For the trade to be profitable, at expiration the stock must be trading above the trader’sbreak-even price. The break-even price for along call is the strike price plus the premium paid—in this example, $23.35 per share. The point here is that if the call is exercised, the effective purchase price of the stock upon exercise is $23.35. The stock is literally bought at the strike price, which is $22.50, but the premium of 0.85 that the trader has paid must be taken into account.
Exhibit 1.1
illustrates this example.
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 alimit of loss, represented by the horizontal line, which in this case is drawn at −0.85. And there is always aline extending upward and to the right, which represents effectively along stock position stemming from the strike.
The trade-offs between along stock position and along 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 acost. If the stock is above the strike at expiration, the call will always underperform the stock by the amount of the premium.
Because of this trade-off, conservative traders will sometimes buy acall rather than the associated stock and sometimes buy the stock rather than the call. Buying acall can be considered more conservative when the volatility of the stock is expected to rise. Traders are willing to risk acomparatively small premium when alarge price decline is feared possible. Instead, in an interest-bearing vehicle, they harbor the capital that would otherwise have been used to purchase the stock. The cost of this protection is acceptable to the trader if high-enough price advances are anticipated. In terms of percentage, much higher returns
and losses
are possible with the long call. If the stock is trading at $27 at expiration, as the trader in this example expected, the trader reaps a 429 percent profit on the $0.85 investment ([$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 alower 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 acall creates the obligation to sell the stock at the strike price. Why is atrader willing to accept this obligation? The answer is option premium. If the position is held until expiration without getting assigned, the entire premium represents aprofit for the trader. If assignment occurs, the trader will be obliged to sell stock at the strike price. If the trader does not have along position in the underlying stock (anaked call), ashort stock position will be created. Otherwise, if stock is owned (acovered call), that stock is sold. Whether the trader has aprofit or aloss depends on the movement of the stock price and how the short call position was constructed.
Consider anaked call example:
Sell 1 TGT October 50 call at 1.45
In this example, Target Corporation (TGT) is trading at $49.42. Atrader, Sam, believes Target will continue to be trading below $50 by October expiration, about two months from now. Sam sells 1 Target two-month 50 call at 1.45, opening ashort position in that series.
Exhibit 1.3
will help explain the expected payout of this naked call position if it is held until expiration.
EXHIBIT 1.3
Naked Target call.
If TGT is trading below the exercise price of 50, the call will expire worthless. Sam keeps the 1.45 premium, and the obligation to sell the stock ceases to exist. If Target is trading above the strike price, the call will be in-the-money. The higher the stock is above the strike price, the more intrinsic value the call will have. As aseller, Sam wants the call to have little or no intrinsic value at expiration. If the stock is below the break-even price at expiration, Sam will still have aprofit. Here


, the call will be in-the-money. The higher the stock is above the strike price, the more intrinsic value the call will have. As aseller, Sam wants the call to have little or no intrinsic value at expiration. If the stock is below the break-even price at expiration, Sam will still have aprofit. Here, the break-even price is $51.45—the strike price plus the call premium. Above the break-even, Sam has aloss. Since stock prices can rise to infinity (although, for the record, Ihave never seen this happen), the naked call position has unlimited risk of loss.
Because ashort stock position may be created, anaked call position must be done in amargin account. For retail traders, many brokerage firms require different levels of approval for different types of option strategies. Because the naked call position has unlimited risk, establishing it will generally require the highest level of approval—and ahigh margin requirement.
Another tactical consideration is what Sam’sobjective was when he entered the trade. His goal was to profit from the stock’sbeing below $50 during this two-month period—not to short the stock. Because equity options are American exercise and can be exercised/assigned any time from the moment the call is sold until expiration, ashort stock position cannot always be avoided. If assigned, the short stock position will extend Sam’speriod of risk—because stock doesn’texpire. Here, he will pay one commission shorting the stock when assignment occurs and one more when he
buys back
the unwanted position. Many traders choose to close the naked call position before expiration rather than risk assignment.
It is important to understand the fundamental difference between buying calls and selling calls. Buying acall option offers limited risk and unlimited reward. Selling anaked call option, however, has limited reward—the call premium—and unlimited risk. This naked call position is not so much bearish as
not bullish
. If Sam thought the stock was going to zero, he would have chosen adifferent strategy.
Now consider acovered call example:
Buy 100 shares TGT at $49.42
Sell 1 TGT October 50 call at 1.45
Unlimited
and
risk
are two words that don’tsit well together with many traders. For that reason, traders often prefer to sell calls as part of aspread. But since spreads are strategies that involve multiple components, they have different risk characteristics from an outright option. Perhaps the most commonly used call-selling spread strategy is the covered call (sometimes called acovered write
or abuy-write
). While selling acall naked is away to take advantage of a “not bullish” forecast, the covered call achieves adifferent set of objectives.
After studying Target Corporation, another trader, Isabel, has aneutral to slightly bullish forecast. With Target at $49.42, she believes the stock will be range-bound between $47 and $51.50 over the next two months, ending with October expiration. Isabel buys 100 shares of Target at $49.42 and sells 1 TGT October 50 call at 1.45. The implications for the covered-call strategy are twofold: Isabel must be content to own the stock at current levels, and—since she sold the right to buy the stock at $50, that is, a 50 call, to another party—she must be willing to sell the stock if the price rises to or through $50 per share.
Exhibit 1.4
shows how this covered call performs if it is held until the call expires.
EXHIBIT 1.4
Target covered call.
The solid kinked line represents the covered call position, and the thin, straight dotted line represents owning the stock outright. At the expiration of the call option, if Target is trading below $50 per share—the strike price—the call expires and Isabel is left with along position of 100 shares
plus
$1.45 per share of expired-option premium. Below the strike, the buy-write always outperforms simply owning the stock by the amount of the premium. The call premium provides limited downside protection; the stock Isabel owns can decline $1.45 in value to $47.97 before the trade is aloser. In the unlikely event the stock collapses and becomes worthless, this limited downside protection is not so comforting. Ultimately, Isabel has $47.97 per share at risk.
The trade-off comes if Target is above $50 at expiration. Here, assignment will likely occur, in which case the stock will be sold. The call can be assigned before expiration, too, causing the stock to be
called away
early. Because the covered call involves this obligation to sell the sock at the strike price, upside potential is limited. In this case, Isabel’sprofit potential is $2.03. The stock can rise from $49.42 to $50—a $0.58 profit—plus $1.45 of option premium.
Isabel does not want the stock to decline too much. Below $47.97, the trade is aloser. If the stock rises too much, the stock is sold prematurely and 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 zer

