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

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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.

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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 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

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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.

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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

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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 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

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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-

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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.

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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 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

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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

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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

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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

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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%

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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 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

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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:

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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

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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 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-

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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-

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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 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.

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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. 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.

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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

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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

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886 Part VI: Measuring and Trading Volatility
that is somewhat variable. But, for the purposes of such a projection, it is acceptable
to use the current volatility. The results of as many as 9 stock prices might be dis
played: every one-half standard deviation from -2 through + 2 (-2.0, -1.5, -1.0,
-0.5, 0, 0.5, 1.0, 1.5, 2.0).
Example: XYZ is at 60 and has a volatility of 35%. A distribution of stock prices 7
days into the future would be determined using the equation:
Future Price = Current Price x eav-ft
where
a corresponds to the constants in the following table: (-2.0 ... 2.0):
# Standard Deviations
-2.0
- 1.5
- 1.0
-0.5
0
0.5
1.0
1.5
2.0
Projected Stack Price
54.46
55.79
57.16
58.56
60.00
61.47
62.98
64.52
66.11
Again, refer to Chapter 28 on mathematical applications for a more in-depth
discussion of this price determination equation.
Note that the formula used to project prices has time as one of its components.
This means that as we look further out in time, the range of possible stock prices will
expand - a necessary and logical component of this analysis. For example, if the
prices were being determined 14 days into the future, the range of prices would be
from 52.31 to 68.82. That is, XYZ has the same probability of being at 54.46 in 7 days
that it has of being at 52.31 in 14 days. At expiration, some 90 days hence, the range
would be quite a bit wider still. Do not make the mistake of trying to evaluate the
position at the same prices for each time period (7 days, 14 days, 1 rnonth, expiration,
etc.). Such an analysis would be wrong.
Once the appropriate stock prices have been determined, the following quanti
ties would be calculated for each stock price: profit or loss, position delta, position
gamma, position theta, and position vega. (Position rho is generally a less important
risk measure for stock and futures short-term options.) Armed with this information,
the strategist can be prepared to face the future. An important item to note: A model

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Chapter 40: Advanced Concepts 887
will necessarily be used to make these projections. As was shown earlier, if there is a
distortion in the current implied volatilities of the options involved in the position,
the strategist should use the current implieds as input to the model for future option
price projections. If he does not, the position may look overly attractive if expensive
options are being sold or cheap ones are being bought. A truer profit picture is
obtained by propagating the current implied volatility structure into the near future.
Using an example similar to the previous one a ratio spread using short stock
to make it delta neutral - the concepts will be described.
Initial Position. XYZ is at 60. The January 70 calls, which have three months
until expiration, are expensive with respect to the January 60 calls. A strategist
expects this discrepancy to disappear when the implied volatility of XYZ options
decreases. He therefore established the following position, which is both gamma
and delta neutral.
Position Delta Gamma
Long 100 January 60 calls 0.57 0.0723
Short 240 January 70 calls 0.20 0.0298
Short 800 XYZ
The risk measures for the entire position are:
Position delta: -38 shares (virtually delta neutral)
Position gamma: + 7 shares (gamma neutral)
Position theta: + $263
Position vega: -$827
Theta Vega
-0.020 0.109
-0.019 0.080
Thus, the position is both gamma and delta neutral. Moreover, it has the attrac
tive feature of making $263 per day because of the positive theta. Finally, as was the
intention of the spreader, it will make money if the volatility of XYZ declines: $827
for each percentage point decrease in implied volatility. Two equations in two
unknowns (gamma and vega) were solved to obtain the quantities to buy and sell. The
resulting position delta was neutralized by selling 800 XYZ.
The following analyses will assume that the relative expensiveness of the April
70 calls persists. These are the calls that were sold in the position. If that overpricing
should disappear, the spread would look more favorable, but there is no guarantee
that they will cheapen - especially over a short time period such as one or two weeks.
How would the position look in 7 days at the stock prices determined above?

