The Effects of Volatility and Time on
Theta
Stock price is not the only factor that affects theta values. Volatility and
time to expiration come into play here as well. The volatility input to the
pricing model has adirect relationship to option values. The higher the
volatility, the higher the value of the option. Higher-valued options decay at
afaster rate than lower-valued options—they have to; their time values will
both be zero at expiration. All else held constant, the higher the volatility
assumption, the higher the theta.
The days to expiration have adirect relationship to option values as well.
As the number of days to expiration decreases, the rate at which an option
decays may change, depending on the relationship of the stock price to the
strike price. ATM options tend to decay at anonlinear rate—that is, they
lose value faster as expiration approaches—whereas the time values of ITM
and OTM options decay at asteadier rate.
Consider ahypothetical stock trading at $70 ashare. Exhibit 2.11 shows
how the theoretical values of the 75-strike call and the 70-strike call decline
with the passage of time, holding all other parameters constant.
EXHIBIT 2.11 Rate of decay: ATM vs. OTM.

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Vega
Over the past decade or so, computers have revolutionized option trading.
Options traded through an online broker are filled faster than you can say,
“Oops! Imeant to click on puts.” Now trading is facilitated almost entirely
online by professional and retail traders alike. Market and trading
information is disseminated worldwide in subseconds, making markets all
the more efficient. And the tools now available to the common retail trader
are very powerful as well. Many online brokers and other web sites offer
high-powered tools like screeners, which allow traders to sift through
thousands of options to find those that fit certain parameters.
Using ascreener to find ATM calls on same-priced stocks—say, stocks
trading at $40 ashare—can yield aresult worth talking about here. One $40
stock can have a 40-strike call trading at around 0.50, while adifferent $40
stock can have a 40 call with the same time to expiration trading at more
like 2.00. Why? The model doesn’tknow the name of the company, what
industry it’sin, or what its price-to-earnings ratio is. It is amathematical
equation with six inputs. If five of the inputs—the stock price, strike price,
time to expiration, interest rate, and dividends—are identical for two
different options but they’re trading at different prices, the difference must
be the sixth variable, which is volatility.

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Implied Volatility (IV) and Vega
The volatility component of option values is called implied volatility (IV).
(For more on implied volatility and how it relates to vega, see Chapter 3.)
IV is apercentage, although in practice the percent sign is often omitted.
This is the value entered into apricing model, in conjunction with the other
variables, that returns the option’stheoretical value. The higher the
volatility input, the higher the theoretical value, holding all other variables
constant. The IV level can change and often does—sometimes dramatically.
When IV rises or falls, option prices rise and fall in line with it. But by how
much?
The relationship between changes in IV and changes in an option’svalue
is measured by the option’svega. Vega is the rate of change of an option’stheoretical value relative to achange in implied volatility . Specifically, if
the IV rises or declines by one percentage point, the theoretical value of the
option rises or declines by the amount of the option’svega, respectively.
For example, if acall with atheoretical value of 1.82 has avega of 0.06 and
IV rises one percentage point from, say, 17 percent to 18 percent, the new
theoretical value of the call will be 1.88—it would rise by 0.06, the amount
of the vega. If, conversely, the IV declines 1 percentage point, from 17
percent to 16 percent, the call value will drop to 1.76—that is, it would
decline by the vega.
Aput with the same expiration month and the same strike on the same
underlying will have the same vega value as its corresponding call. In this
example, raising or lowering IV by one percentage point would cause the
corresponding put value to rise or decline by $0.06, just like the call.
An increase in IV and the consequent increase in option value helps the
P&(L) of long option positions and hurts short option positions. Buying acall or aput establishes along vega position. For short options, the opposite
is true. Rising IV adversely affects P&(L), whereas falling IV helps.
Shorting acall or put establishes ashort vega position.

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The Effect of Moneyness on Vega
Like the other greeks, vega is asnapshot that is afunction of multiple facets
of determinants influencing option value. The stock price’srelationship to
the strike price is amajor determining factor of an option’svega. IV affects
only the time value portion of an option. Because ATM options have the
greatest amount of time value, they will naturally have higher vegas. ITM
and OTM options have lower vega values than those of the ATM options.
Exhibit 2.13 shows an example of 186-day options on AT&T Inc. (T),
their time value, and the corresponding vegas.
EXHIBIT 2.13 AT&Ttheos and vegas (Tat $30, 186 days to Expry, 20%
IV).
Note that the 30-strike calls and puts have the highest time values. This
strike boasts the highest vega value, at 0.085. The lower the time premium,
the lower the vega—therefore, the less incremental IV changes affect the
option. Since higher-priced stocks have higher time premium (in absolute
terms, not necessarily in percentage terms) they will have higher vega.
Incidentally, if this were a $300 stock instead of a $30 stock, the 186-day
ATMs would have a 0.850 vega, if all other model inputs remain the same.

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Put-Call Parity
Put and call values are mathematically bound together by an equation
referred to as put-call parity. In its basic form, put-call parity states:
where
c = call value,
PV(x) = present value of the strike price,
p = put value, and
s = stock price.
The put-call parity assumes that options are not exercised before
expiration (that is, that they are European style). This version of the put-call
parity is for European options on non-dividend-paying stocks. Put-call
parity can be modified to reflect the values of options on stocks that pay
dividends. In practice, equity-option traders look at the equation in aslightly different way:
Traders serious about learning to trade options must know put-call parity
backward and forward. Why? First, by algebraically rearranging this
equation, it can be inferred that synthetically equivalent positions can be
established by simply adding stock to an option. Again, aput is acall; acall
is aput.
and
For example, along call is synthetically equal to along stock position
plus along put on the same strike, once interest and dividends are figured
in. Asynthetic long stock position is created by buying acall and selling aput of the same month and strike. Understanding synthetic relationships is
intrinsic to understanding options. Amore comprehensive discussion of
synthetic relationships and tactical considerations for creating synthetic
positions is offered in Chapter 6.

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Put-call parity also aids in valuing options. If put-call parity shows adifference in the value of the call versus the value of the put with the same
strike, there may be an arbitrage opportunity. That translates as “riskless
profit.” Buying the call and selling it synthetically (short put and short
stock) could allow aprofit to be locked in if the prices are disparate.
Arbitrageurs tend to hold synthetic put and call prices pretty close together.
Generally, only professional traders can capture these types of profit
opportunities, by trading big enough positions to make very small profits (apenny or less per contract sometimes) matter. Retail traders may be able to
take advantage of adisparity in put and call values to some extent, however,
by buying or selling the synthetic as asubstitute for the actual option if the
position can be established at abetter price synthetically.
Another reason that aworking knowledge of put-call parity is essential is
that it helps attain abetter understanding of how changes in the interest rate
affect option values. The greek rho measures this change. Rho is the rate of
change in an option’svalue relative to achange in the interest rate.
Although some modeling programs may display this number differently,
most display arho for the call and arho for the put, both illustrating the
sensitivity to aone-percentage-point change in the interest rate. When the
interest rate rises by one percentage point, the value of the call increases by
the amount of its rho and the put decreases by the amount of its rho.
Likewise, when the interest rate decrease by one percentage point, the value
of the call decreases by its rho and the put increases by its rho. For example,
acall with arho of 0.12 will increase $0.12 in value if the interest rate used
in the model is increased by one percentage point. Of course, interest rates
usually don’trise or fall one percentage point in one day. More commonly,
rates will have incremental changes of 25 basis points. That means acall
with a 0.12 rho will theoretically gain $0.03 given an increase of 0.25
percentage points.
Mathematically, this change in option value as aproduct of achange in
the interest rate makes sense when looking at the formula for put-call parity.
and

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But the change makes sense intuitively, too, when acall is considered as acheaper substitute for owning the stock. For example, compare a $100 stock
with athree-month 60-strike call on that same stock. Being so far ITM,
there would likely be no time value in the call. If the call can be purchased
at parity, which alternative would be asuperior investment, the call for $40
or the stock for $100? Certainly, the call would be. It costs less than half as
much as the stock but has the same reward potential; and the $60 not spent
on the stock can be invested in an interest-bearing account. This interest
advantage adds value to the call. Raising the interest rate increases this
value, and lowering it decreases the interest component of the value of the
call.
Asimilar concept holds for puts. Professional traders often get ashort-
stock rebate on proceeds from ashort-stock sale. This is simply interest
earned on the capital received when the stock is shorted. Is it better to pay
interest on the price of aput for aposition that gives short exposure or to
receive interest on the credit from shorting the stock? There is an interest
disadvantage to owning the put. Therefore, arise in interest rates devalues
puts.
This interest effect becomes evident when comparing ATM call and put
prices. For example, with interest at 5 percent, three-month options on an
$80 stock that pays a $0.25 dividend before option expiration might look
something like this:
The ATM call is higher in theoretical value than the ATM put by $0.75.
That amount can be justified using put-call parity:
(Here, simple interest of $1 is calculated as 80 × 0.05 × [90 / 360] = 1.)
Changes in market conditions are kept in line by the put-call parity. For
example, if the price of the call rises because of an increase in IV, the price
of the put will rise in step. If the interest rate rises by aquarter of apercentage point, from 5 percent to 5.25 percent, the interest calculated for
three months on the 80-strike will increase from $1 to $1.05, causing the

