
Days to Expiration
Figure 4.1 Time Decay: At-the-Money Call
date approaches. But note that the acceleration occurs only in thelast two months prior to expiration; before then the time decay isvery close to linear.The assumptions behind Figure 4.1 were a $50 stock and a $50strike—in other words, an at-the-money option. If we move in- orout-of-the-money, the options start behaving in a different way. Figure 4.2 traces two out-of-the-money options’ values over time: onea 55 call (10% out-of-the-money) and the other a 60 call (20% out-ofthe-money).It would be difﬁcult for time decay to be any more linear thanwhat is illustrated in Figure 4.2. The two options simply becomeworthless at a different point in time, when the option pricing modelestimates that given the time to expiration and the amount by whichthe call is out-of-the-money, the probability of the stock reaching theoption’s strike price is virtually nil.Finally, in-the-money options will behave much like the out-ofthe-money options, except that their values will converge on theirintrinsic amount, not on zero. For instance, the 12-month 45 call(10% in-the-money) is valued at $9.54, which represents $5 of intrinsic value and $4.54 of time premium. The 12-month 55 call waspriced at $4.79. The $4.54 of time premium in the 45 call will decayin a fashion very similar to the $4.79 of time premium of the 55 call,converging on a $5 ﬁnal value (the call’s intrinsic value) instead ofbecoming worthless.
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$5
$3$2
Option Price
$455 C60 C
$1$0300
200
100
0
Days to Expiration
Figure 4.2 Time Decay: Out-of-the-Money Calls
The foregoing analysis was based on options trading at a 30%volatility. The relationships hold for higher-volatility stocks, as isclearly shown in Table 4.6.Under the 0% interest rate assumption, the 2-to-1 price ratiobetween the four- and one-month options holds for the three volatilities illustrated. Under the more realistic 4% interest rate assumption, the theoretical relationship is close for all, and actually closestfor the 60% volatility. With the higher volatilities the interest ratecomponent makes up a smaller and smaller part of the options’ totalprices, and the relationship converges toward the 2-to-1 obtainedunder the 0% interest rate scenario. It could be said that underhigher-volatility assumptions, the volatility component overwhelmsthe interest rate component of the option’s price.To close this section, a few words about puts. We saw earlier thathigher interest rates result in lower put prices. Under a 0% interestTable 4.600030% Volatility000
00045% Volatility000
00060% Volatility000
InterestRate
1-month
4-month
1-month
4-month
1-month
4-month
0%
1.70
3.41
2.55
5.11
3.40
6.80
4%
1.78
3.73
2.67
5.41
3.53
7.09
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assumption the one-month at-the-money put (using our $50 stockand a 30% volatility assumption) is valued at $1.70, while the fourmonth put is at $3.41. Raise the interest rate to 4% and the onemonth put is $1.63, while the four-month is $3.14. Once again, as theinterest rate is increased the theoretical relationship diverges fromthe theoretical 2-to-1 ratio, but not by an amount that would matterto most investors and traders. Puts will lose their time premium ina fashion very close to that illustrated in the foregoing call timedecay tables and graphs.VolatilityMost traders and investors have an instinctive understanding ofvolatility if only because they can feel its effects on their stomach.But this gut understanding needs to be sharpened to gauge the roleof volatility in option pricing. Before delving into the differentmeanings of the word volatility, let’s examine its impact on optionpricing. You may want to return to this discussion after reading thenext section.The convention is to measure volatility on a yearly basis. Whenyou purchase a 13-week Treasury bill yielding 3.8%, the rate you arequoted is an annualized rate, even though this instrument will belong gone before the end of the year. The same applies to volatility.A stock’s volatility over 60 trading days may be given as 32%, or 22day options may be trading with an implied volatility of 44%. Itshould be understood that both of these volatility numbers havebeen annualized, even though the time frame in question is muchshorter.We all agree that the higher a stock’s volatility, the more expensive its options. We try to quantify this in Table 4.7, where theoretTable 4.7VolatilityInterestRate
