atility bid while the calls are offered at 23 vol—atrader can speculate on the skew reverting back to its normal relationship by selling the puts, buying the calls, and hedging the delta by selling the right amount of stock.
This position—long acall, short aput with adifferent strike, and short stock on adelta-neutral ratio—is called arisk reversal. The motive for risk reversals is to capture vega as the skew realigns itself. But there are many risk factors that require careful attention.
First, as in other positions consisting of both long and short strikes, the gamma, theta, and vega of the position will vary from positive to negative depending on the price of the underlying. Risk-reversal traders must be prepared to trade long gamma (and battle time decay) when the stock rallies closer to the long-call strike and trade short gamma (and assume the risk of possible increased realized volatility) when the stock moves closer to the short-put strike.
As for vega, being short implied volatility on the downside and long on the upside is inherently apotentially bad position whichever way the stock moves. Why? As equities decline in price, the implied volatility of their options tends to rise. But the downside is where the risk reversal has its short vega. Furthermore, as equities rally, their IV tends to fall. That means the long vega of the upside hurts as well.
When Delta Neutral Isn’t Direction Indifferent
Many dynamic-volatility option positions, such as the risk reversal, have vega risk from potential IV changes resulting from the stock’smoving. This is indirectly adirectional risk. While having adelta-neutral position hedges against the rather straightforward directional risk of the position delta, this hidden risk of stock movement is left unhedged. In some circumstances, adelta-lean can help abate some of the vega risk of stock-price movement.
Say an option position has fairly flat greeks at the current stock price. Say that given the way this particular position is set up, if the stock rises, the position is still fairly flat, but if the stock falls, short lower-strike options will lead to negative gamma and vega. One way to partially hedge this position is to lean short deltas—that is, instead of maintaining atotally flat delta, have aslightly short delta. That way, if the stock falls, the trade profits some on the short stock to partially offset some of the anticipated vega losses. The trade-off of this hedge is that if the stock rises, the trade loses on the short delta.
Delta leans are more of an art than ascience and should be used as ahedge only by experienced vol traders. They should be one part of awell-orchestrated plan to trade the delta, gamma, theta, and vega of aposition. And, to be sure, adelta lean should be entered into amodel for simulation purposes before executing the trade to study the up-and-down risk of the position. If the lean reduces the overall risk of the position, it should be implemented. But if it creates asituation where there is an anticipated loss if the stock moves in either direction and there is little hope of profiting from the other greeks, the lean is not the answer—closing the position is.
Managing Multiple-Class Risk
Most traders hold option positions in more than one option class. As an aside, Irecommend doing so, capital and experience permitting. In my experience, having positions in multiple classes psychologically allows for acertain level of detachment from each individual position. Most traders can make better decisions if they don’thave all their eggs in one basket.
But holding aportfolio of option positions requires one more layer of risk management. The trader is concerned about the delta, gamma, theta, vega, and rho not only of each individual option class but also of the portfolio as awhole. The trader’sportfolio is actually one big position with alot of moving parts. To keep it running like awell-oiled machine requires monitoring and maintaining each part to make sure they are working together. To have the individual trades work in harmony with one another, it is important to keep awell-balanced series of strategies.
Option trading requires diversification, just like conventional linear stock trading or investing. Diversification of the option portfolio is easily measured by studying the portfolio greeks. By looking at the net greeks of the portfolio, the trader can get some idea of exposure to overall risk in terms of delta, gamma, theta, vega, and rho.


CHAPTER 17
Putting the Greeks into Action
This book was intended to arm the reader with the knowledge of the greeks needed to make better trading decisions. As the preface stated, this book is not so much ahow-to guide as ahow-come tutorial. It is step one in athree-step learning process:
Step One: Study
. First, aspiring option traders must learn as much as possible from books such as this one and from other sources, such as articles, both in print and online, and from classes both in person and online. After completing this book, the reader should have asolid base of knowledge of the greeks.
Step Two: Paper Trade
. Atruly deep understanding requires practice, practice, and more practice! Fortunately, much of this practice can be done without having real money on the line. Paper trading—or simulated trading—in which one trades real markets but with fake money is step two in the learning process. Ihighly recommend paper trading to kick the tires on various types of strategies and to see how they might work differently in reality than you thought they would in theory.
Step Three: Showtime
! Even the most comprehensive academic study or windfall success with paper profits doesn’tgive one atrue feel for how options work in the real world. There are some lessons that must be learned from the black and the blue. When there’sreal money on the line, you will trade differently—at least in the beginning. It’shuman nature to be cautious with wealth. This is not abad thing. But emotions should not override sound judgment. Start small—one or two lots per trade—until you can make rational decisions based on what you have learned, keeping emotions in check.
This simple three-step process can take years of diligent work to get it right. But relax. Getting rich quick is truly apoor motivation for trading options. Option trading is abeautiful thing! It’sabout winning. It’sabout beating the market. It’sabout being smart. Don’tget me wrong—wealth can be anice by-product. I’ve seen many people who have made alot of money trading options, but it takes hard work. For every successful option trader I’ve met, I’ve met many more who weren’twilling to put in the effort, who brashly thought this is easy, and failed miserably.
Trading Option Greeks
Traders must take into account all their collective knowledge and experience with each and every trade. Now that you’re armed with knowledge of the greeks, use it! The greeks come in handy in many ways.
Choosing between Strategies
Avery important use of the greeks is found in selecting the best strategy for agiven situation. Consider asimple bullish thesis on astock. There are plenty of bullish option strategies. But given abullish forecast, which option strategy should atrader choose? The answer is specific to each unique opportunity. Trading is situational.
Example 1
Imagine atrader, Arlo, is studying the following chart of Agilent Technologies Inc. (A). See
Exhibit 17.1
.
EXHIBIT 17.1
Agilent Technologies Inc. daily candles.
Source
: Chart courtesy of Livevol
®
Pro (
www.livevol.com
)
The stock has been in an uptrend for six weeks or so. Close-to-close volatility hasn’tincreased much. But intraday volatility has increased greatly as indicated by the larger candles over the past 10 or so trading sessions. Earnings is coming up in aweek in this example, however implied volatility has not risen much. It is still “cheap” relative to historical volatility and past implied volatility. Arlo is bullish. But how does he play it? He needs to use what he knows about the greeks to guide his decision.
Arlo doesn’twant to hold the trade through earnings, so it will be ashort-term trade. Thus, theta is not much of aconcern. The low-priced volatility guides his strategy selection in terms of vega. Arlo certainly wouldn’twant ashort-vega trade. Not with the prospect of implied volatility potential rising going into earnings. In fact, he’dactually want abig positive vega position. That rules out anaked/cash-secured put, put credit spread and the likes.
He can probably rule out vertical spreads all together. He doesn’tneed to spread off theta. He doesn’twant to spread off vega. Positive gamma is attractive for this sort of trade. He wouldn’twant to spread that off either. Plus, the inherent time component of spreads won’twork well here. As discussed in Chapter 9, the bulk of vertical spreads profits (or losses) take time to come to fruition. The deltas of acall spread are smaller than an outright call. Profits would come from both delta and theta, if the stock rises to the short strike and positive theta kicks in.
The best way for Arlo to play this opportunity is by buying acall. It gives him all the greeks attributes he wants (comparatively big positive delta, gamma and vega) and the detriment (negative theta) is not amajor issue.
He’dthen select among in-the-money (ITM), at-the-money (ATM), and out-of-the-money (OTM) calls and the various available expiration cycles. In this case, because positive gamma

eBooks\Trading Options Greeks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).epub#section:text00029.html