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The Effect of Time on Rho
The more time until expiration, the greater the effect interest rate changes
will have on options. In the previous example, a 25-basis-point change in
the interest rate on the 80-strike based on athree-month period caused achange of 0.05 to the interest component of put-call parity. That is, 80 ×
0.0025 × (90/360) = 0.05. If alonger period were used in the example—say,
one year—the effect would be more profound; it will be $0.20: 80 × 0.0025
× (360/360) = 0.20. This concept is evident when the rhos of options with
different times to expiration are studied.
Exhibit 2.16 shows the rhos of ATM Procter & Gamble Co. (PG) calls
with various expiration months. The 750-day Long-Term Equity
AnticiPation Securities (LEAPS) have arho of 0.858. As the number of
days until expiration decreases, rho decreases. The 22-day calls have arho
of only 0.015. Rho is usually afairly insignificant factor in the value of
short-term options, but it can come into play much more with long-term
option strategies involving LEAPS.
EXHIBIT 2.16 The effect of time on rho (Procter & Gamble @ $64.34)

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Why the Numbers Don’t Don’t Always
Add Up
There will be many times when studying the rho of options in an option
chain will reveal seemingly counterintuitive results. To be sure, the numbers
don’talways add up to what appears logical. One reason for this is
rounding. Another is that traders are more likely to use simple interest in
calculating value, whereas the model uses compound interest. Hard-to-
borrow stocks and stocks involved in mergers and acquisitions may have
put-call parities that don’twork out right. But another, more common and
more significant fly in the ointment is early exercise.
Since the interest input in put-call parity is afunction of the strike price, it
is reasonable to expect that the higher the strike price, the greater the effect
of interest on option prices will be. For European options, this is true to alarge extent, in terms of aggregate impact of interest on the call and put pair.
Strikes below the price where the stock is trading have ahigher rho
associated with the call relative to the put, whereas strikes above the stock
price have ahigher rho associated with the put relative to the call.
Essentially, the more in-the-money an option is, the higher its rho. But with
European options, observing the aggregate of the absolute values of the call
and put rhos would show ahigher combined rho the higher the strike.
With American options, the put can be exercised early. Atrader will
exercise aput before expiration if the alternative—being short stock and
receiving ashort stock rebate—is awiser choice based on the price of the
put. Professional traders may own stock as ahedge against aput. They may
exercise deep ITM puts (1.00-delta puts) to avoid paying interest on capital
charges related to the stock. The potential for early exercise is factored into
models that price American options. Here, when puts get deeper in-the-
money—that is, more apt to be exercised—the rho decreases. When the
strike price is very high relative to the stock price—meaning the put is very
deep ITM—and there is little or no time value left to the call or the put, the
aggregate put-call rho can be zero. Rho is discussed in greater detail in
Chapter 7.

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Where to Find Option Greeks
There are many sources from which to obtain greeks. The Internet is an
excellent resource. Googling “option greeks” will display links to over four
million web pages, many of which have real-time greeks or an option
calculator. An option calculator is asimple interface that accepts the input
of the six variables to the model and yields atheoretical value and the
greeks for asingle option.
Some web sites devoted to option education, such as
MarketTaker.com/option_modeling , have free calculators that can be used
for modeling positions and using the greeks.
In practice, many of the option-trading platforms commonly in use have
sophisticated analytics that involve greeks. Most options-friendly online
brokers provide trading platforms that enable traders to conduct
comprehensive manipulations of the greeks. For example, traders can look
at the greeks for their positions up or down one, two, or three standard
deviations. Or they can see what happens to their position greeks if IV or
time changes. With many trading platforms, position greeks are updated in
real time with changes in the stock price—an invaluable feature for active
traders.

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Caveats with Regard to Online
Greeks
Often, online greeks are one click away, requiring little effort on the part of
the trader. Having greeks calculated automatically online is aquick and
convenient way to eyeball greeks for an option. But there is one major
problem with online greeks: reliability.
For active option traders, greeks are essential. There is no point in using
these figures if their accuracy cannot be assured. Experienced traders can
often spot these inaccuracies aproverbial mile away.
When looking at greeks from an online source that does not require you
to enter parameters into amodel (as would be the case with professional
option-trading platforms), special attention needs to be paid to the
relationship of the option’stheoretical values to the bid and offer. One must
be cautious if the theoretical value of the option lies outside the bid-ask
spread. This scenario can exist for brief periods of time, but arbitrageurs
tend to prevent this from occurring routinely. If several options in achain
all have theoretical values below the bid or above the offer, there is
probably aproblem with one or more of the inputs used in the model.
Remember, an option-pricing model is just that: amodel. It reflects what is
occurring in the market. It doesn’ttell where an option should be trading.
The complex changes that occur intraday in the market—taking the day
or weekend out, changes in stock price, volatility, and the interest rate—are
not always kept current. The user of the model must keep close watch. It’snot reasonable to expect the computer to do the thinking for you.
Automatically calculated greeks can be used as astarting point. But before
using these figures in the decision-making process, the trader may have to
override the parameters that were used in the online calculation to make the
theos line up with market prices. Professional traders will ignore online
greeks altogether. They will use the greeks that are products of the inputs
they entered in their trading software. It comes down to this: if you want
something done right, do it yourself.

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CHAPTER 3
Understanding Volatility
Most option strategies involve trading volatility in one way or another. It’seasy to think of trading in terms of direction. But trading volatility?
Volatility is an abstract concept; it’sadifferent animal than the linear
trading paradigm used by most conventional market players. As an option
trader, it is essential to understand and master volatility.
Many traders trade without asolid understanding of volatility and its
effect on option prices. These traders are often unhappily surprised when
volatility moves against them. They mistake the adverse option price
movements that result from volatility for getting ripped off by the market
makers or some other market voodoo. Or worse, they surrender to the fact
that they simply don’tunderstand why sometimes these unexpected price
movements occur in options. They accept that that’sjust the way it is.
Part of what gets in the way of aready understanding of volatility is
context. The term volatility can have afew different meanings in the
options business. There are three different uses of the word volatility that an
option trader must be concerned with: historical volatility, implied
volatility, and expected volatility.

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Implied Volatility
Volatility is one of the six inputs of an option-pricing model. Some of the
other inputs—strike price, stock price, the number of days until expiration,
and the current interest rate—are easily observable. Past dividend policy
allows an educated guess as to what the dividend input should be. But
where can volatility be found?
As discussed in Chapter 2, the output of the pricing model—the option’stheoretical value—in practice is not necessarily an output at all. When
option traders use the pricing model, they commonly substitute the actual
price at which the option is trading for the theoretical value. Avalue in the
middle of the bid-ask spread is often used. The pricing model can be
considered to be acomplex algebra equation in which any variable can be
solved for. If the theoretical value is known—which it is—it along with the
five known inputs can be combined to solve for the unknown volatility.
Implied volatility (IV) is the volatility input in apricing model that, in
conjunction with the other inputs, returns the theoretical value of an option
matching the market price.
For aspecific stock price, agiven implied volatility will yield aunique
option value. Take astock trading at $44.22 that has the 60-day 45-strike
call at atheoretical value of $1.10 with an 18 percent implied volatility
level. If the stock price remains constant, but IV rises to 19 percent, the
value of the call will rise by its vega, which in this case is about 0.07. The
new value of the call will be $1.17. Raising IV another point, to 20 percent,
raises the theoretical value by another $0.07, to $1.24. The question is:
What would cause implied volatility to change?

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Supply and Demand: Not Just a Good
Idea, It’sthe Law!
Options are an excellent vehicle for speculation. However, the existence of
the options market is better justified by the primary economic purpose of
options: as arisk management tool. Hedgers use options to protect their
assets from adverse price movements, and when the perception of risk
increases, so does demand for this protection. In this context, risk means
volatility—the potential for larger moves to the upside and downside. The
relative prices of options are driven higher by increased demand for
protective options when the market anticipates greater volatility. And option
prices are driven lower by greater supply—that is, selling of options—when
the market expects lower volatility. Like those of all assets, option prices
are subject to the law of supply and demand.
When volatility is expected to rise, demand for options is not limited to
hedgers. Speculative traders would arguably be more inclined to buy acall
than to buy the stock if they are bullish but expect future volatility to be
high. Calls require alower cash outlay. If the stock moves adversely, there
is less capital at risk, but still similar profit potential.
When volatility is expected to be low, hedging investors are less inclined
to pay for protection. They are more likely to sell back the options they may
have bought previously to recoup some of the expense. Options are adecaying asset. Investors are more likely to write calls against stagnant
stocks to generate income in anticipated low-volatility environments.
Speculative traders will implement option-selling strategies, such as short
strangles or iron condors, in an attempt to capitalize on stocks they believe
won’tmove much. The rising supply of options puts downward pressure on
option prices.
Many traders sum up IV in two words: fear and greed . When option
prices rise and fall, not because of changes in the stock price, time to
expiration, interest rates, or dividends, but because of pure supply and
demand, it is implied volatility that is the varying factor. There are many
contributing factors to traders’ willingness to demand or supply options.
Anticipation of events such as earnings reports, Federal Reserve

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announcements, or the release of other news particular to an individual
stock can cause anxiety, or fear, in traders and consequently increase
demand for options that causes IV to rise. IV can fall when there is
complacency in the market or when the anticipated news has been
announced and anxiety wanes. “Buy the rumor, sell the news” is often
reflected in option implied volatility. When there is little fear of market
movement, traders use options to squeeze out more profits—greed.
Arbitrageurs, such as market makers who trade delta neutral—astrategy
that will be discussed further in Chapters 12 and 13—must be relentlessly
conscious of implied volatility. When immediate directional risk is
eliminated from aposition, IV becomes the traded commodity. Arbitrageurs
who focus their efforts on trading volatility (colloquially called vol traders )
tend to think about bids and offers in terms of IV. In the mind of avol
trader, option prices are translated into volatility levels. Atrader may look at
aparticular option and say it is 30 bid at 31 offer. These values do not
represent the prices of the options but rather the corresponding implied
volatilities. The meaning behind the trader’sremark is that the market is
willing to buy implied volatility at 30 percent and sell it at 31 percent. The
actual prices of the options themselves are much less relevant to this type of
trader.

