option must be
analyzed using the greeks. This is true of the variants of the calendar—
double calendars, diagonals, and double diagonals—as well. This chapter
will show how to analyze strategies that involve options with different
expirations and discuss how and when to use them.
Buying the Calendar
The calendar spread and all its variations are commonly associated with
income-generating spreads. Using calendar spreads as income generators is
popular among retail and professional traders alike. The process involves
buying a longer-term at-the-money option and selling a shorter-term at-the-
money (ATM) option. The options must be either both calls or both puts.
Because this transaction results in a net debit—the longer-term option being
purchased has a higher premium than the shorter-term option being sold—
this is referred to as buying the calendar.
The main intent of buying a calendar spread for income is to profit from
the positive net theta of the position. Because the shorter-term ATM option
decays at a faster rate than the longer-term ATM option, the net theta is
positive. As for most income spreads, the ideal outcome occurs when the
underlying is at the short strike (in this case, shared strike) when the
shorter-term option expires. At this strike price, the long option has its
highest value, while the short option expires without the trader’s getting
assigned. As long as the underlying remains close to the strike price, the
value of the spread rises as time passes, because the short option decreases
in value faster than the long option.
For example, a trader, Richard, watches Bed Bath & Beyond Inc. (BBBY)
on a regular basis. Richard believes that Bed Bath & Beyond will trade in a
range around $57.50 a share (where it is trading now) over the next month.
Richard buys the January–February 57.50 call calendar for 0.80. Assuming
January has 25 days until expiration and February has 53 days, Richard will
execute the following trade:
Richard’s best-case scenario occurs when the January calls expire at
expiration and the February calls retain much of their value.
If Richard created an at-expiration P&(L) diagram for his position, he’d
have trouble because of the staggered expiration months. A general
representation would look something like Exhibit 11.1 .
EXHIBIT 11.1 Bed Bath & Beyond January–February 57.50 calendar.
The only point on the diagram that is drawn with definitive accuracy is
the maximum loss to the downside at expiration of the January call. The
maximum loss if Bed Bath & Beyond falls low enough is 0.80—the debit
paid for the spread. If Bed Bath & Beyond is below $57.50 at January
expiration, the January 57.50 call expires worthless, and the February 57.50
call may or may not have residual value. If Bed Bath & Beyond declines
enough, the February 57.50 call can lose all of its value, even with residual
time until expiration. If the stock falls enough, the entire 0.80 debit would
be a loss.
If Bed Bath & Beyond is above $57.50 at January expiration, the January
57.50 call will be trading at parity. It will be a negative-100-delta option,
imitating short stock. If Bed Bath & Beyond is trading high enough, the
February 57.50 call will become a positive-100-delta option trading at
parity plus the interest calculated on the strike. The February deep-in-the-
money option would imitate long stock. At a 2 percent interest rate, interest
on the 57.50 strike is about 0.17. Therefore, Richard would essentially have
a short stock position from $57.50 from the January 57.50 call and would
be essentially long stock from $57.50 plus 0.28 from the February call. The
maximum loss to the upside is about 0.63 (0.80 − 0.17).
The maximum loss if Bed Bath & Beyond is trading over $57.50 at
expiration is only an estimate that assumes there is no time value and that
interest and dividends remain constant. Ultimately, the maximum loss will
be 0.80, the premium paid, if there is no time value or carry considerations.
The maximum profit is gained if Bed Bath & Beyond is at $57.50 at
expiration. At this price, the February 57.50 call is worth the most it can be
worth without having the January 57.50 call assigned and creating negative
deltas to the upside. But how much precisely is the maximum profit?
Richard would have to know what the February 57.50 call would be worth
with Bed Bath & Beyond stock trading at $57.50 at February expiration
before he can know the maximum profit potential. Although Richard can’t
know for sure at what price the calls will be trading, he can use a pricing
model to estimate the call’s value. Exhibit 11.2 shows analytics at January
expiration.
EXHIBIT 11.2 Bed Bath & Beyond January–February 57.50 call calendar
greeks at January expiration.
With an unchanged implied volatility of 23 percent, an interest rate of two
percent, and no dividend payable before February expiration, the February
57.50 calls would be valued at 1.53 at January expiration. In this best-case
scenario, therefore, the spread would go from 0.80, where Richard
purchased it, to 1.53, for a gain of 91 percent. At January expiration, with
Bed Bath & Beyond at $57.50, the January call would expire; thus, the
spread is composed of just the February 57.50 call.
Let’s now go back in time and see how Richard figured this trade. Exhibit
11.3 shows the position when the trade is established.
EXHIBIT 11.3 Bed Bath & Beyond January–February 57.50 call calendar.
A small and steady rise in the stock price with enough time to collect
theta is the recipe for success in this trade. As time passes, delta will flatten
out if Bed Bath & Beyond is still right at-the-money. The delta of the
January call that Richard is short will move closer to exactly −0.50. The
February call delta moves toward exactly +0.50.
Gamma and theta will both rise if Bed Bath & Beyond stays around the
strike. As expiration approaches, there is greater risk if there is movement
and greater reward if there is not.
Vega is positive because the long-term


h & Beyond is still right at-the-money. The delta of the
January call that Richard is short will move closer to exactly −0.50. The
February call delta moves toward exactly +0.50.
Gamma and theta will both rise if Bed Bath & Beyond stays around the
strike. As expiration approaches, there is greater risk if there is movement
and greater reward if there is not.
Vega is positive because the long-term option with the higher vega is the
long leg of the spread. When trading calendars for income, implied
volatility (IV) must be considered as a possible threat. Because it is
Richard’s objective to profit from Bed Bath & Beyond being at $57.50 at
expiration, he will try to avoid vega risk by checking that the implied
volatility of the February call is in the lower third of the 12-month range.
He will also determine if there are any impending events that could cause
IV to change. The less likely IV is to drop, the better.
If there is an increase in IV, that may benefit the profitability of the trade.
But a rise in IV is not really a desired outcome for two reasons. First, a rise
in IV is often more pronounced in the front month than in the months
farther out. If this happens, Richard can lose more on the short call than he
makes on the long call. Second, a rise in IV can indicate anxiety and
therefore a greater possibility for movement in the underlying stock.
Richard doesn’t want IV to rock the boat. “Buy low, stay low” is his credo.
Rho is positive also. A rise in interest rates benefits the position because
the long-term call is helped by the rise more than the short call is hurt. With
only a one-month difference between the two options, rho is very small.
Overall, rho is inconsequential to this trade.
There is something curious to note about this trade: the gamma and the
vega. Calendar spreads are the one type of trade where gamma can be
negative while vega is positive, and vice versa. While it appears—at least
on the surface—that Richard wants higher IV, he certainly wants low
realized volatility.
Bed Bath & Beyond January–February 57.50 Put
Calendar
Richard’s position would be similar if he traded the January–February 57.50
put calendar rather than the call calendar. Exhibit 11.4 shows the put
calendar.
EXHIBIT 11.4 Bed Bath & Beyond January–February 57.50 put calendar.
The premium paid for the put spread is 0.75. A huge move in either
direction means a loss. It is about the same gamma/theta trade as the 57.50
call calendar. At expiration, with Bed Bath & Beyond at $57.50 and IV
unchanged, the value of the February put would be 1.45—a 93 percent gain.
The position is almost exactly the same as the call calendar. The biggest
difference is that the rho is negative, but that is immaterial to the trade. As
with the call spread, being short the front-month option means negative
gamma and positive theta; being long the back month means positive vega.
Managing an Income-Generating
Calendar
Let’s say that instead of trading a one-lot calendar, Richard trades it 20
times. His trade in this case is
His total cash outlay is $1,600 ($80 times 20). The greeks for this trade,
listed in Exhibit 11.5 , are also 20 times the size of those in Exhibit 11.3 .
EXHIBIT 11.5 20-Lot Bed Bath & Beyond January–February 57.50 call
calendar.
Note that Richard has a +0.18 delta. This means he’s long the equivalent
of about 18 shares of stock—still pretty flat. A gamma of −0.72 means that
if Bed Bath & Beyond moves $1 higher, his delta will be starting to get
short; and if it moves $1 lower he will be longer, long 90 deltas.
Richard can use the greeks to get a feel for how much the stock can move
before negative gamma causes a loss. If Bed Bath & Beyond starts trending
in either direction, Richard may need to react. His plan is to cover his deltas
to continue the position.
Say that after one week Bed Bath & Beyond has dropped $1 to $56.50.
Richard will have collected seven days of theta, which will have increased
slightly from $18 per day to $20 per day. His average theta during that time
is about $19, so Richard’s profit attributed to theta is about $133.
With a big-enough move in either direction, Richard’s delta will start
working against him. Since he started with a delta of +0.18 on this 20-lot
spread and a gamma of −0.72, one might think that his delta would increase
to 0.90 with Bed Bath & Beyond a dollar lower (18 − [−0.072 × 1.00]). But
because a week has passed, his delta would actually get somewhat more
positive. The shorter-term call’s delta will get smaller (closer to zero) at a
faster rate compared to the longer-term call because it has less time to
expiration. Thus, the positive delta of the long-term option begins to
outweigh the negative delta of the short-term option as time passes.
In this scenario, Richard would have almost broken even because what
would be lost on stock price movement, is made up for by theta gains.
Richard can sell about 100 shares of Bed Bath & Beyond to eliminate his
immediate directional risk and stem further delta losses. The good news is
that if Bed Bath & Beyond declines more after this hedge, the profit from
the short stock offsets losses from the long delta. The bad news is that if
BBBY rebounds, losses from the short stock offset gains from the long
delta.
After Richard’s hedge trade is executed, his delta would be zero. His
other greeks remain unchanged. The idea is that if Bed Bath & Beyond
stays at its new price level of $56.50, he reaps the benefits of theta
increasing with time from $18 per day. Richard is accepting the new price
level and any profits or losses that have occurred so far. He simply adjusts
his directional exposure to a zero delta.
Rolling and Earning a “Free” Call
Many traders who trade income-generating strategies are conservative.
They are happy to sell low IV for the benefits afforded by low realized
volatility. This is the problem-avoidance philosophy of trading. Due to risk
aversion, it’s common to trade calendar spreads by buying the two-month
opti

