me XYZ gains anoth
er point in price, for it will eventually exceed 1.00 by that calculation, and it is known
that the delta has a maximum of 1.00. Thus, it is obvious that the gamma changes. In
general, the gamma is at its maximum point when the stock is near the strike of the
option. As the stock moves away from the strike in either direction, the gamma
decreases, approaching its minimum value of zero.
Conceptually, this means that a deeply in-the-money or deeply out-of-the
money option has a gamma of nearly zero. This makes sense - it implies that the delta
of a deep 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 a delta 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 a point.
In a similar manner, the January 45 put on XYZ would have a delta of -1.0 with
XYZ at 25. If XYZ moved up one point to 26, the put' s delta 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 a positive number, whether
the option is a put or a call.
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 a day
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
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 a different relationship between gamma and
time remaining. An out-of-the-money option that is about to expire has a very small
delta, and hence a very small gamma. However, if the out-of-the-money option has a
significant amount of time remaining, then it will have a larger 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 a slight 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 a call 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 a more 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
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's and 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 a
slightly higher gamma. Look two strikes higher and the more volatile options have a
vastly higher gamma, both for the January 60's and 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 a slightly 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 a result, the
gamma is high as well, because as the stock moves into-the-money, the increase in
Chapter 40: Advanced Concepts 857
delta will be more dramatic than it would be for a volatile 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 con


pter 40: Advanced Concepts 857
delta will be more dramatic than it would be for a volatile 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 a very 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 a volatile stock will be changing quite a bit for even
a 1-point move in the underlying stock. This is an important point for neutral strate
gists to note, because a position 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 a similar manner. An example of this fol
lows, using the same securities from the preceding example on the delta of a posi
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 a long 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 a small
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
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 a delta long of almost 700 shares. In addition,
one can now see that it has a positive 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 - a fact that makes
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
a long 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, a strategist with a position that has positive gamma has
a net long option position and is generally hoping for large market movements.
Conversely, if one has a position 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 a significant 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 a real 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 a positive number, whether it refers to
a put or a call.
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 volatil

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nges. Vega is always expressed as a positive number, whether it refers to
a put or a call.
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, a review of the term volatility is in order.
Volatility is a measure 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 a per
cent, although relating that percent to actual stock movements can be complicated.
Suffice it to say that a stock that has a 50% volatility is more volatile than a stock with
30% volatility. The stock market generally has a volatility of about 15% overall,
although that may change from time to time (crashes, for example).
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 (a loss 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's options 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 a slightly out-of-the-money option on a very 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 a whole ("position vega"). Since vega is
expressed as a positive 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
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 a positive 10.23 points ($1,023 since each point for these equity
options is worth $100). The fact that the position has a positive 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 a deep in- or out-
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 (a loss 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's options 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-mon


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 a slightly out-of-the-money option on a very 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 a whole ("position vega"). Since vega is
expressed as a positive 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
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 a positive 10.23 points ($1,023 since each point for these equity
options is worth $100). The fact that the position has a positive 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 a deep in- or out-
862 Part VI: Measuring and Trading Volatility
of-the-money option will not be affected much by a change in volatility. In addition,
for at-the-money options, longer-term options have a higher 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, a three-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 a
position with no delta and no gamma (delta neutral and gamma neutral) and still have
exposure to volatility. This does not mean that such a position would be undesirable;
it merely means that if one had such a position, 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 a position. 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's position. Theta is generally expressed
as a negative number, and it is expressed as the amount by which the option value
will change. Thus, if an option has a theta of -0.12, that means the option will lose
12 cents, or about an eighth of a point, per day. This is true for both puts and calls,
although the theta of a put and a call 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's time.
Thus, the theta of a long-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 a daily basis. The theta of options on a highly volatile
stock will be higher than the theta of options on a low-volatility stock. Obviously, the
former options are more expensive (have more time value) and therefore have more
time value to lose on a daily basis, thereby implying that they have a higher theta.
Finally, the decay is not linear - an option will lose a greater 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
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 compute