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Should HV and IV Be the Same?
Most option positions have exposure to volatility in two ways. First, the
profitability of the position is usually somewhat dependent on movement
(or lack of movement) of the underlying security. This is exposure to HV.
Second, profitability can be affected by changes in supply and demand for
the options. This is exposure to IV. In general, along option position
benefits when volatility—both historical and implied—increases. Ashort
option position benefits when volatility—historical and implied—decreases.
That said, buying options is buying volatility and selling options is selling
volatility.
The Relationship of HV and IV
It’sintuitive that there should exist adirect relationship between the HV and
IV. Empirically, this is often the case. Supply and demand for options, based
on the market’sexpectations for asecurity’svolatility, determines IV.
It is easy to see why IV and HV often act in tandem. But, although HV
and IV are related, they are not identical. There are times when IV and HV
move in opposite directions. This is not so illogical, if one considers the key
difference between the two: HV is calculated from past stock price
movements; it is what has happened. IV is ultimately derived from the
market’sexpectation for future volatility.
If astock typically has an HV of 30 percent and nothing is expected to
change, it can be reasonable to expect that in the future the stock will
continue to trade at a 30 percent HV. By that logic, assuming that nothing is
expected to change, IV should be fairly close to HV. Market conditions do
change, however. These changes are often regular and predictable. Earnings
reports are released once aquarter in many stocks, Federal Open Market
Committee meetings happen regularly, and dates of other special
announcements are often disclosed to the public in advance. Although the
outcome of these events cannot be predicted, when they will occur often
can be. It is around these widely anticipated events that HV-IV divergences
often occur.

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Expected Stock Volatility
Option traders must have an even greater focus on volatility, as it plays amuch bigger role in their profitability—or lack thereof. Because options can
create highly leveraged positions, small moves can yield big profits or
losses. Option traders must monitor the likelihood of movement in the
underlying closely. Estimating what historical volatility (standard deviation)
will be in the future can help traders quantify the probability of movement
beyond acertain price point. This leads to better decisions about whether to
enter atrade, when to adjust aposition, and when to exit.
There is no way of knowing for certain what the future holds. But option
data provide traders with tools to develop expectations for future stock
volatility. IV is sometimes interpreted as the market’sestimate of the future
volatility of the underlying security. That makes it aready-made estimation
tool, but there are two caveats to bear in mind when using IV to estimate
future stock volatility.
The first is that the market can be wrong. The market can wrongly price
stocks. This mispricing can lead to acorrection (up or down) in the prices
of those stocks, which can lead to additional volatility, which may not be
priced in to the options. Although there are traders and academics believe
that the option market is fairly efficient in pricing volatility, there is aroom
for error. There is the possibility that the option market can be wrong.
Another caveat is that volatility is an annualized figure—the annualized
standard deviation. Unless the IV of a LEAPS option that has exactly one
year until expiration is substituted for the expected volatility of the
underlying stock over exactly one year, IV is an incongruent estimation for
the future stock volatility. In practice, the IV of an option must be adjusted
to represent the period of time desired.
There is acommon technique for deannualizing IV used by professional
traders and retail traders alike. 1 The first step in this process to deannualize
IV is to turn it into aone-day figure as opposed to one-year figure. This is
accomplished by dividing IV by the square root of the number of trading
days in ayear. The number many traders use to approximate the number of

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trading days per year is 256, because its square root is around number: 16.
The formula is
For example, a $100 stock that has an at-the-money (ATM) call trading at
a 32 percent volatility implies that there is about a 68 percent chance that
the underlying stock will be between $68 and $132 in one year’stime—
that’s $100 ± ($100 × 0.32). The estimation for the market’sexpectation for
the volatility of the stock for one day in terms of standard deviation as apercentage of the price of the underlying is computed as follows:
In one day’stime, based on an IV of 32 percent, there is a 68 percent
chance of the stock’sbeing within 2 percent of the stock price—that’sbetween $98 and $102.
There may be times when it is helpful for traders to have avolatility
estimation for aperiod of time longer than one day—aweek or amonth, for
example. This can be accomplished by multiplying the one-day volatility by
the square root of the number of trading days in the relevant period. The
equation is as follows:
If the period in question is one month and there are 22 business days
remaining in that month, the same $100 stock with the ATM call trading at a
32 percent implied volatility would have aone-month volatility of 9.38
percent.
Based on this calculation for one month, it can be estimated that there is a
68 percent chance of the stock’sclosing between $90.62 and $109.38 based
on an IV of 32 percent.

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Expected Implied Volatility
Although there is agreat deal of science that can be applied to calculating
expected actual volatility, developing expectations for implied volatility is
more of an art. This element of an option’sprice provides more risk and
more opportunity. There are many traders who make their living distilling
direction out of their positions and trading implied volatility. To be
successful, atrader must forecast IV.
Conceptually, trading IV is much like trading anything else. Atrader who
thinks astock is going to rise will buy the stock. Atrader who thinks IV is
going to rise will buy options. Directional stock traders, however, have
many more analysis tools available to them than do vol traders. Stock
traders have both technical analysis (TA) and fundamental analysis at their
disposal.
Technical Analysis
There are scores, perhaps hundreds, of technical tools for analyzing stocks,
but there are not many that are available for analyzing IV. Technical
analysis is the study of market data, such as past prices or volume, which is
manipulated in such away that it better illustrates market activity. TA
studies are usually represented graphically on achart.
Developing TA tools for IV is more of achallenge than it is for stocks.
One reason is that there is simply alot more data to manage—for each
stock, there may be hundreds of options listed on it. The only practical way
of analyzing options from a TA standpoint is to use implied volatility. IV is
more useful than raw historical option prices themselves. Information for
both IV and HV is available in the form of volatility charts, or vol charts.
(Vol charts are discussed in detail in Chapter 14.) Volatility charts are
essential for analyzing options because they give more complete
information.
To get aclear picture of what is going on with the price of an option (the
goal of technical analysis for any asset), just observing the option price does
not supply enough information for atrader to work with. It’sincomplete.
For example, if acall rises in value, why did it rise? What greek contributed

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to its value increase? Was it delta because the underlying stock rose? Or
was it vega because volatility rose? How did time decay factor in? Using avolatility chart in conjunction with aconventional stock chart (and being
aware of time decay) tells the whole, complete, story.
Another reason historical option prices are not used in TA is the option
bid-ask spread. For most stocks, the difference between the bid and the ask
is equal to avery small percentage of the stock’sprice. Because options are
highly leveraged instruments, their bid-ask width can equal amuch higher
percentage of the price.
If atrader uses the last trade to graph an option’sprice, it could look as if
avery large percentage move has occurred when in fact it has not. For
example, if the option trades asmall contract size on the bid (0.80), then on
the offer (0.90) it would appear that the option rose 12.5 percent in value.
This large percentage move is nothing more than market noise. Using
volatility data based off the midpoint-of-the-market theoretical value
eliminates such noise.
Fundamental Analysis
Fundamental analysis can have an important role in developing
expectations for IV. Fundamental analysis is the study of economic factors
that affect the value of an asset in order to determine what it is worth. With
stocks, fundamental analysis may include studying income statements,
balance sheets, and earnings reports. When the asset being studied is IV,
there are fewer hard facts available. This is where the art of analyzing
volatility comes into play.
Essentially, the goal is to understand the psychology of the market in
relation to supply and demand for options. Where is the fear? Where is the
complacency? When are news events anticipated? How important are they?
Ultimately, the question becomes: what is the potential for movement in the
underlying? The greater the chance of stock movement, the more likely it is
that IV will rise. When unexpected news is announced, IV can rise quickly.
The determination of the fundamental relevance of surprise announcements
must be made quickly.
Unfortunately, these questions are subjective in nature. They require the
trader to apply intuition and experience on acase-by-case basis. But there

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are afew observations to be made that can help atrader make better-
educated decisions about IV.
Reversion to the Mean
The IVs of the options on many stocks and indexes tend to trade in arange
unique to those option classes. This is referred to as the mean—or average
—volatility level. Some securities will have smaller mean IV ranges than
others. The range being observed should be established for aperiod long
enough to confirm that it is atypical IV for the security, not just atemporary anomaly. Traders should study IV over the most recent 6-month
period. When IV has changed significantly during that period, a 12-month
study may be necessary. Deviations from this range, either above or below
the established mean range, will occur from time to time. When following abreakout from the established range, it is common for IV to revert back to
its normal range. This is commonly called reversion to the mean among
volatility watchers.
The challenge is recognizing when things change and when they stay the
same. If the fundamentals of the stock change in such away as to give the
options market reason to believe the stock will now be more or less volatile
on an ongoing basis than it typically has been in the recent past, the IV may
not revert to the mean. Instead, anew mean volatility level may be
established.
When considering the likelihood of whether IV will revert to recent levels
after it has deviated or find anew range, the time horizon and changes in
the marketplace must be taken into account. For example, between 1998
and 2003 the mean volatility level of the SPX was around 20 percent to 30
percent. By the latter half of 2006, the mean IV was in the range of 10
percent to 13 percent. The difference was that between 1998 and 2003 was
the buildup of “the tech bubble,” as it was called by the financial media.
Market volatility ultimately leveled off in 2003.
In alater era, between the fall of 2010 and late summer of 2011 SPX
implied volatility settled in to trade mostly between 12 and 20 percent. But
in August 2011, as the European debt crisis heated up, anew, more volatile
range between 24 and 40 percent reigned for some time.

