Chapter 39: Volatility Trading Techniques 841
benefit from the fact that the options are skewed in terms of implied volatility, it is not
aneutral strategy. It requires that the underlying drop in price in order to become
profitable. There is nothing wrong with using adirectional 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 asevere 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 apotentially profitable situation for the strategist.
The backspread strategy would work best for bullish investors, although some
backspreads can be created for credits, so alittle money could also be made if the
index fell. In theory, astrategist could implement both strategies simultaneously,
which would give him an edge over awide 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, aneutral trader would choose either the ratio spread or the
backspread - not both. As ageneral rule of thumb, one would use the ratio spread
strategy if the current level of implied volatility were in ahigh percentile. The back
spread strategy would be used if implied volatility were in alow percentile current
ly. In that way, amovement 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'sexample that one
should probably continue to project his profits using the distorted volatilities that
were present when he establishes aposition. This is aconservative approach, but acorrect one. In the case of these OEX spreads, it may be abenefit.
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 avolatility of 16%; or if it fell to 560, then the 560
puts and calls would have avolatility of 16%. Of course, 16% is just arepresentative
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 aconstant volatility.
Example: Initially, atrader establishes acall 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
Aneutral 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 alater date, but well before expiration.
This is not aparticularly attractive price for this position. Recall that, at expiration, abackspread has its worst result at the striking price of the purchased options. Even
prior to expiration, one would not expect to have aprofit 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 aslightly 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 abenefit. 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 alesser 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 aform 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 amuch 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.
Astrategist 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 areverse (or negative) volatility skew.
First would be acall bull spread, second would be aput backspread, and third would
be acall 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 adirectional 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 alow percentile. Finally, the call ratio spread has agreat 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 ahigh percentile.
As ageneral comment, it should be noted that if the volatility skew disappears
while the trader has the position in place, aprofit 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 amore 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 aposition 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 apositive expected return.
SUMMARY OF VOLATILITY TRADING
The theoretical trading of options, mostly in aneutral 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 aparticular 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 amore 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'trequire one to monitor them continuously. If one owns astraddle,
any major market movements resulting in gaps in prices are abenefit. Hence, mon
itoring of positions as little as just once aday is sufficient, afact that means that the
volatility buyer can have alife apart from watching atrading 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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, 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 apositive 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'sposition, 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'sprice. 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

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Chapter 40: Advanced Concepts 847
Most option strategies fall into one of two categories: as ahedge to astock or
futures strategy (for example, buying puts to protect aportfolio of stocks), or as aprofit venture unto itself. The latter category is where most traders find themselves,
and they often approach it in afairly speculative manner - either by buying options
or by being apremium seller (covered or uncovered). In such strategies, the trader
is taking aview 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 acon
servative strategy, is subject to large losses if the underlying stock drops drastically.
It doesn'thave to be that way. Strategies can be devised that will have achance
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 - aspread, straddle, or some other combination. Often, when
one constructs aneutral 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 amoney-maker; rather, one looks for an
opportunity - perhaps an overpriced series of options - and attempts to capture that
overpricing by constructing aneutral strategy around it. Then, regardless of the
movement of the underlying stock, the strategist has achance 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 ahuge 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 anew
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.

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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 anumber 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 adelta of 0.50 with XYZ at aprice of
49. This means that the call will move 50% as fast as the stock will move. So, if XYZ
jumps to 51, again 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 acall 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
adelta 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 aline that is tangent to the option pricing curve.

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Chapter 40: Advanced Concepts 849
Let us now take alook at how both volatility and time affect the delta of acall
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 ahigher delta, and out-of-the-money
options have alower 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 aprice of 50 in all cases. Note that the graph confirms the fact that alow
volatility stock'sin-the-money options have the higher delta. The opposite holds true
for out-of-the-money options: The high-volatility stock'soptions have the higher delta
in that case. Another way to view this data is that ahigher-volatility stock'soptions 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'soptions 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 ahigher price. Thus, out-of-the
money, the higher-volatility stock'soptions 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'sscales 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 alonger-term at-the-money option is larger than that of ashorter
term option. In fact, the delta shrinks more rapidly as expiration draws nearer. Thus,
even if astock 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 aperiod 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'sdelta x Shares per option x Option quantity

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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'smarkets.
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 awhole 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'sposition will be discussed in alater 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 acall increases as the call moves
from out-of-the-money to in-the-money. The gamma is merely aprecise measure
ment of how fast the delta is increasing.

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854 Part VI: Measuring and Trading Volatility
Example: With XYZ at 49, assume the January 50 call has adelta of 0.50 and agamma 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 apercentage. But in this
case, the increase or decrease applies to the delta.
Example: Again, with XYZ at 49, assume the January 50 call has adelta of 0.50 and agamma 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 amaximum 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 value of zero.
Conceptually, this means that adeeply in-the-money or deeply out-of-the
money option has agamma of nearly zero. This makes sense - it implies that the delta
of adeep in- or deep out-of-the-money option does not change very much at all, even
if the stock moves by one point.
Example: Assume XYZ is 25, and the January 50 call has adelta of virtually zero. If
XYZ moves up one point to 26, the call is still so far out-of-the-money that the delta
will still be zero. Thus, the gamma of this call is zero, since the delta does not change
when the stock increases in price by apoint.
In asimilar manner, the January 45 put on XYZ would have adelta of -1.0 with
XYZ at 25. If XYZ moved up one point to 26, the put' sdelta would not change; it is
still so far in-the-money that it would still be - 1.0. Thus, the gamma of this deeply
in-the-money option is also zero, since the delta remains unchanged in the face of a
I-point rise in the underlying security.
Note that the gamma of any option is expressed as apositive number, whether
the option is aput or acall.
Other properties of gamma are useful to know as well. As expiration nears, the
gamma of at-the-money options increases dramatically. Consider an option with aday
or two of life remaining. If it is at-the-money, the delta is approximately 0.50.
However, if the stock were to move 2 points higher, the delta of the option would jump

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Chapter 40: Advanced Concepts 855
to nearly 1.00 because of the short time remaining until expiration. Thus, the gamma
would be roughly 0.25 (the delta increased by 0.50 when the stock moved 2 points),
as compared to much smaller values of gamma for at-the-money options with several
weeks or months of life remaining. The same 2-point rise in the underlying stock
would not result in much of an increase in the delta of longer-term options at all.
Out-of-the-money options display adifferent relationship between gamma and
time remaining. An out-of-the-money option that is about to expire has avery small
delta, and hence avery small gamma. However, if the out-of-the-money option has asignificant amount of time remaining, then it will have alarger gamma than the
option that is close to expiration.
Figure 40-4 (see Table 40-4) depicts the gammas of three options with varying
amounts of time remaining until expiration. The properties regarding the relation
ship of gamma and time can be observed here. Notice that the short-term options
have very low gammas deeply in- or out-of the-money, but have the highest gamma
at-the-money (at 50). Conversely, the longest-term, one-year option has the highest
gamma of the three time periods for deeply in- or out-of-the-money options. The
data is presented in Table 40-4. This table contains aslight amount of additional data:
the gamma for the at-the-money option at even shorter periods of time remaining
until expiration. Notice how the gamma explodes as time decreases, for the at-the
money option. With only one week remaining, the gamma is over 0.28, meaning that
the delta of such acall would, for example, jump from 0.50 to 0.78 if the stock mere
ly moved up from 50 to 51.
Gamma is dependent on the volatility of the underlying security as well. At-the
money options on less volatile securities will have higher gammas than similar options
on more volatile securities. The following example demonstrates this fact.
Example: Assume XYZ is at 49, as is ABC. Moreover, XYZ is amore volatile stock
(30% implied) as compared to ABC (20% ). Then, similar options on the two stocks
would have significantly different gammas.
XYZ Gammas ABC Gammas
Option (Volatility = 30%) (Volatility= 20%)
January 50 .066 .097
January 55 .045 .039
January 60 .019 .0053
February 50 .055 .081
February 60 .024 .011

