-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
position or it expires. European-exercise options can be exercised and
assigned only at expiration. Exchange-listed equity options are all
American-exercise style. Other kinds of options are commonly European
exercise. Whether an option is American or European has nothing to with
the country in which it’s listed.
ETFs, Indexes, and HOLDRs
So far, we’ve focused on equity options—options on individual stocks. But
investors have other choices for trading securities options. Options on
baskets of stocks can be traded, too. This can be accomplished using
options on exchange-traded funds (ETFs), index options, or options on
holding company depositary receipts (HOLDRs).
ETF Options
Exchange-traded funds are vehicles that represent ownership in a fund or
investment trust. This fund is made up of a basket of an underlying index’s
securities—usually equities. The contract specifications of ETF options are
similar to those of equity options. Let’s look 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 a long 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.
Index Options
Trading options on the Spiders ETF is a convenient way to trade the
Standard & Poor’s (S&P) 500. But it’s not the only way. There are other
option contracts listed on the S&P 500. The SPX is one of the major ones.
The SPX is an index option contract. There are some very important
differences between ETF options like SPY and index options like SPX.
The first difference is the underlying. The underlying for ETF options is
100 shares of the ETF. The underlying for index options is the numerical
value of the index. So if the S&P 500 is at 1303.50, the underlying for SPX
options is 1303.50. When an SPX call option is exercised, instead of getting
100 shares of something, the exerciser gets the ITM cash value of the
option times $100. Again, with SPX at 1303.50, if a 1300 call is exercised,
the exerciser gets $350—that’s 1303.50 minus 1300, times $100. This is
called cash settlement .
Many index options are European, which means no early exercise. At
expiration, any long ITM options in a trader’s inventory result in an account
credit; any short ITMs result in a debit of the ITM value times $100. The
settlement process for determining whether a European-style index option is
in-the-money at expiration is a little different, too. Often, these indexes are
a.m. settled. A.m.-settled index options will have actual expiration on the
conventional Saturday following the third Friday of the month. But the final
trading day is the Thursday before the expiration day. The final settlement
value of the index is determined by the opening prices of the components of
the index on Friday morning.
HOLDR Options
Like ETFs, holding company depositary receipts also represent ownership
in a basket of stocks. The main difference is that investors owning
HOLDRs retain the ownership rights of the individual stocks in the fund,
such as the right to vote shares and the right to receive dividends. Options
on HOLDRs, for all intents and purposes, function much like options on
ETFs.
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 a
call. And there are complex spreads with creative names like jelly roll, guts,
and iron butterfly. A spread is a strategy that involves combining an option
with one or more other options or stock. Each component of the spread is
referred to as a leg. 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’s important to understand that all spreads are made up of a
combination 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’s helpful to see what the option’s payout is if it is held until
expiration.
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’s forecast
and motivations.
Consider a long call example:
Buy 1 INTC June 22.50 call at 0.85.
In this example, a trader is bullish on Intel (INTC). He believes Intel will
rise at least 20 percent, from $22.25 per share to around $27 by June
expir


tock? 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’s forecast
and motivations.
Consider a long call example:
Buy 1 INTC June 22.50 call at 0.85.
In this example, a trader 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—a total investment of $2,225—the trader buys 1 INTC
June 22.50 call at 0.85, for a total 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’s break-even price. The break-
even price for a long 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.
EXHIBIT 1.1 Long Intel call.
Exhibit 1.1 is an at-expiration diagram for the Intel 22.50 call. It shows
the profit and loss, or P&(L), of the option if it is held until expiration. The
X-axis represents the prices at which INTC could be trading at expiration.
The Y-axis represents the associated profit or loss on the position. The at-
expiration diagram of any long call position will always have this same
hockey-stick shape, regardless of the stock or strike. There is always a limit
of loss, represented by the horizontal line, which in this case is drawn at
−0.85. And there is always a line extending upward and to the right, which
represents effectively a long stock position stemming from the strike.
The trade-offs between a long stock position and a long call position are
shown in Exhibit 1.2 .
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 a
cost. 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 a call
rather than the associated stock and sometimes buy the stock rather than the
call. Buying a call can be considered more conservative when the volatility
of the stock is expected to rise. Traders are willing to risk a comparatively
small premium when a large 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
([$27 − 23.35] / $0.85). If Intel is below the strike price at expiration, the
trader loses 100 percent.
This makes call buying an excellent speculative alternative. Those willing
to accept bigger risk can further increase returns by purchasing more calls.
In this example, around 26 Intel calls—representing the rights on 2,600
shares—can be purchased at 85 cents for the cost of 100 shares at $22.25.
This is the kind of leverage that allows for either a lower cash outlay than
buying the stock—reducing risk—or the same cash outlay as buying the
stock but with much greater exposure—creating risk in pursuit of higher
returns.
Sell Call
Selling a call creates the obligation to sell the stock at the strike price. Why
is a trader willing to accept this obligation? The answer is option premium.
If the position is held until expiration without getting assigned, the entire
premium represents a profit for the trader. If assignment occurs, the trader
will be obliged to sell stock at the strike price. If the trader does not have a
long position in the underlying stock (a naked call), a short stock position
will be created. Otherwise, if stock is owned (a covered call), that stock is
sold. Whether the trader has a profit or a loss depends on the movement of
the stock price and how the short call position was constructed.
Consider a naked call example:
Sell 1 TGT October 50 call at 1.45
In this example, Target Corporation (TGT) is trading at $49.42. A trader,
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 a short position in that series. Exhibit 1.3 will help
explain the expected payout of this naked call position if it is held until
expiration.
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