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
a good idea of how much gain or loss he can expect on a daily basis, due to time ero
sion. The following example demonstrates this point. Note that the underlying secu
rity itself has a theta of zero, since it cannot lose any value due to time decay.
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 a large 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 a positive position theta, while long options
have a negative position theta. A negative position theta means the position has risk
due to time, while a positive position theta means time is working for the position.
RHO
Rho is the name given to the price change of an option's value due to a change in
interest rates. Recall that one of the components that contributes to an option's price
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 a reversal
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 a positive number for calls and a negative 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
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 a few long-term options and/or
deeply in-the-money ones. Thus, rho is more important as a consideration 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
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 a position is currently bullish; if the underly
ing security goes up in price, the position should make money. A negative
delta indicates a bearish 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 a pre
ponderance of long options in the position, either puts or calls; negative
gamma indicates written or naked


ntly bullish; if the underly
ing security goes up in price, the position should make money. A negative
delta indicates a bearish 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 a pre
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 a decrease of volatility would be beneficial.
STRATEGY CONSIDERATIONS: USING THE
11
GREEKS"
Before looking at how one operates a particular 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 a general 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.
A few assumptions were made in constructing the table. First, it was assumed
that the strategies where delta is noted as being zero are established in a neutral
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.
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 a lot 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, a fact 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 a profit), vega is positive and rho is negative.
A bull 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, a bear 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; a negative 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.
868 Part VI: Measuring and Trading Volatility
They are useful in establishing a new 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's characteristics 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 a position or in following up on a position 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. A delta 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" (a big 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 a spread
wherein the January 50 calls are bought and a number of January 55's are sold. To
determine how many are to be bought and sold, one merely has to divide the delt


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 a spread
wherein the January 50 calls are bought and a number of January 55's are 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, a delta neutral ratio spread would consist of buying 7 January 50's and 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
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 - a neutral situation.
The resulting position is a ratio 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 a profit, 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 a profit 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 a spread, the ratio of
the two deltas is used. Moreover, the example didn't require 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 a ratio 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 a long straddle of sorts. Recall
that the delta of a put 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
870
FIGURE 40-8.
XYZ ratio spread.
f/)
f/)
3000
2000
.3 1000
~
e a..
(r.>
40
Part VI: Measuring and Trading Volatility
45 55 60
At Expiration
Stock Price
Thus, a delta neutral straddle position would consist of buying 8 J anua: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 a significantly 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 a lot, while the other makes money if the stock moves only a little. (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. A more 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. A delta neutral position may have seriously unneutral characteristics when
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 a straddle 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 op


. 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 a straddle 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 a ratio spread, a naked straddle, or a ratio 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
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, a very small delta. In fact, a person who is
a trader 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's straddle 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 a position - being short 100 straddles - for a small 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 a similar 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. A rise to 90, then, would make the position look like this:
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 a large position, such as being short
100 straddles, can acquire a large delta as the stock moves even a small 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. A small 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 a form 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 a psycho
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 a quick 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 x position delta + position gamma
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


500 ~ 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 a model 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 a price 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 a similar 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.
Chapter 40: Advanced Concepts
p2 = p 1 +delta+ gamma, or substituting from above
p2 = (p0 + delta) + delta + gamma
= Po + 2 x delta + 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 a gain of $9,600 on the sale of 100 puts, for a net unrealized loss
of $800. This verifies the result obtained above using position delta and position
gamma. Again, this confirms the logical fact that a quick stock movement will cause
unrealized losses in a short 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 a negative number, and
since these options are sold, the position theta is a positive number. A positive 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 a theoretical
profit from time decay of $600 per day.
The fact that the sale of a straddle generates profits from time decay is not
earth-shattering. That is a well-known fact. However, the amount of that time decay
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 a volatility of 30% for XYZ. The vega is quite
large. The fact that the call's vega 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 a loss of $1,800. The put has a sim
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 a position 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 a strategy 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.
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