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856 Part VI: Measuring and Trading VolatHity
FIGURE 40-4.
Gamma comparison, with XYZ = 50.
8
7
0 6 0 ..-
X 5
Cl!
E 4 E
Cl! 3 (!)
2 t= 1 year
t= 6 months
t= 3 months 0 '-'----~---__._ ___ _._ ___ _._ ___ ....___
40 45 50 55 60 65
Strike Price
TABLE 40-4.
Gamma comparison - various amounts of time remaining
(with XYZ = 50).
Time Remaining Strike Price
40 45 50 55 60
1 year .015 .029 .039 .04 .033
6 months .011 .037 .058 .051 .030
3 months .003 .039 .086 .057 .015
2 months .108
1 month .166
1 week .288
65
.023
.013
.002
Note that the at-the-money options (January 50'sand February 50's) on ABC,
the less volatile stock, have larger gammas than do their XYZ counterparts. However,
look one strike higher (January 55's), and notice that the more volatile options have aslightly higher gamma. Look two strikes higher and the more volatile options have avastly higher gamma, both for the January 60'sand the February 60's.
This concept makes sense if one thinks about the relationship between volatili
ty and delta. On nonvolatile stocks, one finds that the delta of even aslightly in
the-money option increases rapidly. This is because, since the stock is not volatile,
buyers are not willing to pay much time premium for the option. As aresult, the
gamma is high as well, because as the stock moves into-the-money, the increase in

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Chapter 40: Advanced Concepts 857
delta will be more dramatic than it would be for avolatile stock. Out-of-the-money
options are an entirely different story. Since the nonvolatile stock will have difficulty
moving fast enough to reach an out-of-the-money striking price, the delta of the out
of-the-money option is small and it will not change quickly (that is, the gamma is
small also).
These concepts are summarized in Figure 40-5 (see Table 40-5), which depicts
the gammas for similar options on stocks with differing volatilities. For the purposes
of these graphs, XYZ is equal to 50 and there are three months remaining until expi
ration.
Notice that for avery volatile stock, the gamma is quite stable over nearly all
striking prices when there are 3 months remaining until expiration. This means that
the deltas of all options on such avolatile stock will be changing quite abit for even
a 1-point move in the underlying stock. This is an important point for neutral strate
gists to note, because aposition that starts out as delta neutral may quickly change if
the underlying stock is very volatile. As this table implies, the deltas of the options in
that "neutral" spread may be altered quickly, thereby rendering the spread quite un
neutral. This concept will be discussed in greater detail later in this chapter.
As delta was used to construct the equivalent stock position of an entire option
position or portfolio, gamma can be used in asimilar manner. An example of this fol
lows, using the same securities from the preceding example on the delta of aposi
tion. An important point to note is that the gamma of the underlying security itself is
zero. This is true because the delta of the underlying security (which is always 1.0)
never changes - hence the gamma is zero. The gamma is measuring the amount of
change of the delta; if the delta of the underlying security never changes, the gamma
of the underlying security must be zero.
Example: The following position exists when XYZ is at 31.75. Recall that it resem
bles along straddle ( or backspread) in that there is increased profit potential in either
direction if the stock moves far enough by expiration. In addition to the delta previ
ously listed, the gamma is now shown as well. Note that since gamma is asmall
absolute number, it is sometimes calculated out to three or four decimal places.
Option Position Option Position
Position Delta Delta Gamma Gamma
Short 4,500 XYZ 1.00 - 4,500 0.0000 0
Short l 00 XYZ April 25 calls 0.89 - 8,900 0.0100 -100
Long 50 XYZ April 30 calls 0.76 + 3,800 0.0300 +150
Long 139 XYZ July 30 calls 0.74 +10,286 0.0200 +278
Totals: + 686 +328

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858 Part VI: Measuring and Trading Volatility
FIGURE 40-5.
Gamma comparison, with XYZ = 50, t = three months.
8
7
0 6
0
~ 5
<ti
E 4 E
~ 3
2
TABLE 40-5.
40 45
Low Volatility
Very High Volatility
50 55 60
Strike Price
65
Gamma comparison for varying volatilities (XYZ = 50, t = 3
months).
Gamma Very
Strike Low Volatility High Volatility High Volatility
40 .003 .013 .017
45 .039 .039 .022
50 .086 .057 .024
55 .057 .049 .025
60 .015 .028 .023
65 .002 .012 .020
As before, the position still has adelta long of almost 700 shares. In addition,
one can now see that it has apositive gamma of over 300 shares. This means that the
delta can be expected to change by 328 shares for each point that XYZ moves: If it
moves up 1 point, the delta will increase to +1,014 (the current delta, 686, plus the
gamma of 328). However, ifXYZ moves down by 1 point, then the delta will decrease
to +358 (the current delta, 686, less the gamma of 328).
Note that, in the above example, if XYZ continues higher, the gamma will
remain positive (although it will eventually shrink some), and the delta will continue
to increase. This means the position is getting longer and longer - afact that makes

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Chapter 40: Atlvanced Concepts 859
sense when one notes that there are extra long calls and they would be getting deep
er in-the-money as the stock moves up. Conversely, if XYZ continues to move lower,
the delta will continue to decrease and will quickly become negative, meaning that
the position would become short overall. Hence, the position does indeed resemble
along straddle: It gets longer as the market moves up and it gets shorter as the mar
ket moves down.
Long options, whether puts or calls, have positive gamma, while short options
have negative gamma. Thus, astrategist with aposition that has positive gamma has
anet long option position and is generally hoping for large market movements.
Conversely, if one has aposition with negative gamma, it means he has shorted
options and wants the market to remain fairly stable.
Note that it is possible to be delta neutral, but to have asignificant gamma. (For
example, if one owns puts and calls with offsetting deltas, he would be delta neutral,
but would have positive gamma since both options are long.) If one is delta neutral,
he knows he has no market exposure at this time, but his gamma will show him what
exposure his position will acquire as the market moves. These concepts will be dis
cussed in greater detail later in this chapter.
VEGA OR TAU
There is no letter in the Greek alphabet called "vega." Thus, some strategists, being
purists, prefer to use areal Greek letter, "tau," to refer to this risk measurement. The
term "vega" will be used in this book, but the reader should note that "tau" means
the same thing. Vega is the arrwunt by which the option price changes when the
volatility changes. Vega is always expressed as apositive number, whether it refers to
aput or acall.
It is known that more volatile stocks have more expensive options. Thus, as
volatility increases, the price of an option will rise. If volatility falls, the price of the
option will fall as well. The vega is merely an attempt to quantify how much the
option price will increase or decrease as the volatility moves, all other factors being
equal.
Before considering an example, areview of the term volatility is in order.
Volatility is ameasure of how quickly the underlying security moves around.
Statistically, it is usually calculated as the standard deviation of stock prices over some
period of time, generally annualized. This statistical measure is expressed as aper
cent, although relating that percent to actual stock movements can be complicated.
Suffice it to say that astock that has a 50% volatility is more volatile than astock with
30% volatility. The stock market generally has avolatility of about 15% overall,
although that may change from time to time (crashes, for example).