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf


ed so far. He simply adjusts
his directional exposure to a zero delta.
Rolling and Earning a “Free” Call
Many traders who trade income-generating strategies are conservative.
They are happy to sell low IV for the benefits afforded by low realized
volatility. This is the problem-avoidance philosophy of trading. Due to risk
aversion, it’s common to trade calendar spreads by buying the two-month
option and selling the one-month option. This can allow traders to avoid
buying the calendar in earnings months, and it also means a shorter time
horizon, signifying less time for something unwanted to happen.
But there’s another school of thought among time-spread traders. There
are some traders who prefer to buy a longer-term option—six months to a
year—while selling a one-month option. Why? Because month after month,
the trader can roll the short option to the next month. This is a simple tactic
that is used by market makers and other professional traders as well as
savvy retail traders. Here’s how it works.
XYZ stock is trading at $60 per share. A trader has a neutral outlook over
the next six months and decides to buy a calendar. Assuming that July has
29 days until expiration and December has 180, the trader will take the
following position:
The initial debit here is 2.55. The goal is basically the same as for any
time spread: collect theta without negative gamma spoiling the party. There
is another goal in these trades as well: to roll the spread.
At the end of month one, if the best-case scenario occurs and XYZ is
sitting at $60 at July expiration, the July 60 call expires. The December 60
call will then be worth 3.60, assuming all else is held constant. The positive
theta of the short July call gives full benefits as the option goes from 1.45 to
zero. The lower negative theta of the December call doesn’t bite into profits
quite as much as the theta of a short-term call would.
The profit after month one is 1.05. Profit is derived from the December
call, worth 3.60 at July expiry, minus the 2.55 initial spread debit. This
works out to about a 41 percent return. The profit is hardly as good as it
would have been if a short-term, less expensive August 60 call were the
long leg of this spread.
Rolling the Spread
The July–December spread is different from short-term spreads, however.
When the Julys expire, the August options will have 29 days until
expiration. If volatility is still the same, XYZ is still at $60, and the trader’s
forecast is still neutral, the 29-day August 60 calls can be sold for 1.45. The
trader can either wait until the Monday after July expiration and then sell
the August 60s, or when the Julys are offered at 0.05 or 0.10, he can buy the
Julys and sell the Augusts as a spread. In either case, it is called rolling the
spread. When the August expires, he can sell the Septembers, and so on.
The goal is to get a credit month after month. At some point, the
aggregate credit from the call sales each month is greater than the price
initially paid for the long leg of the spread, thus eliminating the original net
debit. Exhibit 11.6 shows how the monthly credits from selling the one-
month calls aggregate over time.
EXHIBIT 11.6 A “free” call.
After July has expired, 1.45 of premium is earned. After August
expiration, the aggregate increases to 2.90. When the September calls,
which have 36 days until expiration, are sold, another 1.60 is added to the
total premium collected. Over three months—assuming the stock price,
volatility, and the other inputs don’t change—this trader collects a total of
4.50. That’s 0.50 more than the price originally paid for the December 60
call leg of the spread.
At this point, he effectively owns the December call for free. Of course,
this call isn’t really free; it’s earned. It’s paid for with risk and maybe a few
sleepless nights. At this point, even if the stock and, consequently, the
December call go to zero, the position is still a profitable trade because of
the continued month-to-month rolling. This is now a no-lose situation.
When the long call of the spread has been paid for by rolling, there are
three choices moving forward: sell it, hold it, or continue writing calls
against it. If the trader’s opinion calls for the stock to decline, it’s logical to
sell the December call and take the residual value as profit. In this case,
over three months the trade will have produced 4.50 in premium from the
sale of three consecutive one-month calls, which is more than the initial
purchase price of the December call. At September expiration, the premium
that will be received for selling the December call is all profit, plus 0.50,
which is the aggregate premium minus the initial cost of the December call.
If the outlook is for the underlying to rise, it makes sense to hold the call.
Any appreciation in the value of the call resulting from delta gains as the
underlying moves higher is good—$0.50 plus whatever the call can be sold
for.
If the forecast is for XYZ to remain neutral, it’s logical to continue selling
the one-month call. Because the December call has been financed by the
aggregate short call premiums already, additional premiums earned by
writing calls are profit with “free” protection. As long as the short is closed
at its expiration, the risk of loss is eliminated.
This is the general nature of rolling calls in a calendar spread. It’s a
beautiful plan when it works! The problem is that it is incredibly unlikely
that the stock will stay right at $60 per share for five months. It’s almost
inevitable that it will move at some point. It’s like a game of Russian
roulette. At some point it’s going to be a losing proposition—you just don’t
know when. The benefit of rolling is that if the trade works out for a few
months in a row, the long call is paid for and the risk of loss is covered by
aggregate profits.
If we step outside this best-case theoretical world and consider what is
really happening on a day-to-day basis, we can gain insight on how to
manag


an
roulette. At some point it’s going to be a losing proposition—you just don’t
know when. The benefit of rolling is that if the trade works out for a few
months in a row, the long call is paid for and the risk of loss is covered by
aggregate profits.
If we step outside this best-case theoretical world and consider what is
really happening on a day-to-day basis, we can gain insight on how to
manage this type of trade when things go wrong. Effectively, a long
calendar is a typical gamma/theta trade. Negative gamma hurts. Positive
theta helps.
If we knew which way the stock was going, we would simply buy or sell
stock to adjust to get long or short deltas. But, unfortunately, we don’t. Our
only tool is to hedge by buying or selling stock as mentioned above to
flatten out when gamma causes the position delta to get more positive or
negative. 1 The bottom line is that if the effect of gamma creates unwanted
long deltas but the theta/gamma is still a desirable position, selling stock
flattens out the delta. If the effect of gamma creates unwanted short deltas,
buying stock flattens out the delta.
Trading Volatility Term Structure
There are other reasons for trading calendar spreads besides generating
income from theta. If there is skew in the term structure of volatility, which
was discussed in Chapter 3, a calendar spread is a way to trade volatility.
The tactic is to buy the “cheap” month and sell the “expensive” month.
Selling the Front, Buying the Back
If for a particular stock, the February ATM calls are trading at 50 volatility
and the May ATM calls are trading at 35 volatility, a vol-calendar trader
would buy the Mays and sell the Februarys. Sounds simple, right? The devil
is in the details. We’ll look at an example and then discuss some common
pitfalls with vol-trading calendars.
George has been studying the implied volatility of a $164.15 stock.
George notices that front-month volatility has been higher than that of the
other months for a couple of weeks. There is nothing in the news to indicate
immediate risk of extraordinary movement occurring in this example.
George sees that he can sell the 22-day July 165 calls at a 45 percent IV
and buy the 85-day September 165 calls at a 38 percent IV. George would
like to buy the calendar spread, because he believes the July ATM volatility
will drop down to around 38, where the September is trading. If he puts on
this trade, he will establish the following position:
What are George’s risks? Because he would be selling the short-term
ATM option, negative gamma could be a problem. The greeks for this trade,
shown in Exhibit 11.7 , confirm this. The negative gamma means each
dollar of stock price movement causes an adverse change of about 0.09 to
delta. The spread’s delta becomes shorter when the stock rises and longer
when the stock falls. Because the position’s delta is long 0.369 from the
start, some price appreciation may be welcomed in the short term. The stock
advance will yield profits but at a diminishing rate, as negative gamma
reduces the delta.
EXHIBIT 11.7 10-lot July–September 165 call calendar.
But just looking at the net position greeks doesn’t tell the whole story. It
is important to appreciate the fact that long calendar spreads such as this
have long vegas. In this case, the vega is +1.522. But what does this number
really mean? This vega figure means that if IV rises or falls in both the July
and the September calls by the same amount, the spread makes or loses
$152 per vol point.
George’s plan, however, is to see the July’s volatility decline to converge
with the September’s. He hopes the volatilities of the two months will move
independently of each other. To better gauge his risk, he needs to look at the
vega of each option. With the stock at $164.15 the vegas are as follows:
If George is right and July volatility declines 8 points, from 46 to 38, he
will make $1,283 ($1.604 × 100 × 8).
There are a couple of things that can go awry. First, instead of the
volatilities converging, they can diverge further. Implied volatility is a slave
to the whims of the market. If the July IV continues to rise while the
September IV stays the same, George loses $160 per vol point.
The second thing that can go wrong is the September IV declining along
with the July IV. This can lead George into trouble, too. It depends the
extent to which the September volatility declines. In this example, the vega
of the September leg is about twice that of the July leg. That means that if
the July volatility loses eight points while the September volatility declines
four points, profits from the July calls will be negated by losses from the
September calls. If the September volatility falls even more, the trade is a
loser.
IV is a common cause of time-spread failure for market makers. When i
in the front month rises, the volatility of the back-months sometimes does
as well. When this happens, it’s often because market makers who sold
front-month options to retail or institutional buyers buy the back-month
options to hedge their short-gamma risk. If the market maker buys enough
back-month options, he or she will accumulate positive vega. But when the
market sells the front-month volatility back to the market makers, the back
months drop, too, because market makers no longer need the back months
for a hedge.
Traders should study historical implied volatility to avoid this pitfall. As
is always the case with long vega strategies, there is a risk of a decline in
IV. Buying long-term options with implied volatility in the lower third of
the 12-month IV range helps improve the chances of success, since the
volatility being bought is historically cheap.
This can be tricky, however. If a trader looks back on a chart of IV for an
option class and sees that over the past six months it has ranged between 20
and 30 but nine months ago it spiked up to, say, 55, there must be a reason.
This solitary spike could be just an anomaly. To eliminate the noise from
volatility