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stock to decline too much. Below $47.97, the trade is aloser. 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 ashort time, the risk will certainly feel unlimited.) The covered call strategy is for aneutral to moderately bullish outlook.
Sell Put
Selling aput has many similarities to the covered call strategy. We’ll discuss the two positions and highlight the likenesses. Chapter 6 will detail the nuts and bolts of why these similarities exist.
Consider an example of selling aput:
Sell 1 BA January 65 put at 1.20
In this example, trader Sam is neutral to moderately bullish on Boeing (BA) between now and January expiration. He is not bullish enough to buy BA at the current market price of $69.77 per share. But if the shares dropped below $65, he’dgladly scoop some up. Sam sells 1 BA January 65 put at 1.20. The at-expiration diagram in
Exhibit 1.5
shows the P&(L) of this trade if it is held until expiration.
EXHIBIT 1.5
Boeing short put.
At the expiration of this option, if Boeing is above $65, the put expires and Sam retains the premium of $1.20. The obligation to buy stock expires with the option. Below the strike, put owners will be inclined to exercise their option to sell the stock at $65. Therefore, those short the put, as Sam is in this example, can expect assignment. The break-even price for the position is $63.80. That is the strike price minus the option premium. If assigned, this is the effective purchase price of the stock. The obligation to buy at $65 is fulfilled, but the $1.20 premium collected makes the purchase effectively $63.80. Here, again, there is limited profit opportunity ($1.20 if the stock is above the strike price) and seemingly unlimited risk (the risk of potential stock ownership at $63.80) if Boeing is below the strike price.
Why would atrader short aput and willingly assume this substantial risk with comparatively limited reward? There are anumber of motivations that may warrant the short put strategy. In this example, Sam had the twin goals of profiting from aneutral to moderately bullish outlook on Boeing and buying it if it traded below $65. The short put helps him achieve both objectives.
Much like the covered call, if the stock is above the strike at expiration, this trader reaches his maximum profit potential—in this case 1.20. And if the price of Boeing is below the strike at expiration, Sam has ownership of the stock from assignment. Here, astrike price that is lower than the current stock level is used. The stock needs to decline in order for Sam to get assigned and become long the stock. With this strategy, he was able to establish atarget price at which he would buy the stock. Why not use alimit order? If the put is assigned, the effective purchase price is $63.80 even if the stock price is above this price. If the put is not assigned, the premium is kept.
Aconsideration every trader must make before entering the short put position is how the purchase of the stock will be financed in the event the put is assigned. Traders hoping to acquire the stock will often hold enough cash in their trading account to secure the purchase of the stock. This is called acash-secured put
. In this example, Sam would hold $6,380 in his account in addition to the $120 of option premium received. This affords him enough free capital to fund the $6,500 purchase of stock the short put dictates. More speculative traders may be willing to buy the stock on margin, in which case the trader will likely need around 50 percent of the stock’svalue.
Some traders sell puts without the intent of ever owning the stock. They hope to profit from alow-volatility environment. Just as the short call is anot-bullish stance on the underlying, the short put is anot-bearish play. As long as the underlying is above the strike price at expiration, the option 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 aput gives the holder the right to sell stock at the strike price. Of course, puts can be apart of ahost of different spreads, but this chapter discusses the two most basic and common put-buying strategies: the long put and the protective put. The long put is away to speculate on abearish move in the underlying security, and the protective put is away to protect along position in the underlying security.
Consider along put example:
Buy 1 SPY May 139 put at 2.30
In this example, the Spiders have had agood run up to $140.35. Trader Isabel is looking for a 10 percent correction in SPY between now and the end of May, about three months away. She buys 1 SPY May 139 put at 2.30. This put gives her the right to sell 100 shares of SPY at $139 per share.
Exhibit 1.6
shows Isabel’s P&(L) if the


ave had agood run up to $140.35. Trader Isabel is looking for a 10 percent correction in SPY between now and the end of May, about three months away. She buys 1 SPY May 139 put at 2.30. This put gives her the right to sell 100 shares of SPY at $139 per share.
Exhibit 1.6
shows Isabel’s P&(L) if the put is held until expiration.
EXHIBIT 1.6
SPY long put.
If SPY is above the strike price of 139 at expiration, the put will expire and the entire premium of 2.30 will be lost. If SPY is below the strike price at expiration, the put will have value. It can be exercised, creating ashort position in the Spiders at an effective price of $136.70 per share. This price is found by subtracting the premium paid, 2.30, from the strike price, 139. This is the point at which the position breaks even. If SPY is below $136.70 at expiration, Isabel has aprofit. Profits will increase on atick-for-tick basis, with downward movements in SPY down to zero. The long put has limited risk and substantial reward potential.
An alternative for Isabel is to short the ETF at the current price of $140.35. But ashort position in the underlying may not be as attractive to her as along put. The margin requirements for short stock are significantly higher than for along put. Put buyers must post only the premium of the put—that is the most that can be lost, after all.
The margin requirement for short stock reflects unlimited loss potential. Margin requirements aside, risk is avery real consideration for atrader deciding between shorting stock and buying aput. If the trader expects high volatility, he or she may be more inclined to limit upside risk while leveraging downside profit potential by buying aput. In general, traders buy options when they expect volatility to increase and sell them when they expect volatility to decrease. This will be acommon theme throughout this book.
Consider aprotective put example:
This is an example of asituation in which volatility is expected to increase.
Own 100 shares SPY at 140.35
Buy 1 SPY May139 put at 2.30
Although Isabel bought aput because she was bearish on the Spiders, adifferent trader, Kathleen, may buy aput for adifferent reason—she’sbullish but concerned about increasing volatility. In this example, Kathleen has owned 100 shares of Spiders for some time. SPY is currently at $140.35. She is bullish on the market but has concerns about volatility over the next two or three months. She wants to protect her investment. Kathleen buys 1 SPY May 139 put at 2.30. (If Kathleen bought the shares of SPY and the put at the same time, as aspread, the position would be called amarried put.)
Kathleen is buying the right to sell the shares she owns at $139. Effectively, it is an insurance policy on this asset.
Exhibit 1.7
shows the risk profile of this new position.
EXHIBIT 1.7
SPY protective put.
The solid kinked line is the protective put (put and stock), and the thin dotted line is the outright position in SPY alone, without the put. The most Kathleen stands to lose with the protective put is $3.65 per share. SPY can decline from $140.35 to $139, creating aloss of $1.35, plus the $2.30 premium spent on the put. If the stock does not fall and the insuring put hence does not come into play, the cost of the put must be recouped to justify its expense. The break-even point is $142.65.
This position implies that Kathleen is still bullish on the Spiders. When traders believe astock or ETF is going to decline, they sell the shares. Instead, Kathleen sacrifices 1.6 percent of her investment up front by purchasing the put for $2.30. She defers the sale of SPY until the period of perceived risk ends. Her motivation is not to sell the ETF; it is to hedge volatility.
Once the anticipated volatility is no longer aconcern, Kathleen has achoice to make. She can let the option run its course, holding it to expiration, at which point it will either expire or be exercised; or she can sell the option before expiration. If the option is out-of-the-money, it may have residual time value prior to expiration that can be recouped. If it is in-the-money, it will have intrinsic value and maybe time value as well. In this situation, Kathleen can look at this spread as two trades—one that has 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 aparticular option strategy works. They show what the option’sprice will ultimately be at various prices of the underlying. There is, however, acaveat 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 away to develop reasonable expectations as to what the option’sprice will be given changes that can occur in factors affecting the option’


CHAPTER 2
Greek Philosophy
My wife, Kathleen, is not an options trader. Au contraire. However, she, like just about everyone, uses them from time to time—though without really thinking about it. She was on eBay the other day bidding on apair of shoes. The bid was $45 with three days left to go. She was concerned about the price rising too much and missing the chance to buy them at what she thought was agood price. She noticed, though, that someone else was selling the same shoes with abuy-it-now price of $49—agood-enough price in her opinion. Kathleen was effectively afforded acall option. She had the opportunity to buy the shoes at (the strike price of) $49, aright she could exercise until the offer expired.
The biggest difference between the option in the eBay scenario and the sort of options discussed in this book is transferability. Actual options are tradable—they can be bought and sold. And it is the contract itself that has value—there is one more iteration of pricing.
For example, imagine the $49 opportunity was acoupon or certificate that guaranteed the price of $49, which could be passed along from one person to another. And there was the chance that the $49-price guarantee could represent adiscount on the price paid for the shoes—maybe abig discount—should the price of the shoes rise in the eBay auction. The certificate guaranteeing the $49 would have value. Anyone planning to buy the shoes would want the safety of knowing they were guaranteed not to pay more than $49 for the shoes. In fact, some people would even consider paying to buy the certificate itself if they thought the price of the shoes might rise significantly.
Price vs. Value: How Traders Use Option-Pricing Models
Like in the common-life example just discussed, the right to buy or sell an underlying security—that is, an option—can have value, too. The specific value of an option is determined by supply and demand. There are several variables in an option contract, however, that can influence atrader’swillingness to demand (desire to buy) or supply (desire to sell) an option at agiven price. For example, atrader would rather own—that is, there would be higher demand for—an option that has more time until expiration than ashorter-dated option, all else held constant. And atrader would rather own acall with alower strike than ahigher strike, all else kept constant, because it would give the right to buy at alower price.
Several elements contribute to the value of an option. It took academics many years to figure out exactly what those elements are. Fischer Black and Myron Scholes together pioneered research in this area at the University of Chicago. Ultimately, their work led to a Nobel Prize for Myron Scholes. Fischer Black died before he could be honored.
In 1973, Black and Scholes published apaper called “The Pricing of Options and Corporate Liabilities” in the
Journal of Political Economy
, that introduced the Black-Scholes option-pricing model to the world. The Black-Scholes model values European call options on non-dividend-paying stocks. Here, for the first time, was awidely accepted model illustrating what goes into the pricing of an option. Option prices were no longer wild guesswork. They could now be rationalized. Soon, additional models and alterations to the Black-Scholes model were developed for options on indexes, dividend-paying stocks, bonds, commodities, and other optionable instruments. All the option-pricing models commonly in use today have slightly different means but achieve the same end: the option’stheoretical value. For American-exercise equity options, six inputs are entered into any option-pricing model to generate atheoretical value: stock price, strike price, time until expiration, interest rate, dividends, and volatility.
Theoretical value—what aconcept! Atrader plugs six numbers into apricing model, and it tells him what the option is worth, right? Well, in practical terms, that’snot exactly how it works. An option is worth what the market bears. Economists call this price discovery. The price of an option is determined by the forces of supply and demand working in afree and open market. Herein lies an important concept for option traders: the difference between price and value.
Price can be observed rather easily from any source that offers option quotes (web sites, your broker, quote vendors, and so on). Value is calculated by apricing model. But, in practice, the theoretical value is really not an output at all. It is already known: the market determines it. The trader rectifies price and value by setting the theoretical value to fall between the bid and the offer of the option by adjusting the inputs to the model. Professional traders often refer to the theoretical value as the fair value of the option.
At this point, please note the absence of the mathematical formula for the Black-Scholes model (or any other pricing model, for that matter). Although the foundation of trading option greeks is mathema