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CBOE Volatility Index
®
Often traders look to the implied volatility of the market as awhole for
guidance on the IV of individual stocks. Traders use the Chicago Board
Options Exchange (CBOE) Volatility Index® , or VIX® , as an indicator of
overall market volatility.
When people talk about the market, they are talking about abroad-based
index covering many stocks on many diverse industries. Usually, they are
referring to the S&P 500. Just as the IV of astock may offer insight about
investors’ feelings about that stock’sfuture volatility, the volatility of
options on the S&P 500—SPX options—may tell something about the
expected volatility of the market as awhole.
VIX is an index published by the Chicago Board Options Exchange that
measures the IV of ahypothetical 30-day option on the SPX. A 30-day
option on the SPX only truly exists once amonth—30 days before
expiration. CBOE computes ahypothetical 30-day option by means of aweighted average of the two nearest-term months.
When the S&P 500 rises or falls, it is common to see individual stocks
rise and fall in sympathy with the index. Most stocks have some degree of
market risk. When there is aperception of higher risk in the market as awhole, there can consequently be aperception of higher risk in individual
stocks. The rise or fall of the IV of SPX can translate into the IV of
individual stocks rising or falling.

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Implied Volatility and Direction
Who’safraid of falling stock prices? Logically, declining stocks cause
concern for investors in general. There is confirmation of that statement in
the options market. Just look at IV. With most stocks and indexes, there is
an inverse relationship between IV and the underlying price. Exhibit 3.2
shows the SPX plotted against its 30-day IV, or the VIX.
EXHIBIT 3.2 SPX vs. 30-day IV (VIX).
The heavier line is the SPX, and the lighter line is the VIX. Note that as
the price of SPX rises, the VIX tends to decline and vice versa. When the
market declines, the demand for options tends to increase. Investors hedge
by buying puts. Traders speculate on momentum by buying puts and
speculate on aturnaround by buying calls. When the market moves higher,
investors tend to sell their protection back and write covered calls or cash-
secured puts. Option speculators initiate option-selling strategies. There is
less fear when the market is rallying.
This inverse relationship of IV to the price of the underlying is not unique
to the SPX; it applies to most individual stocks as well. When astock

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Calculating Volatility Data
Accurate data are essential for calculating volatility. Many of the volatility
data that are readily available are useful, but unfortunately, some are not.
HV is avalue that is easily calculated from publicly accessible past closing
prices of astock. It’srather straightforward. Traders can access HV from
many sources. Retail traders often have access to HV from their brokerage
firm. Trading firms or clearinghouses often provide professional traders
with HV data. There are some excellent online resources for HV as well.
HV is acalculation with little subjectivity—the numbers add up how they
add up. IV, however, can be abit more ambiguous. It can be calculated
different ways to achieve different desired outcomes; it is user-centric. Most
of the time, traders consider the theoretical value to be between the bid and
the ask prices. On occasion, however, atrader will calculate IV for the bid,
the ask, the last trade price, or, sometimes, another value altogether. There
may be avalid reason for any of these different methods for calculating IV.
For example, if atrader is long volatility and aspires to reduce his position,
calculating the IV for the bid shows him what IV level can be sold to
liquidate his position.
Firms, online data providers, and most options-friendly brokers offer IV
data. Past IV data is usually displayed graphically in what is known as avolatility chart or vol chart. Current IV is often displayed along with other
data right in the option chain. One note of caution: when the current IV is
displayed, however, it should always be scrutinized carefully. Was the bid
used in calculating this figure? What about the ask? How long ago was this
calculation made? There are many questions that determine the accuracy of
acurrent IV, and rarely are there any answers to support the number.
Traders should trust only IV data they knowingly generated themselves
using apricing model.

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Term Structure of Volatility
Term structure of volatility—also called monthly skew or horizontal skew
—is the relationship among the IVs of options in the same class with the
same strike but with different expiration months. IV, again, is often
interpreted as the market’sestimate of future volatility. It is reasonable to
assume that the market will expect some months to be more volatile than
others. Because of this, different expiration cycles can trade at different IVs.
For example, if acompany involved in amajor product-liability lawsuit is
expecting averdict on the case to be announced in two months, the one-
month IV may be low, as the stock is not expected to move much until the
suit is resolved. The two-month volatility may be much higher, however,
reflecting the expectations of abig move in the stock up or down,
depending on the outcome.
The term structure of volatility also varies with the normal ebb and flow
of volatility within the business cycle. In periods of declining volatility, it is
common for the month with the least amount of time until expiration, also
known as the front month, to trade at alower volatility than the back
months, or months with more time until expiration. Conversely, when
volatility is rising, the front month tends to have ahigher IV than the back
months.
Exhibit 3.3 shows historical option prices and their corresponding IVs for
32.5-strike calls on General Motors (GM) during aperiod of low volatility.
EXHIBIT 3.3 GM term structure of volatility.
In this example, no major news is expected to be released on GM, and
overall market volatility is relatively low. The February 32.5 call has the
lowest IV, at 32 percent. Each consecutive month has ahigher IV than the

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previous month. Agraduated increasing or decreasing IV for each
consecutive expiration cycle is typical of the term structure of volatility.
Under normal circumstances, the front month is the most sensitive to
changes in IV. There are two reasons for this. First, front-month options are
typically the most actively traded. There is more buying and selling
pressure. Their IV is subject to more activity. Second, vegas are smaller for
options with fewer days until expiration. This means that for the same
monetary change in an option’svalue, the IV needs to move more for short-
term options.
Exhibit 3.4 shows the same GM options and their corresponding vegas.
EXHIBIT 3.4 GM vegas.
If the value of the September 32.5 calls increases by $0.10, IV must rise
by 1 percentage point. If the February 32.5 calls increase by $0.10, IV must
rise 3 percentage points. As expiration approaches, the vega gets even
smaller. With seven days until expiration, the vega would be about 0.014.
This means IV would have to change about 7 points to change the call value
$0.10.

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Vertical Skew
The second type of skew found in option IV is vertical skew, or strike skew.
Vertical skew is the disparity in IV among the strike prices within the same
month for an option class. The options on most stocks and indexes
experience vertical skew. As ageneral rule, the IV of downside options—
calls and puts with strike prices lower than the at-the-money (ATM) strike
—trade at higher IVs than the ATM IV. The IV of upside options—calls and
puts with strike prices higher than the ATM strike—typically trade at lower
IVs than the ATM IV.
The downside is often simply referred to as puts and the upside as calls.
The rationale for this lingo is that OTM options (puts on the downside and
calls on the upside) are usually more actively traded than the ITM options.
By put-call parity, aput can be synthetically created from acall, and acall
can be synthetically created from aput simply by adding the appropriate
long or short stock position.
Exhibit 3.5 shows the vertical skew for 86-day options on Citigroup Inc.
(C) on atypical day, with IVs rounded to the nearest tenth.
EXHIBIT 3.5 Citigroup vertical skew.
Notice the IV of the puts (downside options) is higher than that of the
calls (upside options), with the 31 strike’svolatility more than 10 points
higher than that of the 38 strike. Also, the difference in IV per unit change

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in the strike price is higher for the downside options than it is for the upside
ones. The difference between the IV of the 31 strike is 2 full points higher
than the 32 strike, which is 1.8 points higher than the 33 strike. But the 36
strike’s IV is only 1.1 points higher than the 37 strike, which is also just 1.1
points higher than the 38 strike.
This incremental difference in the IV per strike is often referred to as the
slope. The puts of most underlyings tend to have agreater slope to their
skew than the calls. Many models allow values to be entered for the upside
slope and the downside slope that mathematically increase or decrease IVs
of each strike incrementally. Some traders believe the slope should be astraight line, while others believe it should be an exponentially sloped line.
If the IVs were graphed, the shape of the skew would vary among asset
classes. This is sometimes referred to as the volatility smile or sneer,
depending on the shape of the IV skew. Although Exhibit 3.5 is atypical
paradigm for the slope for stock options, bond options and other commodity
options would have differently shaped skews. For example, grain options
commonly have calls with higher IVs than the put IVs.
Volatility skew is dependent on supply and demand. Greater demand for
downside protection may cause the overall IV to rise, but it can cause the
IV of puts to rise more relative to the calls or vice versa. There are many
traders who make their living trading volatility skew.

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Long ATM Call
Kim is atrader who is bullish on the Walt Disney Company (DIS) over the
short term. The time horizon of her forecast is three weeks. Instead of
buying 100 shares of Disney at $35.10 per share, Kim decides to buy one
Disney March 35 call at $1.10. In this example, March options have 44
days until expiration. How can Kim profit from this position? How can she
lose?
Exhibit 4.1 shows the profit and loss (P&(L)) for the call at different time
periods. The top line is when the trade is executed; the middle, dotted line is
after three weeks have passed; and the bottom, darker line is at expiration.
Kim wants Disney to rise in price, which is evident by looking at the graph
for any of the three time horizons. She would anticipate aloss if the stock
price declines. These expectations are related to the position’sdelta, but that
is not the only risk exposure Kim has. As indicated by the three different
lines in Exhibit 4.1 , the call loses value over time. This is called theta risk .
She has other risk exposure as well. Exhibit 4.2 lists the greeks for the DIS
March 35 call.
EXHIBIT 4.1 P&(L) of Disney 35 call.
EXHIBIT 4.2 Greeks for 35 Disney call.