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860 Part VI: Measuring and Trading Vo/atiHty
Example: Again, assume XYZ is at 49, and the January 50 call is selling for 3.50. The
vega of the option is 0.25, and the current volatility of XYZ is 30%.
If the volatility increases by one percentage point or 1 % to 31 %, then the vega
indicates that the option will increase in value by 0.25, to 3. 75.
If the volatility had instead decreased by 1 percent to 29%, then the January 50
call would have decreased to 3.25 (aloss of 0.25, the amount of the vega).
If the implied volatility of an option increases, the option price will increase as
well. Consequently, even though XYZ stock may be exhibiting the same historic
movement that it always has, and therefore its (historical) volatility would be
unchanged, if option buyers appear in sufficient quantity, they may drive the implied
volatility ofXYZ'soptions higher. Likewise, an excess of option sellers could drive the
implied volatility lower, even though the historical volatility does not change. So, it
must be concluded that vega measures how much the option price changes as implied
volatility changes.
Vega is related to time. Figure 40-6 (see Table 40-6) shows the vegas for options
with differing times remaining until expiration. The underlying stock is assumed to
be 50 in all cases. Notice that the more time that remains, the higher the vega is. It
is interesting to note that, for very long-term options, the vega of the slightly out-of
the-money calls (strike = 55) is actually higher than that of the at-the-money.
However, this discrepancy disappears as time passes. Not shown, but equally true, is
that the vega of aslightly out-of-the-money option on avery volatile stock may be
higher than that of the at-the-money.
As with the measurements of risk discussed already, vega can refer to the option
itself ("option vega") or to the position as awhole ("position vega"). Since vega is
expressed as apositive number, if one is long options, then his position vega will be
positive. This means he has exposure if volatilities decrease, or can make money if
volatilities increase.
Example: Again, assume that we have the same backspread position as before, with
XYZ at 31.75.
Option Position
Position Vega Vega
Short 4,500 XYZ 0.00 0
Short 100 XYZ April 25 calls 0.02 - 200
Long 50 XYZ April 30 calls 0.05 + 250
Long 139 XYZ July 30 calls 0.07 + 973
Total Vega: + 1,023

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Chapter 40: Advanced Concepts
FIGURE 40-6.
Vega comparison, XYZ = 50.
20
18
16
14
0
0 12 ,--
.2S 10 C1l
Cl
~ 8
6 t= 3 months
4
2
0
40 45 50
Strike Price
TABLE 40-6.
861
55 60 65
Vega comparison for different time periods (with XYZ = 50).
Vega
Strike Price t = 1 Year t = 6 Months t = 3 Months
40 0.12 0.06 0.02
45 0.16 0.11 0.06
50 0.19 0.13 0.09
55 0.20 0.13 0.08
60 0.18 0.11 0.05
65 0.16 0.07 0.02
The vega is apositive 10.23 points ($1,023 since each point for these equity
options is worth $100). The fact that the position has apositive vega means that it is
exposed to variations in volatility. If volatility decreases, the position will lose money:
$1,023 for each one percentage point decrease in volatility. However, if volatility
increases, the position will benefit.
Vega is greatest for at-the-money options and approaches zero as the option is
deeply in- or out-of-the-money. Again, this is common sense, since adeep in- or out-

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860 Part VI: Measuring and Trading Volatility
Example: Again, assume XYZ is at 49, and the January 50 call is selling for 3.50. The
vega of the option is 0.25, and the current volatility of XYZ is 30%.
If the volatility increases by one percentage point or 1 % to 31 %, then the vega
indicates that the option will increase in value by 0.25, to 3. 75.
If the volatility had instead decreased by 1 percent to 29%, then the January 50
call would have decreased to 3.25 (aloss of 0.25, the amount of the vega).
If the implied volatility of an option increases, the option price will increase as
well. Consequently, even though XYZ stock may be exhibiting the same historic
movement that it always has, and therefore its (historical) volatility would be
unchanged, if option buyers appear in sufficient quantity, they may drive the implied
volatility of XYZ'soptions higher. Likewise, an excess of option sellers could drive the
implied volatility lower, even though the historical volatility does not change. So, it
must be concluded that vega measures how much the option price changes as implied
volatility changes.
Vega is related to time. Figure 40-6 (see Table 40-6) shows the vegas for options
with differing times remaining until expiration. The underlying stock is assumed to
be 50 in all cases. Notice that the more time that remains, the higher the vega is. It
is interesting to note that, for very long-term options, the vega of the slightly out-of
the-money calls (strike = 55) is actually higher than that of the at-the-money.
However, this discrepancy disappears as time passes. Not shown, but equally true, is
that the vega of aslightly out-of-the-money option on avery volatile stock may be
higher than that of the at-the-money.
As with the measurements of risk discussed already, vega can refer to the option
itself ("option vega") or to the position as awhole ("position vega"). Since vega is
expressed as apositive number, if one is long options, then his position vega will be
positive. This means he has exposure if volatilities decrease, or can make money if
volatilities increase.
Example: Again, assume that we have the same backspread position as before, with
XYZ at 31.75.
Option Position
Position Vega Vego
Short 4,500 XYZ 0.00 0
Short 100 XYZ April 25 calls 0.02 - 200
Long 50 XYZ April 30 calls 0.05 + 250
Long 139 XYZ July 30 calls 0.07 + 973
Total Vega: + 1,023

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Chapter 40: Advanced Concepts
FIGURE 40-6.
Vega comparison, XYZ = 50.
20
18
16
14
0
0 12 ...
6 10 (13
0)
;g? 8
6
4
2
0
40 45
TABLE 40-6.
861
50 55 60 65
Strike Price
Vega comparison for different time periods (with XYZ = 50).
Vega
Strike Price t = 1 Year t = 6 Months t = 3 Months
40 0.12 0.06 0.02
45 0.16 0.11 0.06
50 0.19 0.13 0.09
55 0.20 0.13 0.08
60 0.18 0.11 0.05
65 0.16 0.07 0.02
The vega is apositive 10.23 points ($1,023 since each point for these equity
options is worth $100). The fact that the position has apositive vega means that it is
exposed to variations in volatility. If volatility decreases, the position will lose money:
$1,023 for each one percentage point decrease in volatility. However, if volatility
increases, the position will benefit.
Vega is greatest for at-the-money options and approaches zero as the option is
deeply in- or out-of-the-money. Again, this is common sense, since adeep in- or out-

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862 Part VI: Measuring and Trading Volatility
of-the-money option will not be affected much by achange in volatility. In addition,
for at-the-money options, longer-term options have ahigher vega than short-term
options. To verify this, think of it in the extreme: An at-the-money option with one
day to expiration will not be overly affected by any change in volatility, due to its
pending expiration. However, athree-month at-the-money option will certainly be
sensitive to changes in volatility.
Vega does not directly correlate with either delta or gamma. One could have aposition with no delta and no gamma (delta neutral and gamma neutral) and still have
exposure to volatility. This does not mean that such aposition would be undesirable;
it merely means that if one had such aposition, he would have removed most of the
market risk from his position and would be concerned only with volatility risk.
In later sections, the use of volatility to establish positions and the use of vega
to monitor them will be discussed.
THETA
Theta measures the time decay of aposition. All option traders know that time is the
enemy of the option holder, and it is the friend of the option writer. Theta is the name
given to the risk measurement of time in one'sposition. Theta is generally expressed
as anegative number, and it is expressed as the amount by which the option value
will change. Thus, if an option has atheta of -0.12, that means the option will lose
12 cents, or about an eighth of apoint, per day. This is true for both puts and calls,
although the theta of aput and acall with the same strike and expiration date are not
equal to each other.
Very long-term options are not subject to much time decay in one day'stime.
Thus, the theta of along-term option is nearly zero. On the other hand, short-term
options, especially at-the-money ones, have the largest absolute theta, since they are
subject to the ravages of time on adaily basis. The theta of options on ahighly volatile
stock will be higher than the theta of options on alow-volatility stock. Obviously, the
former options are more expensive (have more time value) and therefore have more
time value to lose on adaily basis, thereby implying that they have ahigher theta.
Finally, the decay is not linear - an option will lose agreater percent of its daily value
near the end of its life.
Figure 40-7 (see Table 40-7) depicts the relationships of thetas for various strik
ing prices and for differing volatilities on options with three months of life remain
ing. Again, notice that for very volatile stocks, the out-of-the-money options have
thetas as large as the at-the-moneys. This is saying that as each day passes, the prob
ability of the stock reaching that out-of-the-money strike drops and causes the option