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model. Professional traders often refer to the theoretical value as the fair value of the option.
At this point, please note the absence of the mathematical formula for the Black-Scholes model (or any other pricing model, for that matter). Although the foundation of trading option greeks is mathematical, this book will keep the math to aminimum—which is still quite abit. The focus of this book is on practical applications, not academic theory. It’sabout learning to drive the car, not mastering its engineering.
The trader has an equation with six inputs equaling one known output. What good is this equation? An option-pricing model helps atrader understand how market forces affect the value of an option. Five of the six inputs are dynamic; the only constant is the strike price of the option in question. If the price of the option changes, it’sbecause one or more of the five variable inputs has changed. These variables are independent of each other, but they can change in harmony, having either acumulative or net effect on the option’svalue. An option trader needs to be concerned with the relationship of these variables (price, time, volatility, interest). This multidimensional view of asset pricing is unique to option traders.
Delta
The five figures commonly used by option traders are represented by Greek letters: delta, gamma, theta, vega, rho. The figures are referred to as option greeks. Vega, of course, is not an actual letter of the greek alphabet, but in the options vernacular, it is considered one of the greeks.
The greeks are aderivation of an option-pricing model, and each Greek letter represents aspecific sensitivity to influences on the option’svalue. To understand concepts represented by these five figures, we’ll start with delta, which is defined in four ways:
1. The rate of change of an option value relative to achange in the underlying stock price.
2. The derivative of the graph of an option value in relation to the stock price.
3. The equivalent of underlying shares represented by an option position.
4. The estimate of the likelihood of an option expiring in-the-money.
1
Definition 1
: Delta (Δ) is the rate of change of an option’svalue relative to achange in the price of the underlying security. Atrader who is bullish on aparticular stock may choose to buy acall instead of buying the underlying security. If the price of the stock rises by $1, the trader would expect to profit on the call—but by how much? To answer that question, the trader must consider the delta of the option.
Delta is stated as apercentage. If an option has a 50 delta, its price will change by 50 percent of the change of the underlying stock price. Delta is generally written as either awhole number, without the percent sign, or as adecimal. So if an option has a 50 percent delta, this will be indicated as 0.50, or 50. For the most part, we’ll use the former convention in our discussion.
Call values increase when the underlying stock price increases and vice versa. Because calls have this positive correlation with the underlying, they have positive deltas. Here is asimplified example of the effect of delta on an option:
Consider a $60 stock with acall option that has a 0.50 delta and is trading for 3.00. Considering only the delta, if the stock price increases by $1, the theoretical value of the call will rise by 0.50. That’s 50 percent of the stock price change. The new call value will be 3.50. If the stock price decreases by $1, the 0.50 delta will cause the call to decrease in value by 0.50, from 3.00 to 2.50.
Puts have anegative correlation to the underlying. That is, put values decrease when the stock price rises and vice versa. Puts, therefore, have negative deltas. Here is asimplified example of the delta effect on a −0.40-delta put:
As the stock rises from $60 to $61, the delta of −0.40 causes the put value to go from $2.25 to $1.85. The put decreases by 40 percent of the stock price increase. If the stock price instead declined by $1, the put value would increase by $0.40, to $2.65.
Unfortunately, real life is abit more complicated than the simplified examples of delta used here. In reality, the value of both the call and the put will likely be higher with the stock at $61 than was shown in these examples. We’ll expand on this concept later when we tackle the topic of gamma.
Definition 2
: Delta can also be described another way.
Exhibit 2.1
shows the value of acall option with three months to expiration at avariable stock price. As the stock price rises, the call is worth more; as the stock price declines, the call value moves toward zero. Mathematically, for any given point on the graph, the derivative will show the rate of change of the option price.
The delta is the first derivative of the graph of the option price relative to the stock price
.
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 be


te 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 around lot of 100 shares. Acall with a 0.60 delta rises by $0.60 with a $1 increase in the stock. The owner of acall representing rights on 100 shares earns $60 for a $1 increase in the underlying. It’sas if the call owner in this example is long 60 shares of the underlying stock.
Delta is the option’sequivalent of aposition in the underlying shares
.
Atrader who buys five 0.43-delta calls has aposition that is effectively long 215 shares—that’s 5 contracts × 0.43 deltas × 100 shares. In option lingo, the trader is long 215 deltas. Likewise, if the trader were short five 0.43-delta calls, the trader would be short 215 deltas.
The same principles apply to puts. Being long 10 0.59-delta puts makes the trader short atotal of 590 deltas, aposition that profits or loses like being short 590 shares of the underlying stock. Conversely, if the trader were short 10 0.59-delta puts, the trader would theoretically make $590 if the stock were to rise $1 and lose $590 if the stock fell by $1—just like being long 590 shares.
Definition 4
: The final definition of delta is considered the trader’sdefinition. It’smathematically imprecise but is used nonetheless as ageneral rule of thumb by option traders. Atrader would say the
delta is astatistical approximation of the likelihood of the option expiring in-the-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’sdiscuss afew observations about the characteristics of delta.
First, call and put deltas are closely related.
Exhibit 2.2
is apartial option chain of 70-day calls and puts in Rambus Incorporated (RMBS). The stock was trading at $21.30 when this table was created. In
Exhibit 2.2
, the 20 calls have a 0.66 delta.
EXHIBIT 2.2
RMBS Option chain with deltas.
Notice the deltas of the put-call pairs in this exhibit. As ageneral rule, the absolute value of the call delta plus the absolute value of the put delta add up to close to 1.00. The reason for this has to do with amathematical relationship called put-call parity, which is briefly discussed later in this chapter and described in detail in Chapter 6. But with equity options, the put-call pair doesn’talways add up to exactly 1.00.
Sometimes the difference is simply due to rounding. But sometimes there are other reasons. For example, the 30-strike calls and puts in
Exhibit 2.2
have deltas of 0.14 and −0.89, respectively. The absolute values of the deltas add up to 1.03. Because of the possibility of early exercise of American options, the put delta is abit higher than the call delta would imply. When puts have agreater chance of early exercise, they begin to act more like short stock and consequently will have agreater 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 ahigher delta.
Moneyness and Delta
The next observation is the effect of moneyness on the option’sdelta. Moneyness describes the degree to which the option is in- or out-of-the-money. As ageneral rule, options that are in-the-money (ITM) have deltas greater than 0.50. Options that are out-of-the-money (OTM) have deltas less than 0.50. Finally, options that are at-the-money (ATM) have deltas that are about 0.50. The more in-the-money the option is, the closer to 1.00 the delta is. The more out-of-the-money, the closer the delta is to 0.
But ATM options are usually not exactly 0.50. For ATMs, both the call and the put deltas are generally systematically avalue other than 0.50. Typically, the call has ahigher delta than 0.50 and the put has alower absol