ove the chances of success, since the
volatility being bought is historically cheap.
This can be tricky, however. If a trader looks back on a chart of IV for an
option class and sees that over the past six months it has ranged between 20
and 30 but nine months ago it spiked up to, say, 55, there must be a reason.
This solitary spike could be just an anomaly. To eliminate the noise from
volatility charts, it helps to filter the data. News stories from that time
period and historical stock charts will usually tell the story of why volatility
spiked. Often, it is a one-time event that led to the spike. Is it reasonable to
include this unique situation when trying to get a feel for the typical range
of implied volatility? Usually not. This is a judgment call that needs to be
made on a case-by-case basis. The ultimate objective of this exercise is to
determine: “Is volatility cheap or expensive?”
Buying the Front, Selling the Back
All trading is based on the principle of “buy low, sell high”—even volatility
trading. With time spreads, we can do both at once, but we are not limited
to selling the front and buying the back. When short-term options are
trading at a lower IV than long-term ones, there may be an opportunity to
sell the calendar. If the IV of the front month is 17 and the back-month IV is
25, for example, it could be a wise trade to buy the front and sell the back.
But selling time spreads in this manner comes with its own unique set of
risks.
First, a short calendar’s greeks are the opposite of those of a long
calendar. This trade has negative theta with positive gamma. A sideways
market hurts this position as negative theta does its damage. Each day of
carrying the position is paid for with time decay.
The short calendar is also a short-vega trade. At face value, this implies
that a drop in IV leads to profit and that the higher the IV sold in the back
month, the better. As with buying a calendar, there are some caveats to this
logic.
If there is an across-the-board decline in IV, the net short vega will lead to
a profit. But an across-the-board drop in volatility, in this case, is probably
not a realistic expectation. The front month tends to be more sensitive to
volatility. It is a common occurrence for the front month to be “cheap”
while the back month is “expensive.”
The volatilities of the different months can move independently, as they
can when one buys a time spread. There are a couple of scenarios that might
lead to the back-month volatility’s being higher than the front month. One is
high complacency in the short term. When the market collectively sells
options in expectation of lackluster trading, it generally prefers to sell the
short-term options. Why? Higher theta. Because the trade has less time until
expiration, the trade has a shorter period of risk. Because of this, selling
pressure can push down IV in the front-month options more than in the
back. Again, the front month is more sensitive to changes in implied
volatility.
Because volatility has peaks and troughs, this can be a smart time to sell a
calendar. The focus here is in seeing the “cheap” front month rise back up
to normal levels, not so much in seeing the “expensive” back month fall.
This trade is certainly not without risk. If the market doesn’t move, the
negative theta of the short calendar leads to a slow, painful death for
calendar sellers.
Another scenario in which the back-month volatility can trade higher than
the front is when the market expects higher movement after the expiration
of the short-term option but before the expiration of the long-term option.
Situations such as the expectation of the resolution of a lawsuit, a product
announcement, or some other one-time event down the road are
opportunities for the market to expect such movement. This strategy
focuses on the back-month vol coming back down to normal levels, not on
the front-month vol rising. This can be a more speculative situation for a
volatility trade, and more can go wrong.
The biggest volatility risk in selling a time spread is that what goes up can
continue to go up. The volatility disparity here is created by hedgers and
speculators favoring long-term options, hence pushing up the volatility, in
anticipation of a big future stock move. As the likely date of the anticipated
event draws near, more buyers can be attracted to the market, driving up IV
even further. Realized volatility can remain low as investors and traders lie
in wait. This scenario is doubly dangerous when volatility rises and the
stock doesn’t move. A trader can lose on negative theta and lose on negative
vega.
A Directional Approach
Calendar spreads are often purchased when the outlook for the underlying is
neutral. Sell the short-term ATM option; buy the long-term ATM option;
collect theta. But with negative gamma, these trades are never really
neutral. The delta is constantly changing, becoming more positive or
negative. It’s like a rubber band: at times being stretched in either direction
but always demanding a pull back to the strike. When the strike price being
traded is not ATM, calendar spreads can be strategically traded as
directional plays.
Buying a calendar, whether using calls or puts, where the strike price is
above the current stock price is a bullish strategy. With calls, the positive
delta of the long-term out-of-the-money (OTM) call will be greater than the
negative delta of the short-term OTM call. For puts, the positive delta of the
short-term in-the-money (ITM) put will be greater than the negative delta of
the long-term ITM put.
Just the opposite applies if the strike price is below the current stock
price. The negative delta of the short-term ITM call is greater than the
positive delta of the long-term ITM call. The negative delta of the long-term
OTM put is greater than the positive delta of the short-term OTM put.
When the position starts out with either a positive or negative delta,
movement in the direction of the delta is necessar


te applies if the strike price is below the current stock
price. The negative delta of the short-term ITM call is greater than the
positive delta of the long-term ITM call. The negative delta of the long-term
OTM put is greater than the positive delta of the short-term OTM put.
When the position starts out with either a positive or negative delta,
movement in the direction of the delta is necessary for the trade to be
profitable. Negative gamma is also an important strategic consideration.
Stock-price movement is needed, but not too much.
Buying calendar spreads is like playing outfield in a baseball game. To
catch a fly ball, an outfielder must focus on both distance and timing. He
must gauge how far the ball will be hit and how long it will take to get
there. With calendars, the distance is the strike price—that’s where the stock
needs to be—and the time is the expiration day of the short month’s option:
that’s when it needs to be at the target price.
For example, with Wal-Mart (WMT) at $48.50, a trader, Pete, is looking
for a rise to about $50 over the next five or six weeks. Pete buys the
August–September call calendar. In this example, August has 39 days until
expiration and September has 74 days.
Exactly what does 50 cents buy Pete? The stock price sitting below the
strike price means a net positive delta. This long time spread also has
positive theta and vega. Gamma is negative. Exhibit 11.8 shows the
specifics.
EXHIBIT 11.8 10-lot Wal-Mart August–September 50 call calendar.
The delta of this trade, while positive, is relatively small with 39 days left
until August expiration. It’s not rational to expect a quick profit if the stock
advances faster than expected. But ultimately, a rise in stock price is the
goal. In this example, Wal-Mart needs to rise to $50, and timing is
everything. It needs to be at that price in 39 days. In the interim, a move too
big and too fast in either direction hurts the trade because of negative
gamma. Starting with Wal-Mart at $48.50, delta/gamma problems are worse
to the downside. Exhibit 11.9 shows the effects of stock price on delta,
gamma, and theta.
EXHIBIT 11.9 Stock price movement and greeks.
If Wal-Mart moves lower, the delta gets more positive, racking up losses
at a higher rate. To add to Pete’s woes, theta becomes less of a benefit as the
stock drifts lower. At $47 a share, theta is about flat. With Wal-Mart trading
even lower than $47, the positive theta of the August call is overshadowed
by the negative theta of the September. Theta can become negative, causing
the position to lose value as time passes.
A big move to the upside doesn’t help either. If Wal-Mart rises just a bit,
the −0.323 gamma only lessens the benefit of the 0.563 delta. But above
$50, negative gamma begins to cause the delta to become increasingly
negative. Theta begins to wither away at higher stock prices as well.
The place to be is right at $50. The delta is flat and theta is highest. As
long as Wal-Mart finds its way up to this price by the third Friday of
August, life is good for Pete.
The In-or-Out Crowd
Pete could just as well have traded the Aug–Sep 50 put calendar in this
situation. If he’d been bearish, he could have traded either the Aug–Sep 45
call spread or the Aug–Sep 45 put spread. Whether bullish or bearish, as
mentioned earlier, the call calendar and the put calendar both function about
the same. When deciding which to use, the important consideration is that
one of them will be in-the-money and the other will be OTM. Whether you
have an ITM spread or an OTM spread has potential implications for the
success of the trade.
The bid-ask spreads tend to be wider for higher-delta, ITM options.
Because of this, it can be more expensive to enter into an ITM calendar.
Why? Trading options with wider markets requires conceding more edge.
Take the following options series:
By buying the May 50 calls at 3.20, a trader gives up 0.10 of theoretical
edge (3.20 is 0.10 higher than the theoretical value). Buying the put at 1.00
means buying only 0.05 over theoretical.
Because a calendar is a two-legged spread, the double edge given up by
trading the wider markets of two in-the-money options can make the out-of-
the-money spread a more attractive trade. The issue of wider markets is
compounded when rolling the spread. Giving up a nickel or a dime each
month can add up, especially on nominally low-priced spreads. It can cut
into a high percentage of profits.
Early assignment can complicate ITM calendars made up of American
options, as dividends and interest can come into play. The short leg of the
spread could get assigned before the expiration date as traders exercise calls
to capture the dividend. Short ITM puts may get assigned early because of
interest.
Although assignment is an undesirable outcome for most calendar spread
traders, getting assigned on the short leg of the calendar spread may not
necessarily create a significantly different trade. If a long put calendar, for
example, has a short front-month put that is so deep in-the-money that it is
likely to get assigned, it is trading close to a 100 delta. It is effectively a
long stock position already. After assignment, when a long stock position is
created, the resulting position is long stock with a deep ITM long put—a
fairly delta-flat position.
Double Calendars
Definition : A double calendar spread is the execution of two calendar
spreads that have the same months in common but have two different strike
prices.
Example
Sell 1 XYZ February 70 call
Buy 1 XYZ March 70 call
Sell 1 XYZ February 75 call
Buy 1 XYZ March 75 call
Double calendars can be traded for many reasons. They can be vega
plays. If there is a volatility-time skew, a double calendar is a way to take a
position without concentrating delta or gamma/theta risk at a single strike.
This spread can also be a gamma/theta play. In that case, there are two
strikes, so there are two potential focal points to gravitate to (in the case of
a long double calenda