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Chapter 40: Advanced Concepts
FIGURE 40-7.
Theta comparison, with XYZ = 50, t = three months.
-16
Very High Volatility
-14
-12
0 0
T""
X
~
-10
.i:::
I-
-8 Low Volatility
-6
0
40 45 50 55 60
Strike Price
TABLE 40-7.
863
65
Theta comparison for differing volatilities (XYZ = 50, t = 3 months).
Strike low Volatility Medium Volatility High Volatility
40 -0.005 -0.008 -0.013
45 -0.007 -0.010 -0.014
50 -0.008 -0.010 -0.015
55 -0.007 -0.010 -0.016
60 -0.006 -0.009 -0.016
65 -0.004 -0.008 -0.015
to lose value. This does not change the fact that, for very short-term options, the theta
is largest at-the-money.
Normally, the theta of an individual option is of little interest to the strategist.
He generally would be more concerned with delta or gamma. However, as with the
other risk measures, theta can be computed for an entire portfolio of options. This
measure, the "position theta," can be quite important because it gives the strategist
agood idea of how much gain or loss he can expect on adaily basis, due to time ero
sion. The following example demonstrates this point. Note that the underlying secu
rity itself has atheta of zero, since it cannot lose any value due to time decay.

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864 Part VI: Measuring and Trading Volatility
Example: With XYZ at 49, the strategist has the following position in February, so
that the April calls are nearer to expiration than the July calls. This position is similar
to alarge calendar spread position.
Option Position
Position Theta Theta
Short 4,000 XYZ 0.00 0
Short 150 XYZ April 50 calls -0.04 +600
Long 150 XYZ April 30 calls -0.02 -300
Short 78 XYZ July 30 puts -0.02 +156
Total Theta: +456
This position is expected to make $456 per day due to time decay. Note that
short options, whether puts or calls, have apositive position theta, while long options
have anegative position theta. Anegative position theta means the position has risk
due to time, while apositive position theta means time is working for the position.
RHO
Rho is the name given to the price change of an option'svalue due to achange in
interest rates. Recall that one of the components that contributes to an option'sprice
is interest rates. As interest rates rise, call prices will rise, but put prices will fall. The
opposite is true as well: As interest rates fall, call prices decline and put prices rise.
Rho measures the amount by which these prices rise or fall.
This behavior of puts and calls with respect to interest rates may not be imme
diately obvious, but recall that the arbitrage that can be established with in-the
money calls (the "interest play," discussed in Chapter 27 on arbitrage) demonstrates
that arbitrageurs are willing to pay more for an in-the-money call as interest rates rise
because they will be earning more interest on the stock that they sell short against
that in-the-money call. Thus, rising interest rates cause call prices to increase.
The opposite is true for puts: Rising interest rates cause put prices to decline.
Again, an arbitrage can be used to demonstrate the point. Recall that in areversal
arbitrage, the arbitrageur is selling the stock and the put while buying the call. We
have just demonstrated that, as interest rates rise, he is willing to pay more for the
call since he can earn extra interest on the short sale of his stock. This automatically
means that he will be willing to sell the put for less.
Rho is expressed as apositive number for calls and anegative one for puts. Rho
is smallest for deeply out-of-the-money options and is large for deeply-in-the-money
options. It is larger for longer-tenn options and is nearly zero for very short-tenn

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Chapter 40: Advanced Concepts 865
options. The following table of option prices may help to demonstrate these rela
tionships:
Example: With XYZ at 49, the following options have the rho indicated (January is
the near-term expiration):
Month/Strike Call Rho Put Rho
January 35 0.05 -0.01
January 50 0.03 -0.03
January 60 0.00 -0.05
July 35 0.18 -0.02
July 50 0.14 -0.15
July 60 0.07 -0.18
Note that the in-the-money calls (35 strike) have larger rho than the out-of-the
money 60's, in both January and July. Similarly, the in-the-money puts (the 60's) have
larger rho on an absolute basis than the out-of-the-money 35's. Again, this is true for
both January and July.
Furthermore, note that the longer-term July rhos are all larger (again as
absolute numbers) than their shorter-term January counterparts.
Rho can also be calculated for an entire portfolio to obtain a "position rho," sim
ilar to previous examples. Generally, one would not be overly concerned with his
position rho unless his portfolio contained quite afew long-term options and/or
deeply in-the-money ones. Thus, rho is more important as aconsideration when one
is trading LEAPS or warrants, both of which may be extremely long-term vehicles.
Of the risk measures discussed so far, rho is the least used, since many traders tend
to have relatively short-term options in their positions.
THE GA,\1MA OF THE GAMMA
Occasionally, one may hear reference to the "six measures of risk." This is the sixth
one and it is the most arcane. At any point, one knows the delta and gamma of an
option. As the stock moves, the delta changes (by the amount of the gamma), but so
does the gamma. Some traders are interested in knowing how much the gamma will
change when the stock moves. Hence, they will compute the gamma of the gamma,
which is the arrwunt by which the gamma will change when the stock price changes.
This concept will be discussed at the end of this chapter. It is most important for
strategists involved in positions on highly volatile stocks, for if the stock moves far

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866 Part VI: Measuring and Trading VolatOity
enough, the gamma (and therefore the delta) may change dramatically. Thus, one
might want to know how this risk measure affects his profitability.
SUMMARY
Delta: Positive delta indicates that aposition is currently bullish; if the underly
ing security goes up in price, the position should make money. Anegative
delta indicates abearish slant.
Gamma: Positive gamma means that the delta will increase if the underlying secu
rity rises in price. Positive gamma generally implies that there is apre
ponderance of long options in the position, either puts or calls; negative
gamma indicates written or naked options in the position.
Theta: Negative theta means that the position will lose money as time passes
(typical of positions with long options); positive theta implies that time is
working for the position (positions with written options).
Vega: Positive vega means that an increase of (perceived) volatility will benefit
the position - usually true of positions with long options in them; nega
tive vega means that adecrease of volatility would be beneficial.
STRATEGY CONSIDERATIONS: USING THE
11
GREEKS"
Before looking at how one operates aparticular strategy using delta, gamma, etc., it
might be beneficial to see how these factors relate to the individual strategies that
have been described throughout this book. Table 40-8 is ageneral guide to how the
various strategies are exposed to various market factors. It is not an all-purpose or
specific table, because as the stock moves higher or lower, some of the risk measure
ment factors will certainly be affected.
Afew assumptions were made in constructing the table. First, it was assumed
that the strategies where delta is noted as being zero are established in aneutral
stance. The bull spread and bear spread strategies assumed that the stock was mid
way between the striking prices. Two other spread strategies - ratio call and ratio put
- assumed the stock was at the striking price of the option that was sold. In all other
cases, there is only one striking price involved, and it was assumed that the stock was
at the strike.
The table may help to clarify some of the concepts concerning the risk meas
urement factors. First, notice that stock or futures - or any underlying security- have
only delta. None of the other factors pertains to the underlying security itself.

