would give him the
theoretical advantage.
The same sorts of comments that were made about the OEX strategies can be
applied here. The bull spread is a directional strategy and can probably only be
expected to make money if the underlying rises in price, despite the statistical advan-
844 Part VI: Measuring and Trading VolatiRty
tage of the volatility skew. The put backspread is best established when the overall
level of implied volatility is in a low percentile. Finally, the call ratio spread has a
great deal of risk to the upside ( and futures prices can fly to the upside quickly, espe
cially if bad fundamental conditions develop, such as weather in the grain markets).
The call ratio spread would best be used when implied volatilities are already in a
high percentile.
As a general comment, it should be noted that if the volatility skew disappears
while the trader has the position in place, a profit will generally result. It would nor
mally behoove the strategist to take the profit at that time. Otherwise, follow-up
action should adhere to the general kinds of action recommended for the strategies
in question: protective action to prevent large losses in the case of the ratio spreads,
or the taking of partial profits and possibly rolling the long options to a more at-the
money strike in the case of the backspread strategies.
SUMMARY OF VOLATILITY SKEWING
Whenever volatility skewing exists - no matter what market - opportunities arise for
the neutral strategist to establish a position that has advantages. These advantages
arise out of the fact that normal market movements are different from what the
options are implying. Moreover, the options are wrong when there is skewing at all
strikes, from the lowest to the highest. The strategist should be careful to project his
profits prior to expiration using the same skewing, for it may persist for some time to
come. However, at expiration, it must of course disappear. Therefore, the strategist
who is planning to hold the position to expiration will find that volatility skewing has
presented him with an opportunity for a positive expected return.
SUMMARY OF VOLATILITY TRADING
The theoretical trading of options, mostly in a neutral manner, has evolved into one
large branch - volatility trading. This part of the book has attempted to lay out the
foundations, structures, and practices prevalent in this branch of trading. As the read
er can see, there are some sophisticated techniques being applied - not so much in
terms of strategy, but in terms of the ways that one looks at volatility and in the ways
that stocks can move.
Statistical methods are used liberally in trying to determine the ways that either
volatility can move or stocks can move. The probability calculators, stock price dis
tributions, and related topics are all statistical in nature. The volatility trader is intent
on finding situations in which current market implied volatility is incorrect, either in
its absolute value or in the skew that is prevalent in the options on a particular under-
Chapter 39: VolatiDty Trading Techniques 845
lying instrument. Upon finding such discrepancies, the trader attempts to take
advantage by constructing a more or less neutral position, preferring not to predict
price so much, but rather attempting to predict volatility.
Most volatility traders attempt to buy volatility rather than sell it, for the rea
sons that the strategies inherent in doing so have limited risk and large potential
rewards, and don't require one to monitor them continuously. If one owns a straddle,
any major market movements resulting in gaps in prices are a benefit. Hence, mon
itoring of positions as little as just once a day is sufficient, a fact that means that the
volatility buyer can have a life apart from watching a trading screen all day long. In
addition, volatility buyers of stock options can avail themselves of the chaotic move
ments that stocks can make, taking advantage of the occasional fat tail movements.
However, since volatility and prices are so unstable, one cannot predict their
movements with any certaint


cient, a fact that means that the
volatility buyer can have a life apart from watching a trading screen all day long. In
addition, volatility buyers of stock options can avail themselves of the chaotic move
ments that stocks can make, taking advantage of the occasional fat tail movements.
However, since volatility and prices are so unstable, one cannot predict their
movements with any certainty. The vagaries of historical volatility as compared to
implied volatility, the differences between the implied volatility of short- and long
term options, and the difficulty in predicting stock price distributions all compli
cate the process of predicting volatility. Hence, volatility trading is not a "lock," but
its practitioners normally believe that it is by far the best approach to theoretical
option trading available today. Moreover, most option professionals primarily trade
volatility rather than directional positions.
, CHAPTER 40
Advanced Concepts
As the option markets have matured, strategists have been forced to rely more on
mathematics in order to select new positions as well as to discern how their positions
will behave in fluctuating markets. These techniques can be used on simple strategies,
such as bull spreads or ratio spreads, or on far more complex portfolios of options.
First, the concept of implied volatility will be examined in more detail, prima
rily as an aid in choosing new positions that have a positive expected return. Then,
the concept of risk management will be explored. In effect, one can reduce his option
position into several components of risk measurement that can be readily under
stood. This chapter describes the techniques used to evaluate one's position, and
shows how to use this information to reduce the risk in the position. The actual math
ematical calculations required to perform these analyses are included at the end of
the chapter.