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888 Part VI: Measuring and Trading Vo/atillty
Stock
Price P&L Delta Gamma Theta Vega
54.46 1905 - 7.40 1.62 0.94 - 1.57
55.79 1077 - 4.90 2.07 1.18 - 1.96
57.16 606 1.97 2.13 1.53 - 2.90
58.56 528 0.74 1.65 2.00 -4.62
60.00 771 2.38 0.56 2.63 -7.22
61.47 1127 2.07 - 1.01 3.38 -10.63
62.98 1252 - 0.87 - 2.85 4.22 -14.56
64.52 702 - 6.73 - 4.67 5.07 -18.61
66.11 - 1019 -15.42 - 6.21 5.85 -22.31
In a similar manner, the position would have the following characteristics after
14 days had passed:
Stock
Price P&L Delto Gamma Theta Vega
52.31 4221 - 9.10 0.69 0.55 - 0.98
54.14 2731 - 6.93 1.69 0.75 - 0.89
56.02 1782 - 2.87 2.51 1.06 - 1.21
57.98 1717 2.17 2.44 1.61 - 2.69
60.00 2577 5.85 1.00 2.51 -6.00
62.09 3839 5.29 - 1.63 3.73 -11.05
64.26 4361 - 1.55 - 4.61 5.09 -16.90
66.50 2631 -14.80 - 7.02 6.31 -22.17
68.82 - 2799 -32.83 - 8.32 7.18 -25.72
The same information will be presented graphically in Figure 40-13 so that
those who prefer pictures instead of columns of numbers can follow the discussions
easily.
First, the profitability of the spread can be examined. This profit picture
assumes that the volatility of XYZ remains unchanged. Note that in 7 days, there is a
small profit if the stock remains unchanged. This is to be expected, since theta was
positive, and therefore time is working in favor of this spread. Likewise, in 14 days,
there is an even bigger profit if XYZ remains relatively unchanged - again due to the
positive theta. Overall, there is an expected profit of $800 in 7 days, or $2,600 in 14
days, from this position. This indicates that it is an attractive situation statistically, but,
of course, it does not mean that one cannot lose money.

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904 Part VI: Measuring and Trading Volatility
Recall that, in the same example used to describe gamma, the position was delta
long 686 shares and had a positive gamma of 328 shares. Furthermore, we now see
that the gamma itself is going to decrease as the stock moves up ( it is negative) or will
increase as the stock moves down. In fact, it is expected to increase or decrease by
22 shares for each point XYZ moves.
So, if XYZ moves up by 1 point, the following should happen:
a. Delta increases from 686 to 1,014, increasing by the amount of the gamma.
b. Gamma decreases from 328 to 306, indicating that a further upward move by
XYZ will result in a smaller increase in delta.
One can build a general picture of how the gamma of the gamma changes over
different situations - in- or out-of-the-money, or with more or less time remaining
until expiration. The following table of two index calls, the January 350 with one
month of life remaining and the December 350 with eleven months of life remain
ing, shows the delta, gamma, and gamma of the gamma for various stock prices.
Index January 350 call December 350 call
Price Delta Gamma Gamma/Gamma Delta Gamma Gamma/Gamma
310 .0006 .0001 .0000 .3203 .0083 .0000
320 .0087 .0020 .0004 .3971 .0082 .0000
330 .0618 .0100 .0013 .4787 .0080 -.0000
340 .2333 .0744 .0013 .5626 .0078 -.0001
350 .5241 .0309 -.0003 .6360 .0073 -.0001
360 .7957 .0215 -.0014 .6984 .0067 -.0001
370 .9420 .0086 -.0010 .7653 .0060 -.0001
380 .9892 .0021 -.0003 .8213 .0052 -.0001
Several conclusions can be drawn, not all of which are obvious at first glance.
First of all, the gamma of the gamma for long-term options is very small. This should
be expected, since the delta of a long-term option changes very slowly. The next fact
can best be observed while looking at the shorter-term January 350 table. The
gamma of the gamma is near zero for deeply out-of-the-money options. But, as the
option comes closer to being in-the-money, the gamma of the gamma becomes a pos
itive number, reaching its maximum while the option is still out-of-the-money. By the
time the option is at-the-money, the gamma of the gamma has turned negative. It
then remains negative, reaching its most negative point when slightly in-the-money.
From there on, as the option goes even deeper into-the-money, the gamma of the
gamma remains negative but gets closer and closer to zero, eventually reaching
(minus) zero when the option is very far in-the-money.