10%
20%
30%
40%
50%
60%
100%
0%
1.98
3.95
5.92
7.89
9.85
11.81
19.55
4%
2.50
4.45
6.40
8.36
10.31
12.25
19.95
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ical option prices are calculated for a conveniently priced $100 stock.We start by looking at the 91-day at-the-money calls under twointerest rate scenarios.Looking at the prices calculated under the 0% interest rateassumption, we ﬁnd a fairly linear relationship. Except for rounding errors, the price calculated with 20% volatility is double that calculated with 10% volatility, that with 30% is three times, and so on.This should not come as a surprise if volatility is held to be a measure of risk. A stock with twice the volatility has twice the risk, sothe at-the-money options cost twice as much.Looking at the option prices under the 4% interest rate assumption, we ﬁnd the relationship to be close to linear but somewhat distorted by the interest rate component. For example, the 40%volatility option is valued at $8.36, not $8.90, which is what onewould expect based on the $4.45 value of the 20% volatility option.Note that the higher the volatility used, the less important the interest rate component becomes relative to the total option value sincethe interest rate component remains relatively constant (at about$0.50) over the whole range of volatility estimates.Table 4.8 is our ﬁrst look at out-of-the-money options. We look atthe same $100 stock but calculate prices for the 91-day 110 calls.The relationship for this out-of-the-money call is not as straightforward as our previous example. But start with the 50% volatilityassumption and notice what happens as it increases: the call pricegoes up around $2 for every 10% increase in volatility. So above 50%the relationship is linear. Starting with a very low volatility of 10%gives us a nearly worthless call; by increasing the volatility theoption’s value slowly builds until the option’s price starts increasing linearly. The way to think about this is that for 50% or highervolatility, the 110 call is close to being an at-the-money option. Yes,Table 4.8VolatilityInterestRate
10%
20%
30%
40%
50%
60%
0%
0.05
0.96
2.51
4.32
6.12
8.11 10.16 12.19 14.22 16.23
4%
0.09
1.15
2.78
4.63
6.46
8.47 10.52 12.56 14.58 16.59
70%
80%
90% 100%
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The Options Strategist
Table 4.9VolatilityInterestRate
10%
20%
30%
40%
50%
60%
70%
80%
0%
0.00
0.13
0.89
2.17
3.61
5.49
7.36
9.22 11.07 12.91
4%
0.00
0.17
1.02
2.37
3.85
5.76
7.65
9.52 11.38 13.23
90% 100%
it is $10 out-of-the-money, but the more volatile the stock the lesssigniﬁcant $10 becomes. This phenomenon is further illustrated inTable 4.9, where we repeat the exercise for the 120 call, which is evenfurther out-of-the-money.The option prices become linear at volatilities above 70%. Theexplanation is the same as above. If one is dealing with a 70%volatility stock, $20 is not a big deal, and the 120 call, even thoughit is out-of-the-money by 20%, is behaving as though it were at-themoney.It has been said that changes in volatility have the greatest percentage impact on the price of out-of-the-money options. This is easily illustrated by rearranging some of the data from the previousthree tables as we have in Table 4.10, where we examine the impactof volatility rising from 30% to 40% for different strike prices whenthe underlying stock is trading at $100.But remember that the higher the volatility, the more loosely theterms at-the-money and out-of-the-money need to be applied.The impact of volatility changes on in-the-money options will besimilar to the impact on the out-of-the-moneys if one only takes into
Table 4.10Call Prices30% Volatility 40% Volatility
$ Change
% Change
100 Call
6.40
8.36
1.96
30.6%
110 Call
2.78
4.63
1.85
66.5%
120 Call
1.02
2.37
1.35
132.4%
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account the options’ time premium. For instance, the 110 call wasvalued at $2.78 using 30% volatility and 4% interest rates. Under thesame assumptions the 90 call could be estimated to be worth $12.78;that includes $10 of intrinsic value and a symmetrical amount oftime premium at $2.78. Running the actual numbers would give youa $12.66 option value, close enough for most traders.