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


ion, about two months from now. Sam sells 1 Target two-month 50
call at 1.45, opening a short position in that series. Exhibit 1.3 will help
explain the expected payout of this naked call position if it is held until
expiration.
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 a seller, 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 a profit. Here, the break-even price is $51.45
—the strike price plus the call premium. Above the break-even, Sam has a
loss. Since stock prices can rise to infinity (although, for the record, I have
never seen this happen), the naked call position has unlimited risk of loss.
Because a short stock position may be created, a naked call position must
be done in a margin 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 a high margin
requirement.
Another tactical consideration is what Sam’s objective was when he
entered the trade. His goal was to profit from the stock’s being below $50
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, a short stock position cannot
always be avoided. If assigned, the short stock position will extend Sam’s
period of risk—because stock doesn’t expire. 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 a call option offers limited risk and unlimited
reward. Selling a naked 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 a different strategy.
Now consider a covered 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’t sit well together with many
traders. For that reason, traders often prefer to sell calls as part of a spread.
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 a covered write or a buy-write ). While selling a call naked is a way
to take advantage of a “not bullish” forecast, the covered call achieves a
different set of objectives.
After studying Target Corporation, another trader, Isabel, has a neutral 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.
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 a long 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 a loser.
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’s profit 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 a loser. If the stock rises too much, the stock is sold prematurely and
upside opportunity is lost. Limited reward and unlimited risk. (Technically,
the risk is not unlimited—the stock can only go to zero. But if the stock
drops from $49.42 to zero in a short time, the risk will certainly feel
unlimited.) The covered call strategy is for a neutral to moderately bullish
outlook.
Sell Put
Selling a put 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 a put:
Sell 1 BA January 65 put at 1.20
In this example, trader Sam is neutral to moderately bullish on Boeing (BA)
between n


l to moderately bullish
outlook.
Sell Put
Selling a put 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 a put:
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’d gladly 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
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 a trader short a put and willingly assume this substantial risk
with comparatively limited reward? There are a number of motivations that
may warrant the short put strategy. In this example, Sam had the twin goals
of profiting from a neutral 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, a strike 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 a target price at which he would buy the stock. Why not use a limit
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.
A consideration 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 a cash-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’s value.
Some traders sell puts without the intent of ever owning the stock. They
hope to profit from a low-volatility environment. Just as the short call is a
not-bullish stance on the underlying, the short put is a not-bearish play. As
long as the underlying is above the strike price at expiration, the option
premium is all profit. The trader must actively manage the position for fear
of being assigned. Buying the put back to close the position eliminates the
risk of assignment.
Buy Put
Buying a put gives the holder the right to sell stock at the strike price. Of
course, puts can be a part of a host 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 a way to speculate on a bearish
move in the underlying security, and the protective put is a way to protect a
long position in the underlying security.
Consider a long put example:
Buy 1 SPY May 139 put at 2.30
In this example, the Spiders have had a good 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 a short
position in the Spiders at an effective price of $136.70 per share. This price
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 a profit. Profits will increase on a tick-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 a short position in the underlying may not be as attractive to
her as a long put. The margin requirements for short stock are significantly
higher than for a long 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 a very real consideration for a trader
deciding between shorting stock and buying a put. If the trader expects high
volatility, he or she may be more inclined to limit upside risk while
leveraging downside


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 a very real consideration for a trader
deciding between shorting stock and buying a put. If the trader expects high
volatility, he or she may be more inclined to limit upside risk while
leveraging downside profit potential by buying a put. In general, traders buy
options when they expect volatility to increase and sell them when they
expect volatility to decrease. This will be a common theme throughout this
book.
Consider a protective put example:
This is an example of a situation 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 a put because she was bearish on the Spiders, a
different trader, Kathleen, may buy a put for a different reason—she’s
bullish 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 a spread, the position would be called a married
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.
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 a loss 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 a stock 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 a concern, Kathleen has a
choice 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
declined in price, the SPY shares, and one that has risen in price, the put.
Losses on the ETF shares are to some degree offset by gains on the put.
Measuring Incremental Changes in
Factors Affecting Option Prices
At-expiration diagrams are very helpful in learning how a particular option
strategy works. They show what the option’s price will ultimately be at
various prices of the underlying. There is, however, a caveat when using at-
expiration diagrams. According to the Options Industry Council, most
options are closed before they reach expiration. Traders not planning to
hold an option until it expires need to have a way to develop reasonable
expectations as to what the option’s price will be given changes that can
occur in factors affecting the option’s price. The tool option traders use to
aid them in this process is option greeks.
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 a pair 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 a good price. She noticed, though, that someone else was
selling the same shoes with a buy-it-now price of $49—a good-enough
price in her opinion. Kathleen was effectively afforded a call option. She
had the opportunity to buy the shoes at (the strike price of) $49, a right 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 a coupon 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 a discount on the price paid for the shoes—maybe a big 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.
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 a trader’s
willingness to demand (desire to buy) or supply (desire to sell) an option at
a given price. For example, a trader would rather own—that i


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 a trader’s
willingness to demand (desire to buy) or supply (desire to sell) an option at
a given price. For example, a trader would rather own—that is, there would
be higher demand for—an option that has more time until expiration than a
shorter-dated option, all else held constant. And a trader would rather own a
call with a lower strike than a higher strike, all else kept constant, because it
would give the right to buy at a lower 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 a paper 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 a widely 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’s theoretical
value. For American-exercise equity options, six inputs are entered into any
option-pricing model to generate a theoretical value: stock price, strike
price, time until expiration, interest rate, dividends, and volatility.
Theoretical value—what a concept! A trader plugs six numbers into a
pricing model, and it tells him what the option is worth, right? Well, in
practical terms, that’s not 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 a free 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 a pricing 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 a minimum—which is still quite a bit. The focus of
this book is on practical applications, not academic theory. It’s about
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 a trader
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’s because 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 a cumulative or net
effect on the option’s value. 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.
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 a derivation of an option-pricing model, and each Greek
letter represents a specific sensitivity to influences on the option’s value. 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 a change 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’s value relative
to a change in the price of the underlying security. A trader who is bullish
on a particular stock may choose to buy a call 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 a percentage. 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 a whole number, without the percent sign, or as a
decimal. 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