ibutes he wants (comparatively big positive delta, gamma and vega) and the detriment (negative theta) is not amajor issue.
He’dthen select among in-the-money (ITM), at-the-money (ATM), and out-of-the-money (OTM) calls and the various available expiration cycles. In this case, because positive gamma is attractive and theta is not an issue, he’dlean toward afront month (in this case, three week) option. The front month also benefits him in terms of vega. Though the vegas are smaller for short-term options, if there is arise in implied volatility leading up to earnings, the front month will likely rise much more than the rest. Thus, the trader has apossibility for profits from vega.
Example 2
Atrader, Luke, is studying the following chart for United States Steel Corp. (X). See
Exhibit 17.2
.
EXHIBIT 17.2
United States Steel Corp. daily candles.
Source
: Chart courtesy of Livevol
®
Pro (
www.livevol.com
)
This stock is in asteady uptrend, which Luke thinks will continue. Earnings are out and there are no other expected volatility events on the horizon. Luke thinks that over the next few weeks, United States Steel can go from its current price of around $31 ashare to about $34. Volatility is midpriced in this example—not cheap, not expensive.
This scenario is different than the previous one. Luke plans to potentially hold this trade for afew weeks. So, for Luke, theta is an important concern. He cares somewhat about volatility, too. He doesn’tnecessarily want to be long it in case it falls; he doesn’twant to be short it in case it rises. He’dlike to spread it off; the lower the vega, the better (positive or negative). Luke really just wants delta play that he can hold for afew weeks without all the other greeks getting in the way.
For this trade, Luke would likely want to trade adebit call spread with the long call somewhat ITM and the short call at the $34 strike. This way, Luke can start off with nearly no theta or vega. He’ll retain some delta, which will enable the spread to profit if United States Steel rises and as it approaches the 34 strike, positive theta will kick in.
This spread is superior to apure long call because of its optimized greeks. It’ssuperior to an OTM bull put spread in its vega position and will likely produce ahigher profit with the strikes structured as such too, as it would have abigger delta.
Integrating greeks into the process of selecting an option strategy must come natural to atrader. For any given scenario, there is one position that best exploits the opportunity. In any option position, traders need to find the optimal greeks position.
Managing Trades
Once the trade is on, the greeks come in handy for trade management. The most important rule of trading is
Know Thy Risk
. Knowing your risk means knowing the influences that expose your position to profit or peril in both absolute and incremental terms. At-expiration diagrams reveal, in no uncertain terms, what the bottom-line risk points are when the option expires. These tools are especially helpful with simple short-option strategies and some long-option strategies. Then traders need the greeks. After all, that’swhat greeks are: measurements of option risk. The greeks give insight into atrade’sexposure to the other pricing factors. Traders must know the greeks of every trade they make. And they must always know the net-portfolio greeks at all times. These pricing factors ultimately determine the success or failure of each trade, each portfolio, and eventually each trader.
Furthermore, always—and Ido mean always—traders must know their up and down risk, that is, the directional risk of the market moving up or down certain benchmark intervals. By definition, moves of three standard deviations or more are very infrequent. But they happen. In this business anything can happen. Take the “flash crash of 2010 in which the Dow Jones Industrial Average plunged more than 1,000 points in “aflash.” In my trading career, I’ve seen some surprises. Traders have to plan for the worst.
It’snot too hard to tell your significant other, “Sorry I’mlate, but Ihit unexpected traffic. Ijust couldn’tplan for it.” But to say, “Sorry, Ilost our life savings, and the kids’ college fund, and our house because the market made an unexpected move. Icouldn’tplan for it,” won’tgo over so well. The fact is, you
can
plan for it. And as an option trader, you have to. The bottom line is, expect the unexpected because the unexpected will sometimes happen. Traders must use the greeks and up and down risk, instead of relying on other common indicators, such as the HAPI.
The HAPI: The Hope and Pray Index
So you bought acall spread. At the opening bell the next morning, you find that the market for the underlying has moved lower—alot lower. You have aloss on your hands. What do you do? Keep apositive attitude? Wear your lucky shirt? Pray to the options gods? When traders finds themselves hoping and praying—Iswear I’ll never do that again if Ican just get out of this po


find that the market for the underlying has moved lower—alot lower. You have aloss on your hands. What do you do? Keep apositive attitude? Wear your lucky shirt? Pray to the options gods? When traders finds themselves hoping and praying—Iswear I’ll never do that again if Ican just get out of this position!—it is probably time for them to take their losses and move on to the next trade. The Hope and Pray Index is acontraindicator. Typically, the higher it is, the worse the trade.
There are two numbers atrader can control: the entry price and the exit price. All of the other flashing green and red numbers on the screen are out of the trader’scontrol. Savvy traders observe what the market does and make decisions on whether and when to enter aposition and when to exit. Traders who think about their positions in terms of probability make better decisions at both of these critical moments.
In entering atrade, traders must consider their forecast, their assessment of the statistical likelihood of success, the potential payout and loss, and their own tolerance for risk. Having considered these criteria helps the traders stay the course and avoid knee-jerk reactions when the market moves in the wrong direction. Trading is easy when positions make money. It is how traders deal with adverse positions that separates good traders from bad.
Good traders are good at losing money. They take losses quickly and let profits run. Accepting, before entering the trade, the statistical nature of trading can help traders trade their positions with less emotion. It then becomes amatter of competent management of those positions based on their knowledge of the factors affecting option values: the greeks. Learning to think in terms of probability is among the most difficult challenges for anew options trader.
Chapter 5 discussed my Would I Do It Now? Rule, in which atrader asks himself: if Ididn’tcurrently have this position, would Iput it on now at current market prices? This rule is ahandy technique to help traders filter out the noise in their heads that clouds judgment and to help them to make rational decisions on whether to hold aposition, close it out or adjust it.
Adjusting
Sometimes the position atrader starts off with is not the position he or she should have at present. Sometimes positions need to be changed, or adjusted, to reflect current market conditions. Adjusting is very important to option traders. To be good at adjusting, traders need to use the greeks.
Imagine atrader makes the following trade in Halliburton Company (HAL) when the stock is trading $36.85.
Sell 10 February 35–36–38–39 iron condors at 0.45
February has 10 days until expiration in this example. The greeks for this trade are as follows:
Delta: −6.80
Gamma: −119.20
Theta: +21.90
Vega: −12.82
The trader has aneutral outlook, which can be inferred by the near-flat delta. But what if the underlying stock begins to rise? Gamma starts kicking in. The trader can end up with ashort-biased delta that loses exponentially if the stock continues to climb. If Halliburton rises (or falls for that matter) the trader needs to recalibrate his outlook. Surely, if the trader becomes bullish based on recent market activity, he’dwant to close the trade. If the trader is bearish, he’dprobably let the negative delta go in hopes of making back what was lost from negative gamma. But what if the trader is still neutral?
Aneutral trader needs aposition that has greeks which reflect that outlook. The trader would want to get delta back towards zero. Further, depending on how much the stock rises, theta could start to lose its benefit. If Halliburton approaches one of the long strikes, theta could move toward zero, negating the benefit of this sort of trade all together. If after the stock rises, the trader is still neutral at the new underlying price level, he’dlikely adjust to get delta and theta back to desired territory.
Acommon adjustment in this scenario is to roll the call-credit-spread legs of the iron condor up to higher strikes. The trader would buy ten 38 calls and sell ten 39 calls to close the credit spread. Then the trader would buy 10 of the 39 calls as sell 10 of the 40 calls to establish an adjusted position that is short a 10 lot of the February 35–36–39–40 iron condor.
This, of course, is just one possible adjustment atrader can make. But the common theme among all adjustments is that the trader’sgreeks must reflect the trader’soutlook. The position greeks best describe what the position is—that is, how it profits or loses. When the market changes it affects the dynamic greeks of aposition. If the market changes enough to make atrader’sposition greeks no longer represent his outlook, the trader must adjust the position (adjust the greeks) to put it back in line with expectations.
In option trading there are an infinite number of uses for the greeks. From finding trades, to planning execution, to managing and adjusting them, to planning exits; the greeks are tru


sent his outlook, the trader must adjust the position (adjust the greeks) to put it back in line with expectations.
In option trading there are an infinite number of uses for the greeks. From finding trades, to planning execution, to managing and adjusting them, to planning exits; the greeks are truly atrader’sbest resource. They help traders see potential and actual position risk. They help traders project potential and actual trade profitability too. Without the greeks, atrader is at adisadvantage in every aspect of option trading. Use the greeks on each and every trade, and exploit trades to their greatest potential.
Iwish you good luck
!
For me, trading option greeks has been alabor of love through the good trades and the bad. To succeed in the long run at greeks trading—or any endeavor, for that matter—requires enjoying the process. Trading option greeks can be both challenging and rewarding. And remember, although option trading is highly statistical and intellectual in nature, alittle luck never hurt! That said, good luck trading!


ze profit
Standard deviation
and historical volatility
Standard & Poor’s Depositary Receipts (SPDRs or Spiders)
Straddles
long
basic
trading
short
risks with
trading
synthetic
Strangles
long
example
short
premium
risk, limiting
Strategies and At-Expiration Diagrams
buy call
buy put
factors affecting option prices, measuring incremental changes in
sell call
sell put
Strike price
Supply and demand
Synthetic stock
strategies
conversion
market makers
pin risk
reversal
Technical analysis
Teenie buyers
Teenie sellers
Theta
effect of moneyness and stock price on
effects of volatility and time on
positive or negative
risk
taking the day out
Time value
Trading strategies
Value
Vega
effect of implied volatility on
effect of moneyness on
effect of time on
implied volatility (IV) and
Vertical spreads
bear call
bear put
box, building
bull call
bull put
credit and debit
interrelations of
similarities in
and volatility
Volatility
buying and selling
teenie buyers
teenie sellers
calculating data
direction neutral, direction biased, and direction indifferent
expected
CBOE Volatility Index®
implied
stock
historical (HV)
standard deviation
implied (IV)
and direction
HV-IV divergence
inertia
relationship of HV and IV
selling
supply and demand
realized
trading
skew
term structure
vertical
vertical spreads and
Volatility charts, studying
patterns
implied and realized volatility rise
realized volatility falls, implied volatility falls
realized volatility falls, implied volatility remains constant
realized volatility falls, implied volatility rises
realized volatility remains constant, implied volatility falls
realized volatility remains constant, implied volatility remains constant
realized volatility remains constant, implied volatility rises
realized volatility rises, implied volatility falls
realized volatility rises, implied volatility remains constant
Volatility-selling strategies
profit potential
covered call
covered put
gamma-theta relationship
greeks and income generation
naked call
short naked puts
similarities
Would I Do It Now? Rule
Volume
Weeklys
SM
Wing spreads
butterflies
directional
long
short
iron
condors
iron
long
short
greeks and
keys to success
retail trader vs. pro
trades, constructing to maximize profit
Would I Do It Now? Rule

eBooks\Trading Options Greeks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).epub#section:text00031.html