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Chapter 40: Advanced Concepts 867
TABLE 40·8.
General risk exposure of common strategies.
Strategy Delta Gamma Theta Vega RHo
Buy stock + 0 0 0 0
Sell stock short 0 0 0 0
Call buy + + + +
Put buy + +
Straddle buy 0 + + +
Covered write + +
Naked call sale +
Naked put sale + + +
Ratio write
(straddle sale) 0 +
Calendar spread 0 + +
Bull spread + +
Bear spread +
Ratio call spread 0 +
Ratio put spread 0 + +
As might be expected, spread strategies involving both long and short options are
less easily quantified than outright buys or sells. The calendar spread strategy is one
in which the spreader does not want alot of stock movement - he would prefer the
underlying security to remain near the striking price, for that is the area of maximum
profit potential. This is reflected by the fact that gamma is negative. Also, for calendar
spreads, the passage of time is good, afact that is reflected by the fact that theta is pos
itive. Finally, since an increase in implied volatilities or interest rates would boost
prices and widen the spread (creating aprofit), vega is positive and rho is negative.
Abull spread has positive delta, reflecting the bullish nature of the spread, but
it has negative gamma. The reason gamma is negative is that the position becomes
less bullish as the underlying security rises, since the profit potential, and hence the
bullishness of the position, is limited. For similar reasons, abear spread has negative
delta (reflecting bearishness) and negative gamma (reflecting limited bearishness).
Both the bull spread and the bear spread are the same with respect to the other risk
measurements: Theta is negative, reflecting the fact that time decay can hurt the
spread. Less obvious is the fact that these spreads are hurt by an increase in per
ceived volatility; anegative vega tells us this is true, however.
These risk measurement tools are important in that they can quite graphically
depict the risk and reward characteristics of an option position or option portfolio.

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868 Part VI: Measuring and Trading Volatility
They are useful in establishing anew position, because one can see how much expo
sure he is taking on. In addition, they are extremely useful for follow-up action, since
one can see how his position'scharacteristics have developed in the current market
place at the present time. In the following sections, the use of the risk measurement
tools as aids in establishing aposition or in following up on aposition will be dis
cussed in detail.
DELTA NEUTRAL
One popular type of neutral position is to be delta neutral - that is, to have the equiv
alent stock position (ESP) or equivalent futures position (EFP) be zero. Adelta neu
tral position is one in which the sum of the projected price changes of the long options
in the spread is essentially offset by the projected price changes of the short options
in the same spread.
Example: XYZ is trading at 50. The following three options are trading with the
prices and deltas indicated. Furthermore, the "theoretical value" of each option is
shown:
XYZ:50
"Theoretical
Option Price Delta Value"
January 50 call 3.00 0.55 3.50
January 55 call 1.50 0.35 1.48
February 50 put 3.50 -0.40 3.44
Assuming that one can rely upon these "theoretical values" (abig assumption,
by the way), it is obvious that the January 50 call is cheap with respect to the other
options: They are close to their values, while the January 50 is 50 cents under. The
neutral strategist would want to buy the January 50 call and hedge his purchase with
one of the other two options presented. One choice would be to establish aspread
wherein the January 50 calls are bought and anumber of January 55'sare sold. To
determine how many are to be bought and sold, one merely has to divide the deltas
of the two options:
Delta neutral spread ratio = 0.55/0.35 = 11-to-7
Thus, adelta neutral ratio spread would consist of buying 7 January 50'sand sell
ing 11 January 55's. To verify that this spread is neutral with respect to the change in
price ofXYZ, notice that ifXYZ moves up in price 1 point, the January 50 will increase

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Chapter 40: Advanced Concepts 869
in price by 0.55; so seven of them will increase by 7 x 0.55, or 3.85 points total.
Similarly, the January 55 will increase in price by 0.35, so eleven of them would
increase in price by 11 x 0.35, or 3.85 points total. Hence, the long side of the spread
would profit by 3.85 points, while the short side loses 3.85 points - aneutral situation.
The resulting position is aratio spread. The profitability of the spread occurs
between about 51 and 62 at expiration as shown in Figure 40-8, but that is not the
major point. The real attractiveness of the spread to the neutral trader is that if the
underpriced nature of the January 50 call (vis-a-vis the January 55 call) should dis
appear, the spread should produce aprofit, regardless of the short-term market
movement of XYZ. The spread could then be closed if this should occur.
To illustrate this fact, suppose that XYZ actually falls to 49, but the January 50
call returns to "fair value":
XYZ: 49
"Theoretical
Option Price Delta Value"
January 50 call 3.00 0.52 3.00
January 55 call 1.10 0.34 1.13
February 50 put 3.90 -0.42 3.84
Notice that the theoretical values in this table are equal to the theoretical val
ues from the previous table, less the amount of the delta. Since the XYZ January 50
call is no longer underpriced, the position would be removed, and the strategist
would make nothing on his January 50's, but would make .40 on each of the eleven
short January 55's, for aprofit of $440 less commissions.
This example leans heavily on the assumption that one is able to accurately esti
mate the theoretical value and delta of the options. In real life, this chore can be
quite difficult, since the estimate requires one to define the future volatility of the
common stock. This is not easy. However, for the purposes of aspread, the ratio of
the two deltas is used. Moreover, the example didn'trequire that one know the exact
theoretical value of each option; rather, the only knowledge that was required was
that one of the options was cheap with respect to the other options.
As an alternative to aratio spread, another type of delta neutral position could
be established from the previous data: Buy the January 50 call (this is the basis of the
position since it is supposedly the cheap option) and buy the February 50 put - the
only other choice from the data given. This position is along straddle of sorts. Recall
that the delta of aput is negative; so again, the delta neutral ratio can be calculated
by dividing the absolute value of two deltas:
Delta neutral straddle ratio= 0.55/1-0.401 = 11-to-8

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870
FIGURE 40-8.
XYZ ratio spread.
f/)
f/)
3000
2000
.3 1000
~
ea..
(r.>
40
Part VI: Measuring and Trading Volatility
45 55 60
At Expiration
Stock Price
Thus, adelta neutral straddle position would consist of buying 8 Janua:ry 50 calls
and buying 11 Februa:ry 50 puts. The straddle has no market exposure, at least over
the short term. Note that the delta neutral straddle has asignificantly different prof
it picture from the delta neutral ratio spread, but they are both neutral and are both
based on the fact that the Janua:ry 50 call is cheap. The straddle makes money if the
stock moves alot, while the other makes money if the stock moves only alittle. (See
Figure 40-9.)
Can these two vastly different profit pictures be depicting strategies in which
the same thing is to be accomplished ( that is, to capture the underpriced nature of
the XYZ Janua:ry 50 call)? Yes, but in order to decide which strategy is "best," the
strategist would have to take other factors into consideration: the historical volatility
of the underlying security, for example, or how much actual time remains until
Janua:ry expiration, as well as his own psychological attitude toward selling uncovered
calls. Amore precise definition of the other risks of these two positions can be
obtained by looking at their position gammas.
Delta Neutral Is Not Entirely Neutral. In fact, delta neutral means that
one is neutral only with respect to small price changes in the underlying securi
ty. Adelta neutral position may have seriously unneutral characteristics when