ll
change by 50 percent of the change of the underlying stock price. Delta is
generally written as either a whole number, without the percent sign, or as a
decimal. 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 a simplified example of the effect of delta on
an option:
Consider a $60 stock with a call 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 a negative correlation to the underlying. That is, put values
decrease when the stock price rises and vice versa. Puts, therefore, have
negative deltas. Here is a simplified 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 a bit 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 a call option with three months to expiration at a variable 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 .
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 a round lot of 100 shares. A call with a
0.60 delta rises by $0.60 with a $1 increase in the stock. The owner of a call
representing rights on 100 shares earns $60 for a $1 increase in the
underlying. It’s as if the call owner in this example is long 60 shares of the
underlying stock. Delta is the option’s equivalent of a position in the
underlying shares .
A trader who buys five 0.43-delta calls has a position 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 a total of 590 deltas, a position 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’s
definition. It’s mathematically imprecise but is used nonetheless as a
general rule of thumb by option traders. A trader would say the delta is a
statistical approximation of the likelihood of the option expiring in-the-
money . An option with a 0.75 delta would have a 75 percent chance of
being in-the-money at expiration under this definition. An option with a
0.20 delta would be thought of having a 20 percent chance of expiring in-
the-money.
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’s discuss a few observations
about the characteristics of delta.
First, call and put deltas are closely related. Exhibit 2.2 is a partial 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 a general 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 a mathematical
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’t always 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 a bit higher than the call delta would
imply. When puts have a greater chance of early exercise, they begin to act
more like short stock and consequently will have a greater delta. Often,
dividend-paying stocks


xhibit 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 a bit higher than the call delta would
imply. When puts have a greater chance of early exercise, they begin to act
more like short stock and consequently will have a greater delta. Often,
dividend-paying stocks will have higher deltas on some in-the-money calls
than the put in the pair would imply. As the ex-dividend date—the date the
stock begins trading without the dividend—approaches, an in-the-money
call can become more apt to be exercised, because traders will want to own
stock to capture the dividend. Here, the call begins to act more like long
stock, leading to a higher delta.
Moneyness and Delta
The next observation is the effect of moneyness on the option’s delta.
Moneyness describes the degree to which the option is in- or out-of-the-
money. As a general 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 a value other than 0.50.
Typically, the call has a higher delta than 0.50 and the put has a lower
absolute value than 0.50. Incidentally, the call’s theoretical value is
generally greater than the put’s when 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’s theoretical and
delta will be relative to the put.
Effect of Time on Delta
In a close contest, the last few minutes of a football 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’s in the lead wants the game clock to run
down with no interruption to solidify its position. The team that’s losing
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’s definition can help us
gain insight into how time affects option deltas. The more time left until an
option’s expiration, 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—a coin toss. Exhibit 2.3 shows the deltas of a hypothetical
equity call with a strike 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.

As shown in Exhibit 2.3 , the more time until expiration, the closer ITMs
and OTMs move to 0.50. At expiration, of course, the option is either a 100
delta or a 0 delta; it’s either stock or not.
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’s important 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 a call 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 a low 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 a share, the delta
is 1.00. At that same volatility level with the stock at $42 a share, 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 a higher volatility assumption, and OTM option
deltas are bigger with a higher volatility.
Effect of Stock Price on Delta
An option that is $5 in-the-money on a $20 stock will have a higher delta
than an option that is $5 in-the-money on a $200 stock. Proportionately, the
former is more in-the-money. Comparing two options that are in-the-money
by the same percentage yields similar results.
As the stock price changes because the strike price remains stable, the
option’s delta will change. This phenomenon is measured by the option’s
gamma.
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’s gamma, sometimes called
curvature .
Gamma (Γ) is the rate of change of an option’s delta given a change 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 g


ally 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’s delta 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 a rate of
50 percent—the call’s delta at that stock price. But by the time the stock is
at $61, the option value is increasing at a rate 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’s difficult to calculate the
average delta exactly because gamma is not constant; this is discussed in
more detail later in the chapter. A more realistic example of call values in
relation to the stock price would be as follows:
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 a decreasing 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’s delta and thereby its theoretical
value as the stock continues its decline to $58 and beyond.
Puts work the same way, but because they have a negative delta, when
there is a positive 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 a
short 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’s position. When traders buy options,
they acquire positive gamma. Since gamma causes options to gain value at
a faster rate and lose value at a slower rate, (positive) gamma helps the
option buyer. A trader 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 a slower rate. When the underlying moves
adversely, gamma speeds up losses. Selling options yields a negative
gamma position. A trader 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 a bit, making a big 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 a graph of the delta relative to the stock
price. Exhibit 2.5 illustrates the delta of a call relative to the stock price.
EXHIBIT 2.5 Call delta compared with stock price.
Not only does the delta change, but it changes at a changing rate. Gamma
is not constant. Moneyness, time to expiration, and volatility each have an
effect on the gamma of an option.
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’s a
different story. Its delta can change very quickly.
ITM and OTM options have a low gamma.
ATM options have a relatively 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 a constant 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 a small amount in either direction
from the current price of $44, the movement won’t change their deltas
much at all. The chances of their money status changing between now and
expiration would not be significantly different statistically given a small
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 a slightly higher strike price.) Exhibit 2.7 shows a graph of the
corresponding numbers in Exhi


given a small
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 a slightly higher strike price.) Exhibit 2.7 shows a graph of the
corresponding numbers in Exhibit 2.6 .
EXHIBIT 2.7 Option gamma.
A decrease in the time to expiration solidifies the likelihood of ITMs or
OTMs remaining as such. But an ATM option’s moneyness at expiration
remains to the very end uncertain. As expiration draws nearer, the gamma
decreases for ITMs and OTMs and increases for the ATM strikes. Exhibit
2.8 shows the same 92-day QQQ calls plotted against 7-day QQQ calls.
EXHIBIT 2.8 Gamma as time passes.