Volatility or VolatilitiesVolatility is probably the most common word in the language ofoptions. Unfortunately, its exact meaning is often left unexplored.This section looks at the different meanings this word can take andtheir appropriate uses.Historical VolatilityAs its name implies, historical volatility looks at the past and tellsus how volatile a stock (or an index) has been over a given timeframe. There is little room for argument here. A stock has traded theway it has traded, and that’s the end of the story. You can’t arguewith the past.The only place where there is room for debate is in decidingwhich period represents the stock’s history. Options traders live inthe here and now; looking at how volatile a stock was ﬁve years agois of little relevance to today’s marketplace. Should one look at thelast week, the last month, the last quarter, or the last year? There is,of course, no right answer, but our preference is to look at two timeperiods—a longer one (maybe a year) and a shorter one (maybe amonth, or even a week if there has been anything out of the ordinary). By comparing the two periods we can see if the stock is currently more or less volatile than its average. It should be noted thatfor the more established stocks (read blue chips if you will) historical volatility tends to be relatively constant.To learn how to crunch the numbers, see the relevant section inChapter 9. Let’s assume you calculated IBM’s historical volatilityover the last three months to have been 32%. What does this number tell you? The answer, 32%, is actually the standard deviation ofthe stock’s returns. If you hark back to your statistics class you will
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–32%
0%
+32%
Figure 4.3 Expected Returns in One Year
remember that standard deviation is associated with the so-calledbell curve (also known as a normal or Gaussian distribution), androughly two-thirds of observations are expected to fall within plusor minus one standard deviation. Figure 4.3 will provide the necessary ﬂashback.A 32% volatility tells us that if—and this is a big if—the stock’svolatility remains what it has been in the past, one year from nowthere is a 68% probability that the return on this stock will fallbetween ⫺32% and ⫹32%. It also tells us there is a 95% probabilitythat the stock’s return will fall somewhere between ⫺64% and⫹64% (i.e., plus or minus two standard deviations).What if your time frame is shorter than one year? Can a stock’shistorical volatility be of any help? Absolutely: a yearly volatilitynumber can easily be adjusted to a shorter time frame. For example, you can convert annualized volatility to quarterly volatility bymultiplying the yearly number by the square root of the fractionof a year the time period in question represents. So, to convert toa three-month time frame (a quarter of a year), multiply by thesquare root of one-quarter, which is one-half. So 32% annualizedvolatility becomes 16% over a three-month period. This tells us thatthere is a 68% probability that the stock’s return over the next threemonths will fall somewhere between ⫺16% and ⫹16%. To convertto a monthly number, multiply by the square root of one-twelfth(1 ⫼ 3.46) to obtain a monthly volatility of 9.2%. To convert to daily
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volatility, multiply by the square root of 1 ⫼ 256 (1 ⫼ 16) to obtaina daily volatility of 2%. In this last example 256 represents theapproximate number of trading days in a year. Of course, thisassumes that the future will resemble the past.Future VolatilityHow volatile will a stock actually be over the next three months?Unfortunately, we do not know this today, and when we do knowit, in three-months’ time, it will be historical volatility, not futurevolatility. It would be nice to know future volatility, but if we didwe would not be writing about it in an options strategy book.Implied VolatilityOf the ﬁve variables that are used in option pricing equations (stockprice, exercise price, time to expiration, interest rate, and volatility)four are empirically veriﬁable. You can check the stock price on theNYSE or Nasdaq; the exercise price is ﬁxed and will not change during the life of the option; you can count the numbers of days untilexpiration on a calendar or your ﬁngers; the risk-free interest rate isa well-known quantity you can look up in the newspaper. Thatleaves one unknown—volatility.But for listed options there is another known quantity, and thatis the option’s premium. If you want to know how much a DaimlerChrysler August 40 call is worth, you can try to calculate thisyourself, or you can simply look at what price this option is tradingon one of the exchanges. There is no need to reinvent the wheel.This leaves us in a situation where we know the answer (theoption’s premium) and four out of the ﬁve variables that are usedin ﬁnding this answer. In high school, we confronted these types ofproblems known as “solving for x.” We can solve for x using theoption premium and the four observable variables; this is how tocalculate implied volatility.Implied volatility is the number that, when plugged into anoption pricing equation, gives as the answer the option’s currentprice. In other words, it is the volatility number the market is usingto price this option. Implied volatility will be revisited in Chapter 9.
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Forecasted VolatilityImplied volatility can be viewed as the market’s volatility forecast.Of course, anyone can forecast future volatility. Just remember thata forecast is only a best guess and therefore contains an element ofsubjectivity.
Option Pricing VolatilityOne of the inputs to the Black-Scholes model and other option pricing equations is volatility. Which of the four volatilities describedabove should be used as a variable? To determine the “fair” price ofan option we should use future volatility. The price of the optionwill then fully reﬂect the stock’s risk. The only problem is, no oneknows what future volatility will be.Since the future is unknown, traders must use their own forecast.This may be based on historical volatility (under the assumptionthat the future will resemble the past), implied volatility (under theassumption that the marketplace knows best and who are we toargue with professional traders), or various other methods (whichmay include chicken entrails, astrology charts, or that deep feelingin one’s gut).
Overvalued and Undervalued OptionsWe have all heard the terms overvalued and undervalued options.Such expressions assume a comparison is being made between thecurrent price of an option and a benchmark. It could also be saidthat a comparison is being made between the option’s impliedvolatility (a measure of the option’s current price) and someone’sforecasted volatility (used to establish the benchmark).It is important to understand that when a claim is made that theJanuary 70 puts are undervalued this simply means that someone’s(the setter of the benchmark) forecasted volatility is higher than theoption’s implied volatility. Some service providers claim they scanthe options market looking for overvalued and undervaluedoptions. When you hear this, just remember that it’s the serviceprovider’s opinion (i.e., its forecasted volatility) versus the market’sopinion (i.e., the option’s implied volatility).