ll
Double calendars can be traded for many reasons. They can be vega
plays. If there is a volatility-time skew, a double calendar is a way to take a
position without concentrating delta or gamma/theta risk at a single strike.
This spread can also be a gamma/theta play. In that case, there are two
strikes, so there are two potential focal points to gravitate to (in the case of
a long double calendar) or avoid (in the case of a short double calendar).
Selling the two back-month strikes and buying the front-month strikes
leads to negative theta and positive gamma. The positive gamma creates
favorable deltas when the underlying moves. Positive or negative deltas can
be covered by trading the underlying stock. With positive gamma, profits
can be racked up by buying the underlying to cover short deltas and
subsequently selling the underlying to cover long deltas.
Buying the two back-month strikes and selling the front-month strikes
creates negative gamma and positive theta, just as in a conventional
calendar. But the underlying stock has two target price points to shoot for at
expiration to achieve the maximum payout.
Often double calendars are traded as IV plays. Many times when they are
traded as IV plays, traders trade the lower-strike spread as a put calendar
and the higher-strike spread a call calendar. In that case, the spread is
sometimes referred to as a strangle swap . Strangles are discussed in
Chapter 15.
Two Courses of Action
Although there may be many motivations for trading a double calendar,
there are only two courses of action: buy it or sell it. While, for example,
the trader’s goal may be to capture theta, buying a double calendar comes
with the baggage of the other greeks. Fully understanding the
interrelationship of the greeks is essential to success. Option traders must
take a holistic view of their positions.
Let’s look at an example of buying a double calendar. In this example,
Minnesota Mining & Manufacturing (MMM) has been trading in a range
between about $85 and $97 per share. The current price of Minnesota
Mining & Manufacturing is $87.90. Economic data indicate no specific
reasons to anticipate that Minnesota Mining & Manufacturing will deviate
from its recent range over the next month—that is, there is nothing in the
news, no earnings anticipated, and the overall market is stable. August IV is
higher than October IV by one volatility point, and October implied
volatility is in line with 30-day historical volatility. There are 38 days until
August expiration, and 101 days until October expiration.
The Aug–Oct 85–90 double calendar can be traded at the following
prices:
Much like a traditional calendar spread, the price points cannot be
definitively plotted on a P&(L) diagram. What is known for certain is that at
August expiration, the maximum loss is $3,200. While it’s comforting to
know that there is limited loss, losing the entire premium that was paid for
the spread is an outcome most traders would like to avoid. We also know
the maximum gains occur at the strike prices; but not exactly what the
maximum profit can be. Exhibit 11.10 provides an alternative picture of the
position that is useful in managing the trade on a day-to-day basis.
EXHIBIT 11.10 10-lot Minnesota Mining & Manufacturing Aug–Oct 85–
90 double call calendar.
These numbers are a good representation of the position’s risk. Knowing
that long calendars and long double calendars have maximum losses at the
expiration of the short-term option equal to the net premiums paid, the max
loss in this example is 3.20. Break-even prices are not relevant to this
position because they cannot be determined with any certainty. What is
important is to get a feel for how much movement can hurt the position.
To make $19 a day in theta, a −0.468 gamma must be accepted. In the
long run, $1 of movement is irrelevant. In fact, some movement is favorable
because the ideal point for MMM to be at, at August expiration is either $85
or $90. So while small moves are acceptable, big moves are of concern. The
negative gamma is an illustration of this warning.
The other risk besides direction is vega. A positive 1.471 vega means the
calendar makes or loses about $147 with each one-point across-the-board
change in implied volatility. Implied volatility is a risk in all calendar
trades. Volatility was one of the criteria studied when considering this trade.
Recall that the August IV was one point higher than the October and that
the October IV was in line with the 30-day historical volatility at inception
of the trade.
Considering the volatility data is part of the due diligence when
considering a calendar or a double calendar. First, the (slightly) more
expensive options (August) are being sold, and the cheaper ones are being
bought (October). A study of the company reveals no news to lead one to
believe that Minnesota Mining & Manufacturing should move at a higher
realized volatility than it currently is in this example. Therefore, the front
month’s higher IV is not a red flag. Because the volatility of the October
option (the month being purchased) is in line with the historical volatility,
the trader could feel that he is paying a reasonable price for this volatility.
In the end, the trade is evaluated on the underlying stock, realized
volatility, and IV. The trade should be executed only after weighing all the
available data. Trading is both cerebral and statistical in nature. It’s about
gaining a statistically better chance of success by making rational decisions.
Diagonals
Definition : A diagonal spread is an option strategy that involves buying one
option and selling another option with a different strike price and with a
different expiration date. Diagonals are another strategy in the time spread
family.
Diagonals enable a trader to exploit opportunities similar to those
exploited by a calendar spread, but because the options in a diagonal spread
have two different strike prices, the trade is more focused on delta. The
n


involves buying one
option and selling another option with a different strike price and with a
different expiration date. Diagonals are another strategy in the time spread
family.
Diagonals enable a trader to exploit opportunities similar to those
exploited by a calendar spread, but because the options in a diagonal spread
have two different strike prices, the trade is more focused on delta. The
name diagonal comes from the fact that the spread is a combination of a
horizontal spread (two different months) and a vertical spread (two different
strikes).
Say it’s 22 days until January expiration and 50 days until February
expiration. Apple Inc. (AAPL) is trading at $405.10. Apple has been in an
uptrend heading toward the peak of its six-month range, which is around
$420. A trader, John, believes that it will continue to rise and hit $420 again
by February expiration. Historical volatility is 28 percent. The February 400
calls are offered at a 32 implied volatility and the January 420 calls are bid
on a 29 implied volatility. John executes the following diagonal:
Exhibit 11.11 shows the analytics for this trade.
EXHIBIT 11.11 Apple January–February 400–420 call diagonal.

From the presented data, is this a good trade? The answer to this question
is contingent on whether the position John is taking is congruent with his
view of direction and volatility and what the market tells him about these
elements.
John is bullish up to August expiration, and the stock in this example is in
an uptrend. Any rationale for bullishness may come from technical or
fundamental analysis, but techniques for picking direction, for the most
part, are beyond the scope of this book. Buying the lower strike in the
February option gives this trade a more positive delta than a straight
calendar spread would have. The trader’s delta is 0.255, or the equivalent of
about 25.5 shares of Apple. This reflects the trader’s directional view.
The volatility is not as easy to decipher. A specific volatility forecast was
not stated above, but there are a few relevant bits of information that should
be considered, whether or not the trader has a specific view on future
volatility. First, the historical volatility is 28 percent. That’s lower than
either the January or the February calls. That’s not ideal. In a perfect world,
it’s better to buy below historical and sell above. To that point, the February
option that John is buying has a higher volatility than the January he is
selling. Not so good either. Are these volatility observations deal breakers?
A Good Ex-Skews
It’s important to take skew into consideration. Because the January calls
have a higher strike price than the February calls, it’s logical for them to
trade at a lower implied volatility. Is this enough to justify the possibility of
selling the lower volatility? Consider first that there is some margin for
error. The bid-ask spreads of each of the options has a volatility disparity. In
this case, both the January and February calls are 10 cents wide. That means
with a January vega of 0.34 the bid-ask is about 0.29 vol points wide. The
Februarys have a 0.57 vega. They are about 0.18 vol points wide. That
accounts for some of the disparity. Natural vertical skew accounts for the
rest of the difference, which is acceptable as long as the skew is not
abnormally pronounced.
As for other volatility considerations, this diagonal has the rather
unorthodox juxtaposition of positive vega and negative gamma seen with
other time spreads. The trader is looking for a move upward, but not a big
one. As the stock rises and Apple moves closer to the 420 strike, the
positive delta will shrink and the negative gamma will increase. In order to
continue to enjoy profits as the stock rises, John may have to buy shares of
Apple to keep his positive delta. The risk here is that if he buys stock and
Apple retraces, he may end up negative scalping stock. In other words, he
may sell it back at a lower price than he bought it. Using stock to adjust the
delta in a negative-gamma play can be risky business. Gamma scalping is
addressed further in Chapter 13.
Making the Most of Your Options
The trader from the previous example had a time-spread alternative to the
diagonal: John could have simply bought a traditional time spread at the
420 strike. Recall that calendars reap the maximum reward when they are at
the shared strike price at expiration of the short-term option. Why would he
choose one over the other?
The diagonal in that example uses a lower-strike call in the February than
a straight 420 calendar spread and therefore has a higher delta, but it costs
more. Gamma, theta, and vega may be slightly lower with the in-the-money
call, depending on how far from the strike price the ITM call is and how
much time until expiration it has. These, however, are less relevant
differences.
The delta of the February 400 call is about 0.57. The February 420 call,
however, has only a 0.39 delta. The 0.18 delta difference between the calls
means the position delta of the time spread will be only about 0.07 instead
of about 0.25 of the diagonal—a big difference. But the trade-off for lower
delta is that the February 420 call can be bought for 12.15. That means a
lower debit paid—that means less at risk. Conversely, though there is
greater risk with the diagonal, the bigger delta provides a bigger payoff if
the trader is right.
Double Diagonals
A double diagonal spread is the simultaneous trading of two diagonal
spreads: one call spread and one put spread. The distance between the
strikes is the same in both diagonals, and both have the same two expiration
months. Usually, the two long-term options are more out-of-the-money than
the two shorter-term options. For example
Buy 1 XYZ May 70 put
Sell 1 XYZ March 75 put
Sell 1 XYZ March 85 call
Buy 1 XYZ May 90 call
Like many option strategies, the double diagonal can be looked at from a
number of angles. Certainly, this is a trade composed of two diagonal
spreads—th


and both have the same two expiration
months. Usually, the two long-term options are more out-of-the-money than
the two shorter-term options. For example
Buy 1 XYZ May 70 put
Sell 1 XYZ March 75 put
Sell 1 XYZ March 85 call
Buy 1 XYZ May 90 call
Like many option strategies, the double diagonal can be looked at from a
number of angles. Certainly, this is a trade composed of two diagonal
spreads—the March–May 70–75 put and the March–May 85–90 call. It is
also two strangles—buying the May 70–90 strangle and selling the March
75–85 strangle. One insightful way to look at this spread is as an iron
condor in which the guts are March options and the wings are May options.
Trading a double diagonal like this one, rather than a typically positioned
iron condor, can offer a few advantages. The first advantage, of course, is
theta. Selling short-term options and buying long-term options helps the
trader reap higher rates of decay. Theta is the raison d’être of the iron
condor. A second advantage is rolling. If the underlying asset stays in a
range for a long period of time, the short strangle can be rolled month after
month. There may, in some cases, also be volatility-term-structure
discrepancies on which to capitalize.
A trader, Paul, is studying JPMorgan (JPM). The current stock price is
$49.85. In this example, JPMorgan has been trading in a pretty tight range
over the past few months. Paul believes it will continue to do so over the
next month. Paul considers the following trade:

Paul considers volatility. In this example, the JPMorgan ATM call, the
August 50 (which is not shown here), is trading at 22.9 percent implied
volatility. This is in line with the 20-day historical volatility, which is 23
percent. The August IV appears to be reasonably in line with the September
volatility, after accounting for vertical skew. The IV of the August 52.50
calls is 1.5 points above that of the September 55 calls and the August 47.50
put IV is 1.6 points below the September 45 put IV. It appears that neither
month’s volatility is cheap or expensive.
Exhibit 11.12 shows the trade’s greeks.
EXHIBIT 11.12 10-lot JPMorgan August–September 45–47.50–52.50–55
double diagonal.
The analytics of this trade are similar to those of an iron condor.
Immediate directional risk is almost nonexistent, as indicated by the delta.
But gamma and theta are high, even higher than they would be if this were
a straight September iron condor, although not as high as if this were an
August iron condor.
Vega is positive. Surely, if this were an August or a September iron
condor, vega would be negative. In this example, Paul is indifferent as to
whether vega is positive or negative because IV is fairly priced in terms of
historical volatility and term structure. In fact, to play it close to the vest,
Paul probably wants the smallest vega possible, in case of an IV move.
Why take on the risk?
The motivation for Paul’s double diagonal was purely theta. The
volatilities were all in line. And this one-month spread can’t be rolled. If
Paul were interested in rolling, he could have purchased longer-term
options. But if he is anticipating a sideways market for only the next month
and feels that volatility could pick up after that, the one-month play is the
way to go. After August expiration, Paul will have three choices: sell his
Septembers, hold them, or turn them into a traditional iron condor by
selling the September 47.50s and 52.50s. This depends on whether he is
indifferent, expects high volatility, or expects low volatility.
The Strength of the Calendar
Spreads in the calendar-spread family allow traders to take their trading to a
higher level of sophistication. More basic strategies, like vertical spreads
and wing spreads, provide a practical means for taking positions in
direction, realized volatility, and to some extent implied volatility. But
because calendar-family spreads involve two expiration months, traders can
take positions in the same market variables as with these more basic
strategies and also in the volatility spread between different expiration
months. Calendar-family spreads are veritable volatility spreads. This is a
powerful tool for option traders to have at their disposal.
Note
1 . Advanced hedging techniques are discussed in subsequent chapters.
PART III
Volatility
CHAPTER 12
Delta-Neutral Trading
Trading Implied Volatility
Many of the strategies covered so far have been option-selling strategies.
Some had a directional bias; some did not. Most of the strategies did have a
primary focus on realized volatility—especially selling it. These short
volatility strategies require time. The reward of low stock volatility is theta.
In general, most of the strategies previously covered were theta trades in
which negative gamma was an unpleasant inconvenience to be dealt with.
Moving forward, much of the remainder of this book will involve more
in-depth discussions of trading both realized and implied volatility (IV),
with a focus on the harmonious, and sometimes disharmonious, relationship
between the two types. Much attention will be given to how IV trades in the
option market, describing situations in which volatility moves are likely to
occur and how to trade them.
Direction Neutral versus Direction
Indifferent
In the world of nonlinear trading, there are two possible nondirectional
views of the underlying asset: direction neutral and direction indifferent.
Direction neutral means the trader believes the stock will not trend either
higher or lower. The trader is neutral in his or her assessment of the future
direction of the asset. Short iron condors, long time spreads, and out-of-the-
money (OTM) credit spreads are examples of direction-neutral strategies.
These strategies generally have deltas close to zero. Because of negative
gamma, movement is the bane of the direction-neutral trade.
Direction indifferent means the trader may desire movement in the
underlying but is indifferent as to whether that movement is up o


asset. Short iron condors, long time spreads, and out-of-the-
money (OTM) credit spreads are examples of direction-neutral strategies.
These strategies generally have deltas close to zero. Because of negative
gamma, movement is the bane of the direction-neutral trade.
Direction indifferent means the trader may desire movement in the
underlying but is indifferent as to whether that movement is up or down.
Some direction-indifferent trades are almost completely insulated from
directional movement, with a focus on interest or dividends instead.
Examples of these types of trades are conversions, reversals, and boxes,
which are described in Chapter 6, as well as dividend plays, which are
described in Chapter 8.
Other direction-indifferent strategies are long option strategies that have
positive gamma. In these trades, the focus is on movement, but the direction
of that movement is irrelevant. These are plays that are bullish on realized
volatility. Yet other direction-indifferent strategies are volatility plays from
the perspective of IV. These are trades in which the trader’s intent is to take
a bullish or bearish position in IV.
Delta Neutral
To be truly direction neutral or direction indifferent means to have a delta
equal to zero. In other words, there are no immediate gains if the underlying
moves incrementally higher or lower. This zero-delta method of trading is
called delta-neutral trading .
A delta-neutral position can be created from any option position simply
by trading stock to flatten out the delta. A very basic example of a delta-
neutral trade is a long at-the-money (ATM) call with short stock.
Consider a trade in which we buy 20 ATM calls that have a 50 delta and
sell stock on a delta-neutral ratio.
Buy 20 50-delta calls (long 1,000 deltas)
Short 1,000 shares (short 1,000 deltas)
In this position, we are long 1,000 deltas from the calls (20 × 50) and
short 1,000 deltas from the short sale of stock. The net delta of the position
is zero. Therefore, the immediate directional exposure has been eliminated
from the trade. But intuitively, there are other opportunities for profit or loss
with this trade.
The addition of short stock to the calls will affect only the delta, not the
other greeks. The long calls have positive gamma, negative theta, and
positive vega. Exhibit 12.1 is a simplified representation of the greeks for
this trade.
EXHIBIT 12.1 20-lot delta-neutral long call.
With delta not an immediate concern, the focus here is on gamma, theta,
and vega. The +1.15 vega indicates that each one-point change in IV makes
or loses $115 for this trade. Yet there is more to the volatility story. Each
day that passes costs the trader $50 in time decay. Holding the position for
an extended period of time can produce a loser even if IV rises. Gamma is
potentially connected to the success of this trade, too. If the underlying
moves in either direction, profit from deltas created by positive gamma may
offset the losses from theta. In fact, a big enough move in either direction
can produce a profitable trade, regardless of what happens to IV.
Imagine, for a moment, that this trade is held until expiration. If the stock
is below the strike price at this point, the calls expire. The resulting position
is short 1,000 shares of stock. If the stock is above the strike price at
expiration, the calls can be exercised, creating 2,000 shares of long stock.
Because the trade is already short 1,000 shares, the resulting net position is
long 1,000 shares (2,000 − 1,000). Clearly, the more the underlying stock
moves in either direction the greater the profit potential. The underlying has
to move far enough above or below the strike price to allow the beneficial
gains from buying or selling stock to cover the option premium lost from
time decay. If the trade is held until expiration, the underlying needs to
move far enough to cover the entire premium spent on the calls.
The solid lines forming a V in Exhibit 12.2 conceptually illustrate the
profit or loss for this delta-neutral long call at expiration.
EXHIBIT 12.2 Profit-and-loss diagram for delta-neutral long-call trade.
Because of gamma, some deltas will be created by movement of the
underlying before expiration. Gamma may lead to this being a profitable
trade in the short term, depending on time and what happens with IV. The
dotted line illustrates the profit or loss of this trade at the point in time when
the trade is established. Because the options may still have time value at
this point—depending on how far from the strike price the stock is trading
—the value of the position, as a whole, is higher than it will be if the calls
are trading at parity at expiration. Regardless, the plan is for the stock to
make a move in either direction. The bigger the move and the faster it
happens, the better.
Why Trade Delta Neutral?
A few years ago, I was teaching a class on option trading. Before the
seminar began, I was talking with one of the students in attendance. I asked
him what he hoped to learn in the class. He said that he was really
interested in learning how to trade delta neutral. When I asked him why he
was interested in that specific area of trading, he replied, “I hear that’s
where all the big money is made!”
This observation, right or wrong, probably stems from the fact that in the
past most of the trading in this esoteric discipline has been executed by
professional traders. There are two primary reasons why the pros have
dominated this strategy: high commissions and high margin requirements
for retail traders. Recently, these two reasons have all but evaporated.
First, the ultracompetitive world of online brokers has driven
commissions for retail traders down to, in some cases, what some market
makers pay. Second, the oppressive margin requirements that retail option
traders were subjected to until 2007 have given way to portfolio margining.
Portfolio Margining
Customer portfolio margining is a method of calculating customer margin
in which