n
one considers all the other factors.
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's favor.
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 a limited 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 a fairly likely chance that the underlying stock will be jumping
around quite a bit as well. That isn't good either. Thus, it is imperative that the strad
dle seller engage in the strategy only when there is a reasonable 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 a faster 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
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
a little time passes.
The type of analysis shown in the preceding examples gives a much 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't forget) have a description very similar to
straddle selling: Delta is positive, and the other factors are negative. This is a worse
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 a particularly 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't even 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 a large 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
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
a large 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 a few days, it does demonstrate that this position could become ver


his 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 a few 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 a small 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.
A neutral spreader would not use such a large 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 a risk. There is no reason why
a position cannot be constructed as veg a neutral 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 a simple concept until one first attempts to establish a position
that is neutral with respect to more than one risk variable. For example, if one is
880 Part VI: Measuring and Trading VolatiRty
attempting to create a spread that is neutral with respect to both gamma and delta,
he could attempt it in the following way:
Example: XYZ is 60. A spreader wants to establish a spread 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 a spread 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 a gamma 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 a gamma
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
Chapter 40: Advanced Concepts 881
Hence, it is always a simple matter to create a position that is both gamma and delta
neutral. In fact, it is just as simple to create a position that is neutral with respect to
delta and any other risk measure, because all that is necessary is to create a neutral
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 a position 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't be much profit potential in such a posi
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 a
valid one. In fact, if a strategist can determine what he is really attempting to accom
plish, he can often negate other factors and construct a position designed to accom
plish exactly what he wants. Suppose that one thought the implied volatility of a cer
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 a position 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 a negative
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 a gamma neutral spread. This is done in much the
same manner as determining a delta neutral spread, except that g


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 a gamma neutral spread. This is done in much the
same manner as determining a delta neutral spread, except that gamma is used.
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, a gamma neutral position would be created by buying 100 April 50's and 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 a position 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
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 a position that is neutral with respect
to both gamma and delta by first choosing a ratio that makes the gamma zero, and
then using a position 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 a computer. All that is required is
to solve a series 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
A spreader expects volatility to decline and is willing to set up a position 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?
884 Part VI: Measuring and Trading VolatiHty
To answer the question, one must create two equations in two unknowns, x and
y. The unknowns represent the quantities of options to be bought and sold, respec
tively. The constants in the equations are taken from the table above.
The first equation represents gamma neutral:
0.045 X + 0.026 y = 0,
where
xis the number of April 50's in the spread and y is the number of April 60's. Note
that the constants in the equation are the gammas of the two calls involved.
The second equation represents the desired vega risk of making 2.5 points, or $250,
if the volatility decreases:
0.08 X + 0.06 y = - 2.5,
where
x and y are the same quantities as in the first equation, and the constants in this equa
tion are the gammas of the options. Furthermore, note that the vega risk is negative,
since the spreader wants to profit if volatility decreases.
Solving the two equations in two unknowns by algebraic methods yields the fol
lowing results:
Equations:
0.045 X + 0.026 y = 0
0.08 X + 0.06 Y = - 2.5
Solutions:
X = 104.80
y = -181.45
This means that one would buy 105 April 50 calls, since x being positive means that
the options would be bought. He would also sell 181 April 60 calls (y is negative,
which implies that the calls would be sold). This is nearly the same ratio determined
in the previous example. The quantities are slightly higher, since the vega here is
-$250 instead of the -$238 achieved in the previous example.
Finally, one would again determine the amount of stock to buy or sell to neu
tralize the delta by computing the position delta:
Position delta = 105 x 0.47 - 181 x 0.17 = 18.58
Thus 1,858 shares of XYZ would be shorted to neutralize the position.
Chapter 40: Advanced Concepts 885
Note: All the equations cannot be s