Foreword
The past several years have brought about aresurgence in market volatility
and options volume unlike anything that has been seen since the close of the
twentieth century. As markets have become more interdependent,
interrelated, and international, the U.S. listed option markets have solidified
their place as the most liquid and transparent venue for risk management
and hedging activities of the world’slargest economy. Technology,
competition, innovation, and reliability have become the hallmarks of the
industry, and our customer base has benefited tremendously from this
ongoing evolution.
However, these advances can be properly tapped only when the users of
the product continue to expand their knowledge of the options product and
its unique features. Education has always been the driver of growth in our
business, and it will be the steward of the next generation of options traders.
Dan Passarelli’snew and updated book Trading Option Greeks is anecessity for customers and traders alike to ensure that they possess the
knowledge to succeed and attain their objectives in the high-speed, highly
technical arena that the options market has become.
The retail trader of the past has given way to anew retail trader of the
present—one with an increased level of technology, support, capital
treatment, and product selection. The impact of the staggering growth in
such products as the CBOE Holdings’ VIX options and futures, and the
literally dozens of other products tied to it, have made the volatility asset
class anew, unique, and permanent pillar of today’soption markets.
Dan’supdated book continues his mission of supporting, preparing, and
reinforcing the trader’sunderstanding of pricing, volatility, market
terminology, and strategy, in away that few other books have been able.
Using aperspective forged from years as an options market maker,
professional trader, and customer, Dan has once again provided aresource
for those who wish to know best how the option markets behave today, and
how they are likely to continue to behave in the future. It is important to
understand not only what happens in the options space, but also why it

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Preface
I’ve always been fascinated by trading. When Iwas young, I’dsee traders
on television, in their brightly colored jackets, shouting on the seemingly
chaotic trading floor, and I’dmarvel at them. What awonderful job that
must be! These traders seemed to me to be very different from the rest of
us. It’sall so very esoteric.
It is easy to assume that professional traders have closely kept secrets to
their ways of trading—something that secures success in trading for them,
but is out of reach for everyone else. In fact, nothing could be further from
the truth. If there are any “secrets” of professional traders, this book will
expose them.
True enough, in years past there have been some barriers to entry to
trading success that did indeed make it difficult for nonprofessionals to
succeed. For example, commissions, bid-ask spreads, margin requirements,
and information flow all favored the professional trader. Now, these barriers
are gone. Competition among brokers and exchanges—as well as the
ubiquity of information as propagated on the Internet—has torn down those
walls. The only barrier left between the Average Joe and the options pro is
that of knowledge. Those who have it will succeed; those who do not will
fail.
To be sure, the knowledge held by successful traders is not that of what
will happen in the future; it is the knowledge of how to manage the
uncertainty. No matter what our instincts tell us, we do not know what will
happen in the future with regard to the market. As Socrates put it, “The only
true wisdom is in knowing you know nothing.” The masters of option
trading are masters of managing the risk associated with what they don’tknow—the risk of uncertainty.
As an instructor, I’ve talked to many traders who were new to options
who told me, “Imade atrade based on what Ithought was going to happen.
Iwas right, but my position lost money!” Choosing the right strategy makes
all the difference when it comes to mastery of risk management and

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ultimate trading success. Knowing which option strategy is the right
strategy for agiven situation comes with knowledge and experience.
All option strategies are differentiated by their unique risk characteristics.
Some are more sensitive to directional movement of the underlying asset
than others; some are more affected by time passing than others. The exact
exposure positions have to these market influences determines the success
of individual trades and, indeed, the long-term success of the trader who
knows how to exploit these risk characteristics. These option-value
sensitivities can be controlled when atrader understands the option greeks.
Option greeks are metrics used to measure an option’ssensitivity to
influences on its price. This book will provide the reader with an
understanding of these metrics, to help the reader truly master the risk of
uncertainty associated with option trading.
Successful traders strive to create option positions with risk-reward
profiles that benefit them the most in agiven situation. Atrader’sobjectives
will dictate the right strategy for the right situation. Traders can tailor aposition to fit aspecific forecast with respect to the time horizon; the degree
of bullishness, bearishness, neutrality, or volatility in the underlying stock;
and the desired amount of leverage. Furthermore, they can exploit
opportunities unique to options. They can trade option greeks. This opens
the door to many new opportunities.

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Trading Strategies
Buying stock is atrading strategy that most people understand. In practical
terms, traders who buy stock are generally not concerned with the literal
ownership stake in acorporation, just the opportunity to profit if the stock
rises. Although it’simportant for traders to understand that the price of astock is largely tied to the success or failure of the corporation, it’sessential
to keep in mind exactly what the objective tends to be for trading astock: to
profit from changes in its price. Abullish position can also be taken in the
options market. The most basic example is buying acall.
Abearish position can be taken by trading stock or options, as well. If
traders expect the value of astock they own to fall, they will sell the stock.
This eliminates the risk of losses from the stock’sfalling. If the traders do
not own the stock that they think will decline, they can take amore active
stance and short it. The short-seller borrows the stock from aparty that
owns it and then sells the borrowed shares to another party. The goal of
selling stock short is to later repurchase the shares at alower price before
returning the stock to its owner. It is simply reversing the order of “buy
low/sell high.” The risk is that the stock rises and shares have to be bought
at ahigher price than that at which they were sold. Although shorting stock
can lead to profits when the market cooperates, in the options market, there
are alternative ways to profit from falling prices. The most basic example is
buying aput.
Atrader can use options to take abullish or bearish position, given adirectional forecast. Sideways, nontrending stocks and their antithesis,
volatile stocks, can be traded as well. In the later market conditions, profit
or loss can be independent of whether the stock rises or falls. Opportunity
in option trading is not necessarily black and white—not necessarily up and
down. Option trading is nonlinear. Consequently, more opportunities can be
exploited by trading options than by trading stock.
Option traders must consider the time period in question, the volatility
expected during this period, interest rates, and dividends. Along with the
stock price, these factors make up the dynamic components of an option’svalue. These individual factors can be isolated, measured, and exploited.

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This Second Edition of Trading
Option Greeks
This book addresses the complex price behavior of options by discussing
option greeks from both atheoretical and apractical standpoint. There is
some tactical discussion throughout, although the objective of this book is
to provide education to the reader. This book is meant to be less ahow-to
manual than ahow-come tutorial.
This informative guide will give the retail trader alook inside the mind of
aprofessional trader. It will help the professional trader better understand
the essential concepts of his craft. Even the novice trader will be able to
apply these concepts to basic options strategies. Comprehensive knowledge
of the greeks can help traders to avoid common pitfalls and increase profit
potential.
Much of this book is broken down into adiscussion of individual
strategies. Although the nuances of each specific strategy are not relevant,
presenting the material this way allows for adiscussion of very specific
situations in which greeks come into play. Many of the concepts discussed
in asection on one option strategy can be applied to other option strategies.
As in the first edition of Trading Option Greeks , Chapter 1 discusses
basic option concepts and definitions. It was written to be areview of the
basics for the intermediate to advanced trader. For newcomers, it’sessential
to understand these concepts before moving forward.
Adetailed explanation of option greeks begins in Chapter 2. Be sure to
leave abookmark in this chapter, as you will flip to it several times while
reading the rest of the book and while studying the market thereafter.
Chapter 3 introduces volatility. The same bookmark advice can be applied
here, as well. Chapters 4 and 5 explore the minds of option traders. What
are the risks they look out for? What are the opportunities they seek? These
chapters also discuss direction-neutral and direction-indifferent trading. The
remaining chapters take the reader from concept to application, discussing

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Contractual Rights and Obligations
The option buyer is the party who owns the right inherent in the contract.
The buyer is referred to as having along position and may also be called the
holder, or owner, of the option. The right doesn’tlast forever. At some point
the option will expire. At expiration, the owner may exercise the right or, if
the option has no value to the holder, let it expire without exercising it. But
he need not hold the option until expiration. Options are transferable—they
can be traded intraday in much the same way as stock is traded. Because it’suncertain what the underlying stock price of the option will be at expiration,
much of the time this right has value before it expires. The uncertainty of
stock prices, after all, is the raison d’être of the option market.
Along position in an option contract, however, is fundamentally different
from along position in astock. Owning corporate stock affords the
shareholder ownership rights, which may include the right to vote in
corporate affairs and the right to receive dividends. Owning an option
represents strictly the right either to buy the stock or to sell it, depending on
whether it’sacall or aput. Option holders do not receive dividends that
would be paid to the shareholders of the underlying stock, nor do they have
voting rights. The corporation has no knowledge of the parties to the option
contract. The contract is created by the buyer and seller of the option and
made available by being listed on an exchange.
The party to the contract who is referred to as the option seller, also called
the option writer, has ashort position in the option. Instead of having aright
to take aposition in the underlying stock, as the buyer does, the seller
incurs an obligation to potentially either buy or sell the stock. When atrader
who is long an option exercises, atrader with ashort position gets assigned
. Assignment means the trader with the short option position is called on to
fulfill the obligation that was established when the contract was sold.
Shorting an option is fundamentally different from shorting astock.
Corporations have aquantifiable number of outstanding shares available for
trading, which must be borrowed to create ashort position, but establishing
ashort position in an option does not require borrowing; the contract is
simply created. The strategy of shorting stock is implemented statistically

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Open Interest and Volume
Traders use many types of market data to make trading decisions. Two
items that are often studied but sometimes misunderstood are volume and
open interest. Volume, as the name implies, is the total number of contracts
traded during atime period. Often, volume is stated on aone-day basis, but
could be stated per week, month, year, or otherwise. Once anew period
(day) begins, volume begins again at zero. Open interest is the number of
contracts that have been created and remain outstanding. Open interest is arunning total.
When an option is first listed, there are no open contracts. If Trader Aopens along position in anewly listed option by buying aone-lot, or one
contract, from Trader B, who by selling is also opening aposition, acontract is created. One contract traded, so the volume is one. Since both
parties opened aposition and one contract was created, the open interest in
this particular option is one contract as well. If, later that day, Trader Bcloses his short position by buying one contract from Trader C, who had no
position to start with, the volume is now two contracts for that day, but open
interest is still one. Only one contract exists; it was traded twice. If the next
day, Trader Cbuys her contract back from Trader A, that day’svolume is
one and the open interest is now zero.