t evaporated.
First, the ultracompetitive world of online brokers has driven
commissions for retail traders down to, in some cases, what some market
makers pay. Second, the oppressive margin requirements that retail option
traders were subjected to until 2007 have given way to portfolio margining.
Portfolio Margining
Customer portfolio margining is a method of calculating customer margin
in which the margin requirement is based on the “up and down risk” of the
portfolio. Before the advent of portfolio margining, retail traders were
subject to strategy-based margining, also called Reg. T margining, which in
many cases required a significantly higher amount of capital to carry a
position than portfolio margining does.
With portfolio margining, highly correlated securities can be offset
against each other for purposes of calculating margin. For example, SPX
options and SPY options—both option classes based on the Standard &
Poor’s 500 Index—can be considered together in the margin calculation. A
bearish position in one and a bullish position in the other may partially
offset the overall risk of the portfolio and therefore can help to reduce the
overall margin requirement.
With portfolio margining, many strategies are margined in such a way
that, from the point of view of this author, they are subject to a much more
logical means of risk assessment. Strategy-based margining required traders
of some strategies, like a protective put, to deposit significantly more
capital than one could possibly lose by holding the position. The old rules
require a minimum margin of 50 percent of the stock’s value and up to 100
percent of the put premium. A portfolio-margined protective put may
require only a fraction of what it would with strategy-based margining.
Even though Reg. T margining is antiquated and sometimes unreasonable,
many traders must still abide by these constraints. Not all traders meet the
eligibility requirements to qualify for portfolio-based margining. There is a
minimum account balance for retail traders to be eligible for this treatment.
A broker may also require other criteria to be met for the trader to benefit
from this special margining. Ultimately, portfolio margining allows retail
traders to be margined similarly to professional traders.
There are some traders, both professional and otherwise, who indeed have
made “big money,” as the student in my class said, trading delta neutral.
But, to be sure, there are successful and unsuccessful traders in many areas
of trading. The real motivation for trading delta neutral is to take a position
in volatility, both implied and realized.
Trading Implied Volatility
With a typical option, the sensitivity of delta overshadows that of vega. To
try and profit from a rise or fall in IV, one has to trade delta neutral to
eliminate immediate directional sensitivity. There are many strategies that
can be traded as delta-neutral IV strategies simply by adding stock.
Throughout this chapter, I will continue using a single option leg with
stock, since it provides a simple yet practical example. It’s important to note
that delta-neutral trading does not refer to a specific strategy; it refers to the
fact that the trader is indifferent to direction. Direction isn’t being traded,
volatility is.
Volatility trading is fundamentally different from other types of trading.
While stocks can rise to infinity or decline to zero, volatility can’t. Implied
volatility, in some situations, can rise to lofty levels of 100, 200, or even
higher. But in the long-run, these high levels are not sustainable for most
stocks. Furthermore, an IV of zero means that the options have no extrinsic
value at all. Now that we have established that the thresholds of volatility
are not as high as infinity and not as low as zero, where exactly are they?
The limits to how high or low IV can go are not lines in the sand. They are
more like tides that ebb and flow, but normally come up only so far onto the
beach.
The volatility of an individual stock tends to trade within a range that can
be unique to that particular stock. This can be observed by studying a chart
of recent volatility. When IV deviates from the range, it is typical for it to
return to the range. This is called reversion to the mean , which was
discussed in Chapter 3. IV can get stretched in either direction like a rubber
band but then tends to snap back to its original shape.
There are many examples of situations where reversion to the mean enters
into trading. In some, volatility temporarily dips below the typical range,
and in some, it rises beyond the recent range. One of the most common
examples is the rush and the crush.
The Rush and the Crush
In this situation, volatility rises before and falls after a widely anticipated
news announcement, of earnings, for instance, or of a Food and Drug
Administration (FDA) approval. In this situation, option buyers rush in and
bid up IV. The more uncertainty—the more demand for insurance—the
higher vol rises. When the event finally occurs and the move takes place or
doesn’t, volatility gets crushed. The crush occurs when volatility falls very
sharply—sometimes 10 points, 20 points, or more—in minutes. Traders
with large vega positions appreciate the appropriateness of the term crush
all too well. Volatility traders also affectionately refer to this sudden drop in
IV by saying that volatility has gotten “whacked.”
In order to have a feel for whether implied volatility is high or low for a
particular stock, you need to know where it’s been. It’s helpful to have an
idea of where realized volatility is and has been, too. To be sure, one
analysis cannot be entirely separate from the other. Studying both implied
and realized volatility and how they relate is essential to seeing the big
picture.
The Inertia of Volatility
Sir Isaac Newton said that an object in motion tends to stay in motion
unless acted upon by another force. Volatility acts much the same way.
Most stocks tend to trade w


atility is and has been, too. To be sure, one
analysis cannot be entirely separate from the other. Studying both implied
and realized volatility and how they relate is essential to seeing the big
picture.
The Inertia of Volatility
Sir Isaac Newton said that an object in motion tends to stay in motion
unless acted upon by another force. Volatility acts much the same way.
Most stocks tend to trade with a certain measurable amount of daily price
fluctuations. This can be observed by looking at the stock’s realized
volatility. If there is no outside force—some pivotal event that
fundamentally changes how the stock is likely to behave—one would
expect the stock to continue trading with the same level of daily price
movement. This means IV (the market’s expectation of future stock
volatility) should be the same as realized volatility (the calculated past stock
volatility).
But just as in physics, it seems there is always some friction affecting the
course of what is in motion. Corporate earnings, Federal Reserve Board
reports, apathy, lulls in the market, armed conflicts, holidays, rumors, and
takeovers, among other market happenings all provide a catalyst for
volatility changes. Divergences of realized and implied volatility, then, are
commonplace. These divergences can create tradable conditions, some of
which are more easily exploited than others.
To find these opportunities, a trader must conduct a study of volatility.
Volatility charts can help a trader visualize the big picture. This historical
information offers a comparison of what is happening now in volatility with
what has happened in the past. The following examples use a volatility
chart to show how two different traders might have traded the rush and
crush of an earnings report.
Volatility Selling
Susie Seller, a volatility trader, studies semiconductor stocks. Exhibit 12.3
shows the volatilities of a $50 chip stock. The circled area shows what
happened before and after second-quarter earnings were reported in July.
The black line is the IV, and the gray is the 30-day historical.
EXHIBIT 12.3 Chip stock volatility before and after earnings reports.
Source : Chart courtesy of iVolatility.com
In mid-July, Susie did some digging to learn that earnings were to be
announced on July 24, after the close. She was careful to observe the classic
rush and crush that occurred to varying degrees around the last three
earnings announcements, in October, January, and April. In each case, IV
firmed up before earnings only to get crushed after the report. In mid-to-late
July, she watched as IV climbed to the mid-30s (the rush) just before
earnings. As the stock lay in wait for the report, trading came to a
proverbial screeching halt, sending realized volatility lower, to about 13
percent. Susie waited for the end of the day just before the report to make
her move. Before the closing bell, the stock was at $50. Susie sold 20 one-
month 50-strike calls at 2.10 (a 35 volatility) and bought 1,100 shares of the
underlying stock at $50 to become delta neutral.
Exhibit 12.4 shows Susie’s position.
EXHIBIT 12.4 Delta-neutral short ATM call, long stock position.
Her delta was just about flat. The delta for the 50 calls was 0.54 per
contract. Selling a 20-lot creates 10.80 short deltas for her overall position.
After buying 1,100 shares, she was left long 0.20 deltas, about the
equivalence of being long 20 shares. Where did her risk lie? Her biggest
concern was negative gamma. Without even seeing a chart of the stock’s
price, we can see from the volatility chart that this stock can have big
moves on earnings. In October, earnings caused a more than 10-point jump
in realized volatility, to its highest level during the year shown. Whether the
stock rose or fell is irrelevant. Either event means risk for a premium seller.
The positive theta looks good on the surface, but in fact, theta provided
Susie with no significant benefit. Her plan was “in and out and nobody gets
hurt.” She got into the trade right before the earnings announcement and out
as soon as implied volatility dropped off. Ideally, she’d like to hold these
types of trades for less than a day. The true prize is vega.
Susie was looking for about a 10-point drop in IV, which this option class
had following the October and January earnings reports. April had a big
drop in IV, as well, of about eight or nine points. Ultimately, what Susie is
looking for is reversion to the mean.
She gauges the normal level of volatility by observing where it is before
and after the surges caused by earnings. From early November to mid- to
late- December, the stock’s IV bounced around the 25 percent level. In the
month of February, the IV was around 25. After the drop-off following
April earnings and through much of May, the IV was closer to 20 percent.
In June, IV was just above 25. Susie surmised from this chart that when no
earnings event is pending, this stock’s options typically trade at about a 25
percent IV. Therefore, anticipating a 10-point decline from 35 was
reasonable, given the information available. If Susie gets it right, she stands
to make $1,150 from vega (10 points × 1.15 vegas × 100).
As we can see from the right side of the volatility chart in Exhibit 12.3 ,
Susie did get it right. IV collapsed the next morning by just more than ten
points. But she didn’t make $1,150; she made less. Why? Realized volatility
(gamma). The jump in realized volatility shown on the graph is a function
of the fact that the stock rallied $2 the day after earnings. Negative gamma
contributed to negative deltas in the face of a rallying market. This negative
delta affected some of Susie’s potential vega profits.
So what was Susie’s profit? On this trade she made $800. The next
morning at the open, she bought back the 50-strike calls at 2.80 (25 IV) and
sold the stock at $52. To compute her actual profit, she compared the prices
of the spread when entering the trade with the prices of the spread when
exiting. Exhibit 12.5 sh