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 a graph
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.
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’s left 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’s price given a unit 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’s worth of time decay. Some will show thetas representing seven days
of decay. In the case of a one-day theta, the figure may be based on a seven-
day week or on a week 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 a week, each day of
which can see an occurrence with the potential to cause a revaluation in the
stock price (that is, news can come out on Saturday or Sunday). The one-
day theta based on a seven-day week will be used throughout this book.
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’s time. 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’s days 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 a
change in some other input, such as volatility? To some degree, it doesn’t
matter. Remember, the model is used to reflect what the market is doing,
not the other way around. In many cases, it’s logical to presume that small
devaluations in option prices intraday can be attributed to the routine of the
market taking the day out.
Friend or Foe?
Theta can be a good thing or a bad thing, depending on the position. Theta
hurts long option positions; whereas it helps short option positions. Take an
80-strike call with a theoretical value of 3.16 on a stock at $82 a share. The
32-day 80 call has a theta of 0.03. If a trader owned one of these calls, the
trader’s position would theoretically lose 0.03, or $0.03, as the time until
expiration change from 32 to 31 days. This trader has a negative theta
position. A trader short one of these calls would have an overnight
theoretical profit of $0.03 attributed to theta. This trader would have a
positive theta.
Theta affects put traders as well. Using all the same modeling inputs, the
32-day 80-strike put would have a theta of 0.02. A put holder would
theoretically lose $0.02 a day, and a put writer would theoretically make
$0.02. Long options carry with them negative theta; short options carry
positive theta.
A higher 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 begin


itive theta.
A higher 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 a faster 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.
The Effect of Moneyness and Stock Price
on Theta
Theta is not a constant. As variables influencing option values change, theta
can change, too. One such variable is the option’s moneyness. 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 a
share with three months until expiration. The more ITM a call or a put gets,
the higher its theoretical value. But when studying an option’s time decay,
one needs to be concerned only with the option’s time 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 a higher-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 a stock trading at $313,
the 325-strike call would have a theoretical value of 16.39 and a one-day
theta of 0.189, given inputs used otherwise identical to those in the Adobe
example.
The Effects of Volatility and Time on
Theta
Stock price is not the only factor that affects theta values. Volatility and
time to expiration come into play here as well. The volatility input to the
pricing model has a direct relationship to option values. The higher the
volatility, the higher the value of the option. Higher-valued options decay at
a faster rate than lower-valued options—they have to; their time values will
both be zero at expiration. All else held constant, the higher the volatility
assumption, the higher the theta.
The days to expiration have a direct relationship to option values as well.
As the number of days to expiration decreases, the rate at which an option
decays may change, depending on the relationship of the stock price to the
strike price. ATM options tend to decay at a nonlinear rate—that is, they
lose value faster as expiration approaches—whereas the time values of ITM
and OTM options decay at a steadier rate.
Consider a hypothetical stock trading at $70 a share. Exhibit 2.11 shows
how the theoretical values of the 75-strike call and the 70-strike call decline
with the passage of time, holding all other parameters constant.
EXHIBIT 2.11 Rate of decay: ATM vs. OTM.

The OTM 75-strike call has a fairly steady rate of time decay over this
26-week period. The ATM 70-strike call, however, begins to lose its value
at an increasing rate as expiration draws nearer. The acceleration of
premium erosion continues until the option expires. Exhibit 2.12 shows the
thetas for this ATM call during the last 10 days before expiration.
EXHIBIT 2.12 Theta as expiration approaches.
Days to Exp .ATM Theta
10 0.075
9 0.079
8 0.084
7 0.089
6 0.096
5 0.106
4 0.118
3 0.137
2 0.171
1 0.443
Incidentally, in this example, when there is one day to expiration, the
theoretical value of this call is about 0.44. The final day before expiration
ultimately sees the entire time premium erode.
Vega
Over the past decade or so, computers have revolutionized option trading.
Options traded through an online broker are filled faster than you can say,
“Oops! I meant to click on puts.” Now trading is facilitated almost entirely
online by professional and retail traders alike. Market and trading
information is disseminated worldwide in subseconds, making markets all
the more efficient. And the tools now available to the common retail trader
are very powerful as well. Many online brokers and other web sites offer
high-powered tools like screeners, which allow traders to sift through
thousands of options to find those that fit certain parameters.
Using a screener to find ATM calls on same-priced stocks—say, stocks
trading at $40 a share—can yield a result worth talking about here. One $40
stock can have a 40-strike call trading at around 0.50, while a different $40
stock can have a 40 call with the same time to expiration trading at more
like 2.00. Why? The model doesn’t know the name of the company, what
industry it’s in, or what its price-to-earnings ratio is. It is a mathematical
equation with six inputs. If five of the inputs—the stock price, strike price,
time to expiration, interest rate, and dividends—are identical for two
different options but they’re trading at different prices, the difference must
be the sixth variable, which is volatility.
Implied Volatility (IV) and Vega
The volatility component of option values is called implied volatility (IV).
(For more on implied volatility and how it relates to vega, see Chapter 3.)
IV is a percentage, although in practice the percent sign is often omitted.
This is the value entered into a pricing model, in conjunction with the other
variables, that returns the option’s theoretical value. The higher the
volatility input, the higher the theoretical value, holding all other variables
constant. The IV level can change and often does—sometimes dramatically.
When IV rises or falls, option prices rise