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The Options Clearing Corporation
Remember when Wimpy would tell Popeye, “I’ll gladly pay you Tuesday
for ahamburger today.” Did Popeye ever get paid for those burgers? In acontract, it’svery important for each party to hold up his end of the bargain
—especially when there is money at stake. How does atrader know the
party on the other side of an option contract will in fact do that? That’swhere the Options Clearing Corporation (OCC) comes into play.
The OCC ultimately guarantees every options trade. In 2010, that was
almost 3.9 billion listed-options contracts. The OCC accomplishes this
through many clearing members. Here’show it works: When Trader Xbuys
an option through abroker, the broker submits the trade information to its
clearing firm. The trader on the other side of this transaction, Trader Y, who
is probably amarket maker, submits the trade to his clearing firm. The two
clearing firms (one representing Trader X’sbuy, the other representing
Trader Y’ssell) each submit the trade information to the OCC, which
“matches up” the trade.
If Trader Ybuys back the option to close the position, how does that
affect Trader Xif he wants to exercise it? It doesn’t. The OCC, acting as an
intermediary, assigns one of its clearing members with acustomer that is
short the option in question to deliver the stock to Trader X’sclearing firm,
which in turn delivers the stock to Trader X. The clearing member then
assigns one of its customers who is short the option. The clearing member
will assign the trader either randomly or first in, first out. Effectively, the
OCC is the ultimate counterparty to both the exercise and the assignment.

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Standardized Contracts
Exchange-listed options contracts are standardized, meaning the terms of
the contract, or the contract specifications, conform to acustomary
structure. Standardization makes the terms of the contracts intuitive to the
experienced user.
To understand the contract specifications in atypical equity option,
consider an example:
Buy 1 IBM December 170 call at 5.00
Quantity
In this example, one contract is being purchased. More could have been
purchased, but not less—options cannot be traded in fractional units.
Option Series, Option Class, and Contract Size
All calls or puts of the same class, the same expiration month, and the same
strike price are called an option series . For example, the IBM December
170 calls are aseries. Options series are displayed in an option chain on an
online broker’suser interface. An option chain is afull or partial list of the
options that are listed on an underlying.
Option class means agroup of options that represent the same underlying.
Here, the option class is denoted by the symbol IBM—the contract
represents rights on International Business Machines Corp. (IBM) shares.
Buying one contract usually gives the holder the right to buy or to sell 100
shares of the underlying stock. This number is referred to as contract size .
Though this is usually the case, there are times when the contract size is
something other than 100 shares of astock. This situation may occur after
certain types of stock splits, spin-offs, or stock dividends, for example. In
the minority of cases in which the one contract represents rights on
something besides 100 shares, there may be more than one class of options
listed on astock.
Afairly unusual example was presented by the Ford Motor Company
options in the summer of 2000. In June 2000, Ford spun off Visteon

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Corporation. Then, in August 2000, Ford offered shareholders achoice of
converting their shares into (a) new shares of Ford plus $20 cash per share,
(b) new Ford stock plus fractional shares with an aggregate value of $20, or
(c) new Ford stock plus acombination of more new Ford stock and cash.
There were three classes of options listed on Ford after both of these
changes: Frepresented 100 shares of the new Ford stock; XFO represented
100 shares of Ford plus $20 per share ($2,000) plus cash in lieu of $1.24;
and FOD represented 100 shares of new Ford, 13 shares of Visteon, and
$2,001.24.
Sometimes these changes can get complicated. If there is ever aquestion
as to what the underlying is for an option class, the authority is the OCC. Alot of time, money, and stress can be saved by calling the OCC at 888-
OPTIONS and clarifying the matter.
Expiration Month
Options expire on the Saturday following the third Friday of the stated
month, which in this case is December. The final trading day for an option
is commonly the day before expiration—here, the third Friday of
December. There are usually at least four months listed for trading on an
option class. There may be atotal of six months if Long-Term Equity
AnticiPation Securities® or LEAPS® are listed on the class. LEAPS can have
one year to about two-and-a-half years until expiration. Some underlyings
have one-week options called WeeklysSM listed on them.
Strike Price
The price at which the option holder owns the right to buy or to sell the
underlying is called the strike price, or exercise price. In this example, the
holder owns the right to buy the stock at $170 per share. There is method to
the madness regarding how strike prices are listed. Strike prices are
generally listed in $1, $2.50, $5, or $10 increments, depending on the value
of the strikes and the liquidity of the options.
The relationship of the strike price to the stock price is important in
pricing options. For calls, if the stock price is above the strike price, the call
is in-the-money (ITM). If the stock and the strike prices are close, the call is

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at-the-money (ATM). If the stock price is below the strike price the call is
out-of-the-money (OTM). This relationship is just the opposite for puts. If
the stock price is below the strike price, the put is in-the-money. If the stock
price and the strike price are about the same, the put is at-the-money. And,
if the stock price is above the put strike, it is out-of-the-money.
Option Type
There are two types of options: calls and puts. Calls give the holder the
right to buy the underlying and the writer the obligation to sell the
underlying. Puts give the holder the right to sell the underlying and the
writer the obligation to buy the underlying.
Premium
The price of an option is called its premium. The premium of this option is
$5. Like stock prices, option premiums are stated in dollars and cents per
share. Since the option represents 100 shares of IBM, the buyer of this
option will pay $500 when the transaction occurs. Certain types of spreads
may be quoted in fractions of apenny.
An option’spremium is made up of two parts: intrinsic value and time
value. Intrinsic value is the amount by which the option is in-the-money.
For example, if IBM stock were trading at 171.30, this 170-strike call
would be in-the-money by 1.30. It has 1.30 of intrinsic value. The
remaining 3.70 of its $5 premium would be time value.
Options that are out-of-the-money have no intrinsic value. Their values
consist only of time premium. Sometimes options have no time value left.
Options that consist of only intrinsic value are trading at what traders call
parity . Time value is sometimes called premium over parity .
Exercise Style
One contract specification that is not specifically shown here is the exercise
style. There are two main exercise styles: American and European.
American-exercise options can be exercised, and therefore assigned,
anytime after the contract is entered into until either the trader closes the

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ETF Options
Exchange-traded funds are vehicles that represent ownership in afund or
investment trust. This fund is made up of abasket of an underlying index’ssecurities—usually equities. The contract specifications of ETF options are
similar to those of equity options. Let’slook at an example.
One actively traded optionable ETF is the Standard & Poor’s Depositary
Receipts (SPDRs or Spiders). Spider shares and options trade under the
symbol SPY. Exercising one SPY call gives the exerciser along position of
100 shares of Spiders at the strike price of the option. Expiration for ETF
options typically falls on the same day as for equity options—the Saturday
following the third Friday of the month. The last trading day is the Friday
before. ETF options are American exercise. Traders of ETFs should be
aware of the relationship between the price of the ETF shares and the value
of the underlying index. For example, the stated value of the Spiders is
about one tenth the stated value of the S&P 500. The PowerShares QQQ
ETF, representing the Nasdaq 100, is about one fortieth the stated value of
the Nasdaq 100.

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Strategies and At-Expiration
Diagrams
One of the great strengths of options is that there are so many different
ways to use them. There are simple, straightforward strategies like buying acall. And there are complex spreads with creative names like jelly roll, guts,
and iron butterfly. Aspread is astrategy that involves combining an option
with one or more other options or stock. Each component of the spread is
referred to as aleg. Each spread has its own unique risk and reward
characteristics that make it appropriate for certain market outlooks.
Throughout this book, many different spreads will be discussed in depth.
For now, it’simportant to understand that all spreads are made up of acombination of four basic option positions: buy call, sell call, buy put, and
sell put. Understanding complex option strategies requires understanding
these basic positions and their common, practical uses. When learning
options, it’shelpful to see what the option’spayout is if it is held until
expiration.

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Buy Call
Why buy the right to buy the stock when you can simply buy the stock? All
option strategies have trade-offs, and the long call is no different. Whether
the stock or the call is preferable depends greatly on the trader’sforecast
and motivations.
Consider along call example:
Buy 1 INTC June 22.50 call at 0.85.
In this example, atrader is bullish on Intel (INTC). He believes Intel will
rise at least 20 percent, from $22.25 per share to around $27 by June
expiration, about two months from now. He is concerned, however, about
downside risk and wants to limit his exposure. Instead of buying 100 shares
of Intel at $22.25—atotal investment of $2,225—the trader buys 1 INTC
June 22.50 call at 0.85, for atotal of $85.
The trader is paying 0.85 for the right to buy 100 shares of Intel at $22.50
per share. If Intel is trading below the strike price of $22.50 at expiration,
the call will expire and the total premium of 0.85 will be lost. Why? The
trader will not exercise the right to buy the stock at a $22.50 if he can buy it
cheaper in the market. Therefore, if Intel is below $22.50 at expiration, this
call will expire with no value.
However, if the stock is trading above the strike price at expiration, the
call can be exercised, in which case the trader may purchase the stock
below its trading price. Here, the call has value to the trader. The higher the
stock, the more the call is worth. For the trade to be profitable, at expiration
the stock must be trading above the trader’sbreak-even price. The break-
even price for along call is the strike price plus the premium paid—in this
example, $23.35 per share. The point here is that if the call is exercised, the
effective purchase price of the stock upon exercise is $23.35. The stock is
literally bought at the strike price, which is $22.50, but the premium of 0.85
that the trader has paid must be taken into account. Exhibit 1.1 illustrates
this example.