lying market. This negative
delta affected some of Susie’s potential vega profits.
So what was Susie’s profit? On this trade she made $800. The next
morning at the open, she bought back the 50-strike calls at 2.80 (25 IV) and
sold the stock at $52. To compute her actual profit, she compared the prices
of the spread when entering the trade with the prices of the spread when
exiting. Exhibit 12.5 shows the breakdown of the trade.
EXHIBIT 12.5 Profit breakdown of delta-neutral trade.
After closing the trade, Susie knew for sure what she made or lost. But
there are many times when a trader will hold a delta-neutral position for an
extended period of time. If Susie hadn’t closed her trade, she would have
looked at her marks to see her P&(L) at that point in time. Marks are the
prices at which the securities are trading in the actual market, either in real
time or at end of day. With most online brokers’ trading platforms or
options-trading software, real-time prices are updated dynamically and
always at their fingertips. The profit or loss is, then, calculated
automatically by comparing the actual prices of the opening transaction
with the current marks.
What Susie will want to know is why she made $800. Why not more?
Why not less, for that matter? When trading delta neutral, especially with
more complex trades involving multiple legs, a manual computation of each
leg of the spread can be tedious. And to be sure, just looking at the profit or
loss on each leg doesn’t provide an explanation.
Susie can see where her profits or losses came from by considering the
profit or loss for each influence contributing to the option’s value. Exhibit
12.6 shows the breakdown.
EXHIBIT 12.6 Profit breakdown by greek.
Delta
Susie started out long 0.20 deltas. A $2 rise in the stock price yielded a $40
profit attributable to that initial delta.
Gamma
As the stock rose, the negative delta of the position increased as a result of
negative gamma. The delta of the stock remained the same, but the negative
delta of the 50 call grew by the amount of the gamma. Deriving an exact
P&(L) attributable to gamma is difficult because gamma is a dynamic
metric: as the stock price changes, so can the gamma. This calculation
assumes that gamma remains constant. Therefore, the gamma calculation
here provides only an estimate.
The initial position gamma of −1.6 means the delta decreases by 3.2 with
a $2 rise in the stock (–1.60 times the $2 rise in the stock price). Susie, then,
would multiply −3.2 by $2 to find the loss on −3.2 deltas over a $2 rise. But
she wasn’t short 3.2 deltas for the whole $2. She started out with zero deltas
attributable to gamma and ended up being 3.2 shorter from gamma over that
$2 move. Therefore, if she assumes her negative delta from gamma grew
steadily from 0 to −3.2, she can estimate her average delta loss over that
move by dividing by 2.
Theta
Susie held this trade one day. Her total theta contributed 0.75 or $75 to her
position.
Vega
Vega is where Susie made her money on this trade. She was able to buy her
call back 10 IV points lower. The initial position vega was −1.15.
Multiplying −1.15 by the negative 10-point crush of volatility yields a vega
profit of $1,150.
Conclusions
Studying her position’s P&(L) by observing what happened in her greeks
provides Susie with an alternate—and in some ways, better—method to
evaluate her trade. The focus of this delta-neutral trade is less on the price at
which Susie can buy the calls back to close the position than on the
volatility level at which she can buy them back, weighed against the P&(L)
from her other risks. Analyzing her position this way gives her much more
information than just comparing opening and closing prices. Not only does
she get a good estimate of how much she made or lost, but she can
understand why as well.
The Imprecision of Estimation
It is important to notice that the P&(L) found by adding up the P&(L)’s
from the greeks is slightly different from the actual P&(L). There are a
couple of reasons for this. First, the change in delta resulting from gamma is
only an estimate, because gamma changes as the stock price changes. For
small moves in the underlying, the gamma change is less significant, but for
larger moves, the rate of change of the gamma can be bigger, and it can be
nonlinear. For example, as an option moves from being at-the-money
(ATM) to being out-of-the-money (OTM), its gamma decreases. But as the
option becomes more OTM, its gamma decreases at a slower rate.
Another reason that the P&(L) from the greeks is different from the actual
P&(L) is that the greeks are derived from the option-pricing model and are
therefore theoretical values and do not include slippage.
Furthermore, the volatility input in this example is rounded a bit for
simplicity. For example, a volatility of 25 actually yielded a theoretical
value of 2.796, while the call was bought at 2.80. Because some options
trade at minimum price increments of a nickel, and none trade in fractions
of a penny, IV is often rounded.
Caveat Venditor
Reversion to the mean holds the promise of profit in this trade, but Susie
also knows that this strategy does not come without risks of loss. The mean
to which volatility is expected to revert is not a constant. This benchmark
can and does change. In this example, if the company had an unexpectedly
terrible quarter, the stock could plunge sharply. In some cases, this would
cause IV to find a new, higher level at which to reside. If that had happened
here, the trade could have been a big loser. Gamma and vega could both
have wreaked havoc. In trading, there is no sure thing, no matter what the
chart looks like. Remember, every ship on the bottom of the ocean has a
chart!
Volatility Buying
This same earnings event could have been played entirely differently. A
different trader, Bobby Buyer, studied the same volatility chart as Susie. It
is shown again here as Exhibit 12.7 . Bobby also thought there would be a
rush and crush of I


voc. In trading, there is no sure thing, no matter what the
chart looks like. Remember, every ship on the bottom of the ocean has a
chart!
Volatility Buying
This same earnings event could have been played entirely differently. A
different trader, Bobby Buyer, studied the same volatility chart as Susie. It
is shown again here as Exhibit 12.7 . Bobby also thought there would be a
rush and crush of IV, but he decided to take a different approach.
EXHIBIT 12.7 Chip stock volatility before and after earnings reports.
Source : Chart courtesy of iVolatility.com
About an hour before the close of business on July 21, just three days
before earnings announcements, Bobby saw that he could buy volatility at
30 percent. In Bobby’s opinion, volatility seemed cheap with earnings so
close. He believed that IV could rise at least five points over the next three
days. Note that we have the benefit of 20/20 hindsight in the example.
Near the end of the trading day, the stock was at $49.70. Bobby bought 20
33-day 50-strike calls at 1.75 (30 volatility) and sold short 1,000 shares of
the underlying stock at $49.70 to become delta neutral. Exhibit 12.8 shows
Bobby’s position.
EXHIBIT 12.8 Delta-neutral long call, short stock position.
With the stock at $49.70, the calls had +0.51 delta per contract, or +10.2
for the 20-lot. The short sale of 1,000 shares got Bobby as close to delta-
neutral as possible without trading an odd lot in the stock. The net position
delta was +0.20, or about the equivalent of being long 20 shares of stock.
Bobby’s objective in this case is to profit from an increase in implied
volatility leading up to earnings.
While Susie was looking for reversion to the mean, Bobby hoped for a
further divergence. For Bobby, positive gamma looked like a good thing on
the surface. However, his plan was to close the position just before earnings
were released—before the vol crush and before the potential stock-price
move. With realized volatility already starting to drop off at the time the
trade was put on, gamma offered little promise of gain.
As fate would have it, IV did indeed increase. At the end of the day before
the July earnings report, IV was trading at 35 percent. Bobby closed his
trade by selling his 20-lot of the 50 calls at 2.10 and buying his 1,000 shares
of stock back at $50. Exhibit 12.9 shows the P&(L) for each leg of the
spread.
EXHIBIT 12.9 Profit breakdown.

The calls earned Bobby a total of $700, while the stock lost $300. Of
course, with this type of trade, it is not relevant which leg was a winner and
which a loser. All that matters is the bottom line. The net P&(L) on the trade
was a gain of $400. The gain in this case was mostly a product of IV’s
rising. Exhibit 12.10 shows the P&(L) per greek.
EXHIBIT 12.10 Profit breakdown by greek.
Delta
The position began long 0.20 deltas. The 0.30-point rise earned Bobby a
0.06 point gain in delta per contract.
Gamma
Bobby had an initial gamma of +1.8. We will use 1.8 for estimating the P&
(L) in this example, assuming gamma remained constant. A 0.30 rise in the
stock price multiplied by the 1.8 gamma means that with the stock at $50,
Bobby was long an additional 0.54 deltas. We can estimate that over the
course of the 0.30 rise in the stock price, Bobby was long an average of
0.27 (0.54 ÷ 2). His P&(L) due to gamma, therefore, is a gain of about 0.08
(0.27 × 0.30).
Theta
Bobby held this trade for three days. His total theta cost him 1.92 or $192.
Vega
The biggest contribution to Bobby’s profit on this trade was made by the
spike in IV. He bought 30 volatility and sold 35 volatility. His 1.20 position
vega earned him 6.00, or $600.
Conclusions
The $422 profit is not exact, but the greeks provide a good estimate of the
hows and the whys behind it. Whether they are used for forecasting profits
or for doing a postmortem evaluation of a trade, consulting the greeks offers
information unavailable by just looking at the transaction prices.
By thinking about all these individual pricing components, a trader can
make better decisions. For example, about two weeks earlier, Bobby could
have bought an IV level closer to 26 percent. Being conscious of his theta,
however, he decided to wait. The $64-a-day theta would have cost him
$896 over 14 days. That’s much more that the $480 he could have made by
buying volatility four points lower with his 1.20 vega.
Risks of the Trade
Like Susie’s trade, Bobby’s play was not without risk. Certainly theta was a
concern, but in addition to that was the possibility that IV might not have
played out as he planned. First, IV might not have risen enough to cover
three days’ worth of theta. It needed to rise, in this case, about 1.6 volatility
points for the 1.20 vega to cover the 1.92 theta loss. It might even have
dropped. An earlier-than-expected announcement that the earnings numbers
were right on target could have spoiled Bobby’s trade. Or the market simply
might not have reacted as expected; volatility might not have risen at all, or
might have fallen. Remember, IV is a function of the market. It does not
always react as one thinks it should.
CHAPTER 13
Delta-Neutral Trading
Trading Realized Volatility
So far, we’ve discussed many option strategies in which realized volatility
is an important component of the trade. And while the management of these
positions has been the focus of much of the discussion, the ultimate gain or
loss for many of these strategies has been from movement in a single
direction. For example, with a long call, the higher the stock rallies the
better.
But increases or decreases in realized volatility do not necessarily have an
exclusive relationship with direction. Recall that realized volatility is the
annualized standard deviation of daily price movements. Take two similarly
priced stocks that have had a net price change of zero over a one-month
period. Stock A had small daily price changes during that period, rising
$0.10 one day and falling $0.10 the next. Stock B went up or down by