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Delta 0.57
Gamma0.166
Theta −0.013
Vega 0.048
Rho 0.023
Kim’simmediate directional exposure is quantified by the delta, which is
0.57. Delta is immediate directional exposure because it’ssubject to change
by the amount of the gamma. The positive gamma of this position helps
Kim by increasing the delta as Disney rises and decreasing it as it falls.
Kim, however, has time working against her—theta. At this point, she
theoretically loses $0.013 per day. Since her call is close to being at-the-
money, she would anticipate her theta becoming more negative as
expiration approaches if Disney’sshare price remains unchanged. She also
has positive vega exposure. Aone-percentage-point increase in implied
volatility (IV) earns Kim just under $0.05. Aone-point decrease costs her
about $0.05. With so few days until expiration, the 35-strike call has very
little rho exposure. Afull one-percentage-point change in the interest rate
changes her call’svalue by only $0.023.

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Delta
Some of Kim’srisks warrant more concern than others. With this position,
delta is of the greatest concern, followed by theta. Kim expects the call to
rise in value and accepts the risk of decline. Delta exposure was her main
rationale for establishing the position. She expects to hold it for about three
weeks. Kim is willing to accept the trade-off of delta exposure for theta,
which will cost her three weeks of erosion of option premium. If the
anticipated delta move happens sooner than expected, Kim will have less
decay. Exhibit 4.3 shows the value of her 35 call at various stock prices
over time. The left column is the price of Disney. The top row is the number
of days until expiration.
EXHIBIT 4.3 Disney 35 call price–time matrix–value.
The effect of delta is evident as the stock rises or falls. When the position
is established (44 days until expiration), the change in the option price if the
stock were to move from $35 to $36 is 0.62 (1.66 − 1.04). Between stock
prices of $36 and $37, the option gains 0.78 (2.44 −1.66). If the stock were
to decline in value from $35 to $34, the option loses 0.47 (1.04 − 0.57). The
option gains value at afaster rate as the stock rises and loses value at aslower rate as the stock falls. This is the effect of gamma.

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Vega
After delta and theta, vega is the next most influential contributor to Kim’sprofit or peril. With Disney at $35.10, the 1.10 premium for the 35-strike
call represents $1 of time value—all of which is vulnerable to changes in
IV. The option’s 1.10 value returns an IV of about 19 percent, given the
following inputs:
Stock: $35.10
Strike: 35
Days to expiration: 44
Interest: 5.25 percent
No dividend paid during this period
Consequently, the vega is 0.048. What does the 0.048 vega tell Kim?
Given the preceding inputs, for each point the IV rises or falls, the option’svalue gains or loses about $0.05.
Some of the inputs, however, will change. Kim anticipates that Disney
will rise in price. She may be right or wrong. Either way, it is unlikely that
the stock will remain exactly at $35.10 to option expiration. The only
certainty is that time will pass.
Both price and time will change Kim’svega exposure. Exhibit 4.5 shows
the changing vega of the 35 call as time and the underlying price change.
EXHIBIT 4.5 Disney 35 call price–time matrix–vega.

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When comparing Exhibit 4.5 to Exhibit 4.3 , it’seasy to see that as the
time value of the option declines, so does Kim’sexposure to vega. As time
passes, vega gets smaller. And as the call becomes more in- or out-of-the-
money, vega gets smaller. Since she plans to hold the position for around
three weeks, she is not concerned about small fluctuations in IV in the
interim.
If indeed the rise in price that Kim anticipates comes to pass, vega
becomes even less of aconcern. With 23 days to expiration and DIS at $37,
the call value is 2.21. The vega is $0.018. If IV decreases as the stock price
rises—acommon occurrence—the adverse effect of vega will be minimal.
Even if IV declines by 5 points, to ahistorically low IV for DIS, the call
loses less than $0.10. That’sless than 5 percent of the new value of the
option.
If dividend policy changes or the interest rate changes, the value of Kim’scall will be affected as well. Dividends are often fairly predictable.
However, alarge unexpected dividend payment can have asignificant
adverse impact on the value of the call. For example, if asurprise $3
dividend were announced, owning the stock would become greatly
preferable to owning the call. This preference would be reflected in the call
premium. This is ascenario that an experienced trader like Kim will realize
is apossibility, although not aprobability. Although she knows it can

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Rho
For all intents and purposes, rho is of no concern to Kim. In recent years,
interest rate changes have not been amajor issue for option traders. In the
Alan Greenspan years of Federal Reserve leadership, changes in the interest
rate were usually announced at the regularly scheduled Federal Open
Market Committee (FOMC) meetings, with but afew exceptions. Ben
Bernanke, likewise, changed interest rates fairly predictably, when he made
any rate changes at all. In these more stable periods, if there is no FOMC
meeting scheduled during the life of the call, it’sunlikely that rates will
change. Even if they do, the rho with 44 days to expiration is only 0.023.
This means that if rates change by awhole percentage point—which is four
times the most common incremental change—the call value will change by
alittle more than $0.02. In this case, this is an acceptable risk. With 23 days
to expiration, the ATM 35 call has arho of only 0.011.

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Tweaking Greeks
With this position, some risks are of greater concern than others. Kim may
want more exposure to some greeks and less to others. What if she is
concerned that her forecasted price increase will take longer than three
weeks? She may want less exposure to theta. What if she is particularly
concerned about adecline in IV? She may want to decrease her vega.
Conversely, she may believe IV will rise and therefore want to increase her
vega.
Kim has many ways at her disposal to customize her greeks. All of her
alternatives come with trade-offs. She can buy more calls, increasing her
greek positions in exact proportion. She can buy or sell stock or options
against her call, creating aspread. The simplest way to alter her exposure to
option greeks is to choose adifferent call to buy. Instead of buying the ATM
call, Kim can buy acall with adifferent relationship to the current stock
price.

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Long OTM Call
Kim can reduce her exposure to theta and vega by buying an OTM call. The
trade-off here is that she also reduces her immediate delta exposure.
Depending on how much Kim believes Disney will rally, this may or may
not be aviable trade-off. Imagine that instead of buying one Disney March
35 call, Kim buys one Disney March 37.50 call, for 0.20.
There are afew observations to be made about this alternative position.
First, the net premium, and therefore overall risk, is much lower, 0.20
instead of 1.10. From an expiration standpoint, the breakeven at expiration
is $37.70 (the strike price plus the call premium). Since Kim plans on
exiting the position after about three weeks, the exact break-even point at
the expiration of the contract is irrelevant. But the concept is the same: the
stock needs to rise significantly. Exhibit 4.6 shows how Kim’sconcerns
translate into greeks.
EXHIBIT 4.6 Greeks for Disney 35 and 37.50 calls.
35 Call37.50 Call
Delta 0.57 0.185
Gamma0.1660.119
Theta −0.013−0.007
Vega 0.0480.032
Rho 0.0230.007
This table compares the ATM call with the OTM call. Kim can reduce her
theta to half that of the ATM call position by purchasing an OTM. This is
certainly afavorable difference. Her vega is lower with the 37.50 call, too.
This may or may not be afavorable difference. That depends on Kim’sopinion of IV.
On the surface, the disparity in delta appears to be ahighly unfavorable
trade-off. The delta of the 37.50 call is less than one third of the delta of the
35 call, and the whole motive for entering into this trade is to trade
direction! Although this strategy is very delta oriented, its core is more
focused on gamma and theta.

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The gamma of the 37.50 call is about 72 percent that of the 35 call. But
the theta of the 37.50 call is about half that of the 35 call. Kim is improving
her gamma/theta relationship by buying the OTM, but with the call being so
far out-of-the-money and so inexpensive, the theta needs to be taken with agrain of salt. It is ultimately gamma that will make or break this delta play.
The price of the option is 0.20—arather low premium. In order for the
call to gain in value, delta has to go to work with help from gamma. At this
point, the delta is small, only 0.185. If Kim’sforecast is correct and there is
abig move upward, gamma will cause the delta to increase, and therefore
also the premium to increase exponentially. The call’ssensitivity to gamma,
however, is dynamic.
Exhibit 4.7 shows how the gamma of the 37.50 call changes as the stock
price moves over time. At any point in time, gamma is highest when the call
is ATM. However, so is theta. Kim wants to reap as much benefit from
gamma as possible while minimizing her exposure to theta. Ideally, she
wants Disney to rally through the strike price—through the high gamma
and back to the low theta. After three weeks pass, with 23 days until
expiration, if Disney is at $37 ashare, the gamma almost doubles, to 0.237.
When the call is ATM, the delta increases at its fastest rate. As Disney rises
above the strike, the gamma figures in the table begin to decline.
EXHIBIT 4.7 Disney 37.50 call price–time matrix–gamma.