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EXHIBIT 1.2 Long Intel call vs. long Intel stock.
The thin dotted line represents owning 100 shares of Intel at $22.25.
Profits are unlimited, but the risk is substantial—the stock can go to zero.
Herein lies the trade-off. The long call has unlimited profit potential with
limited risk. Whenever an option is purchased, the most that can be lost is
the premium paid for the option. But the benefit of reduced risk comes at acost. If the stock is above the strike at expiration, the call will always
underperform the stock by the amount of the premium.
Because of this trade-off, conservative traders will sometimes buy acall
rather than the associated stock and sometimes buy the stock rather than the
call. Buying acall can be considered more conservative when the volatility
of the stock is expected to rise. Traders are willing to risk acomparatively
small premium when alarge price decline is feared possible. Instead, in an
interest-bearing vehicle, they harbor the capital that would otherwise have
been used to purchase the stock. The cost of this protection is acceptable to
the trader if high-enough price advances are anticipated. In terms of
percentage, much higher returns and losses are possible with the long call.
If the stock is trading at $27 at expiration, as the trader in this example
expected, the trader reaps a 429 percent profit on the $0.85 investment

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Sell Call
Selling acall creates the obligation to sell the stock at the strike price. Why
is atrader willing to accept this obligation? The answer is option premium.
If the position is held until expiration without getting assigned, the entire
premium represents aprofit for the trader. If assignment occurs, the trader
will be obliged to sell stock at the strike price. If the trader does not have along position in the underlying stock (anaked call), ashort stock position
will be created. Otherwise, if stock is owned (acovered call), that stock is
sold. Whether the trader has aprofit or aloss depends on the movement of
the stock price and how the short call position was constructed.
Consider anaked call example:
Sell 1 TGT October 50 call at 1.45
In this example, Target Corporation (TGT) is trading at $49.42. Atrader,
Sam, believes Target will continue to be trading below $50 by October
expiration, about two months from now. Sam sells 1 Target two-month 50
call at 1.45, opening ashort position in that series. Exhibit 1.3 will help
explain the expected payout of this naked call position if it is held until
expiration.

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EXHIBIT 1.3 Naked Target call.
If TGT is trading below the exercise price of 50, the call will expire
worthless. Sam keeps the 1.45 premium, and the obligation to sell the stock
ceases to exist. If Target is trading above the strike price, the call will be in-
the-money. The higher the stock is above the strike price, the more intrinsic
value the call will have. As aseller, Sam wants the call to have little or no
intrinsic value at expiration. If the stock is below the break-even price at
expiration, Sam will still have aprofit. Here, the break-even price is $51.45
—the strike price plus the call premium. Above the break-even, Sam has aloss. Since stock prices can rise to infinity (although, for the record, Ihave
never seen this happen), the naked call position has unlimited risk of loss.
Because ashort stock position may be created, anaked call position must
be done in amargin account. For retail traders, many brokerage firms
require different levels of approval for different types of option strategies.
Because the naked call position has unlimited risk, establishing it will
generally require the highest level of approval—and ahigh margin
requirement.
Another tactical consideration is what Sam’sobjective was when he
entered the trade. His goal was to profit from the stock’sbeing below $50

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during this two-month period—not to short the stock. Because equity
options are American exercise and can be exercised/assigned any time from
the moment the call is sold until expiration, ashort stock position cannot
always be avoided. If assigned, the short stock position will extend Sam’speriod of risk—because stock doesn’texpire. Here, he will pay one
commission shorting the stock when assignment occurs and one more when
he buys back the unwanted position. Many traders choose to close the naked
call position before expiration rather than risk assignment.
It is important to understand the fundamental difference between buying
calls and selling calls. Buying acall option offers limited risk and unlimited
reward. Selling anaked call option, however, has limited reward—the call
premium—and unlimited risk. This naked call position is not so much
bearish as not bullish . If Sam thought the stock was going to zero, he
would have chosen adifferent strategy.
Now consider acovered call example:
Buy 100 shares TGT at $49.42
Sell 1 TGT October 50 call at 1.45
Unlimited and risk are two words that don’tsit well together with many
traders. For that reason, traders often prefer to sell calls as part of aspread.
But since spreads are strategies that involve multiple components, they have
different risk characteristics from an outright option. Perhaps the most
commonly used call-selling spread strategy is the covered call (sometimes
called acovered write or abuy-write ). While selling acall naked is away
to take advantage of a “not bullish” forecast, the covered call achieves adifferent set of objectives.
After studying Target Corporation, another trader, Isabel, has aneutral to
slightly bullish forecast. With Target at $49.42, she believes the stock will
be range-bound between $47 and $51.50 over the next two months, ending
with October expiration. Isabel buys 100 shares of Target at $49.42 and
sells 1 TGT October 50 call at 1.45. The implications for the covered-call
strategy are twofold: Isabel must be content to own the stock at current
levels, and—since she sold the right to buy the stock at $50, that is, a 50
call, to another party—she must be willing to sell the stock if the price rises
to or through $50 per share. Exhibit 1.4 shows how this covered call
performs if it is held until the call expires.

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EXHIBIT 1.4 Target covered call.
The solid kinked line represents the covered call position, and the thin,
straight dotted line represents owning the stock outright. At the expiration
of the call option, if Target is trading below $50 per share—the strike price
—the call expires and Isabel is left with along position of 100 shares plus
$1.45 per share of expired-option premium. Below the strike, the buy-write
always outperforms simply owning the stock by the amount of the
premium. The call premium provides limited downside protection; the stock
Isabel owns can decline $1.45 in value to $47.97 before the trade is aloser.
In the unlikely event the stock collapses and becomes worthless, this
limited downside protection is not so comforting. Ultimately, Isabel has
$47.97 per share at risk.
The trade-off comes if Target is above $50 at expiration. Here, assignment
will likely occur, in which case the stock will be sold. The call can be
assigned before expiration, too, causing the stock to be called away early.
Because the covered call involves this obligation to sell the sock at the
strike price, upside potential is limited. In this case, Isabel’sprofit potential
is $2.03. The stock can rise from $49.42 to $50—a $0.58 profit—plus $1.45
of option premium.
Isabel does not want the stock to decline too much. Below $47.97, the
trade is aloser. If the stock rises too much, the stock is sold prematurely and

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Sell Put
Selling aput has many similarities to the covered call strategy. We’ll
discuss the two positions and highlight the likenesses. Chapter 6 will detail
the nuts and bolts of why these similarities exist.
Consider an example of selling aput:
Sell 1 BA January 65 put at 1.20
In this example, trader Sam is neutral to moderately bullish on Boeing (BA)
between now and January expiration. He is not bullish enough to buy BA at
the current market price of $69.77 per share. But if the shares dropped
below $65, he’dgladly scoop some up. Sam sells 1 BA January 65 put at
1.20. The at-expiration diagram in Exhibit 1.5 shows the P&(L) of this trade
if it is held until expiration.
EXHIBIT 1.5 Boeing short put.
At the expiration of this option, if Boeing is above $65, the put expires
and Sam retains the premium of $1.20. The obligation to buy stock expires
with the option. Below the strike, put owners will be inclined to exercise
their option to sell the stock at $65. Therefore, those short the put, as Sam is
in this example, can expect assignment. The break-even price for the
position is $63.80. That is the strike price minus the option premium. If

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assigned, this is the effective purchase price of the stock. The obligation to
buy at $65 is fulfilled, but the $1.20 premium collected makes the purchase
effectively $63.80. Here, again, there is limited profit opportunity ($1.20 if
the stock is above the strike price) and seemingly unlimited risk (the risk of
potential stock ownership at $63.80) if Boeing is below the strike price.
Why would atrader short aput and willingly assume this substantial risk
with comparatively limited reward? There are anumber of motivations that
may warrant the short put strategy. In this example, Sam had the twin goals
of profiting from aneutral to moderately bullish outlook on Boeing and
buying it if it traded below $65. The short put helps him achieve both
objectives.
Much like the covered call, if the stock is above the strike at expiration,
this trader reaches his maximum profit potential—in this case 1.20. And if
the price of Boeing is below the strike at expiration, Sam has ownership of
the stock from assignment. Here, astrike price that is lower than the current
stock level is used. The stock needs to decline in order for Sam to get
assigned and become long the stock. With this strategy, he was able to
establish atarget price at which he would buy the stock. Why not use alimit
order? If the put is assigned, the effective purchase price is $63.80 even if
the stock price is above this price. If the put is not assigned, the premium is
kept.
Aconsideration every trader must make before entering the short put
position is how the purchase of the stock will be financed in the event the
put is assigned. Traders hoping to acquire the stock will often hold enough
cash in their trading account to secure the purchase of the stock. This is
called acash-secured put . In this example, Sam would hold $6,380 in his
account in addition to the $120 of option premium received. This affords
him enough free capital to fund the $6,500 purchase of stock the short put
dictates. More speculative traders may be willing to buy the stock on
margin, in which case the trader will likely need around 50 percent of the
stock’svalue.
Some traders sell puts without the intent of ever owning the stock. They
hope to profit from alow-volatility environment. Just as the short call is anot-bullish stance on the underlying, the short put is anot-bearish play. As
long as the underlying is above the strike price at expiration, the option