although in practice the percent sign is often omitted.
This is the value entered into a pricing model, in conjunction with the other
variables, that returns the option’s theoretical value. The higher the
volatility input, the higher the theoretical value, holding all other variables
constant. The IV level can change and often does—sometimes dramatically.
When IV rises or falls, option prices rise and fall in line with it. But by how
much?
The relationship between changes in IV and changes in an option’s value
is measured by the option’s vega. Vega is the rate of change of an option’s
theoretical value relative to a change in implied volatility . Specifically, if
the IV rises or declines by one percentage point, the theoretical value of the
option rises or declines by the amount of the option’s vega, respectively.
For example, if a call with a theoretical value of 1.82 has a vega of 0.06 and
IV rises one percentage point from, say, 17 percent to 18 percent, the new
theoretical value of the call will be 1.88—it would rise by 0.06, the amount
of the vega. If, conversely, the IV declines 1 percentage point, from 17
percent to 16 percent, the call value will drop to 1.76—that is, it would
decline by the vega.
A put with the same expiration month and the same strike on the same
underlying will have the same vega value as its corresponding call. In this
example, raising or lowering IV by one percentage point would cause the
corresponding put value to rise or decline by $0.06, just like the call.
An increase in IV and the consequent increase in option value helps the
P&(L) of long option positions and hurts short option positions. Buying a
call or a put establishes a long vega position. For short options, the opposite
is true. Rising IV adversely affects P&(L), whereas falling IV helps.
Shorting a call or put establishes a short vega position.
The Effect of Moneyness on Vega
Like the other greeks, vega is a snapshot that is a function of multiple facets
of determinants influencing option value. The stock price’s relationship to
the strike price is a major determining factor of an option’s vega. IV affects
only the time value portion of an option. Because ATM options have the
greatest amount of time value, they will naturally have higher vegas. ITM
and OTM options have lower vega values than those of the ATM options.
Exhibit 2.13 shows an example of 186-day options on AT&T Inc. (T),
their time value, and the corresponding vegas.
EXHIBIT 2.13 AT&T theos and vegas (T at $30, 186 days to Expry, 20%
IV).
Note that the 30-strike calls and puts have the highest time values. This
strike boasts the highest vega value, at 0.085. The lower the time premium,
the lower the vega—therefore, the less incremental IV changes affect the
option. Since higher-priced stocks have higher time premium (in absolute
terms, not necessarily in percentage terms) they will have higher vega.
Incidentally, if this were a $300 stock instead of a $30 stock, the 186-day
ATMs would have a 0.850 vega, if all other model inputs remain the same.
The Effect of Implied Volatility on Vega
The distribution of vega values among the strike prices shown in Exhibit
2.13 holds for a specific IV level. The vegas in Exhibit 2.13 were calculated
using a 20 percent IV. If a different IV were used in the calculation, the
relationship of the vegas to one another might change. Exhibit 2.14 shows
what the vegas would be at different IV levels.
EXHIBIT 2.14 Vega and IV.
Note in Exhibit 2.14 that at all three IV levels, the ATM strike maintains a
similar vega value. But the vegas of the ITM and OTM options can be
significantly different. Lower IV inputs tend to cause ITM and OTM vegas
to decline. Higher IV inputs tend to cause vegas to increase for ITMs and
OTMs.
The Effect of Time on Vega
As time passes, there is less time premium in the option that can be affected
by changes in IV. Consequently, vega gets smaller as expiration approaches.
Exhibit 2.15 shows the decreasing vega of a 50-strike call on a $50 stock
with a 25 percent IV as time to expiration decreases. Notice that as the
value of this ATM option decreases at its nonlinear rate of decay, the vega
decreases in a similar fashion.
EXHIBIT 2.15 The effect of time on vega.

Rho
One of my early jobs in the options business was clerking on the floor of
the Chicago Board of Trade in what was called the bond room. On one of
my first days on the job, the trader I worked for asked me what his position
was in a certain strike. I told him he was long 200 calls and long 300 puts.
“I’m long 500 puts?” he asked. “No,” I corrected, “you’re long 200 calls
and 300 puts.” At this point, he looked at me like I was from another planet
and said, “That’s 500. A put is a call; a call is a put.” That lesson was the
beginning of my journey into truly understanding options.
Put-Call Parity
Put and call values are mathematically bound together by an equation
referred to as put-call parity. In its basic form, put-call parity states:
where
c = call value,
PV(x) = present value of the strike price,
p = put value, and
s = stock price.
The put-call parity assumes that options are not exercised before
expiration (that is, that they are European style). This version of the put-call
parity is for European options on non-dividend-paying stocks. Put-call
parity can be modified to reflect the values of options on stocks that pay
dividends. In practice, equity-option traders look at the equation in a
slightly different way:
Traders serious about learning to trade options must know put-call parity
backward and forward. Why? First, by algebraically rearranging this
equation, it can be inferred that synthetically equivalent positions can be
established by simply adding stock to an option. Again, a put is a call; a call
is a put.
and
For example, a long call is synthetically equal to a long stock position
plus a long put on the same strike, once interest and dividends are figured
in. A synthetic long stock position is created by buyi


braically rearranging this
equation, it can be inferred that synthetically equivalent positions can be
established by simply adding stock to an option. Again, a put is a call; a call
is a put.
and
For example, a long call is synthetically equal to a long stock position
plus a long put on the same strike, once interest and dividends are figured
in. A synthetic long stock position is created by buying a call and selling a
put of the same month and strike. Understanding synthetic relationships is
intrinsic to understanding options. A more comprehensive discussion of
synthetic relationships and tactical considerations for creating synthetic
positions is offered in Chapter 6.
Put-call parity also aids in valuing options. If put-call parity shows a
difference in the value of the call versus the value of the put with the same
strike, there may be an arbitrage opportunity. That translates as “riskless
profit.” Buying the call and selling it synthetically (short put and short
stock) could allow a profit to be locked in if the prices are disparate.
Arbitrageurs tend to hold synthetic put and call prices pretty close together.
Generally, only professional traders can capture these types of profit
opportunities, by trading big enough positions to make very small profits (a
penny or less per contract sometimes) matter. Retail traders may be able to
take advantage of a disparity in put and call values to some extent, however,
by buying or selling the synthetic as a substitute for the actual option if the
position can be established at a better price synthetically.
Another reason that a working knowledge of put-call parity is essential is
that it helps attain a better understanding of how changes in the interest rate
affect option values. The greek rho measures this change. Rho is the rate of
change in an option’s value relative to a change in the interest rate.
Although some modeling programs may display this number differently,
most display a rho for the call and a rho for the put, both illustrating the
sensitivity to a one-percentage-point change in the interest rate. When the
interest rate rises by one percentage point, the value of the call increases by
the amount of its rho and the put decreases by the amount of its rho.
Likewise, when the interest rate decrease by one percentage point, the value
of the call decreases by its rho and the put increases by its rho. For example,
a call with a rho of 0.12 will increase $0.12 in value if the interest rate used
in the model is increased by one percentage point. Of course, interest rates
usually don’t rise or fall one percentage point in one day. More commonly,
rates will have incremental changes of 25 basis points. That means a call
with a 0.12 rho will theoretically gain $0.03 given an increase of 0.25
percentage points.
Mathematically, this change in option value as a product of a change in
the interest rate makes sense when looking at the formula for put-call parity.
and

