it dropped to $39.With either method, you should also consider selling half of yourposition if the underlying stock reaches your upside target (and seta second target for the other half) while exiting the whole positionif the downside sell point is reached. This is a form of the age-oldtechnique of letting your proﬁts run but cutting your losses.Finally, if you purchased a two-month option when you wereexpecting the underlying stock to make a move over the next month,and if your plan was to sell the option one month before expiration,then stick to your guns. Sell the call after one month, even if it meanstaking a loss.
Buying PutsBuying puts is the mirror image of buying calls. Put buyers expectthe price of a stock to fall over a short or extended period of time.Most of the foregoing discussion on selecting strike prices and expiration dates for call options applies to puts as well.Shorting StocksBefore the advent of options, a trader who believed that the sharesof Such & Such (SNS) were about to head south had but one avenueopen to her: short shares of SNS. As we saw in Chapter 1, in shorting a stock a trader sells shares she does not own in the hope thattheir price will drop so she can purchase them later at a lower price.This trader is reversing the normal order of the two stock trades.Instead of buying a stock, waiting for it to go up, and then sellingit, the trader sells the stock short, waits for it to go down, and thenbuys it. Sell high, buy low instead of buy low, sell high.How can a trader sell shares she does not own? She simply borrows these from her broker. (Brokers are more than willing to lend
Basic Trading Strategies
69
stock because they usually charge a borrowing fee.) For instance, atrader borrows 500 shares of SNS and sells these short at $44. If herforecast is correct and SNS drops to $34, she can then cover her position by purchasing SNS at $34 and returning these shares to her broker. Her proﬁt is $10 per share.Of course there is always the possibility that SNS will misbehaveand start rallying. At some point this trader will want to cut herlosses and cover her position. If she buys back 500 SNS at $50, shewould have a loss of $6 per share (buying at $50 and selling at $44,not in that order).There are two types of problems associated with shorting stocks.The ﬁrst type is ﬁnancial. A broker will not let a trader short stockwithout her margining her position; that is, depositing cash or securities in her account to guarantee potential losses on the trade. If theprice of the shorted stock starts to rise, the broker will increase themargin required as the unrealized losses start accruing. In theory,since there is no limit as to how high the price of a stock can rise,the losses on a short stock position could be unlimited. In additionto the initial and maybe additional margin, a short seller could facea buy-in. This happens if the broker is no longer able to lend stockto the trader. A buy-in could occur when too many investors ask fortheir stock certiﬁcates or, more likely, simply want to sell the sharesthey rightfully own. A third more minor cost is the dividend liability. If a trader is short SNS on the day it goes ex-dividend, she mustpay this dividend to her broker, who in turn pays the investor fromwhom he borrowed the shares.The second order of problems with a short stock position is psychological. Maintaining a short position is nerve-wracking. Thestock goes up a dollar and you start worrying. It goes up anotherdollar and you wonder if you should cover, and so on until you can’tsleep at night. On average, retail investors maintain their short stockpositions for only short periods of time, usually not long enough fortheir forecast to come to reality. That’s why a lot of bears haveturned to put options.Buying Puts: An ExampleA trader has turned bearish on the shares of American InternationalGroup (AIG) and expects these to tumble over the next threemonths. An Internet search produces the following quotes:
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The Options Strategist
AIG:May 55 put:May 60 put:May 65 put:May 70 put:May 75 put:
$71.12$0.40–$0.55$0.85–$1.00$1.85–$1.95$3.40–$3.60$5.90–$6.10