the vega here is
-$250 instead of the -$238 achieved in the previous example.
Finally, one would again determine the amount of stock to buy or sell to neu
tralize the delta by computing the position delta:
Position delta = 105 x 0.47 - 181 x 0.17 = 18.58
Thus 1,858 shares of XYZ would be shorted to neutralize the position.
Chapter 40: Advanced Concepts 885
Note: All the equations cannot be set equal to zero, or the solution will be all
zeros. This is easily handled by setting at least one equation equal to a small, nonzero
quantity, such as 0.1. As long as at least one of the risk factors is nonzero, one can
determine the neutral ratio for all other factors merely by solving these simultaneous
equations. There are plenty of low-cost computer programs that can solve simultane
ous equations such as these.
This concept can be carried to greater lengths in order to determine the best
spread to create in order to achieve the desired results. One might even try to use
three different options, using the third option to neutralize delta, so that he wouldn't
have to neutralize with stock. The third equation would use deltas as constants and
would be set to equal zero, representing delta neutral. Solving this would require
solving three equations in three unknowns, a simple matter for a computer.
As long as at least one of the risk factors is nonzero, one can determine the neu
tral ratio for all other factors merely by solving these simultaneous equations. Even
more importantly, the computer can scan many combinations of options that produce
a position that is both gamma and delta neutral and has a specific position vega
(-$238, for example). One would then choose the "best" spread of the available pos
sibilities by logical methods including, if possible, choosing one with positive theta,
so time is working in his favor.
To summarize, one can neutralize all variables, or he can specify the risk that he
wants to accept in any of them. Merely write the equations and solve them. It is best
to use a computer to do this, but the fact that it can be done adds an entirely new,
broad dimension to option spreading and risk-reducing strategies.
EVALUATING A POSITION USING THE RISK MEASURES
The previous sections have dealt with establishing a new position and determining its
neutrality or lack thereof. However, the most important use of these risk measures is
to predict how a position will perform into the future. At a minimum, a serious strate
gist should use a computer to print out a projection of the profits and losses and posi
tion risk at future expected prices. Moreover, this type of analysis should be done for
several future times in order to give the strategist an idea of how the passage of time
and the resultant larger movements by the underlying security would affect the posi
tion.
First, one would choose an appropriate time period - say, 7 days hence - for the
first analysis. Then he should use the statistical projection of stock prices (see Chapter
28 on mathematical applications) to determine probable prices for the underlying
security at that time. Obviously, this stock price projection needs to use volatility, and
886 Part VI: Measuring and Trading Volatility
that is somewhat variable. But, for the purposes of such a projection, it is acceptable
to use the current volatility. The results of as many as 9 stock prices might be dis
played: every one-half standard deviation from -2 through + 2 (-2.0, -1.5, -1.0,
-0.5, 0, 0.5, 1.0, 1.5, 2.0).
Example: XYZ is at 60 and has a volatility of 35%. A distribution of stock prices 7
days into the future would be determined using the equation:
Future Price = Current Price x eav-ft
where
a corresponds to the constants in the following table: (-2.0 ... 2.0):
# Standard Deviations
-2.0
- 1.5
- 1.0
-0.5
0
0.5
1.0
1.5
2.0
Projected Stack Price
54.46
55.79
57.16
58.56
60.00
61.47
62.98
64.52
66.11
Again, refer to Chapter 28 on mathematical applications for a more in-depth
discussion of this price determination equation.
Note that the formula used to project prices has time as one of its components.
This means that as we look further out in time, the range of possible stock prices will
expand - a necessary and logical component of this analysis. For example, if the
prices were being determined 14 days into the future, the range of prices would be
from 52.31 to 68.82. That is, XYZ has the same probability of being at 54.46 in 7 days
that it has of being at 52.31 in 14 days. At expiration, some 90 days hence, the range
would be quite a bit wider still. Do not make the mistake of trying to evaluate the
position at the same prices for each time period (7 days, 14 days, 1 rnonth, expiration,
etc.). Such an analysis would be wrong.
Once the appropriate stock prices have been determined, the following quanti
ties would be calculated for each stock price: profit or loss, position delta, position
gamma, position theta, and position vega. (Position rho is generally a less important
risk measure for stock and futures short-term options.) Armed with this information,
the strategist can be prepared to face the future. An important item to note: A model
Chapter 40: Advanced Concepts 887
will necessarily be used to make these projections. As was shown earlier, if there is a
distortion in the current implied volatilities of the options involved in the position,
the strategist should use the current implieds as input to the model for future option
price projections. If he does not, the position may look overly attractive if expensive
options are being sold or cheap ones are being bought. A truer profit picture is
obtained by propagating the current implied volatility structure into the near future.
Using an example similar to the previous one a ratio spread using short stock
to make it delta neutral - the concepts will be described.
Initial Position. XYZ is at 60. The January 70 calls, which h