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Buy Put
Buying aput gives the holder the right to sell stock at the strike price. Of
course, puts can be apart of ahost of different spreads, but this chapter
discusses the two most basic and common put-buying strategies: the long
put and the protective put. The long put is away to speculate on abearish
move in the underlying security, and the protective put is away to protect along position in the underlying security.
Consider along put example:
Buy 1 SPY May 139 put at 2.30
In this example, the Spiders have had agood run up to $140.35. Trader
Isabel is looking for a 10 percent correction in SPY between now and the
end of May, about three months away. She buys 1 SPY May 139 put at 2.30.
This put gives her the right to sell 100 shares of SPY at $139 per share.
Exhibit 1.6 shows Isabel’s P&(L) if the put is held until expiration.
EXHIBIT 1.6 SPY long put.
If SPY is above the strike price of 139 at expiration, the put will expire
and the entire premium of 2.30 will be lost. If SPY is below the strike price
at expiration, the put will have value. It can be exercised, creating ashort
position in the Spiders at an effective price of $136.70 per share. This price

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is found by subtracting the premium paid, 2.30, from the strike price, 139.
This is the point at which the position breaks even. If SPY is below $136.70
at expiration, Isabel has aprofit. Profits will increase on atick-for-tick
basis, with downward movements in SPY down to zero. The long put has
limited risk and substantial reward potential.
An alternative for Isabel is to short the ETF at the current price of
$140.35. But ashort position in the underlying may not be as attractive to
her as along put. The margin requirements for short stock are significantly
higher than for along put. Put buyers must post only the premium of the put
—that is the most that can be lost, after all.
The margin requirement for short stock reflects unlimited loss potential.
Margin requirements aside, risk is avery real consideration for atrader
deciding between shorting stock and buying aput. If the trader expects high
volatility, he or she may be more inclined to limit upside risk while
leveraging downside profit potential by buying aput. In general, traders buy
options when they expect volatility to increase and sell them when they
expect volatility to decrease. This will be acommon theme throughout this
book.
Consider aprotective put example:
This is an example of asituation in which volatility is expected to
increase.
Own 100 shares SPY at 140.35
Buy 1 SPY May139 put at 2.30
Although Isabel bought aput because she was bearish on the Spiders, adifferent trader, Kathleen, may buy aput for adifferent reason—she’sbullish but concerned about increasing volatility. In this example, Kathleen
has owned 100 shares of Spiders for some time. SPY is currently at
$140.35. She is bullish on the market but has concerns about volatility over
the next two or three months. She wants to protect her investment. Kathleen
buys 1 SPY May 139 put at 2.30. (If Kathleen bought the shares of SPY and
the put at the same time, as aspread, the position would be called amarried
put.)
Kathleen is buying the right to sell the shares she owns at $139.
Effectively, it is an insurance policy on this asset. Exhibit 1.7 shows the risk
profile of this new position.

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EXHIBIT 1.7 SPY protective put.
The solid kinked line is the protective put (put and stock), and the thin
dotted line is the outright position in SPY alone, without the put. The most
Kathleen stands to lose with the protective put is $3.65 per share. SPY can
decline from $140.35 to $139, creating aloss of $1.35, plus the $2.30
premium spent on the put. If the stock does not fall and the insuring put
hence does not come into play, the cost of the put must be recouped to
justify its expense. The break-even point is $142.65.
This position implies that Kathleen is still bullish on the Spiders. When
traders believe astock or ETF is going to decline, they sell the shares.
Instead, Kathleen sacrifices 1.6 percent of her investment up front by
purchasing the put for $2.30. She defers the sale of SPY until the period of
perceived risk ends. Her motivation is not to sell the ETF; it is to hedge
volatility.
Once the anticipated volatility is no longer aconcern, Kathleen has achoice to make. She can let the option run its course, holding it to
expiration, at which point it will either expire or be exercised; or she can
sell the option before expiration. If the option is out-of-the-money, it may
have residual time value prior to expiration that can be recouped. If it is in-
the-money, it will have intrinsic value and maybe time value as well. In this
situation, Kathleen can look at this spread as two trades—one that has

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CHAPTER 2
Greek Philosophy
My wife, Kathleen, is not an options trader. Au contraire. However, she,
like just about everyone, uses them from time to time—though without
really thinking about it. She was on eBay the other day bidding on apair of
shoes. The bid was $45 with three days left to go. She was concerned about
the price rising too much and missing the chance to buy them at what she
thought was agood price. She noticed, though, that someone else was
selling the same shoes with abuy-it-now price of $49—agood-enough
price in her opinion. Kathleen was effectively afforded acall option. She
had the opportunity to buy the shoes at (the strike price of) $49, aright she
could exercise until the offer expired.
The biggest difference between the option in the eBay scenario and the
sort of options discussed in this book is transferability. Actual options are
tradable—they can be bought and sold. And it is the contract itself that has
value—there is one more iteration of pricing.
For example, imagine the $49 opportunity was acoupon or certificate that
guaranteed the price of $49, which could be passed along from one person
to another. And there was the chance that the $49-price guarantee could
represent adiscount on the price paid for the shoes—maybe abig discount
—should the price of the shoes rise in the eBay auction. The certificate
guaranteeing the $49 would have value. Anyone planning to buy the shoes
would want the safety of knowing they were guaranteed not to pay more
than $49 for the shoes. In fact, some people would even consider paying to
buy the certificate itself if they thought the price of the shoes might rise
significantly.

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Price vs. Value: How Traders Use
Option-Pricing Models
Like in the common-life example just discussed, the right to buy or sell an
underlying security—that is, an option—can have value, too. The specific
value of an option is determined by supply and demand. There are several
variables in an option contract, however, that can influence atrader’swillingness to demand (desire to buy) or supply (desire to sell) an option at
agiven price. For example, atrader would rather own—that is, there would
be higher demand for—an option that has more time until expiration than ashorter-dated option, all else held constant. And atrader would rather own acall with alower strike than ahigher strike, all else kept constant, because it
would give the right to buy at alower price.
Several elements contribute to the value of an option. It took academics
many years to figure out exactly what those elements are. Fischer Black and
Myron Scholes together pioneered research in this area at the University of
Chicago. Ultimately, their work led to a Nobel Prize for Myron Scholes.
Fischer Black died before he could be honored.
In 1973, Black and Scholes published apaper called “The Pricing of
Options and Corporate Liabilities” in the Journal of Political Economy ,
that introduced the Black-Scholes option-pricing model to the world. The
Black-Scholes model values European call options on non-dividend-paying
stocks. Here, for the first time, was awidely accepted model illustrating
what goes into the pricing of an option. Option prices were no longer wild
guesswork. They could now be rationalized. Soon, additional models and
alterations to the Black-Scholes model were developed for options on
indexes, dividend-paying stocks, bonds, commodities, and other optionable
instruments. All the option-pricing models commonly in use today have
slightly different means but achieve the same end: the option’stheoretical
value. For American-exercise equity options, six inputs are entered into any
option-pricing model to generate atheoretical value: stock price, strike
price, time until expiration, interest rate, dividends, and volatility.

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Theoretical value—what aconcept! Atrader plugs six numbers into apricing model, and it tells him what the option is worth, right? Well, in
practical terms, that’snot exactly how it works. An option is worth what the
market bears. Economists call this price discovery. The price of an option is
determined by the forces of supply and demand working in afree and open
market. Herein lies an important concept for option traders: the difference
between price and value.
Price can be observed rather easily from any source that offers option
quotes (web sites, your broker, quote vendors, and so on). Value is
calculated by apricing model. But, in practice, the theoretical value is really
not an output at all. It is already known: the market determines it. The trader
rectifies price and value by setting the theoretical value to fall between the
bid and the offer of the option by adjusting the inputs to the model.
Professional traders often refer to the theoretical value as the fair value of
the option.
At this point, please note the absence of the mathematical formula for the
Black-Scholes model (or any other pricing model, for that matter).
Although the foundation of trading option greeks is mathematical, this book
will keep the math to aminimum—which is still quite abit. The focus of
this book is on practical applications, not academic theory. It’sabout
learning to drive the car, not mastering its engineering.
The trader has an equation with six inputs equaling one known output.
What good is this equation? An option-pricing model helps atrader
understand how market forces affect the value of an option. Five of the six
inputs are dynamic; the only constant is the strike price of the option in
question. If the price of the option changes, it’sbecause one or more of the
five variable inputs has changed. These variables are independent of each
other, but they can change in harmony, having either acumulative or net
effect on the option’svalue. An option trader needs to be concerned with the
relationship of these variables (price, time, volatility, interest). This
multidimensional view of asset pricing is unique to option traders.