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Chapter 40: Advanced Concepts 905
Can one possibly reason this risk measurement out without making severe
mathematical calculations? Well, possibly. Note that the delta of an option starts as a
small number when the option is out-of-the-money. It then increases, slowly at first,
then more quickly, until it is just below 0.60 for an at-the-money option. From there
on, it will continue to increase, but much more slowly as the option becomes in-the
money. This movement of the delta can be observed by looking at gamma: It is the
change in the delta, so it starts slowly, increases as the stock nears the strike, and then
begins to decrease as the option is in-the-money, always remaining a positive num
ber, since delta can only change in the positive direction as the stock rises. Finally,
the gamma of the gamma is the change in the gamma, so it in tum starts as a positive
number as gamma grows larger; but then when gamma starts tapering off, this is
reflected as a negative gamma of the gamma.
In general, the gamma of the gamma is used by sophisticated traders on large
option positions where it is not obvious what is going to happen to the gamma as the
stock changes in price. Traders often have some feel for their delta. They may even
have some feel for how that delta is going to change as the stock moves (i.e., they
have a feel for gamma). However, sophisticated traders know that even positions that
start out with zero delta and zero gamma may eventually acquire some delta. The
gamma of the gamma tells the trader how much and how soon that eventual delta will
be acquired.
MEASURING THE DIFFERENCE OF IMPLIED VOLATILITIES
Recall that when the topic of implied volatility was discussed, it was shown that if one
could identify situations in which the various options on the same underlying securi
ty had substantially different implied volatilities, then there might be an attractive
neutral spread available. The strategist might ask how he is to determine if the dis
crepancies between the individual options are significantly large to warrant attention.
Furthermore, is there a quick way (using a computer, of course) to determine this?
A logical way to approach this is to look at each individual implied volatility and
compute the standard deviation of these numbers. This standard deviation can be
converted to a percentage by dividing it by the overall implied volatility of the stock.
This percentage, if it is large enough, alerts the strategist that there may be opportu
nities to spread the options of this underlying security against each other. An exam
ple should clarify this procedure.
Example: XYZ is trading at 50, and the following options exist with the indicated
implied volatilities. We can calculate a standard deviation of these implieds, called
implied deviation, via the formula:

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906 Part VI: Measuring and Trading Volatility
Implied deviation = sqrt (sum of differences from mean) 2/(# options - 1)
XYZ:50
Implied
Option Volatility
October 45 call 21%
November 45 call 21%
January 45 call 23%
October 50 call 32%
November 50 call 30%
January 50 call 28%
October 55 call 40%
November 55 call 37%
January 55 call 34%
Average: 30.44%
Sum of ( difference from avg)2 = 389.26
Implied deviation = sqrt (sum of diff)2/(# options - 1)
= sqrt (389.26 I 8)
= 6.98
Difference
from Average
-9.44
-9.44
-7.44
+ 1.56
-0.44
-2.44
+9.56
+6.56
+3.56
This figure represents the raw standard deviation of the implied volatilities. To
convert it into a useful number for comparisons, one must divide it by the average
implied volatility.
P d . . Implied deviation ercent eV1at10n = A . 1. d verage imp ie
= 6.98/30.44
= 23%
This "percent deviation" number is usually significant if it is larger than 15%.
That is, if the various options have implied volatilities that are different enough from
each other to produce a result of 15% or greater in the above calculation, then the
strategist should take a look at establishing neutral spreads in that security or futures
contract.
The concept presented here can be refined further by using a weighted average
of the implieds ( taking into consideration such factors as volume and distance from the
striking price) rather than just using the raw average. That task is left to the reader.