erly attractive if expensive
options are being sold or cheap ones are being bought. A truer profit picture is
obtained by propagating the current implied volatility structure into the near future.
Using an example similar to the previous one a ratio spread using short stock
to make it delta neutral - the concepts will be described.
Initial Position. XYZ is at 60. The January 70 calls, which have three months
until expiration, are expensive with respect to the January 60 calls. A strategist
expects this discrepancy to disappear when the implied volatility of XYZ options
decreases. He therefore established the following position, which is both gamma
and delta neutral.
Position Delta Gamma
Long 100 January 60 calls 0.57 0.0723
Short 240 January 70 calls 0.20 0.0298
Short 800 XYZ
The risk measures for the entire position are:
Position delta: -38 shares (virtually delta neutral)
Position gamma: + 7 shares (gamma neutral)
Position theta: + $263
Position vega: -$827
Theta Vega
-0.020 0.109
-0.019 0.080
Thus, the position is both gamma and delta neutral. Moreover, it has the attrac
tive feature of making $263 per day because of the positive theta. Finally, as was the
intention of the spreader, it will make money if the volatility of XYZ declines: $827
for each percentage point decrease in implied volatility. Two equations in two
unknowns (gamma and vega) were solved to obtain the quantities to buy and sell. The
resulting position delta was neutralized by selling 800 XYZ.
The following analyses will assume that the relative expensiveness of the April
70 calls persists. These are the calls that were sold in the position. If that overpricing
should disappear, the spread would look more favorable, but there is no guarantee
that they will cheapen - especially over a short time period such as one or two weeks.
How would the position look in 7 days at the stock prices determined above?
888 Part VI: Measuring and Trading Vo/atillty
Stock
Price P&L Delta Gamma Theta Vega
54.46 1905 - 7.40 1.62 0.94 - 1.57
55.79 1077 - 4.90 2.07 1.18 - 1.96
57.16 606 1.97 2.13 1.53 - 2.90
58.56 528 0.74 1.65 2.00 -4.62
60.00 771 2.38 0.56 2.63 -7.22
61.47 1127 2.07 - 1.01 3.38 -10.63
62.98 1252 - 0.87 - 2.85 4.22 -14.56
64.52 702 - 6.73 - 4.67 5.07 -18.61
66.11 - 1019 -15.42 - 6.21 5.85 -22.31
In a similar manner, the position would have the following characteristics after
14 days had passed:
Stock
Price P&L Delto Gamma Theta Vega
52.31 4221 - 9.10 0.69 0.55 - 0.98
54.14 2731 - 6.93 1.69 0.75 - 0.89
56.02 1782 - 2.87 2.51 1.06 - 1.21
57.98 1717 2.17 2.44 1.61 - 2.69
60.00 2577 5.85 1.00 2.51 -6.00
62.09 3839 5.29 - 1.63 3.73 -11.05
64.26 4361 - 1.55 - 4.61 5.09 -16.90
66.50 2631 -14.80 - 7.02 6.31 -22.17
68.82 - 2799 -32.83 - 8.32 7.18 -25.72
The same information will be presented graphically in Figure 40-13 so that
those who prefer pictures instead of columns of numbers can follow the discussions
easily.