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Chapter 40: Advanced Concepts
FIGURE 40-9.
XYZ straddle buy.
Cl)
8000
7000
6000
5000
4000
~ 3000
~ 2000
! 1000
01------------------------
-1000
-2000
-3000 At January Expiration
Stock Price
871
some of the other risk measurements are considered. Consequently, one cannot
blithely go around establishing delta neutral positions and ignoring them, for
they may have significant market risk as certain factors change.
For example, it is obvious to the naked eye that the two positions described in
the previous section - the ratio spread and the long straddle - are not alike at all,
but both are delta neutral. If one incorporates the usage of some of the other risk
measurements into his position, he will be able to quantify the differences between
"neutral" strategies. The sale of astraddle will be used to examine how these vari
ous factors work.
Positions with naked options in them have negative position gamma. This
means that as the underlying security moves, the position will acquire traits opposite
to that movement: If the security rises, the position becomes short; if it falls, the posi
tion becomes long. This description generally fits any position with naked options,
such as aratio spread, anaked straddle, or aratio write.
Example: XYZ is at 88. There are three months remaining until July expiration, and
the volatility of XYZ is 30%. Suppose 100 July 90 straddles are sold for 10 points -
the put and the call each selling for 5. Initially, this position is nearly delta neutral, as

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872 Part VI: Measuring and Trading Volatility
shown in Table 40-9. However, since both options are sold, each sale places negative
gamma in the position.
The usefulness of calculating gamma is shown by this example. The initial posi
tion is NET short only 100 shares of XYZ, avery small delta. In fact, aperson who is
atrader of small amounts of stock might actually be induced into believing that he
could sell these 100 straddles, because that is equivalent to being short merely 100
shares of the stock.
TABLE 40-9.
Position delta and gamma of straddle sale. XYZ = 88.
Option Position Option Position
Position Delto Delta Gamma Gamma
Sell l 00 July 90 calls 0.505 -5,050 0.03 -300
Sell 1 00 July 90 puts 0.495 +4,950 0.03 -300
Total shares - 100 -600
Calculating the gamma quickly dispels those notions. The gamma is large: 600
shares of negative gamma. Hence, if the stock moves only 2 points lower, this trad
er'sstraddle position can be expected to behave as if it were now long 1,100 shares
(the original 100 shares short plus 1,200 that the gamma tells us we can expect to get
long)! The position might look like this after the stock drops 2 points:
XYZ: 86
Position
Sold 1 00 July 90 calls
Sold 100 July 90 puts
Option
Delta
0.44
0.55
Position
Delta
-4,400
+5,500
+ 1 , 100 shares
Hence, a 2-point drop in the stock means that the position is already acquiring
a "long" look. Further drops will cause the position to become even "longer." This is
certainly not aposition - being short 100 straddles - for asmall trader to be in, even
though it might have erroneously appeared that way when one observed only the
delta of the position. Paying attention to gamma more fully discloses the real risks.
In asimilar manner, if the stock had risen 2 points to 90, the position would
quickly have become delta short. In fact, one could expect it to be short 1,300 shares
in that case: the original short 100 shares plus the 1,200 indicated by the negative
gamma. Arise to 90, then, would make the position look like this:

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Chapter 40: Advanced Concepts
XYZ:90
Position
Sold 100 July 90 calls
Sold 1 00 July 90 puts
Option
Delto
0.56
0.43
Position
Delta
-5,600
+4,300
873
1,300 shares
These examples demonstrate how quickly alarge position, such as being short
100 straddles, can acquire alarge delta as the stock moves even asmall distance.
Extrapolating the moves is not completely correct, because the gamma changes as
the stock price changes, but it can give the trader some feel for how much his delta
will change.
It is often useful to calculate this information in advance, to some point in the
near future. Figure 40-10 depicts what the delta of this large short straddle position
will be, two weeks after it was first sold. The points on the horizontal axis are stock
prices. The quickness with which the neutrality of the position disappears is alarm
ing. Asmall move up to 93 - only one standard deviation - in two weeks makes the
overall position short the equivalent of about 3,300 shares of XYZ. Figure 40-10 real
ly shows nothing more than the effect that gamma is having on the position, but it is
presented in aform that may be preferable for some traders.
What this means is that the position is "fighting" the market: As the market goes
up, this position becomes shorter and shorter. That can be an unpleasant situation,
both from the point of view of creating unrealized losses as well as from apsycho
logical viewpoint. The position delta and gamma can be used to estimate the amount
of unrealized loss that will occur: Just how much can this position be expected to lose
if there is aquick move in the underlying stock? The answer is quickly obtained from
the delta and gamma: With the first point that XYZ moves, from 88 to 89, the posi
tion acts as if it is short 100 shares (the position delta), so it would lose $100. With
the next point that XYZ rises, from 89 to 90, the position will act as if it is short the
original 100 shares (the position delta), plus another 600 shares (the position
gamma). Hence, during that second point of movement by XYZ, the entire position
will act as if it is short 700 shares, and therefore lose another $700. Therefore, an
immediate 2-point jump in XYZ will cause an unrealized loss of $800 in the position.
Summarizing:
Loss, first point of stock movement = position delta
Loss, second point of stock movement = position delta + gamma
Total loss for 2 points of stock movement
= 2 xposition delta + position gamma

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874 Part VI: Measuring and Trading Volatility
FIGURE 40· 1 O.
Proiected delta, in 14 days.
6000
4500
3000
Cl) 1500 ~ ro .c
(/)
0 'E
(1)
80 85 ~ ·5 -1500
95
XYZ Stock Price
C"
UJ
-3000
-4500
Using the example data:
Loss, XYZ moves from 88 to 89: -$100 (the position delta)
Loss, XYZ moves from 89 to 90: -$100 (delta) - $600 (gamma)
: -$700
Total loss, XYZ moves from 88 to 90: -$100 x 2 - $600 = -$800
This can be verified by looking at the prices of the call and put after XYZ has jumped
from 88 to 90. One could use amodel to calculate expected prices if that happened.
However, there is another way. Consider the following statements:
If the stock goes up by 1 point, the call will then have aprice of:
p 1 = Po + delta
5.505 = 5.00 + 0.505 (if XYZ goes to 89 in the example)
If the stock goes up 2 points, the call will have an increase of the above amount
plus asimilar increase for the next point of stock movement. The delta for that sec
ond point of stock movement is the original delta plus the gamma, since gamma tells
one how much his delta is going to change.

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Chapter 40: Advanced Concepts
p2 = p 1 +delta+ gamma, or substituting from above
p2 = (p0 + delta) + delta + gamma
= Po + 2 xdelta + gamma
6.04 = 5.00 + 2 x 0.505 + 0.03 (in the example if XYZ = 90)
875
By the same calculation, the put in the example will be priced at 4.04 if XYZ imme
diately jumps to 90:
4.04 = 5.00 - 2 X 0.495 + 0.03
So, overall, the call will have increased by 1.04, but the put will only have
decreased by 0.96. The unrealized loss would then be computed as -$10,400 for the
100 calls, offset by again of $9,600 on the sale of 100 puts, for anet unrealized loss
of $800. This verifies the result obtained above using position delta and position
gamma. Again, this confirms the logical fact that aquick stock movement will cause
unrealized losses in ashort straddle position.
Continuing on, let us look at some of the other factors affecting the sale of this
straddle. The straddle seller has time working in his favor. After the position is estab
lished, there will not be as much decay in the first two-week period as there will be
when expiration draws near. The exact amount of time decay to expect can be calcu
lated from the theta of the position:
XYZ: 88
Position
Sold l 00 July 90 calls
Sold l 00 July 90 puts
Option
Theta
-0.03
-0.03
Position
Theta
+$300
+$300
+$600
This is how the position looked with respect to time decay when it was first
established (XYZ at 88 and three months remaining until expiration). The theta of the
put and the call are essentially the same, and indicate that each option is losing about
3 cents of value each day. Note that the theta is expressed as anegative number, and
since these options are sold, the position theta is apositive number. Apositive posi
tion theta means time decay is working in your favor. One could expect to make $300
per day from the sale of the 100 calls. He could expect to make another $300 per day
from the sale of the 100 puts. Thus, his overall position is generating atheoretical
profit from time decay of $600 per day.
The fact that the sale of astraddle generates profits from time decay is not
earth-shattering. That is awell-known fact. However, the amount of that time decay