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916 Part VI: Measuring and Trading Volatllity
Thus, this covered writer has a net gain of $1,125 and it is a long-term gain because
the stock was held for more than one year (from September 1st of the year in which
he bought it, to October expiration of the next year, when the stock was called away).
Note that in a similar situation in which the stock had been held for less than
one year before being called away, the gain would be short-term.
Let us now look at the other two rules. They are related in that their differen
tiation relies on the definition of "too deeply in-the-money." They come into play
only if the stock was not already held long-term when the call was written. If the writ
ten call is too deeply in-the-money, it can eliminate the holding period of short-term
stock. Otherwise, it can suspend it. If the call is in-the-money, but not too deeply in
the-money, it is referred to as a qualified covered call. There are several rules regard
ing the determination of whether an in-the-money call is qualified or not. Before
actually getting to that definition, which is complicated, let us look at two examples
to show the effect of the call being qualified or not qualified.
Example: Qualified Covered Write: On March 1st, an investor buys 100 XYZ at 35.
He holds the stock for 3% months, and, on July 15th, the stock has risen to 43. This
time he sells an in-the-money call, the October 40 call for 6. By October expiration,
the stock has declined and the call expires worthless.
He would now have the following situation: a $575 short-term gain from the
sale of the call, plus he is long 100 XYZ with a holding period of only 3% months.
Thus, the sale of the October call suspended his holding period, but did not elimi
nate it.
He could now hold the stock for another 8½ months and then sell it as a long
term item.
If the stock in this example had stayed above 40 and been called away, the net
result would have been that the option proceeds would have been added to the stock
sale price as in previous examples, and the entire net gain would have been short
term due to the fact that the writing of the qualified covered call had suspended the
holding period of the stock at 3½ months.
That example was one of writing a call which was not too deeply in-the-money.
If, however, one writes a call on stock that is not yet held long-term and the call is too
deeply in-the-money, then the holding period of the stock is eliminated. That is, if the
call is subsequently bought back or expires worthless, the stock must then be held for
another year in order to qualify as a long-term investment. This rule can work to an
investor's advantage. If one buys stock and it goes down and he is in jeopardy of hav
ing a long-term loss, but he really does not want to sell the stock, he can sell a call

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Chapter 41: Taxes 917
that is too deeply in-the-money (if one exists), and eliminate the holding period on
the stock
Qualified Covered Call. The preceding examples and discussion summa
rize the covered writing rules. Let us now look at what is a qualified covered call.
The following rules are the literal interpretation. Most investors work from
tables that are built from these rules. Such a table may be found in Appendix E.
(Be aware that these rules may change, and consult a tax advisor for the latest
figures.) A covered call is qualified if:
1. the option has more than 30 days of life remaining when it is written, and
2. the strike of the written call is not lower than the following benchmarks:
a. First determine the applicable stock price (ASP). That is normally the closing
price of the stock on the previous day. However, if the stock opens more than
ll0% higher than its previous close, then the applicable stock price is that
higher opening.
b. If the ASP is less than $25, then the benchmark strike is 85% of ASP. So any
call written with a strike lower than 85% of ASP would not be qualified. (For
example, if the stock was at 12 and one wrote a call with a striking price of 10,
it would not be qualified- it is too deeply in-the-money.)
c. If the ASP is between 25.13 and 60, then the benchmark is the next lowest
strike. Thus, if the stock were at 39 and one wrote a call with a strike of 35, it
would be qualified.
d. If the ASP is greater than 60 and not higher than 150, and the call has more
than 90 days of life remaining, the benchmark is two strikes below the ASP.
There is a further condition here that the benchmark cannot be more than 10
points lower than the ASP. Thus, if a stock is trading at 90, one could write a
call with a strike of 80 as long as the call had more than 90 days remaining
until expiration, and still be qualified.
e. If the ASP is greater than 150 and the call has more than 90 days of life remain
ing, the benchmark is two strikes below the ASP. Thus, if there are 10-point
striking price intervals, then one could write a call that was 20 points in-the
money and still be qualified. Of course, if there are 5-point intervals, then one
could not write a call deeper than 10 points in-the-money and still be qualified.
These rules are complicated. That is why they are summarized in Appendix E.
In addition, they are always subject to change, so if an investor is considering writing
an in-the-money covered call against stock that is still short-term in nature, he should
check with his tax advisor and/or broker to determine whether the in-the-money call
is qualified or not.