ility do not necessarily have an
exclusive relationship with direction. Recall that realized volatility is the
annualized standard deviation of daily price movements. Take two similarly
priced stocks that have had a net price change of zero over a one-month
period. Stock A had small daily price changes during that period, rising
$0.10 one day and falling $0.10 the next. Stock B went up or down by $5
each day for a month. In this rather extreme example, Stock B was much
more volatile than Stock A, regardless of the fact that the net price change
for the period for both stocks was zero.
A stock’s volatility—either high or low volatility—can be capitalized on
by trading options delta neutral. Simply put, traders buy options delta
neutral when they believe a stock will have more movement and sell
options delta neutral when they believe a stock will move less.
Delta-neutral option sellers profit from low volatility through theta. Every
day that passes in which the loss from delta/gamma movement is less than
the gain from theta is a winning day. Traders can adjust their deltas by
hedging. Delta-neutral option buyers exploit volatility opportunities through
a trading technique called gamma scalping.
Gamma Scalping
Intraday trading is seldom entirely in one direction. A stock may close
higher or lower, even sharply higher or lower, on the day, but during the day
there is usually not a steady incremental rise or fall in the stock price. A
typical intraday stock chart has peaks and troughs all day long. Delta-
neutral traders who have gamma don’t remain delta neutral as the
underlying price changes, which inevitably it will. Delta-neutral trading is
kind of a misnomer.
In fact, it is gamma trading in which delta-neutral traders engage. For
long-gamma traders, the position delta gets more positive as the underlying
moves higher and more negative as the underlying moves lower. An upward
move in the underlying increases positive deltas, resulting in exponentially
increasing profits. But if the underlying price begins to retrace downward,
the gain from deltas can be erased as quickly as it was racked up.
To lock in delta gains, a trader can adjust the position to delta neutral
again by selling short stock to cover long deltas. If the stock price declines
after this adjustment, losses are curtailed thanks to the short stock. In fact,
the delta will become negative as the underlying price falls, leading to
growing profits. To lock in profits again, the trader buys stock to cover
short deltas to once again become delta neutral.
The net effect is a stock scalp. Positive gamma causes the delta-neutral
trader to sell stock when the price rises and buy when the stock falls. This
adds up to a true, realized profit. So positive gamma is a money-making
machine, right? Not so fast. As in any business, the profits must be great
enough to cover expenses. Theta is the daily cost of running this gamma-
scalping business.
For example, a trader, Harry, notices that the intraday price swings in a
particular stock have been increasing. He takes a bullish position in realized
volatility by buying 20 off the 40-strike calls, which have a 50 delta, and
selling stock on a delta-neutral ratio.
Buy 20 40-strike calls (50 delta) (long 1,000 deltas)
Short 1,000 shares at $40 (short 1,000 deltas)
The immediate delta of this trade is flat, but as the stock moves up or
down, that will change, presenting gamma-scalping opportunities. Gamma
scalping is the objective here. The position greeks in Exhibit 13.1 show the
relationship of the two forces involved in this trade: gamma and theta.
EXHIBIT 13.1 Greeks for 20-lot delta-neutral long call.
The relationship of gamma to theta in this sort of trade is paramount to its
success. Gamma-scalping plays are not buy-and-hold strategies. This is
active trading. These spreads need to be monitored intraday to take
advantage of small moves in the underlying security. Harry will sell stock
when the underlying rises and buy it when the underlying falls, taking a
profit with each stock trade. The goal for each day that passes is to profit
enough from positive gamma to cover the day’s theta. But that’s not always
as easy as it sounds. Let’s study what happens the first seven days after this
hypothetical trade is executed. For the purposes of this example, we assume
that gamma remains constant and that the trader is content trading odd lots
of stock.
Day One
The first day proves to be fairly volatile. The stock rallies from $40 to $42
early in the day. This creates a positive position delta of 5.60, or the
equivalent of being long about 560 shares. At $42, Harry covers the
position delta by selling 560 shares of the underlying stock to become delta
neutral again.
Later in the day, the market reverses, and the stock drops back down to
$40 a share. At this point, the position is short 5.60 deltas. Harry again
adjusts the position, buying 560 shares to get flat. The stock then closes
right at $40.
The net result of these two stock transactions is a gain of $1,070. How?
The gamma scalp minus the theta, as shown below.
The volatility of day one led to it being a profitable day. Harry scalped 560
shares for a $2 profit, resulting from volatility in the stock. If the stock
hadn’t moved as much, the delta would have been smaller, and the dollar
amount scalped would have been smaller, leading to an exponentially
smaller profit. If there had been more volatility, profits would have been
exponentially larger. It would have led to a bigger bite being taken out of
the market.
Day Two
The next day, the market is a bit quieter. There is a $0.40 drop in the price
of the stock, at which point the position delta is short 1.12. Harry buys 112
shares at $39.60 to get delta neutral.
Following Harry’s purchase, the stock slowly drifts back up and is trading
at $40 near the close. Harry decides to cover his deltas and sell 112 shares
at $40. It is common to cover all deltas at the end of the day to get back to
being delta


ieter. There is a $0.40 drop in the price
of the stock, at which point the position delta is short 1.12. Harry buys 112
shares at $39.60 to get delta neutral.
Following Harry’s purchase, the stock slowly drifts back up and is trading
at $40 near the close. Harry decides to cover his deltas and sell 112 shares
at $40. It is common to cover all deltas at the end of the day to get back to
being delta neutral. Remember, the goal of gamma scalping is to trade
volatility, not direction. Starting the next trading day with a delta, either
positive or negative, means an often unwanted directional bias and
unwanted directional risk. Tidying up deltas at the end of the day to get
neutral is called going home flat.
Today was not a banner day. Harry did not quite have the opportunity to
cover the decay.

Day Three
On this day, the market trends. First, the stock rises $0.50, at which point
Harry sells 140 shares of stock at $40.50 to lock in gains from his delta and
to get flat. However, the market continues to rally. At $41 a share, Harry is
long another 1.40 deltas and so sells another 140 shares. The rally
continues, and at $41.50 he sells another 140 shares to cover the delta.
Finally, at the end of the day, the stock closes at $42 a share. Harry sells a
final 140 shares to get flat.
There was not any literal scalping of stock today. It was all selling.
Nonetheless, gamma trading led to a profitable day.
As the stock rose from $40 to $40.50, 140 deltas were created from
positive gamma. Because the delta was zero at $40 and 140 at $40.50, the
estimated average delta is found by dividing 140 in half. This estimated
average delta multiplied by the $0.50 gain on the stock equals a $35 profit.
The delta was zero after the adjustment made at $40.50, when 140 shares
were sold. When the stock reached $41, another $35 was reaped from the
average delta of 70 over the $0.50 move. This process was repeated every
time the stock rose $0.50 and the delta was covered.
Day Four
Day four offers a pleasant surprise for Harry. That morning, the stock opens
$4 lower. He promptly covers his short delta of 11.2 by buying 1,120 shares
of the stock at $38 a share. The stock barely moves the rest of the day and
closes at $38.
An exponentially larger profit was made because there was $4 worth of
gains on the growing delta when the stock gapped open. The whole position
delta was covered $4 lower, so both the delta and the dollar amount gained
on that delta had a chance to grow. Again, Harry can estimate the average
delta over the $4 move to be half of 11.20. Multiplying that by the $4 stock
advance gives him his gamma profit of $2,240. After accounting for theta,
the net profit is $2,190.
Days Five and Six
Days five and six are the weekend; the market is closed.

Day Seven
This is a quiet day after the volatility of the past week. Today, the stock
slowly drifts up $0.25 by the end of the day. Harry sells 70 shares of stock
at $38.25 to cover long deltas.
This day was a loser for Harry, as profits from gamma were not enough to
cover his theta.
Art and Science
Although this was a very simplified example, it was typical of how a
profitable week of gamma scalping plays out. This stock had a pretty
volatile week, and overall the week was a winner: there were four losing
days and three winners. The number of losing days includes the weekends.
Weekends and holidays are big hurdles for long-gamma traders because of
the theta loss. The biggest contribution to this being a winning week was
made by the gap open on day four. Part of the reason was the sheer
magnitude of the move, and part was the fact that the deltas weren’t covered
too soon, as they had been on day three.
In a perfect world, a long-gamma trader will always buy the low of the
day and sell the high of the day when covering deltas. This, unfortunately,
seldom happens. Long-gamma traders are very often wrong when trading
stock to cover deltas.
Being wrong can be okay on occasion. In fact, it can even be rewarding.
Day three was profitable despite the fact that 140 shares were sold at
$40.50, $41, and $41.50. The stock closed at $42; the first three stock trades
were losers. Harry sold stock at a lower price than the close. But the
position still made money because of his positive gamma. To be sure, Harry
would like to have sold all 560 shares at $42 at the end of the day. The day’s
profits would have been significantly higher.
The problem is that no one knows where the stock will move next. On
day three, if the stock had topped out at $40.50 and Harry did not sell stock
because he thought it would continue higher, he would have missed an
opportunity. Gamma scalping is not an exact science. The art is to pick
spots that capture the biggest moves possible without missing opportunities.
There are many methods traders have used to decide where to cover
deltas when gamma scalping: the daily standard deviation, a fixed
percentage of the stock price, a fixed nominal value, covering at a certain
time of day, “market feel.” No system appears to be absolutely better than
another. This is where it gets personal. Finding what works for you, and
what works for the individual stocks you trade, is the art of this science.
Gamma, Theta, and Volatility
Clearly, more volatile stocks are more profitable for gamma scalping, right?
Well . . . maybe. Recall that the higher the implied volatility, the lower the
gamma and the higher the theta of at-the-money (ATM) options. In many
cases, the more volatile a stock, the higher the implied volatility (IV). That
means that a volatile stock might have to move more for a trader to scalp
enough stock to cover the higher theta.
Let’s look at the gamma-theta relationship from another perspective. In
this example, for 0.50 of theta, Harry could buy 2.80 gamma. This
relationship is based on an assumed 25 percent implied volatility. If IV were
50 percent, theta for this 20 lot would be higher, and the gamma would be
lower. At a volatility of 50, Harry could