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The sooner the price rise occurs, the better. It means less time for theta to
eat away profits. If Kim must hold the position for the entire three weeks,
she needs agood pop in the stock to make it worth her while. At a $37 share
price, the call is worth about 0.50, assuming all other market influences
remain constant. That’sabout a 150 percent profit. At $38, Exhibit 4.9
reveals the call value to be 1.04. That’sa 420 percent profit.
On one hand, it’shard for atrader like Kim not to get excited about the
prospect of making 420 percent on an 8 percent move in astock. On the
other hand, Kim has to put things in perspective. When the position is
established, the call has a 0.185 delta. By the trader’sdefinition of delta,
that means the call is estimated to have about an 18.5 percent chance of
expiring in-the-money. More than four out of five times, this position will
be trading below the strike at expiration.
Although Kim is not likely to hold the position until expiration, this
observation tells her something: she’sstarting in the hole. She is more likely
to lose than to win. She needs to be compensated well for her risk on the
winners to make up for the more prevalent losers.
Buying OTM calls can be considered more speculative than buying ITM
or ATM calls. Unlike what the at-expiration diagrams would lead one to
believe, OTM calls are not simply about direction. There’sabit more to it.
They are really about gamma, time, and the magnitude of the stock’smove

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Long ITM Call
Kim also has the alternative to buy an ITM call. Instead of the 35 or 37.50
call, she can buy the 32.50. The 32.50 call shares some of the advantages
the 37.50 call has over the 35 call, but its overall greek characteristics make
it avery different trade from the two previous alternatives. Exhibit 4.10
shows acomparison of the greeks of the three different calls.
EXHIBIT 4.10 Greeks for Disney 32.50, 35, and 37.50 calls.
Like the 37.50 call, the 32.50 has alower gamma, theta, and vega than the
ATM 35-strike call. Because the call is ITM, it has ahigher delta: 0.862. In
this example, Kim can buy the 32.50 call for 3. That’s 0.40 over parity (3 −
[35.10 − 32.50] = 0.40). There is not much time value, but more than the
37.50 call has. Thus, theta is of some concern. Ultimately, the ITMs have
0.40 of time value to lose compared with the 0.20 of the OTM calls. Vega is
also of some concern, but not as much as in the other alternatives because
the vega of the 32.50 is lower than the 35s or the 37.50s. Gamma doesn’thelp much as the stock rallies—it will get smaller as the stock price rises.
Gamma will, however, slow losses somewhat if the stock declines by
decreasing delta at an increasing rate.
In this case, the greek of greatest consequence is delta—it is amore
purely directional play than the other alternatives discussed. Exhibit 4.11
shows the matrix of the delta of the 32.50 call.
EXHIBIT 4.11 Disney 32.50 call price–time matrix–delta.

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Long ATM Put
The beauty of the free market is that two people can study all the available
information on the same stock and come up with completely different
outlooks. First of all, this provides for entertaining television on the
business-news channels when the network juxtaposes an outspoken bullish
analyst with an equally unreserved bearish analyst. But differing opinions
also make for arobust marketplace. Differing opinions are the oil that
greases the machine that is price discovery. From amarket standpoint, it’swhat makes the world go round.
It is possible that there is another trader, Mick, in the market studying
Disney, who arrives at the conclusion that the stock is overpriced. Mick
believes the stock will decline in price over the next three weeks. He
decides to buy one Disney March 35 put at 0.80. In this example, March has
44 days to expiration.
Mick initiates this long put position to gain downside exposure, but along
with his bearish position comes option-specific risk and opportunity. Mick
is buying the same month and strike option as Kim did in the first example
of this chapter: the March 35 strike. Despite the different directional bias,
Mick’sposition and Kim’sposition share many similarities. Exhibit 4.13
offers acomparison of the greeks of the Disney March 35 call and the
Disney March 35 put.
EXHIBIT 4.13 Greeks for Disney 35 call and 35 put.
Call Put
Delta 0.57 −0.444
Gamma0.1660.174
Theta −0.013−0.009
Vega 0.0480.048
Rho 0.023−0.015
The first comparison to note is the contrasting deltas. The put delta is
negative, in contrast to the call delta. The absolute value of the put delta is
close to 1.00 minus the call delta. The put is just slightly OTM, so its delta

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is just under 0.50, while that of the call is just over 0.50. The disparate, yet
related deltas represent the main difference between these two trades.
The difference between the gamma of the 35 put and that of the
corresponding call is fairly negligible: 0.174 versus 0.166, respectively. The
gamma of this ATM put will enter into the equation in much the same way
as the gamma of the ATM call. The put’snegative delta will become more
negative as the stock declines, drawing closer to −1.00. It will get less
negative as the stock price rises, drawing closer to zero. Gamma is
important here, because it helps the delta. Delta, however, still remains the
most important greek. Exhibit 4.14 illustrates how the 35 put delta changes
as time and price change.
EXHIBIT 4.14 Disney 35 put price–time matrix–delta.
Since this put is ATM, it starts out with abig enough delta to offer the
directional exposure Mick desires. The delta can change, but gamma
ensures that it always changes in Mick’sfavor. Exhibit 4.15 shows how the
value of the 35 put changes with the stock price.
EXHIBIT 4.15 Disney 35 put price–time matrix–value.

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Over time, adecline of only 10 percent in the stock yields high
percentage returns. This is due to the leveraged directional nature of this
trade—delta.
While the other greeks are not of primary concern, they must be
monitored. At the onset, the 0.80 premium is all time value and, therefore
subject to the influences of time decay and volatility. This is where trading
greeks comes into play.
Conventional trading wisdom says, “Cut your losses early, and let your
profits run.” When trading astock, that advice is intellectually easy to
understand, although psychologically difficult to follow. Buyers of options,
especially ATM options, must follow this advice from the standpoint of
theta. Options are decaying assets. The time premium will be zero at
expiration. ATMs decay at an increasing nonlinear rate. Exiting along
position before getting too close to expiration can cut losses caused by an
increasing theta. When to cut those losses, however, will differ from trade
to trade, situation to situation, and person to person.
When buying options, accepting some loss of premium due to time decay
should be part of the trader’splan. It comes with the territory. In this
example, Mick is willing to accept about three weeks of erosion. Mick
needs to think about what his put will be worth, not just if the underlying
rises or falls but also if it doesn’tmove at all. At the time the position is

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established, the theta is 0.009, just under apenny. If Disney share price is
unchanged when three weeks pass, his theta will be higher. Exhibit 4.16
shows how thetas and theoretical values change over time if DIS stock
remains at $35.10.
EXHIBIT 4.16 Disney 35 put—thetas and theoretical values.
Mick needs to be concerned not only about what the theta is now but what
it will be when he plans on exiting the position. His plan is to exit the trade
in about three weeks, at which point the put theta will be −0.013. If he
amortizes his theta over this three-week period, he theoretically loses an
average of about 0.01 aday during this time if nothing else changes. The
average daily theta is calculated here by subtracting the value of the put at
23 days to expiration from its value when the trade was established to find
the loss of premium attributed to time decay, then dividing by the number
of days until expiration.
Since the theta doesn’tchange much over the first three weeks, Mick can
eyeball the theta rather easily. As expiration approaches and theta begins to
grow more quickly, he’ll need to do the math.
At nine days to expiration, the theoretical value of Mick’sput is about
0.35, assuming all other variables are held constant. By that time, he will
have lost 0.45 (0.80 − 0.35) due to erosion over the 35-day period he held
the position if the stock hasn’tmoved. Mick’saverage daily theta during
that period is about 0.0129 (0.45 ÷ 35). The more time he holds the trade,
the greater aconcern is theta. Mick must weigh his assessment of the
likelihood of the option’sgaining value from delta against the risk of
erosion. If he holds the trade for 35 days, he must make 0.0129 on average
per day from delta to offset theta losses. If the forecast is not realized within
the expected time frame or if the forecast changes, Mick needs to act fast to
curtail average daily theta losses.

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Finding the Right Risk
Mick could lower the theta of his position by selecting aput with agreater
number of days to expiration. This alternative has its own set of trade-offs:
lower gamma and higher vega than the 44-day put. He could also select an
ITM put or an OTM put. Like Kim’scall alternatives, the OTM put would
have less exposure to time decay, lower vega, lower gamma, and alower
delta. It would have alower premium, too. It would require abigger price
decline than the ATM put and would be more speculative.
The ITM put would also have lower theta, vega, and gamma, but it would
have ahigher delta. It would take on more of the functionality of ashort
stock position in much the same way that Kim’s ITM call alternative did for
along stock position. In its very essence, however, an option trade, ITM or
otherwise, is still fundamentally different than astock trade.
Stock has a 1.00 delta. The delta of astock never changes, so it has zero
gamma. Stock is not subject to time decay and has no volatility component
to its pricing. Even though ITM options have deltas that approach 1.00 and
other greeks that are relatively low, they have two important differences
from an equity. The first is that the greeks of options are dynamic. The
second is the built-in leverage feature of options.
The relationship of an option’sstrike price to the stock price can change
constantly. Options that are ITM now may be OTM tomorrow and vice
versa. Greeks that are not in play at the moment may be later. Even if there
is no time value in the option now because it is so far away-from-the-
money, there is the potential for time premium to become acomponent of
the option’sprice if the stock moves closer to the strike price. Gamma,
theta, and vega always have the potential to come into play.
Since options are leveraged by nature, small moves in the stock can
provide big profits or big losses. Options can also curtail big losses if used
for hedging. Long option positions can reap triple-digit percentage gains
quickly with afavorable move in the underlying. Even though 100 percent
of the premium can be lost just as easily, one option contract will have far
less nominal exposure than asimilar position in the stock.