er to 1.00 the delta is. The more out-of-the-money, the closer the delta is to 0.
But ATM options are usually not exactly 0.50. For ATMs, both the call and the put deltas are generally systematically avalue other than 0.50. Typically, the call has ahigher delta than 0.50 and the put has alower absolute value than 0.50. Incidentally, the call’stheoretical value is generally greater than the put’swhen the options are right at-the-money as well. One reason for this disparity between exactly at-the-money calls and puts is the interest rate. The more time until expiration, the more effect the interest rate will have, and, therefore, the higher the call’stheoretical and delta will be relative to the put.
Effect of Time on Delta
In aclose contest, the last few minutes of afootball game are often the most exciting—not because the players run faster or knock heads harder but because one strategic element of the game becomes more and more important: time. The team that’sin the lead wants the game clock to run down with no interruption to solidify its position. The team that’slosing uses its precious time-outs strategically. The more playing time left, the less certain defeat is for the losing team.
Although mathematically imprecise, the trader’sdefinition can help us gain insight into how time affects option deltas. The more time left until an option’sexpiration, the less certain it is whether the option will be ITM or OTM at expiration. The deltas of both the ITM and the OTM options reflect that uncertainty. The more time left in the life of the option, the closer the deltas tend to gravitate to 0.50. A 0.50 delta represents the greatest level of uncertainty—acoin toss.
Exhibit 2.3
shows the deltas of ahypothetical equity call with astrike price of 50 at various stock prices with different times until expiration. All other parameters are held constant.
EXHIBIT 2.3
Estimated delta of 50-strike call—impact of time.
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’seither 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’simportant to address it here as it relates to the concept of delta.
Exhibit 2.4
shows how changing the volatility percentage (explained further in Chapter 3), as opposed to the time to expiration, affects option deltas. In this table, the delta of acall with 91 days until expiration is studied.
EXHIBIT 2.4
Estimated delta of 50-strike call—impact of volatility.
Notice the effect that volatility has on the deltas of this option with the underlying stock at various prices. In this table, at alow volatility with the call deep in- or out-of-the-money, the delta is very large or very small, respectively. At 10 percent volatility with the stock at $58 ashare, the delta is 1.00. At that same volatility level with the stock at $42 ashare, the delta is 0.
But at higher volatility levels, the deltas change. With the stock at $58, a 45 percent volatility gives the 50-strike call a 0.79 delta—much smaller than it was at the low volatility level. With the stock at $42, a 45-percent volatility returns a 0.30 delta for the call. Generally speaking, ITM option deltas are smaller given ahigher volatility assumption, and OTM option deltas are bigger with ahigher volatility.
Effect of Stock Price on Delta
An option that is $5 in-the-money on a $20 stock will have ahigher 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’sdelta will change. This phenomenon is measured by the option’sgamma.
Gamma
The strike price is the only constant in the pricing model. When the stock price moves relative to this constant, the option in question becomes more in-the-money or out-of-the-money. This means the delta changes. This isolated change is measured by the option’sgamma, sometimes called
curvature
.
Gamma (Γ) is the rate of change of an option’sdelta given achange in the price of the underlying security
. Gamma is conventionally stated in terms of deltas per dollar move. The simplified examples above under Definition 1 of delta, used to describe the effect of delta, had one important piece of the puzzle missing: gamma. As the stock price moved higher in those examples, the delta would not remain constant. It would change due to the effect of gamma. The following example shows how the delta would change given a 0.04 gamma attributed to the call option.
The call in this example starts as a 0.50-delta option. When the stock price increases by $1, the delta increases by the amount of the gamma. In this example, delta increases from 0.50 to 0.54, adding 0.04 deltas. As the s


how the delta would change given a 0.04 gamma attributed to the call option.
The call in this example starts as a 0.50-delta option. When the stock price increases by $1, the delta increases by the amount of the gamma. In this example, delta increases from 0.50 to 0.54, adding 0.04 deltas. As the stock price continues to rise, the delta continues to move higher. At $62, the call’sdelta is 0.58.
This increase in delta will affect the value of the call. When the stock price first begins to rise from $60, the option value is increasing at arate of 50 percent—the call’sdelta at that stock price. But by the time the stock is at $61, the option value is increasing at arate of 54 percent of the stock price. To estimate the theoretical value of the call at $61, we must first estimate the average change in the delta between $60 and $61. The average delta between $60 and $61 is roughly 0.52. It’sdifficult to calculate the average delta exactly because gamma is not constant; this is discussed in more detail later in the chapter. Amore realistic example of call values in relation to the stock price would be as follows:
Each $1 increase in the stock shows an increase in the call value about equal to the average delta value between the two stock prices. If the stock were to decline, the delta would get smaller at adecreasing rate.
As the stock price declines from $60 to $59, the option delta decreases from 0.50 to 0.46. There is an average delta of about 0.48 between the two stock prices. At $59 the new theoretical value of the call is 2.52. The gamma continues to affect the option’sdelta and thereby its theoretical value as the stock continues its decline to $58 and beyond.
Puts work the same way, but because they have anegative delta, when there is apositive stock-price movement the gamma makes the put delta less negative, moving closer to 0. The following example clarifies this.
As the stock price rises, this put moves more and more out-of-the-money. Its theoretical value is decreasing by the rate of the changing delta. At $60, the delta is −0.40. As the stock rises to $61, the delta changes to −0.36. The average delta during that move is about −0.38, which is reflected in the change in the value of the put.
If the stock price declines and the put moves more toward being in-the-money, the delta becomes more negative—that is, the put acts more like ashort stock position.
Here, the put value rises by the average delta value between each incremental change in the stock price.
These examples illustrate the effect of gamma on an option without discussing the impact on the trader’sposition. When traders buy options, they acquire positive gamma. Since gamma causes options to gain value at afaster rate and lose value at aslower rate, (positive) gamma helps the option buyer. Atrader buying one call or put in these examples would have +0.04 gamma. Buying 10 of these options would give the trader a +0.4 gamma.
When traders sell options, gamma works against them. When options lose value, they move toward zero at aslower rate. When the underlying moves adversely, gamma speeds up losses. Selling options yields anegative gamma position. Atrader selling one of the above calls or puts would have −0.04 gamma per option.
The effect of gamma is less significant for small moves in the underlying than it is for bigger moves. On proportionately large moves, the delta can change quite abit, making abig difference in the position’s P&(L). In
Exhibit 2.1
, the left side of the diagram showed the call price not increasing at all with advances in the stock—a 0 delta. The right side showed the option advancing in price 1-to-1 with the stock—a 1.00 delta. Between the two extremes, the delta changes. From this diagram another definition for gamma can be inferred: gamma is the second derivative of the graph of the option price relative to the stock price. Put another way, gamma is the first derivative of agraph of the delta relative to the stock price.
Exhibit 2.5
illustrates the delta of acall relative to the stock price.
EXHIBIT 2.5
Call delta compared with stock price.
Not only does the delta change, but it changes at achanging rate. Gamma is not constant. Moneyness, time to expiration, and volatility each have an effect on the gamma of an option.
Dynamic Gamma
When options are far in-the-money or out-of-the-money, they are either 1.00 delta or 0 delta. At the extremes, small changes in the stock price will not cause the delta to change much. When an option is at-the-money, it’sadifferent story. Its delta can change very quickly.
ITM and OTM options have alow gamma.
ATM options have arelatively high gamma.
Exhibit 2.6
is an example of how moneyness translates into gamma on QQQ calls.
EXHIBIT 2.6
Gamma of QQQ calls with QQQ at $44.
With QQQ at $44, 92 days until expiration, and aconstant volatility input of 19 percent, the 36- and 54-strike calls are far enough in- and out-of-the-money, respectively, that if the Qs move asmall amount in either di