But the change makes sense intuitively, too, when a call is considered as a
cheaper substitute for owning the stock. For example, compare a $100 stock
with a three-month 60-strike call on that same stock. Being so far ITM,
there would likely be no time value in the call. If the call can be purchased
at parity, which alternative would be a superior investment, the call for $40
or the stock for $100? Certainly, the call would be. It costs less than half as
much as the stock but has the same reward potential; and the $60 not spent
on the stock can be invested in an interest-bearing account. This interest
advantage adds value to the call. Raising the interest rate increases this
value, and lowering it decreases the interest component of the value of the
call.
A similar concept holds for puts. Professional traders often get a short-
stock rebate on proceeds from a short-stock sale. This is simply interest
earned on the capital received when the stock is shorted. Is it better to pay
interest on the price of a put for a position that gives short exposure or to
receive interest on the credit from shorting the stock? There is an interest
disadvantage to owning the put. Therefore, a rise in interest rates devalues
puts.
This interest effect becomes evident when comparing ATM call and put
prices. For example, with interest at 5 percent, three-month options on an
$80 stock that pays a $0.25 dividend before option expiration might look
something like this:
The ATM call is higher in theoretical value than the ATM put by $0.75.
That amount can be justified using put-call parity:
(Here, simple interest of $1 is calculated as 80 × 0.05 × [90 / 360] = 1.)
Changes in market conditions are kept in line by the put-call parity. For
example, if the price of the call rises because of an increase in IV, the price
of the put will rise in step. If the interest rate rises by a quarter of a
percentage point, from 5 percent to 5.25 percent, the interest calculated for
three months on the 80-strike will increase from $1 to $1.05, causing the
difference between the call and put price to widen. Another variable that
affects the amount of interest and therefore option prices is the time until
expiration.
The Effect of Time on Rho
The more time until expiration, the greater the effect interest rate changes
will have on options. In the previous example, a 25-basis-point change in
the interest rate on the 80-strike based on a three-month period caused a
change of 0.05 to the interest component of put-call parity. That is, 80 ×
0.0025 × (90/360) = 0.05. If a longer period were used in the example—say,
one year—the effect would be more profound; it will be $0.20: 80 × 0.0025
× (360/360) = 0.20. This concept is evident when the rhos of options with
different times to expiration are studied.
Exhibit 2.16 shows the rhos of ATM Procter & Gamble Co. (PG) calls
with various expiration months. The 750-day Long-Term Equity
AnticiPation Securities (LEAPS) have a rho of 0.858. As the number of
days until expiration decreases, rho


be more profound; it will be $0.20: 80 × 0.0025
× (360/360) = 0.20. This concept is evident when the rhos of options with
different times to expiration are studied.
Exhibit 2.16 shows the rhos of ATM Procter & Gamble Co. (PG) calls
with various expiration months. The 750-day Long-Term Equity
AnticiPation Securities (LEAPS) have a rho of 0.858. As the number of
days until expiration decreases, rho decreases. The 22-day calls have a rho
of only 0.015. Rho is usually a fairly insignificant factor in the value of
short-term options, but it can come into play much more with long-term
option strategies involving LEAPS.
EXHIBIT 2.16 The effect of time on rho (Procter & Gamble @ $64.34)