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Real-World VolatilitiesWe have discussed the various meanings of volatility somewhatabstractly, but now let’s look at the real world. What should weexpect when calculating implied volatilities? What’s the range ofnumbers that can realistically be seen in the marketplace? We havedone a limited survey of equity options and offer our results in Table4.11. Keep in mind that this is one sampling done at one point intime and that these numbers will change over time.One fact should not be too surprising—the order in which thesestocks fell: relatively sedate consumer stocks and utilities at the lowend, the more mature technology stocks in the middle, and theTable 4.11Stock
Option Implied Volatility
Johnson & Johnson
16%
Procter & Gamble
20%
TXU Corp.
22%
Verizon Communications
22%
General Electric
27%
International Business Machines
28%
Microsoft
30%
Amgen
31%
Intel
36%
AOL Time Warner
41%
Cisco Systems
49%
Genzyme
53%
Siebel Systems
61%
Broadcom
71%
Amazon.com
72%
PMC-Sierra
73%
JDS Uniphase
75%
I2 Technologies
94%
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The Options Strategist
newer up-and-coming (or down-and-going?) equities at the highend. If nothing else, when you see implied volatility of 40% on astock about which you don’t know anything, you now have a reference point. It could be an interesting exercise to recalculate theimplied volatilities for these same stocks and see how the markethas reassessed their relative risk. Yes, implied volatility can bevolatile. As an example, International Business Machines used tohave one of the most constant implied volatilities (ranging from themid- to high-20s); the stock then decided to go from $160 to the low$50s in a relatively short period of time and all of a sudden itsoptions’ implied volatility rose to the low- to mid-40s.Note that we quote our volatility numbers without fractions.Some computer programs will spit out 40.52% as an option’simplied volatility. The numbers to the right of the decimal point arepure rubbish. Write down 41% as the implied volatility andmove on.
The Greeks: DeltaAlphabet Soup Stock (ABCDE) is trading at $50. Some of its calloptions are shown in Table 4.12.If ABCDE rises in price by $1 over the course of the trading day(which means the options will still have 45 days until expiration),by how much do you expect each of the various options to increasein price? We can easily recalculate the values of these options withABCDE at $51, as shown in Table 4.13.As ABCDE went from $50 to $51, call prices increased. This muchwe expected. The options’ prices also increased at different rates,which should not come as a surprise to those who have dabbled in
Table 4.12 ABCDE @ $50
45-day 45 Call
$5.81
45-day 50 Call
$2.55
45-day 55 Call
$0.86
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Table 4.13 ABCDE @ $51Call Price
Change in Price
45-day 45 Call
$6.66
$0.85
45-day 50 Call
$3.16
$0.61
45-day 55 Call
$1.15
$0.29
the market. This change in an option’s price is known as the option’sdelta. If you are about to purchase a call on ABCDE, or on any otherstock for that matter, is this a piece of information you would liketo have? In all likelihood yes, to either help in selecting an option(choosing the one with a higher delta to capture a greater percentage of the stock’s move, for example), or to set expectations (whenthe 55 call only increases by $0.30 when the underlying goes up $1you should not be surprised).If you want to know the delta of an option, there is no need torepeat the above exercise. Most option pricing software (includingthe program used to calculate the preceding prices, which can befound under the Trading Tools tab at www.cboe.com) will give youdeltas along with price. When we calculated the prices for Table 4.12,we also obtained the deltas shown in Table 4.14.Why the discrepancy between the deltas we calculated and thosegiven by the option pricing software? Because delta is only valid foran inﬁnitesimal change in the price of the stock, something that isimpossible to observe, even with stocks trading in pennies. A moreinteresting observation is that deltas change. All of the calls havehigher deltas at $51 than they had at $50. The industry lingo is thatthe calls are “picking up deltas” as the price of the stock increases.
Table 4.14Delta (ABCDE @ $50)
Delta (ABCDE @ $51)
45-day 45 Call
0.83
.87
45-day 50 Call
0.54
.60
45-day 55 Call
0.25
.30
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The Options Strategist
But let’s get practical and examine what this means for optionstraders and investors. First, in many instances there is no need touse pricing software to estimate an option’s delta, keeping in mindthe following guidelines:■
At-the-money call options will have deltas of approximately 0.50,no matter the time to expiration or the stock’s volatility.
■
Out-of-the-money options will have deltas lower than 0.50, andthe further out-of-the-money an option is, the closer its delta willbe to 0.00.
■
In-the-money options will have deltas greater than 0.50, and thefurther in-the-money the option, the closer delta will approach1.00, the highest value delta can take.