NEUTRALITY
In many of the examples in previous chapters, it was generally assumed that one
would take a "neutral" position in order to capture the pricing or volatility differen
tial. Why this concentration on neutrality? Neutrality, as it applies to option positions,
means that one is noncommittal with respect to at least one of the factors that influ
ence an option's price. Simply put, this means that one can design an option position
in which he can profit, no matter which way the underlying security moves.
846
Chapter 40: Advanced Concepts 847
Most option strategies fall into one of two categories: as a hedge to a stock or
futures strategy (for example, buying puts to protect a portfolio of stocks), or as a
profit venture unto itself. The latter category is where most traders find themselves,
and they often approach it in a fairly speculative manner - either by buying options
or by being a premium seller (covered or uncovered). In such strategies, the trader
is taking a view of the market; he needs certain price action from the underlying
security in order to profit. Even covered call writing, which is considered to be a con
servative strategy, is subject to large losses if the underlying stock drops drastically.
It doesn't have to be that way. Strategies can be devised that will have a chance
to profit regardless of price changes in the underlying stock, as well as because of
them. Such strategies are neutral strategies and they always require at least two
options in the position - a spread, straddle, or some other combination. Often, when
one constructs a neutral strategy, he is neutral with respect to price changes in the
underlying security. It is also possible, and often wise, to be neutral with respect to
the rate of price change of the underlying security, with respect to the volatility of
the security, or with respect to time decay. This is not to imply that any option spread
that is neutral will automatically be a money-maker; rather, one looks for an
opportunity - perhaps an overpriced series of options - and attempts to capture that
overpricing by constructing a neutral strategy around it. Then, regardless of the
movement of the underlying stock, the strategist has a chance of making money if the
overpricing disappears.
Note that the neutral approach is distinctly different from the speculator's, who,
upon determining that he has discovered an underpriced call, would merely buy the
call, hoping for the stock to increase in price. He would not make money if XYZ fell
in price unless there was a huge expansion in implied volatility - not something to
count on. The next section of this chapter deals with how one determines his neu
trality. In effect, if he is not neutral, then he has risk of some sort. The following sec
tions outline various measures of risk that the strategist can use to establish a new
position or manage an existing one.
The most important of these risk measurements is how much market exposure
the position currently has. This has previously been described as the "delta." Of near
ly equal importance to the strategist is how much the option strategy will change with
respect to the rate of change in the price of the underlying security. Also of interest
are how changes in volatility, in time remaining until expiration, or even in the risk
free interest rate will affect the position. Once the components of the option position
are defined, the strategist can then take action to reduce the risk associated with any
of the factors, should he so desire.
848 Part VI: Measuring and Trading Volatmty
THE "GREEKS"
Risk measurements have generally been given the names of actual or contrived
Greek letters. For example, "delta" was discussed in previous chapters. It has become
common practice to refer to the exposure of an option position merely by describing
it in terms of this "Greek" nomenclature. For example, "delta long 200 shares" means
that the entire option position behaves as if the strategist were merely long 200 shares
of the underlying stock


ames of actual or contrived
Greek letters. For example, "delta" was discussed in previous chapters. It has become
common practice to refer to the exposure of an option position merely by describing
it in terms of this "Greek" nomenclature. For example, "delta long 200 shares" means
that the entire option position behaves as if the strategist were merely long 200 shares
of the underlying stock. In all, there are six components, but only four are heavily
used.
DELTA
The first risk measurement that concerns the option strategist is how much current
exposure his option position has as the underlying security moves. This is called the
"delta." In fact, the term delta is commonly used in at least two different contexts: to
express the amount by which an option changes for a I-point move in the underlying
security, or to describe the equivalent stock position of an entire option portfolio.
Reviewing the definition of the delta of an individual option (first described in
Chapter 3), recall that the delta is a number that ranges between 0.0 and 1.0 for calls,
and between -1.0 and 0.0 for puts. It is the amount by which the option will move if
the underlying stock moves 1 point; stated another way, it is the percentage of any
stock price change that will be reflected in the change of price of the option.
Example: Assume an XYZ January 50 call has a delta of 0.50 with XYZ at a price of
49. This means that the call will move 50% as fast as the stock will move. So, if XYZ
jumps to 51, a gain of 2 points, then the January 50 call can be expected to increase
in price by 1 point (50% of the stock increase).
In another context, the delta of a call is often thought of as the probability of the
call being in-the-money at expiration. That is, ifXYZ is 50 and the January 55 call has
a delta of 0.40, then there is a 40% probability that XYZ will be over 55 at January
expiration.