Why the Numbers Don’t Don’t Always
Add Up
There will be many times when studying the rho of options in an option
chain will reveal seemingly counterintuitive results. To be sure, the numbers
don’t always add up to what appears logical. One reason for this is
rounding. Another is that traders are more likely to use simple interest in
calculating value, whereas the model uses compound interest. Hard-to-
borrow stocks and stocks involved in mergers and acquisitions may have
put-call parities that don’t work out right. But another, more common and
more significant fly in the ointment is early exercise.
Since the interest input in put-call parity is a function of the strike price, it
is reasonable to expect that the higher the strike price, the greater the effect
of interest on option prices will be. For European options, this is true to a
large extent, in terms of aggregate impact of interest on the call and put pair.
Strikes below the price where the stock is trading have a higher rho
associated with the call relative to the put, whereas strikes above the stock
price have a higher rho associated with the put relative to the call.
Essentially, the more in-the-money an option is, the higher its rho. But with
European options, observing the aggregate of the absolute values of the call
and put rhos would show a higher combined rho the higher the strike.
With American options, the put can be exercised early. A trader will
exercise a put before expiration if the alternative—being short stock and
receiving a short stock rebate—is a wiser choice based on the price of the
put. Professional traders may own stock as a hedge against a put. They may
exercise deep ITM puts (1.00-delta puts) to avoid paying interest on capital
charges related to the stock. The potential for early exercise is factored into
models that price American options. Here, when puts get deeper in-the-
money—that is, more apt to be exercised—the rho decreases. When the
strike price is very high relative to the stock price—meaning the put is very
deep ITM—and there is little or no time value left to the call or the put, the
aggregate put-call rho can be zero. Rho is discussed in greater detail in
Chapter 7.
THE GREEKS DEFINED
Delta (Δ) is:
1. The rate of change in an option’s value relative to a change in the underlying asset
price.
2. The derivative of the graph of an option’s value in relation to the underlying asset
price.
3. The equivalent of underlying asset represented by an option position.
4. The estimate of the likelihood of an option’s expiring in-the-money.
Gamma (Γ) is the rate of change in an option’s delta given a change in the price of the
underlying asset.
Theta (θ) is the rate of change in an option’s value given a unit change in the time to
expiration.
Vega is the rate of change in an option’s value relative to a change in implied volatility.
Rho (ρ) is the rate of change in an option’s value relative to a change in the interest rate.
Where to Find Option Greeks
There are many sources from which to obtain greeks. The Internet is an
excellent resource. Googling “option greeks” will display links to over four
million web pages, many of which have real-time greeks or an option
calculator. An option calculator is a simple interface that accepts the input
of the six variables to the model and yields a theoretical value and the
greeks for a single option.
Some web sites devoted to option education, such as
MarketTaker.com/option_modeling , have free calculators that can be used
for modeling positions and using the greeks.
In practice, many of the option-trading platforms commonly in use have
sophisticated analytics that involve greeks. Most options-friendly online
brokers provide trading platforms that enable traders to conduct
comprehensive manipulations of the greeks. For example, traders can look
at the greeks for their positions up or down one, two, or three standard
deviations. Or they can see what happens to their position greeks if IV or
time changes. With many trading platforms, position greeks are updated in
real time with changes in the stock price—an invaluable feature for active
traders.
Caveats with Regard to Online
Greeks
Often, online greeks are one click away, requiring little effort on the part of
the trader. Having greeks calculated automatically online is a quick and
convenient way to eyeball greeks for an option. But there is one major
problem with online greeks: reliability.
For active option traders, greeks are essential. There is no point in using
these figures if their accuracy cannot be assured. Experienced traders can
often spot these inaccuracies a proverbial mile away.
When looking at greeks from an online source that does not require you
to enter parameters into a model (as would be the case with professional
option-trading platforms), special attention needs to be paid to the
relationship of the option’s theoretical values to the bid and offer. One must
be cautious if the theoretical value of the option lies outside the bid-ask
spread. This scenario can exist for brief periods of time, but arbitrageurs
tend to prevent this from occurring routinely. If several options in a chain
all have theoretical values below the bid or above the offer, there is
probably a problem with one or more of the inputs used in the model.
R


o the bid and offer. One must
be cautious if the theoretical value of the option lies outside the bid-ask
spread. This scenario can exist for brief periods of time, but arbitrageurs
tend to prevent this from occurring routinely. If several options in a chain
all have theoretical values below the bid or above the offer, there is
probably a problem with one or more of the inputs used in the model.
Remember, an option-pricing model is just that: a model. It reflects what is
occurring in the market. It doesn’t tell where an option should be trading.
The complex changes that occur intraday in the market—taking the day
or weekend out, changes in stock price, volatility, and the interest rate—are
not always kept current. The user of the model must keep close watch. It’s
not reasonable to expect the computer to do the thinking for you.
Automatically calculated greeks can be used as a starting point. But before
using these figures in the decision-making process, the trader may have to
override the parameters that were used in the online calculation to make the
theos line up with market prices. Professional traders will ignore online
greeks altogether. They will use the greeks that are products of the inputs
they entered in their trading software. It comes down to this: if you want
something done right, do it yourself.
Thinking Greek
The challenge of trading option greeks is to adapt to thinking in terms of
delta, gamma, theta, vega, and rho. One should develop a feel for how
greeks react to changing market conditions. Greeks need to be monitored as
closely as and in some cases more closely than the option’s price itself. This
greek philosophy forms the foundation of option trading for active traders.
It offers a logical way to monitor positions and provides a medium in and of
itself to trade.
Notes
1 . Please note that definition 4 is not necessarily mathematically accurate.
This “trader’s definition” is included in the text because many option
traders use delta as a quick rule of thumb for estimating probability
without regard to the mathematical shortcomings of doing so.
2 . Note that the interest input in the equation is the interest, in dollars and
cents, on the strike. Technically, this would be calculated as compounded
interest, but in practice many traders use simple interest as a quick and
convenient way to do the calculation.
CHAPTER 3
Understanding Volatility
Most option strategies involve trading volatility in one way or another. It’s
easy to think of trading in terms of direction. But trading volatility?
Volatility is an abstract concept; it’s a different animal than the linear
trading paradigm used by most conventional market players. As an option
trader, it is essential to understand and master volatility.
Many traders trade without a solid understanding of volatility and its
effect on option prices. These traders are often unhappily surprised when
volatility moves against them. They mistake the adverse option price
movements that result from volatility for getting ripped off by the market
makers or some other market voodoo. Or worse, they surrender to the fact
that they simply don’t understand why sometimes these unexpected price
movements occur in options. They accept that that’s just the way it is.
Part of what gets in the way of a ready understanding of volatility is
context. The term volatility can have a few different meanings in the
options business. There are three different uses of the word volatility that an
option trader must be concerned with: historical volatility, implied
volatility, and expected volatility.
Historical Volatility
Imagine there are two stocks: Stock A and Stock B. Both are trading at
around $100 a share. Over the past month, a typical end-of-day net change
in the price of Stock A has been up or down $5 to $7. During that same
period, a typical daily move in Stock B has been something more like up or
down $1 or $2. Stock A has tended to move more than Stock B as a
percentage of its price, without regard to direction. Therefore, Stock A is
more volatile—in the common usage of the word—than Stock B. In the
options vernacular, Stock A has a higher historical volatility than Stock B.
Historical volatility (HV) is the annualized standard deviation of daily
returns. Also called realized volatility, statistical volatility , or stock
volatility , HV is a measure of how volatile the price movement of a
security has been during a certain period of time. But exactly how much
higher is Stock A’s HV than Stock B’s?
In order to objectively compare the volatilities of two stocks, historical
volatility must be quantified. HV relates this volatility information in an
objective numerical form. The volatility of a stock is expressed in terms of
standard deviation.
Standard Deviation
Although knowing the mathematical formula behind standard deviation is
not entirely necessary, understanding the concept is essential. Standard
deviation, sometimes represented by the Greek letter sigma (σ), is a
mathematical calculation that measures the dispersion of data from a mean
value. In this case, the mean is the average stock price over a certain period
of time. The farther from the mean the dispersion of occurrences (data) was
during the period, the greater the standard deviation.
Occurrences, in this context, are usually the closing prices of the stock.
Some utilizers of volatility data may use other inputs (a weighted average
of high, low, and closing prices, for example) in calculating standard
deviation. Close-to-close price data are the most commonly used.
The number of occurrences, a function of the time period, used in
calculating standard deviation may vary. Many online purveyors of this data
use the closing prices from the last 30 consecutive trading days to calculate
HV. Weekends and holidays are not factored into the equation since there is
no trading, and therefore no volatility, when the market isn’t open. After
each day, the oldest price is taken out of the calculation