First, the profitability of the spread can be examined. This profit picture
assumes that the volatility of XYZ remains unchanged. Note that in 7 days, there is a
small profit if the stock remains unchanged. This is to be expected, since theta was
positive, and therefore time is working in favor of this spread. Likewise, in 14 days,
there is an even bigger profit if XYZ remains relatively unchanged - again due to the
positive theta. Overall, there is an expected profit of $800 in 7 days, or $2,600 in 14
days, from this position. This indicates that it is an attractive situation statistically, but,
of course, it does not mean that one cannot lose money.
Chapter 40: Advanced Concepts 889
Continuing to look at the profit picture, the downside is favorable to the spread
since the short stock in the position would contribute to ever larger profits in the case
that XYZ tumbles dramatically (see Figure 40-12). The upside is where problems
could develop. In 7 days, the position breaks even at about 65 on the upside; in 14
days, it breaks even at about 67.50.
The reader may be asking, "Why is there such a dramatic risk to the upside? I
thought the position was delta neutral and gamma neutral." True, the position was
originally neutral with respect to both those variables. That neutrality explains the
flatness of the profit curves about the current stock price of 60. However, once the
stock has moved 1.50 standard deviations to the upside, the neutrality begins to dis
appear. To see this, let us look at Figures 40-13 and 40-14 that show both the posi
tion delta and position gamma 7 days and 14 days after the spread was established.
Again, these are the same numbers listed in the previous tables.
First, look at the position delta in 7 days (Figure 40-13). Note that the position
remains relatively delta neutral with XYZ between 57 and 63. This is because the
gamma was initially neutral. However, the position begins to get quite delta short if
XYZ rises above 63 or falls below 57 in 7 days. What is happening to gamma while
this is going on? Since we just observed that the delta eventually changes, that has to
mean that the position is acquiring some gamma.
FIGURE 40-12.
XYZ ratio spread, gamma and delta neutral.
4300
3400
2500
1600
~ 700 a..
0
-200 53 55 57 59 61 63
.-1100
-2000
Stock Price
890
FIGURE 40-13.
XYZ ratio spread, position delta.
-300
-800
fu -1300 w
-1800
-2300
-2800
FIGURE 40-14.
Stock Price
XYZ ratio spread, position gamma.
100
Part VI: Measuring and Trading Volatility
67
0t----....----....----,---.---'"""".,------,----,---r--
-100
«l
E E -300
«l
0
-500
-700
63 65 67
Stock Price
Chapter 40: Advanced Concepts 891
Figure 40-14 depicts the fact that gamma is not very stable, considering that it
started at nearly zero. If XYZ falls, gamma increases a little, reflecting the fact that
the position will get somewhat shorter as XYZ falls. But since there are only calls cou
pled with short stock in this po