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It’s All About Volatility
What are Kim and Mick really trading? Volatility. The motivation for
buying an option as opposed to buying or shorting the stock is volatility. To
some degree, these options have exposure to both flavors of volatility—
implied volatility and historical volatility (HV). The positions in each of the
examples have positive vega. Their values are influenced, in part, by IV.
Over time, IV begins to lose its significance if the option is no longer close
to being at-the-money.
The main objective of each of these trades is to profit from the volatility
of the stock’sprice movement, called future stock volatility or future
realized volatility. The strategies discussed in this chapter are contingent on
volatility being one directional. The bigger the move in the trader’sforecasted direction the better. Volatility in the form of an adverse
directional move results in adecline in premium. The gamma in these long
option positions makes volatility in the right direction more beneficial and
volatility in the wrong direction less costly.
This phenomenon is hardly unique to the long call and the long put.
Although some basic strategies, such as the ones studied in this chapter,
depend on aparticular direction, many don’t. Except for interest rate
strategies and perhaps some arbitrage strategies, all option trades are
volatility trades in one way or another. In general, option strategies can be
divided into two groups: volatility-buying strategies and volatility-selling
strategies. The following is abreakdown of common option strategies into
categories of volatility-buying strategies and volatility-selling strategies:
Volatility-Selling Strategies Volatility-Buying Strategies
Short Call, Short Put, Covered Call, Covered Put,
Bull Call Spread, Bear Call Spread, Bull Put
Spread, Bear Put Spread, Short Straddle, Short
Strangle, Guts, Ratio Call Spread, Calendar,
Butterfly, Iron Butterfly, Broken-Wing Butterfly,
Condor, Iron Condor, Diagonals, Double Diagonals,
Risk Reversals/Collars.
Long Call, Long Put, Bull Call Spread, Bear
Call Spread, Bull Put Spread, Bear Put Spread,
Long Straddle, Long Strangle, Guts, Back
Spread, Calendar, Butterfly, Iron Butterfly,
Broken-Wing Butterfly, Condor, Iron Condor,
Diagonals, Double Diagonals, Risk
Reversals/Collars.
Long option strategies appear in the volatility-buying group because they
have positive gamma and positive vega. Short option strategies appear in

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Direction Neutral, Direction Biased, and
Direction Indifferent
As typically traded, volatility-selling option strategies are direction neutral.
This means that the position has the greatest results if the underlying price
remains in arange—that is, neutral. Although some option-selling strategies
—for example, anaked put—may have apositive or negative delta in the
short term, profit potential is decidedly limited. This means that if traders
are expecting abig move, they are typically better off with option-buying
strategies.
Option-buying strategies can be either direction biased or direction
indifferent. Direction-biased strategies have been shown throughout this
chapter. They are delta trades. Direction-indifferent strategies are those that
benefit from increased volatility in the underlying but where the direction of
the move is irrelevant to the profitability of the trade. Movement in either
direction creates awinner.

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Are You a Buyer or a Seller?
The question is: which is better, selling volatility or buying volatility? Ihave attended option seminars with instructors (many of whom Iregard
with great respect) teaching that volatility-selling strategies, or income-
generating strategies, are superior to buying options. Ialso know option
gurus that tout the superiority of buying options. The answer to the question
of which is better is simple: it’sall amatter of personal preference.
When Ibegan trading on the floor of Chicago Board Options Exchange
(CBOE) in the 1990s, Iquickly became aware of adichotomy among my
market-making peers. Those making markets on the floor of the exchange at
that time were divided into two groups: teenie buyers and teenie sellers.
Teenie Buyers
Before options traded in decimals (dollars and cents) like they do today, the
lowest price increment in which an option could be traded was one
sixteenth of adollar—ateenie . Teenie buyers were market makers who
would buy back OTM options at one sixteenth to eliminate short positions.
They would sometimes even initiate long OTM option positions at ateenie,
too. The focus of the teenie-buyer school of thought was the fact that long
options have unlimited reward, while short options have unlimited risk. An
option purchased so far OTM that it was offered at one sixteenth is unlikely
to end up profitable, but it’san inexpensive lottery ticket. At worst, the
trader can only lose ateenie. Teenie buyers felt being short OTM options
that could be closed by paying asixteenth was an unreasonable risk.
Teenie Sellers
Teenie sellers, however, focused on the fact that options offered at one
sixteenth were far enough OTM that they were very likely to expire
worthless. This appears to be free money, unless the unexpected occurs, in
which case potential losses can be unlimited. Teenie sellers would routinely
save themselves $6.25 (one sixteenth of adollar per contract representing
100 shares) by selling their long OTMs at ateenie to close the position.
They sometimes would even initiate short OTM contracts at one sixteenth.

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These long-option or short-option biases hold for other types of strategies
as well. Volatility-selling positions, such as the iron condor, can be
constructed to have limited risk. The paradigm for these strategies is they
tend to produce winners more often than not. But when the position loses,
the trader loses more than he would stand to profit if the trade worked out
favorably.
Herein lies the issue of preference. Long-option traders would rather trade
Babe Ruth–style. For years, Babe Ruth was the record holder for the most
home runs. At the same time, he was also the record holder for the most
strikeouts. The born fighters that are option buyers accept the fact that they
will have more strikeouts, possibly many more strikeouts, than winning
trades. But the strategy dictates that the profit on one winner more than
makes up for the string of small losers.
Short-option traders, conversely, like to have everything cool and
copacetic. They like the warm and fuzzy feeling they get from the fact that
month after month they tend to generate winners. The occasional loser that
nullifies afew months of profits is all part of the game.

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Options and the Fair Game
There may be astatistical advantage to buying stock as opposed to shorting
stock, because the market has historically had apositive annualized return
over the long run. Astatistical advantage to being either an option buyer or
an option seller, however, should not exist in the long run, because the
option market prices IV. Assuming an overall efficient market for pricing
volatility into options, there should be no statistical advantage to
systematically buying or selling options. 1
Consider agame consisting of one six-sided die. Each time aone, two, or
three is rolled, the house pays the player $1. Each time afour, five, or six is
rolled, the house pays zero. What is the most aplayer would be willing to
pay to play this game? If the player paid nothing, the house would be at atremendous disadvantage, paying $1 50 percent of the time and nothing the
other 50 percent of the time. This would not be afair game from the house’sperspective, as it would collect no money. If the player paid $1, the player
would get his dollar back when one, two, or three came up. Otherwise, he
would lose his dollar. This is not afair game from the player’sperspective.
The chances of winning this game are 3 out of 6, or 50–50. If this game
were played thousands of times, one would expect to receive $1 half the
time and receive nothing the other half of the time. The average return per
roll one would expect to receive would be $0.50, that’s ($1 × 50 percent +
$0 × 50 percent). This becomes afair game with an entrance fee of $0.50.
Now imagine asimilar game in which asix-sided die is rolled. This time
if aone is rolled, the house pays $1. If any other number is rolled, the house
pays nothing. What is afair price to play this game? The same logic and the
same math apply. There is apercent chance of aone coming up and the
player receiving $1. And there is apercent chance of each of the other
five numbers being rolled and the player receiving nothing. Mathematically,
this translates to:
 percent
 percent). Fair value for achance to play this game is about $0.1667 per roll.
The fair game concept applies to option prices as well. The price of the
game, or in this case the price of the option, is determined by the market in

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CHAPTER 5
An Introduction to Volatility-Selling Strategies
Along with death and taxes, there is one other fact of life we can all count
on: the time value of all options ultimately going to zero. What an alluring
concept! In abusiness where expected profits can be thwarted by an
unexpected turn of events, this is one certainty traders can count on. Like all
certainties in the financial world, there is away to profit from this fact, but
it’snot as easy as it sounds. Alas, the potential for profit only exists when
there is risk of loss.
In order to profit from eroding option premiums, traders must implement
option-selling strategies, also known as volatility-selling strategies. These
strategies have their own set of inherent risks. Selling volatility means
having negative vega—the risk of implied volatility rising. It also means
having negative gamma—the risk of the underlying being too volatile. This
is the nature of selling volatility. The option-selling trader does not want the
underlying stock to move—that is, the trader wants the stock to be less
volatile. That is the risk.

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Greeks and Income Generation
With volatility-selling strategies (sometimes called income-generating
strategies), greeks are often overlooked. Traders simply dismiss greeks as
unimportant to this kind of trade. There is some logic behind this reasoning.
Time decay provides the profit opportunity. In order to let all of time
premium erode, the position must be held until expiration. Interim changes
in implied volatility are irrelevant if the position is held to term. The
gamma-theta loses some significance if the position is held until expiration,
too. The position has either passed the break-even point on the at-expiration
diagram, or it has not. Incremental daily time decay–related gains are not
the ultimate goal. The trader is looking for all the time premium, not
portions of it.
So why do greeks matter to volatility sellers? Greeks allow traders to be
flexible. Consider short-term-momentum stock traders. The traders buy astock because they believe it will rise over the next month. After one week,
if unexpected bearish news is announced causing the stock to break through
its support lines, the traders have adecision to make. Short-term speculative
traders very often choose to cut their losses and exit the position early rather
than risk alarger loss hoping for arecovery.
Volatility-selling option traders are often faced with the same dilemma. If
the underlying stays in line with the traders’ forecast, there is little to worry
about. But if the environment changes, the traders have to react. Knowing
the greeks for aposition can help traders make better decisions if they plan
to close the position before expiration.