anslates into gamma on QQQ calls.
EXHIBIT 2.6
Gamma of QQQ calls with QQQ at $44.
With QQQ at $44, 92 days until expiration, and aconstant volatility input of 19 percent, the 36- and 54-strike calls are far enough in- and out-of-the-money, respectively, that if the Qs move asmall amount in either direction from the current price of $44, the movement won’tchange their deltas much at all. The chances of their money status changing between now and expiration would not be significantly different statistically given asmall stock price change. They have the smallest gammas in the table.
The highest gammas shown here are around the ATM strike prices. (Note that because of factors not yet discussed, the strike that is exactly at-the-money may not have the highest gamma. The highest gamma is likely to occur at aslightly higher strike price.)
Exhibit 2.7
shows agraph of the corresponding numbers in
Exhibit 2.6
.
EXHIBIT 2.7
Option gamma.
Adecrease in the time to expiration solidifies the likelihood of ITMs or OTMs remaining as such. But an ATM option’smoneyness 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 agraph of the gamma if the volatility is doubled.
EXHIBIT 2.9
Gamma as volatility changes.
Raising the volatility assumption flattens the curve, causing ITM and OTM to have higher gamma while lowering the gamma for ATMs.
Short-term ATM options with low volatility have the highest gamma. Lower gamma is found in ATMs when volatility is higher and it is lower for ITMs and OTMs and in longer-dated options.
Theta
Option prices can be broken down into two parts: intrinsic value and time value. Intrinsic value is easily measurable. It is simply the ITM part of the premium. Time value, or extrinsic value, is what’sleft over—the premium paid over parity for the option. All else held constant, the more time left in the life of the option, the more valuable it is—there is more time for the stock to move. And as the useful life of an option decreases, so does its time value.
The decline in the value of an option because of the passage of time is called time decay, or erosion. Incremental measurements of time decay are represented by the Greek letter theta (θ).
Theta is the rate of change in an option’sprice given aunit change in the time to expiration
. What exactly is the
unit
involved here? That depends.
Some providers of option greeks will display thetas that represent one day’sworth of time decay. Some will show thetas representing seven days of decay. In the case of aone-day theta, the figure may be based on aseven-day week or on aweek counting only trading days. The most common and, arguably, most useful display of this figure is the one-day theta based on the seven-day week. There are, after all, seven days in aweek, each day of which can see an occurrence with the potential to cause arevaluation in the stock price (that is, news can come out on Saturday or Sunday). The one-day theta based on aseven-day week will be used throughout this book.
Taking the Day Out
When the number of days to expiration used in the pricing model declines from, say, 32 days to 31 days, the price of the option decreases by the amount of the theta, all else held constant. But when is the day “taken out”? It is intuitive to think that after the market closes, the model is changed to reflect the passing of one day’stime. But, in fact, this change is logically anticipated and may be priced in early.
In the earlier part of the week, option prices can often be observed getting cheaper relative to the stock price sometime in the middle of the day. This is because traders will commonly take the day out of their model during trading hours after the underlying stabilizes following the morning business. On Fridays and sometimes Thursdays, traders will take all or part of the weekend out. Commonly, by Friday afternoon, traders will be using Monday’sdays to value their options.
When option prices are seen getting cheaper on, say, a Friday, how can one tell whether this is the effect of the market taking the weekend out or achange in some other input, such as volatility? To some degree, it doesn’tmatter. Remember, the model is used to reflect what the market is doing, not the other way a


ices are seen getting cheaper on, say, a Friday, how can one tell whether this is the effect of the market taking the weekend out or achange in some other input, such as volatility? To some degree, it doesn’tmatter. Remember, the model is used to reflect what the market is doing, not the other way around. In many cases, it’slogical to presume that small devaluations in option prices intraday can be attributed to the routine of the market taking the day out.
Friend or Foe?
Theta can be agood thing or abad thing, depending on the position. Theta hurts long option positions; whereas it helps short option positions. Take an 80-strike call with atheoretical value of 3.16 on astock at $82 ashare. The 32-day 80 call has atheta of 0.03. If atrader owned one of these calls, the trader’sposition would theoretically lose 0.03, or $0.03, as the time until expiration change from 32 to 31 days. This trader has anegative theta position. Atrader short one of these calls would have an overnight theoretical profit of $0.03 attributed to theta. This trader would have apositive theta.
Theta affects put traders as well. Using all the same modeling inputs, the 32-day 80-strike put would have atheta of 0.02. Aput holder would theoretically lose $0.02 aday, and aput writer would theoretically make $0.02. Long options carry with them negative theta; short options carry positive theta.
Ahigher theta for the call than for the put of the same strike price is common when an interest rate greater than zero is used in the pricing model. As will be discussed in greater detail in the section on rho, interest causes the time value of the call to be higher than that of the corresponding put. At expiration, there is no time value left in either option. Because the call begins with more time value, its premium must decline at afaster rate than that of the put. Most modeling software will attribute the disparate rates of decline in value all to theta, whereas some modeling interfaces will make clear the distinction between the effect of time decay and the effect of interest on the put-call pair.
The Effect of Moneyness and Stock Price on Theta
Theta is not aconstant. As variables influencing option values change, theta can change, too. One such variable is the option’smoneyness.
Exhibit 2.10
shows theoretical values (theos), time values, and thetas for 3-month options on Adobe (ADBE). In this example, Adobe is trading at $31.30 ashare with three months until expiration. The more ITM acall or aput gets, the higher its theoretical value. But when studying an option’stime decay, one needs to be concerned only with the option’stime value, because intrinsic value is not subject to time decay.
EXHIBIT 2.10
Adobe theos and thetas (Adobe at $31.30).
The ATM options shown here have higher time value than ITM or OTM options. Hence, they have more time premium to lose in the same three-month period. ATM options have the highest rate of decay, which is reflected in higher thetas. As the stock price changes, the theta value will change to reflect its change in moneyness.
If this were ahigher-priced stock, say, 10 times the stock price used in this example, with all other inputs held constant, the option values, and therefore the thetas, would be higher. If this were astock trading at $313, the 325-strike call would have atheoretical value of 16.39 and aone-day 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 adirect relationship to option values. The higher the volatility, the higher the value of the option. Higher-valued options decay at afaster rate than lower-valued options—they have to; their time values will both be zero at expiration. All else held constant, the higher the volatility assumption, the higher the theta.
The days to expiration have adirect relationship to option values as well. As the number of days to expiration decreases, the rate at which an option decays may change, depending on the relationship of the stock price to the strike price. ATM options tend to decay at anonlinear rate—that is, they lose value faster as expiration approaches—whereas the time values of ITM and OTM options decay at asteadier rate.
Consider ahypothetical stock trading at $70 ashare.
Exhibit 2.11
shows how the theoretical values of the 75-strike call and the 70-strike call decline with the passage of time, holding all other parameters constant.
EXHIBIT 2.11
Rate of decay: ATM vs. OTM.
The OTM 75-strike call has afairly 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 day


eady 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! Imeant to click on puts.” Now trading is facilitated almost entirely online by professional and retail traders alike. Market and trading information is disseminated worldwide in subseconds, making markets all the more efficient. And the tools now available to the common retail trader are very powerful as well. Many online brokers and other web sites offer high-powered tools like screeners, which allow traders to sift through thousands of options to find those that fit certain parameters.
Using ascreener to find ATM calls on same-priced stocks—say, stocks trading at $40 ashare—can yield aresult worth talking about here. One $40 stock can have a 40-strike call trading at around 0.50, while adifferent $40 stock can have a 40 call with the same time to expiration trading at more like 2.00. Why? The model doesn’tknow the name of the company, what industry it’sin, or what its price-to-earnings ratio is. It is amathematical equation with six inputs. If five of the inputs—the stock price, strike price, time to expiration, interest rate, and dividends—are identical for two different options but they’re trading at different prices, the difference must be the sixth variable, which is volatility.
Implied Volatility (IV) and Vega
The volatility component of option values is called implied volatility (IV). (For more on implied volatility and how it relates to vega, see Chapter 3.) IV is apercentage, although in practice the percent sign is often omitted. This is the value entered into apricing model, in conjunction with the other variables, that returns the option’stheoretical value. The higher the volatility input, the higher the theoretical value, holding all other variables constant. The IV level can change and often does—sometimes dramatically. When IV rises or falls, option prices rise and fall in line with it. But by how much?
The relationship between changes in IV and changes in an option’svalue is measured by the option’svega.
Vega is the rate of change of an option’stheoretical value relative to achange in implied volatility
. Specifically, if the IV rises or declines by one percentage point, the theoretical value of the option rises or declines by the amount of the option’svega, respectively. For example, if acall with atheoretical value of 1.82 has avega of 0.06 and IV rises one percentage point from, say, 17 percent to 18 percent, the new theoretical value of the call will be 1.88—it would rise by 0.06, the amount of the vega. If, conversely, the IV declines 1 percentage point, from 17 percent to 16 percent, the call value will drop to 1.76—that is, it would decline by the vega.
Aput with the same expiration month and the same strike on the same underlying will have the same vega value as its corresponding call. In this example, raising or lowering IV by one percentage point would cause the corresponding put value to rise or decline by $0.06, just like the call.
An increase in IV and the consequent increase in option value helps the P&(L) of long option positions and hurts short option positions. Buying acall or aput establishes along vega position. For short options, the opposite is true. Rising IV adversely affects P&(L), whereas falling IV helps. Shorting acall or put establishes ashort vega position.
The Effect of Moneyness on Vega
Like the other greeks, vega is asnapshot that is afunction of multiple facets of determinants influencing option value. The stock price’srelationship to the strike price is amajor determining factor of an option’svega. IV affects only the time value portion of an option. Because ATM options have the greatest amount of time value, they will naturally have higher vegas. ITM and OTM options have lower vega values than those of the ATM options.
Exhibit 2.13
shows an example of 186-day options on AT&T Inc. (T), their time value, and the corresponding vegas.
EXHIBIT 2.13
AT&Ttheos and vegas (Tat $30, 186 days to Expry, 20% IV).
Note that the 30-strike calls and puts have the highest time values. This strike boasts the highest vega value, at 0.085. The lower the time premium, the lower the vega—therefore, the less incremental IV changes affect the option. Since higher-pr