A quick note on put options. Since delta is the expected pricechange in the value of an option, when the price of the underlyinggoes up, puts will have negative deltas. The delta of at-the-moneyputs will be approximately ⫺0.50. Deltas of out-of-the money putswill be greater than ⫺0.50 and will approach 0.00 as the puts movefurther and further out-of-the-money. In-the-money puts will havedeltas lower than ⫺0.50 and tending toward ⫺1.00 as the puts gofurther and further in-the-money.We have already noted that deltas change as the price of the stockmoves up and down. This is represented in Figure 4.4, which depictsthe deltas of the 45-day 50 call for various stock prices; also includedare the deltas of the same option when it has only 15 days remaining until expiration.Figure 4.4 illustrates two points. First, deltas go from 0.00 (whenthe option is far out-of-the-money) to 1.00 (when it is deep in-themoney), but they do not do so in a linear fashion. Delta changesrelatively slowly at ﬁrst, then more rapidly as it becomes an at-themoney option, and ﬁnally more slowly once again. Second, we canobserve the impact the passage of time has on deltas. The deltaremains relatively close to 0.50 when the stock is trading around $50(and the option is at-the-money); it tends toward 0.00 when thestock is below $50 (and the option is out-of-the-money); and itmoves toward 1.00 when the underlying is above $50 (and the callis in-the-money).
Options 401: Option Pricing, Volatility, and a First Greek Letter
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1
Call Delta
0.80.6
45 Days15 Days
0.40.20$40
$45
$50
$55
$60
Stock Price
Figure 4.4 Deltas: 50 Call
At expiration, a call’s delta will either be 0.00 (if the option is outof-the-money and about to become worthless) or 1.00 (if it is in-themoney and about to turn into the underlying stock). This lastcomment may need some clariﬁcation. If a call is in-the-moneyimmediately prior to expiration, we say it is about to turn into stockbecause a long call position that is exercised becomes a long position in the underlying stock. And stock has a delta of 1.00, since ifa stock goes up $1.00, then, by deﬁnition, the value of this stock hasgone up $1.00. As an in-the-money option approaches expiration, itstarts behaving more and more like the stock it is about to become,and its delta tends toward the delta of the stock, 1.00.What does all this mean for the options trader? If you are goingto purchase a call, you should know its delta because that will tellyou how your option will behave if the price of the underlyingmoves in the anticipated direction. You should also be aware that ifyou purchased an out-of-the-money call, for each day that goes by,that option’s delta will be slightly lower (the option is losing deltas).This is not good because your call is becoming less sensitive tomoves in the price of the underlying. Conversely, an in-the-moneycall will be picking up deltas on a daily basis, a deﬁnite positive. Butdo not confuse picking up deltas and losing deltas with changes inthe value of an option. An option can pick up deltas and fall in valueeven if the price of the underlying remains unchanged. Rememberwhat we said earlier in this chapter about how options lose valueover time.
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The Options Strategist
Table 4.15-------------Calls-------------
-------------Puts-------------
Quantity
Delta
Quantity
Delta
45
⫹17
0.83
⫺22
⫺0.17
50
⫺32
0.54
⫺32
⫺0.46
55
⫹37
0.25
⫹19
⫺0.76
Strike Price
Delta and Complex PositionsOne characteristic of deltas is that they are additive—one option’sdelta can be added to that of another. Take a market maker inABCDE options. Over the course of a trading day she has boughtand sold various option series and is now long and short the putsand calls shown in Table 4.15.This market maker is long some calls and puts, but she is alsoshort some calls and puts. What does her overall position look like?Should she be bullish, bearish, or indifferent? How can she protectherself if she does not like the overall position? Table 4.16 calculatesthis trader’s net delta. This is done by simply multiplying the number of contracts by each option’s delta, paying close attention to theplus and minus signs. Remember, puts have negative deltas, and ashort position generates a minus sign.So this trader’s net delta is ⫹608 for the calls and ⫹402 for theputs for a total net delta of ⫹1010 (we multiplied the deltas by 100because there are 100 shares underlying each option). This tells us
Table 4.16-------------------Calls-------------------
--------------------Puts--------------------
Quantity
Delta
Q ⫻ Delta
Quantity
Delta
Q ⫻ Delta
45
⫹17
0.83
1411
⫺22
⫺0.17
374
50
⫺32
0.54
⫺1728
⫺32
⫺0.46
1472
55
⫹37
0.25
925
⫹19
⫺0.76
⫺1444
Total Call Deltas
608
Total Put Deltas
402
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that if ABCDE goes up $1