Put deltas are expressed as negative numbers to indicate that put prices move
in the opposite direction from the underlying security. Recall that deltas of out-of
the-money options are smaller numbers, tending toward 0 as the option becomes
very far out-of-the-money. Conversely, deeply in-the-money calls have deltas
approaching 1.0, while deeply in-the-money puts have deltas approaching -1.0.
Note: Mathematically, the delta of an option is the partial derivative of the
Black-Scholes equation ( or whatever formula one is using) with respect to stock
price. Graphically, it is the slope of a line that is tangent to the option pricing curve.
Chapter 40: Advanced Concepts 849
Let us now take a look at how both volatility and time affect the delta of a call
option. Much of the data to be presented in this chapter will be in both tabular and
graphical form, since some readers prefer one style or the other.
The volatility of the underlying stock has an effect on delta. If the stock is not
volatile, then in-the-money options have a higher delta, and out-of-the-money
options have a lower delta. Figure 40-1 and Table 40-1 depict the deltas of various
calls on two stocks with differing volatilities. The deltas are shown for various strike
prices, with the time remaining to expiration equal to 3 months and the underlying
stock at a price of 50 in all cases. Note that the graph confirms the fact that a low
volatility stock's in-the-money options have the higher delta. The opposite holds true
for out-of-the-money options: The high-volatility stock's options have the higher delta
in that case. Another way to view this data is that a higher-volatility stock's options will
always have more time value premium than the low-volatility stock's. In-the-money,
these options with more time value will not track the underlying stock price move
ment as closely as ones with little or no time value. Thus, in-the-money, the low
volatility stock's options have the higher delta, since they track the underlying stock
price movements more closely. Out-of-the-money, the entire price of the option is
composed of time value premium. The ones with higher time value (the ones on the
high-volatility stock) will move more since they have a higher price. Thus, out-of-the
money, the higher-volatility stock's options have the greater delta.
Time also affects delta. Figures 40-2 (see Table 40-2) and 40-4 show the rela
tionships between time and delta. Figure 40-2's scales are similar to those in Figure
40-2, delta vs. volatility: The deltas are shown for various striking prices, with XYZ
assumed to be equal to 50 in all cases. Notice that in-the-money, the shorter-term
options have the higher delta. Again, this is because they have the least time value
premium. Out-of-the-money, the opposite is true: The longer-term options have the
higher deltas, since these options have the most time value premium.
Figure 40-3 (see Table 40-3) depicts the delta for an XYZ January 50 call with
XYZ equal to 50. The horizontal axis in this graph is "weeks until expiration." Note
that the delta of a longer-term at-the-money option is larger than that of a shorter
term option. In fact, the delta shrinks more rapidly as expiration draws nearer. Thus,
even if a stock remains unchanged and its volatility is constant, the delta of its options
will be altered as time passes. This is an important point to note for the strategist,
since he is constantly monitoring the risk characteristics of his position. He cannot
assume that his position is the same just because the stock has remained at the same
price for a period of time.
Position Delta. Another usage of the term delta is what has previously been
referred to as the equivalent stock position (ESP); for futures options, it would be
850
FIGURE 40-1.
Delta comparison, with XYZ = 50.
100
75
$
<ii 50
0 ',, .. ..........
Low-Volatility Stock ,
Part VI: Measuring and Trading Volatility
25 ', ', High-Volatility Stock
', ',, ..............
---..............
0 L..L _______ .....L _____


sage of the term delta is what has previously been
referred to as the equivalent stock position (ESP); for futures options, it would be
850
FIGURE 40-1.
Delta comparison, with XYZ = 50.
100
75
$
<ii 50
0 ',, .. ..........
Low-Volatility Stock ,
Part VI: Measuring and Trading Volatility
25 ', ', High-Volatility Stock
', ',, ..............
---..............
0 L..L _______ .....L _______ __:::::i~~ ..... -..:L-_
40 45 50
TABLE 40-1.
55
Strike Price
60 65
Delta comparison for different volatilities with XYZ = 50 and
time = 3 months.
Delta
Strike Price Low-Volatility Stock High-Volatility Stock
40 100 94
45 93 78
50 51 53
55 11 29
60 1 13
65 0 5
referred to as EFP (equivalent futures position). To differentiate between the two
terms, the delta of the option is generally referred to as "option delta," while the ESP
or EFP is called "position delta." Recall that the position delta is computed accord
ing to the following simple equation:
Position delta = Option's delta x Shares per option x Option quantity
Chapter 40: Advanced Concepts
FIGURE 40-2.