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918 Part VI: Measuring and Trading Volatility
There is one further rule in connection with qualified calls. Recall that we stat
ed that the above rules apply only if the stock is not yet held long-term when the call
is written. If the stock is already long-term when the call is written, then it is consid
ered long-term when called away, regardless of the position of the striking price when
the call was written. However, if one sells an in-the-money call on stock already held
long-term, and then subsequently buys that call back at a loss, the loss on the call
must be taken as a long-term loss because the stock was long-term.
Overall, a rising market is the best, taxwise, for the covered call writer. If he
writes out-of-the-money calls and the stock rises, he could have a short-term loss on
the calls plus a long-term gain on the stock.
Example: On January 2nd of a particular year, an investor bought 100 shares of XYZ
at 32, paying $75 in commissions, and simultaneously wrote a July 35 call for 2 points.
The July 35 expired worthless, and the investor then wrote an October 35 call for 3
points. In October, with XYZ at 39, the investor bought back the October 35 call for
6 points (it was in-the-money) and sold a January 40 call for 4 points. In January, on
the expiration day, the stock was called away at 40. The investor would have a long
term capital gain on his stock, because he had held it for more than one year. He
would also have two short-term capital transactions from the July 35 and October 35
calls. Tables 41-2 and 41-3 show his net tax treatment from operating this covered
writing strategy. The option commission on each trade was $25.
Things have indeed worked out quite well, both profit-wise and tax-wise, for this
covered call writer. Not only has he made a net profit of $850 from his transactions on
the stock and options over the period of one year, but he has received very favorable
tax treatment. He can take a short-term loss of $175 from the combined July and
October option transactions, and is able to take the $1,025 gain as a long-term gain.
TABLE 41-2.
Summary of trades.
January 2
July
October
January
Bought 100 XYZ at 32
Sold 1 July 35 call at 2
July call expired worthless (XYZ at 32)
Sold 1 October 35 call at 3
Bought back October 35 call for 6 points (XYZ at 39)
Sold 1 January 40 call for 4 points
(of the following year)
1 00 XYZ called away at 40

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Chapter 41: Taxes
TABLE 41-3.
Tax treatment of trades.
Short-term capital items:
July 35 call: Net proceeds ($200 - $25)
Net cost {expired worthless)
Short-term capital gain
October 35 call: Net proceeds ($300 - $25)
Net cost ($600 + $25)
Short-term capital loss
919
$175
0
$175
$275
- 625
($350)
Long-term capital item:
100 shares XYZ: Purchased January 2 of one year and sold at January
expiration of the following year. Therefore, held for
more than one year, qualifying for long-term treatment.
Net sale proceeds of stock {assigned call):
January 40 call sale proceeds
($400 - $25)
Sold 1 00 XYZ at 40 strike
{$4,000 $75)
Net cost of stock (January 2 trade):
Bought 100 at 32 {$3,200 + $75)
Long-term capital gain
$375
+ 3,925
$4,300
- 3,275
$1,025
This example demonstrates an important tax consequence for the covered call
writer: His optimum scenario tax-wise is a rising market, for he may be able to
achieve a long-term gain on the underlying stock if he holds it for at least one year,
while simultaneously subtracting short-term losses from written calls that were
closed out at higher prices. Unfortunately, in a declining market, the opposite result
could occur: short-term option gains coupled with the possibility of a long-term loss
on the underlying stock. There are ways to avoid long-term stock losses, such as buy
ing a put ( discussed later in the chapter) or going short against the box before the
stock becomes long-term. However, these maneuvers would interrupt the covered
writing strategy, which may not be a wise tactic.
In summary, then, the covered call writer who finds himself with an in-the
money call written and expiration date drawing near may have several alternatives
open to him. If the stock is not yet held long-term, he might elect to buy back the
written call and to write another call whose expiration date is beyond the date
required for a long-term holding period on the stock. This is apparently what the
hypothetical investor in the preceding example did with his October 35 call. Since