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876 Part VI: Measuring and Trading Volatility
is quantified by using theta. Furthermore, it serves to show that this position, which
is delta neutral, is not neutral with respect to the passage of time.
Finally, let us examine the position with respect to changes in volatility. This is
done by calculating the position vega.
XYZ:88
Position
Sold 1 00 July 90 calls
Sold 100 July 90 puts
Option
Vega
0.18
0.18
Position
Vega
-$1,800
-$1,800
-$3,600
Again, this information is displayed at the time the position was established,
three months to expiration, and with avolatility of 30% for XYZ. The vega is quite
large. The fact that the call'svega is 0.18 means that the call price is expected to
increase by 18 cents if the implied volatility of the option increases by one percent
age point, from 30% to 31 %. Since the position is short 100 calls, an increase of 18
cents in the price of the call would translate into aloss of $1,800. The put has asim
ilar vega, so the overall position would lose $3,600 if the options trade with an
increase in volatility of just one percentage point. Of course, the position would make
$3,600 if the volatility decreased by one percentage point, to 29%.
This volatility risk, then, is the greatest risk in this short straddle position. As
before, it is obvious that an increase in volatility is not good for aposition with naked
options in it. The use of vega quantifies this risk and shows how important it is to
attempt to sell overpriced options when establishing such positions. One should not
adhere to any one strategy all the time. For example, one should not always be sell
ing naked puts. If the implied volatilities of these puts are below historical norms,
such astrategy is much more likely to encounter the risk represented by the posi
tion vega. There have been several times in the recent past - mostly during market
crashes - when the implied volatilities of both index and equity options have leaped
tremendously. Those times were not kind to sellers of options. However, in almost
every case, the implied volatility of index options was quite low before the crash
occurred. Thus, any trader who was examining his vega risk would not have been
inclined to sell naked options when they were historically "cheap."
In summary then, this "neutral" position is, in reality, much more complex when
one considers all the other factors.

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Chapter 40: Advanced Concepts
Position summary
Risk Factor
Position delta = l 00
Position gamma = -600
Position theta = +$600
Position vega = -$3,600
877
Comment
Neutral; no immediate exposure to small
market movements; lose $100 for 1
point move in underlying.
Fairly negative; position will react
inversely to market movements, causing
losses of $700 for second point of
movement by underlying.
Favorable; the passage of time works in the
position'sfavor.
Very negative; position is extremely
subject to changes in implied volatility.
This straddle sale has only one thing guaranteed to work for it initially: time
decay. (The risk factors will change as price, time, and volatility change.) Stock price
movements will not be helpful, and there will always be stock price movements, so
one can expect to feel the negative effect of those price changes. Volatility is the big
unknown. If it decreases, the straddle seller will profit handsomely. Realistically,
however, it can only decrease by alimited amount. If it increases, very bad things will
happen to the profitability of the position. Even worse, if the implied volatility is
increasing, there is afairly likely chance that the underlying stock will be jumping
around quite abit as well. That isn'tgood either. Thus, it is imperative that the strad
dle seller engage in the strategy only when there is areasonable expectation that
volatilities are high and can be expected to decrease. If there is significant danger of
the opposite occurring, the strategy should be avoided.
If volatility remains relatively stable, one can anticipate what effects the passage
of time will have on the position. The delta will not change much, since the options
are nearly at-the-money. However, the gamma will increase, indicating that nearer to
expiration, short-term price movements will have more exaggerated effects on the
unrealized profits of the position. The theta will grow even more, indicating that time
will be an even better friend for the straddle writer. Shorter-term options tend to
decay at afaster rate than do longer-term ones. Finally, the vega will decrease some
as well, so that the effect of an increase in implied volatility alone will not be as dam
aging to the position when there is significantly less time remaining. So, the passage

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878 Part VI: Measuring and Trading Volatility
of time generally will improve most aspects of this naked straddle sale. However, that
does not mitigate the current situation, nor does it imply that there will be no risk if
alittle time passes.
The type of analysis shown in the preceding examples gives amuch more in
depth look than merely envisioning the straddle sale as being delta short 100 shares
or looking at how the position will do at expiration. In the previous example, it is
known that the straddle writer will profit if XYZ is between 80 and 100 in three
months, at expiration. However, what might happen in the interim is another matter
entirely. The delta, gamma, theta, and vega are useful for the purpose of defining
how the position will behave or misbehave at the current point in time.
Refer back to the table of strategies at the beginning of this section. Notice that
ratio writing or straddle selling ( they are equivalent strategies) have the characteris
tics that have been described in detail: Delta is 0, and several other factors are neg
ative. It has been shown how those negative factors translate into potential profits or
losses. Observing other lines in the same table, note that covered writing and naked
put selling ( they are also equivalent, don'tforget) have adescription very similar to
straddle selling: Delta is positive, and the other factors are negative. This is aworse
situation than selling naked straddles, for it entails all the same risks, but in addition
will suffer losses on immediate downward moves by the underlying stock. The point
to be made here is that if one felt that straddle selling is not aparticularly attractive
strategy after he had observed these examples, he then should feel even less inclined
to do covered writing, for it has all the same risk factors and isn'teven delta neutral.
An example that was given in the chapter on futures options trading will be
e,,'Panded as promised at this time. To review, one may often find volatility skewing
in futures options, but it was noted that one should not normally buy an at-the-money
call (the cheapest one) and sell alarge quantity of out-of-the-money calls just because
that looks like the biggest theoretical advantage. The following example was given. It
will now be expanded to include the concept of gamma.
Example: Heavy volatility skewing exists in the prices of January soybean options:
The out-of-the-money calls are much more expensive than the at-the-money calls.
The following data is known:
January soybeans: 583
Option Price Implied Volatility Delta Gamma
575 call 19.50 15% 0.55 .0100
675 call 2.25 23% 0.09 .0026

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Chapter 40: Advanced Concepts
Using deltas, the following spread appears to be neutral:
Buy l January bean 57 5 call at 19 .50
Sell 6 January bean 675 calls at 2.25
Net position:
19.50 DB
13.50 CR
6 Debit
879
At the time the original example was presented, it was demonstrated through
the use of the profit picture that the ratio was too steep and problems could result in
alarge rally.
Now that one has the concept of gamma at his disposal, he can quantify what
those problems are.
The position gamma of this spread is quite negative:
Position gamma = .01 - 6 x .0026 = -0.0056
That is, for every 10 points that January soybeans rally, the position will become
short about 1/2 of one futures contract. The maximum profit point, 675, is 92 points
above the current price of 583. While beans would not normally rally 92 points in
only afew days, it does demonstrate that this position could become very short if
beans quickly rallied to the point of maximum profit potential. Rest assured there
would be no profit if that happened.
Even asmall rally of 20 cents (points) in soybeans - less than the daily limit -
would begin to make this tiny spread noticeably short. If one had established the
spread in some quantity, say buying 100 and selling 600, he could become seriously
short very fast.
Aneutral spreader would not use such alarge ratio in this spread. Rather, he
would neutralize the gamma and then attempt to deal with the resulting delta. The
next section deals with ways to accomplish that.
CREATING MULTIFACETED NEUTRALITY
So what is the strategist to do? He can attempt to construct positions that are neutral
with respect to the other factors if he perceives them as arisk. There is no reason why
aposition cannot be constructed as veg aneutral rather than delta neutral, if he wants
to eliminate the risk of volatility increases or decreases. Or, maybe he wants to elim
inate the risk of stock price movements, in which case he would attempt to be gamma
neutral as well as delta neutral.
This seems like asimple concept until one first attempts to establish aposition
that is neutral with respect to more than one risk variable. For example, if one is