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Naked Call
Anaked call is when atrader shorts acall without having stock or other
options to cover or protect it. Since the call is uncovered, it is one of the
riskier trades atrader can make. Recall the at-expiration diagram for the
naked call from Chapter 1, Exhibit 1.3 : Naked TGT Call. Theoretically,
there is limited reward and unlimited risk. Yet there are times when
experienced traders will justify making such atrade. When astock has been
trading in arange and is expected to continue doing so, traders may wait
until it is near the top of the channel, where there is resistance, and then
short acall.
For example, atrader, Brendan, has been studying achart of Johnson &
Johnson (JNJ). Brendan notices that for afew months the stock has trading
been in achannel between $60 and $65. As he observes Johnson & Johnson
beginning to approach the resistance level of $65 again, he considers selling
acall to speculate on the stock not rising above $65. Before selling the call,
Brendan consults other technical analysis tools, like ADX/DMI, to confirm
that there is no trend present. ADX/DMI is used by some traders as afilter
to determine the strength of atrend and whether the stock is overbought or
oversold. In this case, the indicator shows no strong trend present. Brendan
then performs due diligence. He studies the news. He looks for anything
specific that could cause the stock to rally. Is the stock atakeover target?
Brendan finds nothing. He then does earnings research to find out when
they will be announced, which is not for almost two more months.
Next, Brendan pulls up an option chain on his computer. He finds that
with the stock trading around $64 per share, the market for the November
65 call (expiring in four weeks) is 0.66 bid at 0.68 offer. Brendan considers
when Johnson & Johnson’searnings report falls. Although recent earnings
have seldom been amajor concern for Johnson & Johnson, he certainly
wants to sell an option expiring before the next earnings report. The
November fits the mold. Brendan sells ten of the November 65 calls at the
bid price of 0.66.
Brendan has arather straightforward goal. He hopes to see Johnson &
Johnson shares remain below $65 between now and expiration. If he is

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right, he stands to make $660. If he is wrong? Exhibit 5.1 shows how
Brendan’scalls hold up if they are held until expiration.
EXHIBIT 5.1 Naked Johnson & Johnson call at expiration.
Considering the risk/reward of this trade, Brendan is rightfully concerned
about abig upward move. If the stock begins to rally, he must be prepared
to act fast. Brendan must have an idea in advance of what his pain threshold
is. In other words, at what price will he buy back his calls and take aloss if
Johnson & Johnson moves adversely?
He decides he will buy all 10 of his calls back at 1.10 per contract if the
trade goes against him. (1.10 is an arbitrary price used for illustrative
purposes. The actual price will vary, based on the situation and the risk
tolerance of the trader. More on when to take profits and losses is discussed
in future chapters.) He may choose to enter agood-till-canceled (GTC)
stop-loss order to buy back his calls. Or he may choose to monitor the stock
and enter the order when he sees the calls offered at 1.10—amental stop
order. What Brendan needs to know is: How far can the stock price advance
before the calls are at 1.10?
Brendan needs to examine the greeks of this trade to help answer this
question. Exhibit 5.2 shows the hypothetical greeks for the position in this
example.
EXHIBIT 5.2 Greeks for short Johnson & Johnson 65 call (per contract).

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Delta −0.34
Gamma−0.15
Theta 0.02
Vega −0.07
The short call has anegative delta. It also has negative gamma and vega,
but it has positive time decay (theta). As Johnson & Johnson ticks higher,
the delta increases the nominal value of the call. Although this is not adirectional trade per se, delta is acrucial element. It will have abig impact
on Brendan’sexpectations as to how high the stock can rise before he must
take his loss.
First, Brendan considers how much the option price can move before he
covers. The market now is 0.66 bid at 0.68 offer. To buy back his calls at
1.10, they must be offered at 1.10. The difference between the offer now
and the offer price at which Brendan will cover is 0.42 (that’s 1.10 − 0.68).
Brendan can use delta to convert the change in the ask prices into astock
price change. To do so, Brendan divides the change in the option price by
the delta.
The −0.34 delta indicates that if JNJ rises $1.24, the calls should be
offered at 1.10.
Brendan takes note that the bid-ask spreads are typically 0.01 to 0.03
wide in near-term Johnson & Johnson options trading under 1.00. This is
not necessarily the case in other option classes. Less liquid names have
wider spreads. If the spreads were wider, Brendan would have more
slippage. Slippage is the difference between the assumed trade price and the
actual price of the fill as aproduct of the bid-ask spread. It’sthe difference
between theory and reality. If the bid-ask spread had atypical width of, say,
0.70, the market would be something more like 0.40 bid at 1.10 offer. In
this case, if the stock moved even afew cents higher, Brendan could not
buy his calls back at his targeted exit price of 1.10. The tighter markets
provide lower transaction costs in the form of lower slippage. Therefore,
there is more leeway if the stock moves adversely when there are tighter
bid-ask option spreads.

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But just looking at delta only tells apart of the story. In reality, the delta
does not remain constant during the price rise in Johnson & Johnson but
instead becomes more negative. Initially, the delta is −0.34 and the gamma
is −0.15. After arise in the stock price, the delta will be more negative by
the amount of the gamma. To account for the entire effect of direction,
Brendan needs to take both delta and gamma into account. He needs to
estimate the average delta based on gamma during the stock price move.
The formula for the change in stock price is
Taking into account the effect of gamma as well as delta, Johnson &
Johnson needs to rise only $1.01, in order for Brendan’scalls to be offered
at his stop-loss price of 1.10.
While having apredefined price point to cover in the event the underlying
rises is important, sometimes traders need to think on their feet. If material
news is announced that changes the fundamental outlook for the stock,
Brendan will have to adjust his plan. If the news leads Brendan to become
bullish on the stock, he should exit the trade at once, taking asmall loss
now instead of the bigger loss he would expect later. If the trader is
uncertain as to whether to hold or close the position, the Would I Do It
Now? rule is auseful rule of thumb.

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Would I Do It Now? Rule
To follow this rule, ask yourself, “If Idid not already have this position,
would Ido it now? Would Iestablish the position at the current market
prices, given the current market scenario?” If the answer is no, then the
solution is simple: Exit the trade.
For example, if after one week material news is released and Johnson &
Johnson is trading higher, at $64.50 per share, and the November 65 call is
trading at 0.75, Brendan must ask himself, based on the price of the stock
and all known information, “If Iwere not already short the calls, would Ishort them now at the current price of 0.75, with the stock trading at
$64.50?”
Brendan’sopinion of the stock is paramount in this decision. If, for
example, based on the news that was announced he is now bullish, he
would likely not want to sell the calls at 0.75—he only gets $0.09 more in
option premium and the stock is 0.50 closer to the strike. If, however, he is
not bullish, there is more to consider.
Theta can be of great use in decision making in this situation. As the
number of days until expiration decreases and the stock approaches $65
(making the option more at-the-money), Brendan’stheta grows more
positive. Exhibit 5.3 shows the theta of this trade as the underlying rises
over time.
EXHIBIT 5.3 Theta of Johnson & Johnson.
When the position is first established, positive theta comforts Brendan by
showing that with each passing day he gets alittle closer to his goal—to
have the 65 calls expire out-of-the-money (OTM) and reap aprofit of the
entire 66-cent premium. Theta becomes truly useful if the position begins to
move against him. As Johnson & Johnson rises, the trade gets more

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precarious. His negative delta increases. His negative gamma increases. His
goal becomes more out of reach. In conjunction with delta and gamma,
theta helps Brendan decide whether the risk is worth the reward.
In the new scenario, with the stock at $64.50, Brendan would collect $18
aday (1.80 × 10 contracts). Is the risk of loss in the short run worth earning
$18 aday? With Johnson & Johnson at $64.50, would Brendan now short
10 calls at 0.75 to collect $18 aday, knowing that each day may bring acontinued move higher in the stock? The answer to this question depends on
Brendan’sassessment of the risk of the underlying continuing its ascent. As
time passes, if the stock remains closer to the strike, the daily theta rises,
providing more reward. Brendan must consider that as theta—the reward—
rises, so does gamma: arisk factor.
Asmall but noteworthy risk is that implied volatility could rise. The
negative vega of this position would, then, adversely affect the profitability
of this trade. It will make Brendan’s 1.10 cover-point approach faster
because it makes the option more expensive. Vega is likely to be of less
consequence because it would ultimately take the stock’srising though the
strike price for the trade to be aloser at expiration.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:172


Short Naked Puts
Another trader, Stacie, has also been studying Johnson & Johnson. Stacie
believes Johnson & Johnson is on its way to test the $65 resistance level yet
again. She believes it may even break through $65 this time, based on
strong fundamentals. Stacie decides to sell naked puts. Anaked put is ashort put that is not sold in conjunction with stock or another option.
With the stock around $64, the market for the November 65 put is 1.75
bid at 1.80. Stacie likes the fact that the 65 puts are slightly in-the-money
(ITM) and thus have ahigher delta. If her price rise comes sooner than
expected, the high delta may allow her to take aprofit early. Stacie sells 10
puts at 1.75.
In the best-case scenario, Stacie retains the entire 1.75. For that to happen,
she will need to hold this position until expiration and the stock will have to
rise to be trading above the 65 strike. Logically, Stacie will want to do an
at-expiration analysis. Exhibit 5.4 shows Stacie’snaked put trade if she
holds it until expiration.
EXHIBIT 5.4 Naked Johnson & Johnson put at expiration.
While harvesting the entire premium as aprofit sounds attractive, if
Stacie can take the bulk of her profit early, she’ll be happy to close the
position and eliminate her risk—nobody ever went broke taking aprofit.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:173