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Delta
The five figures commonly used by option traders are represented by Greek
letters: delta, gamma, theta, vega, rho. The figures are referred to as option
greeks. Vega, of course, is not an actual letter of the greek alphabet, but in
the options vernacular, it is considered one of the greeks.
The greeks are aderivation of an option-pricing model, and each Greek
letter represents aspecific sensitivity to influences on the option’svalue. To
understand concepts represented by these five figures, we’ll start with delta,
which is defined in four ways:
1. The rate of change of an option value relative to achange in the
underlying stock price.
2. The derivative of the graph of an option value in relation to the stock
price.
3. The equivalent of underlying shares represented by an option
position.
4. The estimate of the likelihood of an option expiring in-the-money. 1
Definition 1 : Delta (Δ) is the rate of change of an option’svalue relative
to achange in the price of the underlying security. Atrader who is bullish
on aparticular stock may choose to buy acall instead of buying the
underlying security. If the price of the stock rises by $1, the trader would
expect to profit on the call—but by how much? To answer that question, the
trader must consider the delta of the option.
Delta is stated as apercentage. If an option has a 50 delta, its price will
change by 50 percent of the change of the underlying stock price. Delta is
generally written as either awhole number, without the percent sign, or as adecimal. So if an option has a 50 percent delta, this will be indicated as
0.50, or 50. For the most part, we’ll use the former convention in our
discussion.
Call values increase when the underlying stock price increases and vice
versa. Because calls have this positive correlation with the underlying, they
have positive deltas. Here is asimplified example of the effect of delta on
an option:

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Consider a $60 stock with acall option that has a 0.50 delta and is trading
for 3.00. Considering only the delta, if the stock price increases by $1, the
theoretical value of the call will rise by 0.50. That’s 50 percent of the stock
price change. The new call value will be 3.50. If the stock price decreases
by $1, the 0.50 delta will cause the call to decrease in value by 0.50, from
3.00 to 2.50.
Puts have anegative correlation to the underlying. That is, put values
decrease when the stock price rises and vice versa. Puts, therefore, have
negative deltas. Here is asimplified example of the delta effect on a −0.40-
delta put:
As the stock rises from $60 to $61, the delta of −0.40 causes the put value
to go from $2.25 to $1.85. The put decreases by 40 percent of the stock
price increase. If the stock price instead declined by $1, the put value would
increase by $0.40, to $2.65.
Unfortunately, real life is abit more complicated than the simplified
examples of delta used here. In reality, the value of both the call and the put
will likely be higher with the stock at $61 than was shown in these
examples. We’ll expand on this concept later when we tackle the topic of
gamma.
Definition 2 : Delta can also be described another way. Exhibit 2.1 shows
the value of acall option with three months to expiration at avariable stock
price. As the stock price rises, the call is worth more; as the stock price
declines, the call value moves toward zero. Mathematically, for any given
point on the graph, the derivative will show the rate of change of the option
price. The delta is the first derivative of the graph of the option price
relative to the stock price .

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EXHIBIT 2.1 Call value compared with stock price.
Definition 3 : In terms of absolute value (meaning that plus and minus
signs are ignored), the delta of an option is between 1.00 and 0. Its price can
change in tandem with the stock, as with a 1.00 delta; or it cannot change at
all as the stock moves, as with a 0 delta; or anything in between. By
definition, stock has a 1.00 delta—it is the underlying security. A $1 rise in
the stock yields a $100 profit on around lot of 100 shares. Acall with a
0.60 delta rises by $0.60 with a $1 increase in the stock. The owner of acall
representing rights on 100 shares earns $60 for a $1 increase in the
underlying. It’sas if the call owner in this example is long 60 shares of the
underlying stock. Delta is the option’sequivalent of aposition in the
underlying shares .
Atrader who buys five 0.43-delta calls has aposition that is effectively
long 215 shares—that’s 5 contracts × 0.43 deltas × 100 shares. In option
lingo, the trader is long 215 deltas. Likewise, if the trader were short five
0.43-delta calls, the trader would be short 215 deltas.
The same principles apply to puts. Being long 10 0.59-delta puts makes
the trader short atotal of 590 deltas, aposition that profits or loses like
being short 590 shares of the underlying stock. Conversely, if the trader
were short 10 0.59-delta puts, the trader would theoretically make $590 if
the stock were to rise $1 and lose $590 if the stock fell by $1—just like
being long 590 shares.
Definition 4 : The final definition of delta is considered the trader’sdefinition. It’smathematically imprecise but is used nonetheless as ageneral rule of thumb by option traders. Atrader would say the delta is astatistical approximation of the likelihood of the option expiring in-the-

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Dynamic Inputs
Option deltas are not constants. They are calculated from the dynamic
inputs of the pricing model—stock price, time to expiration, volatility, and
so on. When these variables change, the changes affect the delta. These
changes can be mathematically quantified—they are systematic.
Understanding these patterns and other quirks as to how delta behaves can
help traders use this tool more effectively. Let’sdiscuss afew observations
about the characteristics of delta.
First, call and put deltas are closely related. Exhibit 2.2 is apartial option
chain of 70-day calls and puts in Rambus Incorporated (RMBS). The stock
was trading at $21.30 when this table was created. In Exhibit 2.2 , the 20
calls have a 0.66 delta.
EXHIBIT 2.2 RMBS Option chain with deltas.
Notice the deltas of the put-call pairs in this exhibit. As ageneral rule, the
absolute value of the call delta plus the absolute value of the put delta add
up to close to 1.00. The reason for this has to do with amathematical
relationship called put-call parity, which is briefly discussed later in this
chapter and described in detail in Chapter 6. But with equity options, the
put-call pair doesn’talways add up to exactly 1.00.
Sometimes the difference is simply due to rounding. But sometimes there
are other reasons. For example, the 30-strike calls and puts in Exhibit 2.2
have deltas of 0.14 and −0.89, respectively. The absolute values of the
deltas add up to 1.03. Because of the possibility of early exercise of
American options, the put delta is abit higher than the call delta would

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Moneyness and Delta
The next observation is the effect of moneyness on the option’sdelta.
Moneyness describes the degree to which the option is in- or out-of-the-
money. As ageneral rule, options that are in-the-money (ITM) have deltas
greater than 0.50. Options that are out-of-the-money (OTM) have deltas
less than 0.50. Finally, options that are at-the-money (ATM) have deltas that
are about 0.50. The more in-the-money the option is, the closer to 1.00 the
delta is. The more out-of-the-money, the closer the delta is to 0.
But ATM options are usually not exactly 0.50. For ATMs, both the call
and the put deltas are generally systematically avalue other than 0.50.
Typically, the call has ahigher delta than 0.50 and the put has alower
absolute value than 0.50. Incidentally, the call’stheoretical value is
generally greater than the put’swhen the options are right at-the-money as
well. One reason for this disparity between exactly at-the-money calls and
puts is the interest rate. The more time until expiration, the more effect the
interest rate will have, and, therefore, the higher the call’stheoretical and
delta will be relative to the put.

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Effect of Time on Delta
In aclose contest, the last few minutes of afootball game are often the most
exciting—not because the players run faster or knock heads harder but
because one strategic element of the game becomes more and more
important: time. The team that’sin the lead wants the game clock to run
down with no interruption to solidify its position. The team that’slosing
uses its precious time-outs strategically. The more playing time left, the less
certain defeat is for the losing team.
Although mathematically imprecise, the trader’sdefinition can help us
gain insight into how time affects option deltas. The more time left until an
option’sexpiration, the less certain it is whether the option will be ITM or
OTM at expiration. The deltas of both the ITM and the OTM options reflect
that uncertainty. The more time left in the life of the option, the closer the
deltas tend to gravitate to 0.50. A 0.50 delta represents the greatest level of
uncertainty—acoin toss. Exhibit 2.3 shows the deltas of ahypothetical
equity call with astrike price of 50 at various stock prices with different
times until expiration. All other parameters are held constant.
EXHIBIT 2.3 Estimated delta of 50-strike call—impact of time.

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Effect of Volatility on Delta
The level of volatility affects option deltas as well. We’ll discuss volatility
in more detail in future chapters, but it’simportant to address it here as it
relates to the concept of delta. Exhibit 2.4 shows how changing the
volatility percentage (explained further in Chapter 3), as opposed to the
time to expiration, affects option deltas. In this table, the delta of acall with
91 days until expiration is studied.
EXHIBIT 2.4 Estimated delta of 50-strike call—impact of volatility.
Notice the effect that volatility has on the deltas of this option with the
underlying stock at various prices. In this table, at alow volatility with the
call deep in- or out-of-the-money, the delta is very large or very small,
respectively. At 10 percent volatility with the stock at $58 ashare, the delta
is 1.00. At that same volatility level with the stock at $42 ashare, the delta
is 0.
But at higher volatility levels, the deltas change. With the stock at $58, a
45 percent volatility gives the 50-strike call a 0.79 delta—much smaller
than it was at the low volatility level. With the stock at $42, a 45-percent
volatility returns a 0.30 delta for the call. Generally speaking, ITM option
deltas are smaller given ahigher volatility assumption, and OTM option
deltas are bigger with ahigher volatility.