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920 Part VI: Measuring and Trading VolatiRty
that call was in-the-money, he could have elected to let the call be assigned and to
take his profit on the position at that time. However, this would have produced a
short-term gain, since the stock had not yet been held for one year, so he elected
instead to terminate the October 35 call through a closing purchase transaction and
to simultaneously write a call whose expiration date exceeded the one year period
required to make the stock a long-term item. He thus wrote the January 40 call,
expiring in the next year. Note that this investor not only decided to hold the stock
for a long-term gain, but also decided to try for more potential profits: He rolled the
call up to a higher striking price. This lets the holding period continue. An in-the
money write would have suspended it.
DELIVERING .,.,NEW" STOCK TO AVOID A LARGE LONG· TERM GAIN
Some covered call writers may not want to deliver the stock that they are using to
cover the written call, if that call is assigned. For example, if a covered writer were
writing against stock that had an extremely low cost basis, he might not be willing to
take the tax consequences of selling that particular stock holding. Thus, the writer of
a call that is assigned may sometimes wish to buy stock in the open market to deliv
er against his assignment, rather than deliver the stock he already owns. Recall that
it is completely in accordance with the Options Clearing Corporation rules for a call
writer to buy stock in the open market to deliver against an assignment. For tax pur
poses, the confirmation that the investor receives from his broker for the sale of the
stock via assignment should clearly specify which particular shares of stock are being
sold. This is usually accomplished by having the confirmation read "Versus Purchase"
and listing the purchase date of the stock being sold. This is done to clearly identify
that the "new" stock, and not the older long-term stock, is being delivered against the
assignment. The investor must give these instructions to his broker, so that the
brokerage firm puts the proper notation on the confirmation itself. If the investor
realizes that his stock might be in danger of being called away and he wants to avail
himself of this procedure, he should discuss it with his broker beforehand, so that the
proper procedures can be enacted when the stock is actually called away.
Example: An investor owns 100 shares ofXYZ and his cost basis, after multiple stock
splits and stock dividends over the years, is $2 per share. With XYZ at 50, this investor
decides to sell an XYZ July 50 call for 5 points to bring in some income to his port
folio. Subsequently, the call is assigned, but the investor does not want to deliver his
XYZ, which he owns at a cost basis of $2 per share, because he would have to pay cap
ital gains on a large profit. He may go into the open market and buy another 100
shares of XYZ at its current market price for delivery against the assignment notice.