theos and vegas (Tat $30, 186 days to Expry, 20% IV).
Note that the 30-strike calls and puts have the highest time values. This strike boasts the highest vega value, at 0.085. The lower the time premium, the lower the vega—therefore, the less incremental IV changes affect the option. Since higher-priced stocks have higher time premium (in absolute terms, not necessarily in percentage terms) they will have higher vega. Incidentally, if this were a $300 stock instead of a $30 stock, the 186-day ATMs would have a 0.850 vega, if all other model inputs remain the same.
The Effect of Implied Volatility on Vega
The distribution of vega values among the strike prices shown in
Exhibit 2.13
holds for aspecific IV level. The vegas in
Exhibit 2.13
were calculated using a 20 percent IV. If adifferent 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 asimilar 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 asimilar 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 Iworked for asked me what his position was in acertain strike. Itold him he was long 200 calls and long 300 puts. “I’mlong 500 puts?” he asked. “No,” Icorrected, “you’re long 200 calls and 300 puts.” At this point, he looked at me like Iwas from another planet and said, “That’s 500. Aput is acall; acall is aput.” 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 aslightly different way:
Traders serious about learning to trade options must know put-call parity backward and forward. Why? First, by algebraically rearranging this equation, it can be inferred that synthetically equivalent positions can be established by simply adding stock to an option. Again, aput is acall; acall is aput.
and
For example, along call is synthetically equal to along stock position plus along put on the same strike, once interest and dividends are figured in. Asynthetic long stock position is created by buying acall and selling aput of the same month and strike. Understanding synthetic relationships is intrinsic to understanding options. Amore comprehensive discussion of synthetic relationships and tactical considerations for creating synthetic positions is offered in Chapter 6.
Put-call parity also aids in valuing options. If put-call parity shows adifference in the value of the call versus the value of the put with the same strike, there may be an arbitrage opportunity. That translates as “riskless profit.” Buying the call and selling it synthetically (short put and short stock) could allow aprofit to be locked in if the prices are disparate. Arbitrageurs tend to hold synthetic put and call prices pretty close together. Generally, only professional traders can capture these types of profit opportunities, by trading big enough positions to make very small profits (apenny or less per contract sometimes) matter. Retail traders may be able to take advantage of adisparity in put and call values to some extent, however, by buying or selling the synthetic as asubstitute for the actual option if the position can be established at abetter price synthetically.
Another reason that aworking knowledge of put-call parity is essential is that it helps attain abetter understanding of how changes in the interest rate affect option values. The greek rho measures this change. Rho is the rate of change in an option’svalue relative to achange in the interest rate.
Although some modeling programs may displ


f put-call parity is essential is that it helps attain abetter understanding of how changes in the interest rate affect option values. The greek rho measures this change. Rho is the rate of change in an option’svalue relative to achange in the interest rate.
Although some modeling programs may display this number differently, most display arho for the call and arho for the put, both illustrating the sensitivity to aone-percentage-point change in the interest rate. When the interest rate rises by one percentage point, the value of the call increases by the amount of its rho and the put decreases by the amount of its rho. Likewise, when the interest rate decrease by one percentage point, the value of the call decreases by its rho and the put increases by its rho. For example, acall with arho of 0.12 will increase $0.12 in value if the interest rate used in the model is increased by one percentage point. Of course, interest rates usually don’trise or fall one percentage point in one day. More commonly, rates will have incremental changes of 25 basis points. That means acall with a 0.12 rho will theoretically gain $0.03 given an increase of 0.25 percentage points.
Mathematically, this change in option value as aproduct of achange in the interest rate makes sense when looking at the formula for put-call parity.
and
But the change makes sense intuitively, too, when acall is considered as acheaper substitute for owning the stock. For example, compare a $100 stock with athree-month 60-strike call on that same stock. Being so far ITM, there would likely be no time value in the call. If the call can be purchased at parity, which alternative would be asuperior investment, the call for $40 or the stock for $100? Certainly, the call would be. It costs less than half as much as the stock but has the same reward potential; and the $60 not spent on the stock can be invested in an interest-bearing account. This interest advantage adds value to the call. Raising the interest rate increases this value, and lowering it decreases the interest component of the value of the call.
Asimilar concept holds for puts. Professional traders often get ashort-stock rebate on proceeds from ashort-stock sale. This is simply interest earned on the capital received when the stock is shorted. Is it better to pay interest on the price of aput for aposition that gives short exposure or to receive interest on the credit from shorting the stock? There is an interest disadvantage to owning the put. Therefore, arise in interest rates devalues puts.
This interest effect becomes evident when comparing ATM call and put prices. For example, with interest at 5 percent, three-month options on an $80 stock that pays a $0.25 dividend before option expiration might look something like this:
The ATM call is higher in theoretical value than the ATM put by $0.75. That amount can be justified using put-call parity:
(Here, simple interest of $1 is calculated as 80 × 0.05 × [90 / 360] = 1.)
Changes in market conditions are kept in line by the put-call parity. For example, if the price of the call rises because of an increase in IV, the price of the put will rise in step. If the interest rate rises by aquarter of apercentage point, from 5 percent to 5.25 percent, the interest calculated for three months on the 80-strike will increase from $1 to $1.05, causing the 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 athree-month period caused achange of 0.05 to the interest component of put-call parity. That is, 80 × 0.0025 × (90/360) = 0.05. If alonger period were used in the example—say, one year—the effect would be more profound; it will be $0.20: 80 × 0.0025 × (360/360) = 0.20. This concept is evident when the rhos of options with different times to expiration are studied.
Exhibit 2.16
shows the rhos of ATM Procter & Gamble Co. (PG) calls with various expiration months. The 750-day Long-Term Equity AnticiPation Securities (LEAPS) have arho of 0.858. As the number of days until expiration decreases, rho decreases. The 22-day calls have arho of only 0.015. Rho is usually afairly insignificant factor in the value of short-term options, but it can come into play much more with long-term option strategies involving LEAPS.
EXHIBIT 2.16
The effect of time on rho (Procter & Gamble @ $64.34)
Why the Numbers Don’t Don’t Always Add Up
There will be many times when studying the rho of options in an option chain will reveal seemingly counterintuitive results. To be sure, the numbers don’talways add up to what appears logical. One reason for this is rounding. Another is that traders are more likely to use simple interest in calculating value, whereas t