---,---r--
-100
«l
E E -300
«l
0
-500
-700
63 65 67
Stock Price
Chapter 40: Advanced Concepts 891
Figure 40-14 depicts the fact that gamma is not very stable, considering that it
started at nearly zero. If XYZ falls, gamma increases a little, reflecting the fact that
the position will get somewhat shorter as XYZ falls. But since there are only calls cou
pled with short stock in this position, there is no risk to the downside. Positive
gamma, even a small positive gamma like this one, is beneficial to stock movement.
The upside is another matter entirely. The gamma begins to become seriously
negative above a stock price of 63 in 7 days. Recall that negative gamma means that
one's position is about to react poorly to price changes in the market - the position
will soon be "fighting the market." As the stock goes even higher, the gamma
becomes even more negative. These observations apply to stock price movements in
either 7 days or 14 days; in fact, the effect on gamma does not seem to be particu
larly dependent on time in this example, since the two lines on Figure 40-15 are very
close to each other.
The above information depicts in detailed form the fact that this position will
not behave well if the stock rises too far in too short a time. However, stable stock
prices will produce profits, as will falling prices. These are not earth-shattering con
clusions since, by simple observation, one can see that there are extra short calls plus
some short stock in the position. However, the point of calculating this information
in advance is to be able to anticipate where to make adjustments and how much to
adjust.
Follow-Up Action. How should the strategist use this information? A sim
plistic approach is to adjust the delta as it becomes non-neutral. This won't do
anything for gamma, however, and may therefore not necessarily be the best
approach. If one were to adjust only the delta, he would do it in the following
manner: The chart of delta (Figure 40-13) shows that the position will be
approximately delta short 800 shares if XYZ rises to 64.50 in a week. One sim
ple plan would be to cover the 800 shares of XYZ that are short if the stock rises
to 64.50. Covering the 800 shares would return the position to delta neutral at
that time. Note that if the stock rises at a slower pace, the point at which the
strategist would cover the 800 shares moves higher. For example, the delta in 14
days (again in Figure 40-13) shows that XYZ would have to be at about 65.50 for
the position to be delta short 800 shares. Hence, if it took two weeks for XYZ to
begin rising, one could wait until 65.50 before covering the 800 shares and
returning the position to delta neutral.
In either case, the purchase of the 800 shares does not take care of the negative
gamma that is creeping into the position as the stock rises. The only way to counter
negative gamma is to buy options, not stock. To return a position to neutrality with
892 Part VI: Measuring and Trading Volatility
respect to more than one risk variable requires one to approach the problem as he
did when the position was established: Neutralize the gamma first, and then use stock
to adjust the delta. Note the difference between this approach and the one described
in the previous paragraph. Here, we are trying to adjust gamma first, and will get to
delta later.
In order to add some positive gamma, one might want to buy back (cover) some
of the January 70 calls that are currently short. Suppose that the decision is made to
cover when XYZ reaches 65.50 in 14 days. From the graph above, one can see that
the position would be approximately gamma short 700 shares at the time. Suppose
that the gamma of the January 70 calls is 0.07. Then, one would have to cover 100
January 70 calls to add 700 shares of positive gamma to the position, returning it to
gamma neutral. This purchase would, of course, make the position delta long, so
some stock would have to be sold short as well in order to make the position delta
neutral once again.
Thus, the procedure for follow-up action is somewhat similar to that for estab
lishing the position: First, neutralize the gamma and then eliminate the resulting
delta by using the common stock. The resulting profit graph will not be shown for
this follow-up adjustment, since the process could go on and on. However, a few
observations are pertinent. First, the purchase of calls to reduce the negative gamma
hurts the original thesis of the position - to have negative vega and positive theta, if
possible. Buying calls will add vega to and subtract theta from the position, which is
not desirable. However, it is more desirable than letting losses build up in the posi
tion as the stock continues to run to the upside. Second, one might choose to rerrwve
the position if it is profitable. This might happen if the volatility did decrease as
expected. Then, when the stock rallies, producing negative gamma, one might actu
ally have a profit, because his assumption concerning volatility had been right. If he
does not see much further potential gains from decreasing volatility, he might use the
point at which negative gamma starts to build up as the exit point from his position.
Third, one might choose to accept the acquired gamma risk. Rather than jeopardize
his initial thesis, one may just want to adjust the delta and let the gamma build up.
This is no longer a neutral strategy, but one may have reasons for approaching the
position this way. At least he has calculated the risk and is aware of it. If he chooses
to accept it rather than eliminate it, that is his decision.
Finally, it is obvious that the process is dynamic. As factors change (stock price,
volatility, time), the position itself changes and the strategist is presented with new
choices. There is no absolutely correct adjustment. The process is more of an art than
a science at times. Moreover, the strategist should conti