unction of the time period, used in
calculating standard deviation may vary. Many online purveyors of this data
use the closing prices from the last 30 consecutive trading days to calculate
HV. Weekends and holidays are not factored into the equation since there is
no trading, and therefore no volatility, when the market isn’t open. After
each day, the oldest price is taken out of the calculation and replaced by the
most recent closing price. Using a shorter or longer period can yield
different results and can be useful in studying a stock’s volatility.
Knowing the number of days used in the calculation is crucial to
understanding what the output represents. For example, if the last 5 trading
days were extremely volatile, but the 25 days prior to that were
comparatively calm, the 5-day standard deviation would be higher than the
30-day standard deviation.
Standard deviation is stated as a percentage move in the price of the asset.
If a $100 stock has a standard deviation of 15 percent, a one-standard-
deviation move in the stock would be either $85 or $115—a 15 percent
move in either direction. Standard deviation is used for comparison
purposes. A stock with a standard deviation of 15 percent has experienced
bigger moves—has been more volatile—during the relevant time period
than a stock with a standard deviation of 6 percent.
When the frequency of occurrences are graphed, the result is known as a
distribution curve. There are many different shapes that a distribution curve
can take, depending on the nature of the data being observed. In general,
option-pricing models assume that stock prices adhere to a lognormal
distribution.
The shape of the distribution curve for stock prices has long been the
topic of discussion among traders and academics alike. Regardless of what
the true shape of the curve is, the concept of standard deviation applies just
the same. For the purpose of illustrating standard deviation, a normal
distribution is used here.
When the graph of data adheres to a normal distribution, the result is a
symmetrical bell-shaped curve. Standard deviation can be shown on the bell
curve to either side of the mean. Exhibit 3.1 represents a typical bell curve
with standard deviation.
EXHIBIT 3.1 Standard deviation.
Large moves in a security are typically less frequent than small ones.
Events that cause big changes in the price of a stock, like a company’s
being acquired by another or discovering its chief financial officer cooking
the books, are not a daily occurrence. Comparatively smaller price
fluctuations that reflect less extreme changes in the value of the corporation
are more typically seen day to day. Statistically, the most probable outcome
for a price change is found around the midpoint of the curve. What
constitutes a large move or a small move, however, is unique to each
individual security. For example, a two percent move in an index like the
Standard & Poor’s (S&P) 500 may be considered a big one-day move,
while a two percent move in a particularly active tech stock may be a daily
occurrence. Standard deviation offers a statistical explanation of what
constitutes a typical move.
In Exhibit 3.1 , the lines to either side of the mean represent one standard
deviation. About 68 percent of all occurrences will take place between up
one standard deviation and down one standard deviation. Two- and three-
standard-deviation values could be shown on the curve as well. About 95
percent of data occur between up and down two standard deviations and
about 99.7 percent between up and down three standard deviations. One
standard deviation is the relevant figure in determining historical volatility.
Standard Deviation and Historical
Volatility
When standard deviation is used in the context of historical volatility, it is
annualized to state what the one-year volatility would be. Historical
volatility is the annualized standard deviation of daily returns. This means
that if a stock is trading at $100 a share and its historical volatility is 10
percent, then about 68 percent of the occurrences (closing prices) are
expected to fall between $90 and $110 during a one-year period (based on
recent past performance).
Simply put, historical volatility shows how volatile a stock has been
based on price movements that have occurred in the past. Although option
traders may study HV to make informed decisions as to the value of options
traded on a stock, it is not a direct function of option prices. For this, we
must look to implied volatility.
Implied Volatility
Volatility is one of the six inputs of an option-pricing model. Some of the
other inputs—strike price, stock price, the number of days until expiration,
and the current interest rate—are easily observable. Past dividend policy
allows an educated guess as to what the dividend input should be. But
where can volatility be found?
As discussed in Chapter 2, the output of the pricing model—the option’s
theoretical value—in practice is not necessarily an output at all. When
option traders use the pricing model, they commonly substitute the actual
price at which the option is trading for the theoretical value. A value in the
middle of the bid-ask spread is often used. The pricing model can be
considered to be a complex algebra equation in which any variable can be
solved for. If the theoretical value is known—which it is—it along with the
five known inputs can be combined to solve for the unknown volatility.
Implied volatility (IV) is the volatility input in a pricing model that, in
conjunction with the other inputs, returns the theoretical value of an option
matching the market price.
For a specific stock price, a given implied volatility will yield a unique
option value. Take a stock trading at $44.22 that has the 60-day 45-strike
call at a theoretical value of $1.10 with an 18 percent implied volatility
level. If the stock price remains constant, but IV rises to 19 percent, the
value of the call will rise by its vega, which in this case is ab