Delta comparison, with XYZ = 50.
100
75
~ ai 50
0
25
40
TABLE 40-2.
45 50 55
Strike Price
60 65
Delta comparison - varying time remaining with XYZ = 50.
Delta
Strike Price t = 1 year t = 6 months
40 92 97
45 79 83
50 61 57
55 41 30
60 25 12
65 14 4
851
t = 3 months
99
90
55
18
3
0
For futures options, the term "shares per option" would be replaced by "shares
per contract," which is always 1. This is the risk measurement of how much market
exposure the option position has. Whether called position delta, ESP, or EFP, one
uses the deltas of the individual options in his portfolio to calculate the overall expo
sure. By summing the calculations for each item in a position, or even in an entire
option portfolio, one can approximate how much market exposure the entire option
852
FIGURE 40-3.
Delta as a function of time.
57
56
55
.s
8 54
53
52
TABLE 40-3.
2 4 6
Delta as a function of time.
8 10 12
Weeks Remaining
Weeks Remaining
20
18
15
13
10
8
6
4
2
Part VI: Measuring and Trading Volatility
14 16 18 20
Delta
.566
.562
.557
.553
.547
.543
.538
.531
.521
.515
position has. The next example, reprinted from the chapter on mathematical appli
cations, shows how one computes the net exposure of a complicated position.
Example: The following position exists when XYZ is at 31. 75. It resembles a long
straddle (or backspread), in that there is increased profit potential in either direction
Chapter 40: Advanced Concepts 853
if the stock moves far enough by expiration. Many market-makers and professional
traders attempt to structure these types of positions, if possible, in order to take
advantage of the sudden volatility that is inherent in today's markets.
Position
Position Delta Delta
Short 4,500 XYZ 1.00 - 4,500
Short 1 00 XYZ April 25 calls 0.89 - 8,900
Long 50 XYZ April 30 calls 0.76 + 3,800
Long 139 XYZ July 30 calls 0.74 + 10,286
Total ESP: + 686
This position, though complicated to the naked eye, reduces to being long only
approximately 700 shares of XYZ. This is commonly referred to as being "delta long
700 shares." Thus, the terms "delta," when referring to the sum of the deltas of a
whole position, and "equivalent stock position" are synonymous.
This position has some exposure to the market since it is delta long. If the posi
tion delta were zero, it would be referred to as being delta neutral and would, theo
retically, have no exposure to the market at that time.
Note that one can derive some general characteristics of his delta by just exam
ining his portfolio by eye: Short calls or long puts will introduce negative delta into
the position; long calls or short puts will introduce positive delta. Furthermore, it is
obvious that being long the underlying security adds to the long delta of the position,
while being short the underlying security places more negative delta in the position.
The use of this information to adjust the delta of one's position will be discussed in a
later section of this chapter.
Obviously, the delta of this entire position will change as the stock price moves
up or down as time passes. The figure given is merely an instantaneous look at how
the position is structured. It is the need to know how the position will change when
other factors change that has led strategists to employ the following concepts.
GAMMA
Simply stated, the gamma is how fast the delta changes with respect to changes in the
underlying stock price. It is known that the delta of a call increases as the call moves
from out-of-the-money to in-the-money. The gamma is merely a precise measure
ment of how fast the delta is increasing.
854 Part VI: Measuring and Trading Volatility
Example: With XYZ at 49, assume the January 50 call has a delta of 0.50 and a
gamma of 0.05. If XYZ moves up one point to 50, the delta of the call will increase
by the amount of the gamma: It will increase from 0.50 to 0.55.
As with the delta, the gamma can also be expressed as a percentage. But in this
case, the increase or decrease applies to the delta.
Example: Again, with XYZ at 49, assume the January 50 call has a delta of 0.50 and a
gamma of 0.05. If XYZ moves up 2 points to 51, the delta of the call will increase by
5% of the stock rrwve, because the gamma is 0.05, or 5 percent. Five percent of the
stock move is 0.05 x 2, or 0.10. Thus, the delta will increase by 0.10, from 0.50 to 0.60.
Obviously, the delta cannot keep increasing by 0.05 each time XYZ gains anoth
er point in price, for it will eventually exceed 1.00 by that calculation, and it is known
that the delta has a maximum of 1.00. Thus, it is obvious that the gamma changes. In
general, the gamma is at its maximum point when the stock is near the strike of the
option. As the stock moves away from the strike in either direction, the gamma
decreases, approaching its minimum val