it. If he chooses
to accept it rather than eliminate it, that is his decision.
Finally, it is obvious that the process is dynamic. As factors change (stock price,
volatility, time), the position itself changes and the strategist is presented with new
choices. There is no absolutely correct adjustment. The process is more of an art than
a science at times. Moreover, the strategist should continue to recalculate these prof
it pictures and risk measures as the stock moves and time passes, or if there is a
Chapter 40: Advanced Concepts 893
change in the securities involved in the position. There is one absolute truism and
that is that the serious strategist should be aware of the risk his position has with
respect to at least the four basic measures of delta, gamma, theta, and vega. To be
ignorant of the risk is to be delinquent in the management of the position.
TRADING GAMMA FROM THE LONG SIDE
The strategist who is selling overpriced options and hedging that purchase with other
options or stock will often have a position similar to the one described earlier. Large
stock movements - at least in one direction will typically be a problem for such
positions. The opposite of this strategy would be to have a position that is long
gamma. That is, the position does better if the stock moves quickly in one direction.
While this seems pleasing to the psyche, these types of positions have their own
brand of risk.
The simplest position with long gamma is a long straddle, or a backspread
(reverse ratio spread). Another way to construct a position with long gamma is to
invert a calendar spread - to buy the near-term option and to sell a longer-term one.
Since a near-term option has a higher gamma than a longer-term one with the same
strike, such a position has long gamma. In fact, traders who expect violent action in
a stock often construct such a position for the very reason that the public will come
in behind them, bid up the short-term calls (increasing their implied volatility), and
make the spread more profitable for the trader.
Unfortunately, all of these positions often involve being long just about every
thing else, including theta and vega as well. This means that time is working against
the position, and that swings in implied volatility can be helpful or harmful as well.
Can one construct a position that is long gamma, but is not so subject to the other
variables? Of course he can, but what would it look like? The answer, as one might
suspect, is not an ironclad one.
For the following examples, assume these prices exist:
XYZ: 60
Option
March 60 call
June 60 call
Price
3.25
5.50
Delta
0.54
0.57
Gamma
0.0510
0.0306
Theta
0.033
0.021
Vega
0.089
0.147
Example: Suppose that a strategist wants to create a position that is gamma long, but
is neutral with respect to both delta and vega. He thinks the stock will move, but is
not sure of the price direction, and does not want to have any risk with respect to
894 Part VI: Measuring and Trading Volatility
quick changes in volatility. In order to quantify the statement that he "wants to be
gamma long," let us assume that he wants to be gamma long 1,000 shares or 10 con
tracts.
It is known that delta can always be neutralized last, so let us concentrate on the
other two variables first. The two equations below are used to determine the quanti
ties to buy in order to make gamma long and vega neutral:
0.0510x + 0.0306y = 10 (gamma, expressed in# of contracts)
0.089x + 0.147y = 0 (vega)
The solution to these equations is:
X = 308, y = -186
Thus, one would buy 308 March 60 calls and would sell 186 June 60 calls. This is the
reverse calendar spread that was discussed: Near-term calls are bought and longer
term calls are sold.
Finally, the delta must be neutralized. To do this, calculate the position delta
using the quantities just determined:
Position delta= 0.54 x 308 - 0.57 x 186 = 60.30
So, the position is long 60 contracts, or 6,000 shares. It can be made delta neutral by
selling short 6,000 shares of XYZ.
The overall position would look like this:
Position
Short 6,000 XYZ
Long 308 March 60 calls
Short 186 June 60 calls
Its risk measurements are:
Delta
1.00
0.54
0.57
Position delta: long 30 shares (neutral)
Position vega: $7 (neutral)
Position gamma: long 1,001 shares
Gamma
0
0.0510
0.0306
Vega
0
0.089
0.147
This position then satisfies the initial objectives of wanting to be gamma long
1,000 shares, but delta and vega neutral.
Finally, note that theta = -$625. The position will lose $625 per day from time
decay.
The strategist must go further than this analysis, especially if one is dealing with
positions that are not simple constructions. He should calculate a profit picture as
Chapter 40: Advanced Concepts 895
well as look at how the risk measures behave as time passes and the stock price
changes.
Figure 40-15 (see Tables 40-10, 40-11, and 40-12) shows the profit potential in
7 days, in 14 days, and at March expiration. Figure 40-16 shows the position vega at
the 7- and 14-day time intervals. Before discussing these items, the data will be pre
sented in tabular form at three different times: in 7 days, in 14 days, and at March
expiration.
The data in Table 40-10 depict the position in 7 days.
Table 40-11 represents the results in 14 days.
Finally, the position as it looks at March expiration should be known as well (see
Table 40-12).
In each case, note that the stock prices are calculated in accordance with the
statistical formula shown in the last section. The more time that passes, the further it
is possible for the stock to roam from the current price.
The profit picture (Figure 40-15) shows that this position looks much like a long
straddle would: It makes large, symmetric profits if the stock goes either way up or
way down. Moreover, the losses if the stock remains relatively unchanged can be
large. These losses tend to mount right away