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Gamma
The strike price is the only constant in the pricing model. When the stock
price moves relative to this constant, the option in question becomes more
in-the-money or out-of-the-money. This means the delta changes. This
isolated change is measured by the option’sgamma, sometimes called
curvature .
Gamma (Γ) is the rate of change of an option’sdelta given achange in
the price of the underlying security . Gamma is conventionally stated in
terms of deltas per dollar move. The simplified examples above under
Definition 1 of delta, used to describe the effect of delta, had one important
piece of the puzzle missing: gamma. As the stock price moved higher in
those examples, the delta would not remain constant. It would change due
to the effect of gamma. The following example shows how the delta would
change given a 0.04 gamma attributed to the call option.
The call in this example starts as a 0.50-delta option. When the stock
price increases by $1, the delta increases by the amount of the gamma. In
this example, delta increases from 0.50 to 0.54, adding 0.04 deltas. As the
stock price continues to rise, the delta continues to move higher. At $62, the
call’sdelta is 0.58.
This increase in delta will affect the value of the call. When the stock
price first begins to rise from $60, the option value is increasing at arate of
50 percent—the call’sdelta at that stock price. But by the time the stock is
at $61, the option value is increasing at arate of 54 percent of the stock
price. To estimate the theoretical value of the call at $61, we must first
estimate the average change in the delta between $60 and $61. The average
delta between $60 and $61 is roughly 0.52. It’sdifficult to calculate the
average delta exactly because gamma is not constant; this is discussed in
more detail later in the chapter. Amore realistic example of call values in
relation to the stock price would be as follows:

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Each $1 increase in the stock shows an increase in the call value about
equal to the average delta value between the two stock prices. If the stock
were to decline, the delta would get smaller at adecreasing rate.
As the stock price declines from $60 to $59, the option delta decreases
from 0.50 to 0.46. There is an average delta of about 0.48 between the two
stock prices. At $59 the new theoretical value of the call is 2.52. The
gamma continues to affect the option’sdelta and thereby its theoretical
value as the stock continues its decline to $58 and beyond.
Puts work the same way, but because they have anegative delta, when
there is apositive stock-price movement the gamma makes the put delta
less negative, moving closer to 0. The following example clarifies this.
As the stock price rises, this put moves more and more out-of-the-money.
Its theoretical value is decreasing by the rate of the changing delta. At $60,
the delta is −0.40. As the stock rises to $61, the delta changes to −0.36. The
average delta during that move is about −0.38, which is reflected in the
change in the value of the put.
If the stock price declines and the put moves more toward being in-the-
money, the delta becomes more negative—that is, the put acts more like ashort stock position.
Here, the put value rises by the average delta value between each
incremental change in the stock price.
These examples illustrate the effect of gamma on an option without
discussing the impact on the trader’sposition. When traders buy options,
they acquire positive gamma. Since gamma causes options to gain value at

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afaster rate and lose value at aslower rate, (positive) gamma helps the
option buyer. Atrader buying one call or put in these examples would have
+0.04 gamma. Buying 10 of these options would give the trader a +0.4
gamma.
When traders sell options, gamma works against them. When options lose
value, they move toward zero at aslower rate. When the underlying moves
adversely, gamma speeds up losses. Selling options yields anegative
gamma position. Atrader selling one of the above calls or puts would have
−0.04 gamma per option.
The effect of gamma is less significant for small moves in the underlying
than it is for bigger moves. On proportionately large moves, the delta can
change quite abit, making abig difference in the position’s P&(L). In
Exhibit 2.1 , the left side of the diagram showed the call price not
increasing at all with advances in the stock—a 0 delta. The right side
showed the option advancing in price 1-to-1 with the stock—a 1.00 delta.
Between the two extremes, the delta changes. From this diagram another
definition for gamma can be inferred: gamma is the second derivative of the
graph of the option price relative to the stock price. Put another way,
gamma is the first derivative of agraph of the delta relative to the stock
price. Exhibit 2.5 illustrates the delta of acall relative to the stock price.
EXHIBIT 2.5 Call delta compared with stock price.
Not only does the delta change, but it changes at achanging rate. Gamma
is not constant. Moneyness, time to expiration, and volatility each have an
effect on the gamma of an option.

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Dynamic Gamma
When options are far in-the-money or out-of-the-money, they are either
1.00 delta or 0 delta. At the extremes, small changes in the stock price will
not cause the delta to change much. When an option is at-the-money, it’sadifferent story. Its delta can change very quickly.
ITM and OTM options have alow gamma.
ATM options have arelatively high gamma.
Exhibit 2.6 is an example of how moneyness translates into gamma on
QQQ calls.
EXHIBIT 2.6 Gamma of QQQ calls with QQQ at $44.
With QQQ at $44, 92 days until expiration, and aconstant volatility input
of 19 percent, the 36- and 54-strike calls are far enough in- and out-of-the-
money, respectively, that if the Qs move asmall amount in either direction
from the current price of $44, the movement won’tchange their deltas
much at all. The chances of their money status changing between now and
expiration would not be significantly different statistically given asmall
stock price change. They have the smallest gammas in the table.
The highest gammas shown here are around the ATM strike prices. (Note
that because of factors not yet discussed, the strike that is exactly at-the-
money may not have the highest gamma. The highest gamma is likely to
occur at aslightly higher strike price.) Exhibit 2.7 shows agraph of the
corresponding numbers in Exhibit 2.6 .

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At seven days until expiration, there is less time for price action in the
stock to change the expected moneyness at expiration of ITMs or OTMs.
ATM options, however, continue to be in play. Here, the ATM gamma is
approaching 0.35. But the strikes below 41 and above 48 have 0 gamma.
Similarly-priced securities that tend to experience bigger price swings
may have strikes $3 away-from-the-money with seven-day gammas greater
than zero. The volatility of the underlying will affect gamma, too. Exhibit
2.9 shows the same 19 percent volatility QQQ calls in contrast with agraph
of the gamma if the volatility is doubled.
EXHIBIT 2.9 Gamma as volatility changes.
Raising the volatility assumption flattens the curve, causing ITM and
OTM to have higher gamma while lowering the gamma for ATMs.
Short-term ATM options with low volatility have the highest gamma.
Lower gamma is found in ATMs when volatility is higher and it is lower for
ITMs and OTMs and in longer-dated options.

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Theta
Option prices can be broken down into two parts: intrinsic value and time
value. Intrinsic value is easily measurable. It is simply the ITM part of the
premium. Time value, or extrinsic value, is what’sleft over—the premium
paid over parity for the option. All else held constant, the more time left in
the life of the option, the more valuable it is—there is more time for the
stock to move. And as the useful life of an option decreases, so does its time
value.
The decline in the value of an option because of the passage of time is
called time decay, or erosion. Incremental measurements of time decay are
represented by the Greek letter theta (θ). Theta is the rate of change in an
option’sprice given aunit change in the time to expiration . What exactly is
the unit involved here? That depends.
Some providers of option greeks will display thetas that represent one
day’sworth of time decay. Some will show thetas representing seven days
of decay. In the case of aone-day theta, the figure may be based on aseven-
day week or on aweek counting only trading days. The most common and,
arguably, most useful display of this figure is the one-day theta based on the
seven-day week. There are, after all, seven days in aweek, each day of
which can see an occurrence with the potential to cause arevaluation in the
stock price (that is, news can come out on Saturday or Sunday). The one-
day theta based on aseven-day week will be used throughout this book.

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Taking the Day Out
When the number of days to expiration used in the pricing model declines
from, say, 32 days to 31 days, the price of the option decreases by the
amount of the theta, all else held constant. But when is the day “taken out”?
It is intuitive to think that after the market closes, the model is changed to
reflect the passing of one day’stime. But, in fact, this change is logically
anticipated and may be priced in early.
In the earlier part of the week, option prices can often be observed getting
cheaper relative to the stock price sometime in the middle of the day. This is
because traders will commonly take the day out of their model during
trading hours after the underlying stabilizes following the morning
business. On Fridays and sometimes Thursdays, traders will take all or part
of the weekend out. Commonly, by Friday afternoon, traders will be using
Monday’sdays to value their options.
When option prices are seen getting cheaper on, say, a Friday, how can
one tell whether this is the effect of the market taking the weekend out or achange in some other input, such as volatility? To some degree, it doesn’tmatter. Remember, the model is used to reflect what the market is doing,
not the other way around. In many cases, it’slogical to presume that small
devaluations in option prices intraday can be attributed to the routine of the
market taking the day out.

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Friend or Foe?
Theta can be agood thing or abad thing, depending on the position. Theta
hurts long option positions; whereas it helps short option positions. Take an
80-strike call with atheoretical value of 3.16 on astock at $82 ashare. The
32-day 80 call has atheta of 0.03. If atrader owned one of these calls, the
trader’sposition would theoretically lose 0.03, or $0.03, as the time until
expiration change from 32 to 31 days. This trader has anegative theta
position. Atrader short one of these calls would have an overnight
theoretical profit of $0.03 attributed to theta. This trader would have apositive theta.
Theta affects put traders as well. Using all the same modeling inputs, the
32-day 80-strike put would have atheta of 0.02. Aput holder would
theoretically lose $0.02 aday, and aput writer would theoretically make
$0.02. Long options carry with them negative theta; short options carry
positive theta.
Ahigher theta for the call than for the put of the same strike price is
common when an interest rate greater than zero is used in the pricing
model. As will be discussed in greater detail in the section on rho, interest
causes the time value of the call to be higher than that of the corresponding
put. At expiration, there is no time value left in either option. Because the
call begins with more time value, its premium must decline at afaster rate
than that of the put. Most modeling software will attribute the disparate
rates of decline in value all to theta, whereas some modeling interfaces will
make clear the distinction between the effect of time decay and the effect of
interest on the put-call pair.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:81


The Effect of Moneyness and Stock Price
on Theta
Theta is not aconstant. As variables influencing option values change, theta
can change, too. One such variable is the option’smoneyness. Exhibit 2.10
shows theoretical values (theos), time values, and thetas for 3-month
options on Adobe (ADBE). In this example, Adobe is trading at $31.30 ashare with three months until expiration. The more ITM acall or aput gets,
the higher its theoretical value. But when studying an option’stime decay,
one needs to be concerned only with the option’stime value, because
intrinsic value is not subject to time decay.
EXHIBIT 2.10 Adobe theos and thetas (Adobe at $31.30).
The ATM options shown here have higher time value than ITM or OTM
options. Hence, they have more time premium to lose in the same three-
month period. ATM options have the highest rate of decay, which is
reflected in higher thetas. As the stock price changes, the theta value will
change to reflect its change in moneyness.
If this were ahigher-priced stock, say, 10 times the stock price used in
this example, with all other inputs held constant, the option values, and
therefore the thetas, would be higher. If this were astock trading at $313,
the 325-strike call would have atheoretical value of 16.39 and aone-day

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:82