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880 Part VI: Measuring and Trading VolatiRty
attempting to create aspread that is neutral with respect to both gamma and delta,
he could attempt it in the following way:
Example: XYZ is 60. Aspreader wants to establish aspread that is neutral with
respect to both gamma and delta, using the following prices:
Option
October 60 call
October 70 call
Delta
0.60
0.25
Gamma
0.050
0.025
The secret to determining aspread that is neutral with respect to both risk meas
ures is to neutralize gamma first, for delta can always be neutralized by taking an off
setting position in the underlying security, whether it be stock or futures. First, deter
mine agamma neutral spread by dividing the two gammas:
Gamma neutral ratio= 0.050/0.025 = 2-to-l
So, buying one October 60 and selling two October 70 calls would be agamma
neutral spread. Now, the position delta of that spread is computed:
Position
Long 1 October 60 call
Short 2 October 70 calls
Net position delta:
Delta
0.60
0.25
Position Delta
+60 shares
-50 shares
+ 10 shares
Hence, this gamma neutral ratio is making the position delta long by 10 shares
of stock for each l-by-2 spread that is established. For example, if one bought 100
October 60 calls and sold 200 October 70 calls, his position delta would be long 1,000
shares.
This position delta is easily neutralized by selling short 1,000 shares of the stock.
The resulting position is both gamma neutral and delta neutral:
Option Position Option Position
Position Delta Delta Gamma Gamma
Short 1,000 XYZ 1.00 -1,000 0 0
Long 1 00 October 60 calls 0.60 +6,000 0.050 + 500
Short 200 October 70 calls 0.25 -5,000 0.025 - 500
Net Position: 0 0

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Chapter 40: Advanced Concepts 881
Hence, it is always asimple matter to create aposition that is both gamma and delta
neutral. In fact, it is just as simple to create aposition that is neutral with respect to
delta and any other risk measure, because all that is necessary is to create aneutral
ratio of the other risk measure (gamma, vega, theta, etc.) and then eliminate the
resulting position delta by using the underlying.
In theory, one could construct aposition that was neutral with respect to all five
risk measures (or six, if you really want to go overboard and include "gamma of the
gamma" as well). Of course, there wouldn'tbe much profit potential in such aposi
tion, either. But such constructions are actually employed, or at least attempted, by
traders such as market-makers who try to make their profits from the difference
between the bid and off er of an option quote, and not from assuming market risk
Still, the concept of being neutral with respect to more than one risk factor is avalid one. In fact, if astrategist can determine what he is really attempting to accom
plish, he can often negate other factors and construct aposition designed to accom
plish exactly what he wants. Suppose that one thought the implied volatility of acer
tain set of options was too high. He could just sell straddles and attempt to capture
that volatility. However, he is then exposed to movements by the underlying stock He
would be better served to construct aposition with negative vega to reflect his expec
tation on volatility, but then also have the position be delta neutral and gamma
neutral, so that there would be little risk to the position from market movements. This
can normally be done quite easily. An example will demonstrate how.
Example: XYZ is 48. There are three months to expiration, and the volatility of XYZ
and its options is 35%. The following information is also known:
XYZ:48
Option Price Delta Gamma Vega
April 50 call 2.50 0.47 0.045 0.08
April 60 call l.00 0.17 0.026 0.06
For whatever reasons - perhaps the historical volatility is much lower - the
strategist decides that he wants to sell volatility. That is, he wants to have anegative
position vega so that when the volatility decreases, he will make money. This can
probably be accomplished by buying some April 50 calls and selling more April 60
calls. However, he does not want any risk of price movement, so some analysis must
be done.
First, he should determine agamma neutral spread. This is done in much the
same manner as determining adelta neutral spread, except that gamma is used.

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882 Part VI: Measuring and Trading Volatmty
Merely divide the two gammas to determine the neutral ratio to be used. In this case,
assume that the April 50 call and the April 60 call are to be used:
Gamma neutral ratio: 0.045/0.026 = 1.73-to-l
Thus, agamma neutral position would be created by buying 100 April 50'sand sell
ing 173 April 60's. Alternatively, buying 10 and selling 17 would be close to gamma
neutral as well. The larger position will be used for the remainder of this example.
Now that this ratio has been chosen, what is the effect on delta and vega?
Option Position Option Position Option Position
Position Delta Delta Gamma Gamma Vega Vega
Long 1 00 April 50 0.47 +4,700 0.045 +450 0.08 + $800
Short 173 April 60 0.17 -2,941 0.026 -450 0.06 -1,038
Total: + 1,759 0 - $238
The position delta is long 1,759 shares of XYZ. This can easily be "cured" by
shorting 1,700 or 1,800 shares ofXYZ to neutralize the delta. Consequently, the com
plete position, including the short 1,700 shares, would be neutral with respect to both
delta and gamma, and would have the desired negative vega.
The actual profit picture at expiration is shown in Figure 40-11. Bear in mind,
however, that the strategist would normally not intend to hold aposition like this until
expiration. He would close it out if his expectations on volatility decline were fulfilled
( or proved false).
FIGURE 40-11.
Spread with negative vega; gamma and delta neutral.
40000...., ....
10000
50 55 60
XVZ :Stock Price

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Chapter 40: Advanced Concepts 883
One other point should be made: The fact that gamma and delta are neutral to
begin with does not mean that they will remain neutral indefinitely as the stock
moves (or even as volatility changes). However, there will be little or no effect of
stock price movements on the position in the short run.
In summary, then, one can always create aposition that is neutral with respect
to both gamma and delta by first choosing aratio that makes the gamma zero, and
then using aposition in the underlying security to neutralize the delta that is created
by the chosen ratio. This type of position would always involve two options and some
stock. The resulting position will not necessarily be neutral with respect to the other
risk factors.
THE MATHEMATICAL APPROACH
The strategist should be aware that the process of determining neutrality in several
of the risk variables can be handled quite easily by acomputer. All that is required is
to solve aseries of simultaneous equations.
In the preceding example, the resulting vega was negative: -$238. For each
decline of 1 percentage point in volatility from .the current level of 35%, one could
expect to make $238. This result could have been reached by another method, as
long as one were willing to spell out in advance the amount of vega risk he wants to
accept. Then, he can also assume the gamma is zero and solve for the quantity of
options to trade in the spread. The delta would be neutralized, as above, by using the
common stock.
Example: Prices are the same as in the preceding example. XYZ is 48. There are
three months to expiration, and the volatility of XYZ and its options is 35%. The fol
lowing information is also the same:
Option
April 50 call
April 60 call
Price
2.50
1.01
Delta
0.47
0.17
Gamma
0.045
0.026
Vega
0.08
0.06
Aspreader expects volatility to decline and is willing to set up aposition where
by he will profit by $250 for each one percentage decrease in volatility. Moreover, he
wants to be gamma and delta neutral. He knows that he can eventually neutralize any
delta by using XYZ common stock, as in the previous example. How many options
should be spread to achieve the desired result?

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