imes when studying the rho of options in an option chain will reveal seemingly counterintuitive results. To be sure, the numbers don’talways add up to what appears logical. One reason for this is rounding. Another is that traders are more likely to use simple interest in calculating value, whereas the model uses compound interest. Hard-to-borrow stocks and stocks involved in mergers and acquisitions may have put-call parities that don’twork out right. But another, more common and more significant fly in the ointment is early exercise.
Since the interest input in put-call parity is afunction of the strike price, it is reasonable to expect that the higher the strike price, the greater the effect of interest on option prices will be. For European options, this is true to alarge extent, in terms of aggregate impact of interest on the call and put pair. Strikes below the price where the stock is trading have ahigher rho associated with the call relative to the put, whereas strikes above the stock price have ahigher rho associated with the put relative to the call. Essentially, the more in-the-money an option is, the higher its rho. But with European options, observing the aggregate of the absolute values of the call and put rhos would show ahigher combined rho the higher the strike.
With American options, the put can be exercised early. Atrader will exercise aput before expiration if the alternative—being short stock and receiving ashort stock rebate—is awiser choice based on the price of the put. Professional traders may own stock as ahedge against aput. They may exercise deep ITM puts (1.00-delta puts) to avoid paying interest on capital charges related to the stock. The potential for early exercise is factored into models that price American options. Here, when puts get deeper in-the-money—that is, more apt to be exercised—the rho decreases. When the strike price is very high relative to the stock price—meaning the put is very deep ITM—and there is little or no time value left to the call or the put, the aggregate put-call rho can be zero. Rho is discussed in greater detail in Chapter 7.
THE GREEKS DEFINED
Delta
(Δ) is:
1. The rate of change in an option’svalue relative to achange in the underlying asset price.
2. The derivative of the graph of an option’svalue 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’sexpiring in-the-money.
Gamma
(Γ) is the rate of change in an option’sdelta given achange in the price of the underlying asset.
Theta
(θ) is the rate of change in an option’svalue given aunit change in the time to expiration.
Vega
is the rate of change in an option’svalue relative to achange in implied volatility.
Rho
(ρ) is the rate of change in an option’svalue relative to achange 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 asimple interface that accepts the input of the six variables to the model and yields atheoretical value and the greeks for asingle option.
Some web sites devoted to option education, such as
MarketTaker.com/option_modeling
, have free calculators that can be used for modeling positions and using the greeks.
In practice, many of the option-trading platforms commonly in use have sophisticated analytics that involve greeks. Most options-friendly online brokers provide trading platforms that enable traders to conduct comprehensive manipulations of the greeks. For example, traders can look at the greeks for their positions up or down one, two, or three standard deviations. Or they can see what happens to their position greeks if IV or time changes. With many trading platforms, position greeks are updated in real time with changes in the stock price—an invaluable feature for active traders.
Caveats with Regard to Online Greeks
Often, online greeks are one click away, requiring little effort on the part of the trader. Having greeks calculated automatically online is aquick and convenient way to eyeball greeks for an option. But there is one major problem with online greeks: reliability.
For active option traders, greeks are essential. There is no point in using these figures if their accuracy cannot be assured. Experienced traders can often spot these inaccuracies aproverbial mile away.
When looking at greeks from an online source that does not require you to enter parameters into amodel (as would be the case with professional option-trading platforms), special attention needs to be paid to the relationship of the option’stheoretical values to the bid and offer. One must be cautious if the theoretical value of the option lies outside the bid-ask spread. This scenario can exist for brief periods of time, but arbitrageurs ten


tion-trading platforms), special attention needs to be paid to the relationship of the option’stheoretical values to the bid and offer. One must be cautious if the theoretical value of the option lies outside the bid-ask spread. This scenario can exist for brief periods of time, but arbitrageurs tend to prevent this from occurring routinely. If several options in achain all have theoretical values below the bid or above the offer, there is probably aproblem with one or more of the inputs used in the model. Remember, an option-pricing model is just that: amodel. It reflects what is occurring in the market. It doesn’ttell where an option should be trading.
The complex changes that occur intraday in the market—taking the day or weekend out, changes in stock price, volatility, and the interest rate—are not always kept current. The user of the model must keep close watch. It’snot reasonable to expect the computer to do the thinking for you. Automatically calculated greeks can be used as astarting point. But before using these figures in the decision-making process, the trader may have to override the parameters that were used in the online calculation to make the theos line up with market prices. Professional traders will ignore online greeks altogether. They will use the greeks that are products of the inputs they entered in their trading software. It comes down to this: if you want something done right, do it yourself.
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 afeel 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’sprice itself. This greek philosophy forms the foundation of option trading for active traders. It offers alogical way to monitor positions and provides amedium in and of itself to trade.
Notes
1
. Please note that definition 4 is not necessarily mathematically accurate. This “trader’sdefinition” is included in the text because many option traders use delta as aquick 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 aquick and convenient way to do the calculation.


CHAPTER 3
Understanding Volatility
Most option strategies involve trading volatility in one way or another. It’seasy to think of trading in terms of direction. But trading volatility? Volatility is an abstract concept; it’sadifferent animal than the linear trading paradigm used by most conventional market players. As an option trader, it is essential to understand and master volatility.
Many traders trade without asolid understanding of volatility and its effect on option prices. These traders are often unhappily surprised when volatility moves against them. They mistake the adverse option price movements that result from volatility for getting ripped off by the market makers or some other market voodoo. Or worse, they surrender to the fact that they simply don’tunderstand why sometimes these unexpected price movements occur in options. They accept that that’sjust the way it is.
Part of what gets in the way of aready understanding of volatility is context. The term
volatility
can have afew different meanings in the options business. There are three different uses of the word
volatility
that an option trader must be concerned with: historical volatility, implied volatility, and expected volatility.
Historical Volatility
Imagine there are two stocks: Stock Aand Stock B. Both are trading at around $100 ashare. Over the past month, atypical end-of-day net change in the price of Stock Ahas been up or down $5 to $7. During that same period, atypical daily move in Stock Bhas been something more like up or down $1 or $2. Stock Ahas tended to move more than Stock Bas apercentage of its price, without regard to direction. Therefore, Stock Ais more volatile—in the common usage of the word—than Stock B. In the options vernacular, Stock Ahas ahigher 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 ameasure of how volatile the price movement of asecurity has been during acertain 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 astock 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 amathematical calculation that measures the dispersion of data from amean value. In this case, the mean is the average stock price over acertain 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 (aweighted 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, afunction 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’topen. After each day, the oldest price is taken out of the calculation and replaced by the most recent closing price. Using ashorter or longer period can yield different results and can be useful in studying astock’svolatility.
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 apercentage move in the price of the asset. If a $100 stock has astandard deviation of 15 percent, aone-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. Astock with astandard deviation of 15 percent has experienced bigger moves—has been more volatile—during the relevant time period than astock with astandard deviation of 6 percent.
When the frequency of occurrences are graphed, the result is known as adistribution curve. There are many different shapes that adistribution curve can take, depending on the nature of the data being observed. In general, option-pricing models assume that stock prices adhere to alognormal 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 sh

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pending on the nature of the data being observed. In general, option-pricing models assume that stock prices adhere to alognormal 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, anormal distribution is used here.
When the graph of data adheres to anormal distribution, the result is asymmetrical bell-shaped curve. Standard deviation can be shown on the bell curve to either side of the mean.
Exhibit 3.1
represents atypical bell curve with standard deviation.
EXHIBIT 3.1
Standard deviation.
Large moves in asecurity are typically less frequent than small ones. Events that cause big changes in the price of astock, like acompany’sbeing acquired by another or discovering its chief financial officer cooking the books, are not adaily 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 aprice change is found around the midpoint of the curve. What constitutes alarge move or asmall move, however, is unique to each individual security. For example, atwo percent move in an index like the Standard & Poor’s (S&P) 500 may be considered abig one-day move, while atwo percent move in aparticularly active tech stock may be adaily occurrence. Standard deviation offers astatistical explanation of what constitutes atypical 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 astock is trading at $100 ashare 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 aone-year period (based on recent past performance).
Simply put, historical volatility shows how volatile astock 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 astock, it is not adirect 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’stheoretical value—in practice is not necessarily an output at all. When option traders use the pricing model, they commonly substitute the actual price at which the option is trading for the theoretical value. Avalue in the middle of the bid-ask spread is often used. The pricing model can be considered to be acomplex algebra equation in which any variable can be solved for. If the theoretical value is known—which it is—it along with the five known inputs can be combined to solve for the unknown volatility.
Implied volatility (IV) is the volatility input in apricing model that, in conjunction with the other inputs, returns the theoretical value of an option matching the market price.
For aspecific stock price, agiven implied volatility will yield aunique option value. Take astock trading at $44.22 that has the 60-day 45-strike call at atheoretical value of $1.10 with an 18 percent implied volatility level. If the stock price remains constant, but IV rises to 19 percent, the value of the call will rise by its vega, which in this case is about 0.07. The new value of the call will be $1.17. Raising IV another point, to 20 percent, raises the theoretical value by another $0.07, to $1.24. The question is: What would cause implied volatility to change?
Supply and Demand: Not Just a Good Idea, It’sthe Law!
Options are an excellent vehicle for speculation. However, the existence of the options market is better justified by the primary economic purpose of options: as arisk management tool. Hedgers use options to protect their assets from adverse price movements, and when the percepti