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Chapter 41: Taxes 92S
Another usage of the put purchase, for tax purposes, might be to avoid a long
term loss on a stock position. If an investor owns a stock that has declined in price
and also is about to become a long-term holding, he can buy a put on that stock to
eliminate the holding period. This avoids having to take a long-term loss. Once the
put is removed, either by its sale or by its expiring worthless, the stock holding peri
od would begin all over again and it would be a short-term position. In addition, if
the investor should decide to exercise the put that he purchased, the result would be
a short-term loss. The sale basis of the stock upon exercise of the put would be equal
to the striking price of the put less the amount of premium paid for the put, less all
commission costs. Furthermore, note that this strategy does not lock in the loss on
the underlying stock. If the stock rallies, the investor would be able to participate in
that rally, although he would probably lose all of the premium that he paid for the
put. Note that both of these long-term strategies can be accomplished via the sale of
a deeply in-the-money call as well.
SUMMARY
This concludes the section of the tax chapter dealing with listed option trades and
their direct consequences on option strategies. In addition to the basic tax treatment
for option traders of liquidation, expiring worthless, or assignment or exercise, sev
eral other useful tax situations have been described. The call buyer should be aware
of the wash sale rule. The put buyer must be aware of the short sale rules involving
both put and stock ownership. The call writer should realize the beneficial effects of
selling an in-the-money call to protect the underlying stock, while waiting for a real
ization of profit in the following tax year. The put writer may be able to avoid a wash
sale by utilizing an in-the-money put write, while still retaining profit potential from
a rally by the underlying stock.
TAX PLANNING STRATEGIES FOR EQUITY OPTIONS
DEFERRING A SHORT· TERM CALL GAIN
The call holder may be interested in either deferring a gain until the following year
or possibly converting a short-term gain on the call into a long-term gain on the stock.
It is much easier to do the former than the latter. A holder of a profitable call that is
due to expire in the following year can take any of three possible actions that might
let him retain his profit while deferring the gain until the following tax year. One way
in which to do this would be to buy a put option. Obviously, he would want to buy an

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926 Part VI: Measuring and Trading Volatillty
in-the-money put for this purpose. By so doing, he would be spending as little as pos
sible in the way of time value premium for the put option and he would also be lock
ing in his gain on the call. The gains and losses from the put and call combination
would nearly equal each other from that time forward as the stock moves up or down,
unless the stock rallies strongly, thereby exceeding the striking price of the put. This
would be a happy event, however, since even larger gains would accrue. The combi
nation could be liquidated in the following tax year, thus achieving a gain.
Example: On September 1st, an investor bought an XYZ January 40 call for 3 points.
The call is due to expire in the following year. XYZ has risen in price by December
1st, and the call is selling for 6 points. The call holder might want to take his 3-point
gain on the call, but would also like to defer that gain until the following year. He
might be able to do this by buying an XYZ January 50 put for 5 points, for example.
He would then hold this combination until after the first of the new year. At that
time, he could liquidate the entire combination for at least 10 points, since the strik
ing price of the put is 10 points greater than that of the call. In fact, if the stock should
have climbed to or above 50 by the first of the year, or should have fallen to or below
40 by the first of the year, he would be able to liquidate the combination for more
than 10 points. The increase in time value premium at either strike would also be a
benefit. In any case, he would have a gain - his original cost was 8 points (3 for the
call and 5 for the put). Thus, he has effectively deferred taking the gain on the orig
inal call holding until the next tax year. The risk that the call holder incurs in this type
of transaction is the increased commission charges of buying and selling the put as
well as the possible loss of any time value premium in the put itself. The investor
must decide for himself whether these risks, although they may be relatively small,
outweigh the potential benefit from deferring his tax gain into the next year.
Another way in which the call holder might be able to defer his tax gain into the
next year would be to sell another XYZ call against the one that he currently holds.
That is, he would create a spread. To assure that he retains as much of his current
gain as possible, he should sell an in-the-money call. In fact, he should sell an in-the
money call with a lower striking price than the call held long, if possible, to ensure
that his gain remains intact even if the underlying stock should collapse substantial
ly. Once the spread has been established, it could be held until the following tax year
before being liquidated. The obvious risk in this means of deferring gain is that one
could receive an assignment notice on the short call. This is not a remote possibility,
necessarily, since an in-the-money call should be used as protection for the current
gain. Such an assignment would result in large commission costs on the resultant pur
chase and sale of the underlying stock, and could substantially reduce one's gain.