Idea, It’sthe Law!
Options are an excellent vehicle for speculation. However, the existence of the options market is better justified by the primary economic purpose of options: as arisk management tool. Hedgers use options to protect their assets from adverse price movements, and when the perception of risk increases, so does demand for this protection. In this context, risk means volatility—the potential for larger moves to the upside and downside. The relative prices of options are driven higher by increased demand for protective options when the market anticipates greater volatility. And option prices are driven lower by greater supply—that is, selling of options—when the market expects lower volatility. Like those of all assets, option prices are subject to the law of supply and demand.
When volatility is expected to rise, demand for options is not limited to hedgers. Speculative traders would arguably be more inclined to buy acall than to buy the stock if they are bullish but expect future volatility to be high. Calls require alower cash outlay. If the stock moves adversely, there is less capital at risk, but still similar profit potential.
When volatility is expected to be low, hedging investors are less inclined to pay for protection. They are more likely to sell back the options they may have bought previously to recoup some of the expense. Options are adecaying asset. Investors are more likely to write calls against stagnant stocks to generate income in anticipated low-volatility environments. Speculative traders will implement option-selling strategies, such as short strangles or iron condors, in an attempt to capitalize on stocks they believe won’tmove much. The rising supply of options puts downward pressure on option prices.
Many traders sum up IV in two words:
fear
and
greed
. When option prices rise and fall, not because of changes in the stock price, time to expiration, interest rates, or dividends, but because of pure supply and demand, it is implied volatility that is the varying factor. There are many contributing factors to traders’ willingness to demand or supply options. Anticipation of events such as earnings reports, Federal Reserve announcements, or the release of other news particular to an individual stock can cause anxiety, or fear, in traders and consequently increase demand for options that causes IV to rise. IV can fall when there is complacency in the market or when the anticipated news has been announced and anxiety wanes. “Buy the rumor, sell the news” is often reflected in option implied volatility. When there is little fear of market movement, traders use options to squeeze out more profits—greed.
Arbitrageurs, such as market makers who trade delta neutral—astrategy that will be discussed further in Chapters 12 and 13—must be relentlessly conscious of implied volatility. When immediate directional risk is eliminated from aposition, IV becomes the traded commodity. Arbitrageurs who focus their efforts on trading volatility (colloquially called
vol traders
) tend to think about bids and offers in terms of IV. In the mind of avol trader, option prices are translated into volatility levels. Atrader may look at aparticular option and say it is 30 bid at 31 offer. These values do not represent the prices of the options but rather the corresponding implied volatilities. The meaning behind the trader’sremark is that the market is willing to buy implied volatility at 30 percent and sell it at 31 percent. The actual prices of the options themselves are much less relevant to this type of trader.
Should HV and IV Be the Same?
Most option positions have exposure to volatility in two ways. First, the profitability of the position is usually somewhat dependent on movement (or lack of movement) of the underlying security. This is exposure to HV. Second, profitability can be affected by changes in supply and demand for the options. This is exposure to IV. In general, along option position benefits when volatility—both historical and implied—increases. Ashort option position benefits when volatility—historical and implied—decreases. That said, buying options is buying volatility and selling options is selling volatility.
The Relationship of HV and IV
It’sintuitive that there should exist adirect relationship between the HV and IV. Empirically, this is often the case. Supply and demand for options, based on the market’sexpectations for asecurity’svolatility, determines IV.
It is easy to see why IV and HV often act in tandem. But, although HV and IV are related, they are not identical. There are times when IV and HV move in opposite directions. This is not so illogical, if one considers the key difference between the two: HV is calculated from past stock price movements; it is what has happened. IV is ultimately derived from the market’sexpectation for future volatility.
If astock typically has an HV of 30 percent and nothing is expected to change, it can be reasonable to expect that in th


difference between the two: HV is calculated from past stock price movements; it is what has happened. IV is ultimately derived from the market’sexpectation for future volatility.
If astock typically has an HV of 30 percent and nothing is expected to change, it can be reasonable to expect that in the future the stock will continue to trade at a 30 percent HV. By that logic, assuming that nothing is expected to change, IV should be fairly close to HV. Market conditions do change, however. These changes are often regular and predictable. Earnings reports are released once aquarter in many stocks, Federal Open Market Committee meetings happen regularly, and dates of other special announcements are often disclosed to the public in advance. Although the outcome of these events cannot be predicted, when they will occur often can be. It is around these widely anticipated events that HV-IV divergences often occur.
HV-IV Divergence
An HV-IV divergence occurs when HV declines and IV rises or vice versa. The classic example is often observed before acompany’squarterly earnings announcement, especially when there is lack of consensus among analysts’ estimates. This scenario often causes HV to remain constant or decline while IV rises. The reason? When there is agreat deal of uncertainty as to what the quarterly earnings will be, investors are reluctant to buy
or
sell the stock until the number is released. When this happens, the stock price movement (volatility) consolidates, causing the calculated HV to decline. IV, however, can rise as traders scramble to buy up options—bidding up their prices. When the news is out, the feared (or hoped for) move in the stock takes place (or doesn’t), and HV and IV tend to converge again.
Expected Volatility
Whether trading options or stocks, simple or complex strategies, traders must consider volatility. For basic buy-and-hold investors, taking apotential investment’svolatility into account is innate behavior. Do Ibuy conservative (nonvolatile) stocks or more aggressive (volatile) stocks? Taking into account volatility, based not just on agut feeling but on hard numbers, can lead to better, more objective trading decisions.
Expected Stock Volatility
Option traders must have an even greater focus on volatility, as it plays amuch bigger role in their profitability—or lack thereof. Because options can create highly leveraged positions, small moves can yield big profits or losses. Option traders must monitor the likelihood of movement in the underlying closely. Estimating what historical volatility (standard deviation) will be in the future can help traders quantify the probability of movement beyond acertain price point. This leads to better decisions about whether to enter atrade, when to adjust aposition, and when to exit.
There is no way of knowing for certain what the future holds. But option data provide traders with tools to develop expectations for future stock volatility. IV is sometimes interpreted as the market’sestimate of the future volatility of the underlying security. That makes it aready-made estimation tool, but there are two caveats to bear in mind when using IV to estimate future stock volatility.
The first is that the market can be wrong. The market can wrongly price stocks. This mispricing can lead to acorrection (up or down) in the prices of those stocks, which can lead to additional volatility, which may not be priced in to the options. Although there are traders and academics believe that the option market is fairly efficient in pricing volatility, there is aroom for error. There is the possibility that the option market can be wrong.
Another caveat is that volatility is an annualized figure—the annualized standard deviation. Unless the IV of a LEAPS option that has exactly one year until expiration is substituted for the expected volatility of the underlying stock over exactly one year, IV is an incongruent estimation for the future stock volatility. In practice, the IV of an option must be adjusted to represent the period of time desired.
There is acommon technique for deannualizing IV used by professional traders and retail traders alike.
1
The first step in this process to deannualize IV is to turn it into aone-day figure as opposed to one-year figure. This is accomplished by dividing IV by the square root of the number of trading days in ayear. The number many traders use to approximate the number of trading days per year is 256, because its square root is around number: 16. The formula is
For example, a $100 stock that has an at-the-money (ATM) call trading at a 32 percent volatility implies that there is about a 68 percent chance that the underlying stock will be between $68 and $132 in one year’stime—that’s $100 ± ($100 × 0.32). The estimation for the market’sexpectation for the volatility of the stock for one day in terms of standard deviation as apercentage of the price of the underlying is computed as follows:
In one day’stime, based on an IV of 32 pe