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Chapter 41: Taxes 927
Thus, the risk in this strategy is greater than that in the previous one (buying a put),
but it may be the only alternative available if puts are not traded on the underlying
stock in question.
Example: An investor bought an XYZ February 50 call for 3 points in August. In
December, the stock is at 65 and the call is at 15. The holder would like to "lock in"
his 12-point call profit, but would prefer deferring the actual gain into the following
tax year. He could sell an XYZ February 45 call for approximately 20 points to do this.
If no assignment notice is received, he will be able to liquidate the spread at a cost
of 5 points with the stock anywhere above 50 at February expiration. Thus, in the end
he would still have a 12-point gain - having received 20 points for the sale of the
February 45 and having paid out 3 points for the February 50 plus 5 points to liqui
date the spread to take his gain. If the stock should fall below 50 before February
expiration, his gain would be even larger, since he would not have to pay out the
entire 5 points to liquidate the spread.
The third way in which a call holder could lock in his gain and still defer the gain
into the following tax year would be to sell the stock short while continuing to hold
the call. This would obviously lock in the gain, since the short sale and the call pur
chase will offset each other in profit potential as the underlying stock moves up or
down. In fact, if the stock should plunge downward, large profits could accrue.
However, there is risk in using this strategy as well. The commission costs of the short
sale will reduce the call holder's profit. Furthermore, if the underlying stock should
go ex-dividend during the time that the stock is held short, the strategist will be liable
for the dividend as well. In addition, more margin will be required for the short stock.
The three tactics discussed above showed how to defer a profitable call gain into
the following tax year. The gain would still be short-term when realized. The only way
in which a call holder could hope to convert his gain into a long-term gain would be
to exercise the call and then hold the stock for more than one year. Recall that the
holding period for stock acquired through exercise begins on the day of exercise - the
option's holding period is lost. If the investor chooses this alternative, he of course is
spending some of his gains for the commissions on the stock purchase as well as sub
jecting himself to an entire year's worth of market risk. There are ways to protect a
stock holding while letting the holding period accrue - for example, writing out-of
the-money calls - but the investor who chooses this alternative should carefully
weigh the risks involved against the possible benefits of eventually achieving a long
term gain. The investor should also note that he will have to advance considerably
more money to hold the stock.

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Glossary 967
Cycle: the expiration dates applicable to various classes of options. There are three
cycles: January/April/July/October, February/May/ August/November, and
March/J une/Septem ber/Decem ber.
Debit: an expense, or money paid out from an account. A debit transaction is one in
which the net cost is greater than the net sale proceeds. See also Credit.
Deliver: to take securities from an individual or firm and transfer them to another
individual or firm. A call writer who is assigned must deliver stock to the call hold
er who exercised. A put holder who exercises must deliver stock to the put writer
who is assigned.
Delivery: the process of satisfying an equity call assignment or an equity put exer
cise. In either case, stock is delivered. For futures, the process of transferring the
physical commodity from the seller of the futures contract to the buyer.
Equivalent delivery refers to a situation in which delivery may be made in any of
various, similar entities that are equivalent to each other (for example, Treasury
bonds with differing coupon rates).
Delta: (1) the amount by which an option's price will change for a corresponding 1-
point change in price by the underlying entity. Call options have positive deltas,
while put options have negative deltas. Technically, the delta is an instantaneous
measure of the option's price change, so that the delta will be altered for even frac
tional changes by the underlying entity. Consequently, the terms "up delta" and
"down delta" may be applicable. They describe the option's change after a full 1-
point change in price by the underlying security, either up or down. The "up delta"
may be larger than the "down delta" for a call option, while the reverse is true for
put options. (2) the percent probability of a call being in-the-money at expiration.
See also Hedge Ratio.
Delta Neutral Spread: a ratio spread that is established as a neutral position by uti
lizing the deltas of the options involved. The neutral ratio is determined by divid
ing the delta of the purchased option by the delta of the written option. See also
Delta, Ratio Spread.
Depository Trust Corporation (OTC): a corporation that will hold securities for
member institutions. Generally used by option writers, the DTC facilitates and
guarantees delivery of underlying securities when assignment is made against
securities held in DTC.
Diagonal Spread: any spread in which the purchased options have a longer matu
rity than do the written options, as well as having different striking prices. Typical

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