Note that the May options have 86 days until expiration. Ourtrader is willing to risk $2,000 on a bearish trade. He looks at the 60puts, thinking: “If I purchase these, I am only risking $1 a share. Thisisn’t a lot of money, so maybe I should purchase 20 contracts.”Wrong reasoning. If he is thinking in terms of $1 a share, why notreduce the risk to $0.50 per share and try to purchase 40 of the 55puts at $0.50? We prefer to think of the dollar amount that is beingcommitted to the trade. This trader is not risking $1 or $0.50 a share,but $2,000, and his aim should be to earn the highest return possible on this amount.We turn to Table 3.8 to ﬁnd out how to best position ourselves fora fall in the price of AIG.From this table we can calculate the expiration date price rangeover which each option generates the highest returns. This is summarized in Table 3.9.Table 3.9 shows that if AIG rallies above $75 buying puts is not agood idea because all the puts expire worthless. On any pullbackdown to $62.80 (which represents an 11.7% drop in the price of AIG)
Table 3.800055 Put000 00060 Put000 00065 Put000 00070 Put000 00075 Put000AIGPrice Value Return Value Return Value Return Value Return Value Return$50
$5
809%
$10
900%
$15
669%
$20
456%
$25
310%
$55
$0
(100%)
$5
400%
$10
413%
$15
317%
$20
228%
$60
$0
(100%)
$0
(100%)
$5
156%
$10
178%
$15
146%
$65
$0
(100%)
$0
(100%)
$0
(100%)
$5
39%
$10
64%
$70
$0
(100%)
$0
(100%)
$0
(100%)
$0
(100%)
$5
(18%)
$75
$0
(100%)
$0
(100%)
$0
(100%)
$0
(100%)
$0
(100%)
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Basic Trading Strategies
Table 3.9From
To
75 Put
$74.95
$62.80
70 Put
$62.80
$59.20
65 Put
$59.15
$54.75
60 Put
$54.75
$48.90
55 Put
$48.85
$0.00
the 75 puts generate the highest returns. For the “it’s just a dollar”60 put to top the chart AIG needs to fall to at least $54.75, or a 23%drop. It boils down to how much of a bear our investor is in this particular instance.When looking at the data in Table 3.9, remember that because anoption has a relatively narrow range over which it offers the highest return does not mean it is not proﬁtable outside of that range.For example, the 70 put (which most traders would classify as atthe-money) is the best option from $62.80 down to $59.20, a rangeof only $3.60. If a trader buys this put and the stock moves signiﬁcantly below $59.20, this purchase will remain very proﬁtable,although not as proﬁtable as a 65 or a 60 put. One last calculationthat the put buyer should make is that of the puts’ downsidebreakeven. How far down does AIG have to go before the tradergets his money back? Breakeven at expiration is simply the option’sstrike price less the premium paid. Table 3.10 summarizes these forAIG.
Table 3.10Put Option
Premium
Downside Breakeven
May 55
$0.55
$54.45
May 60
$1.00
$59.00
May 65
$1.95
$63.05
May 70
$3.60
$66.40
May 75
$6.10
$68.90
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The Options Strategist
We believe that choosing which put option to purchase with thehelp of tables such as the three developed in this section will reducethe tendency too many option traders have of buying more lowpriced options rather than fewer higher-priced ones. Remember thatthe aim is not to impress your peers by trading large quantities ofoptions, but to dazzle them with the proﬁts you are generating.Puts’ Sweet SpotLike call options, each put option has its sweet spot. We see no needto repeat the calls’ full analysis, but the most important pointsshould be underscored:■
Out-of-the-money puts are suitable only if the price of the underlying stock drops soon after the position is initiated or if it fallsby a signiﬁcant amount by the expiration date.
■
In-the-money puts will hold more of their values if the stockremains unchanged or if it declines by a relatively small percentage.
■
At-the-money options have a sweet spot that is initially narrow,but it widens as the expiration date approaches.
Puts’ Proﬁtability BoundaryAs with calls, we can illustrate the proﬁtability boundary of the different options from our example. This is presented in Figure 3.3.
$75
$65$60$55$5080
60
40
20
Days to Expiration
Figure 3.3 AIG Puts: Proﬁtability Boundaries
0
AIG Price
$70
75 P70 P65 P60 P55 P
Basic Trading Strategies
73
Referring to the boundaries graph, the put buyer can ask himselfhow likely it is that AIG will trade below the boundary by anamount sufﬁcient to generate a good trading proﬁt. Although calculating proﬁtability boundaries before every put purchase wouldbe ideal, it may not be practical. But a trader can draw a quick anddirty proﬁtability boundary by following these three easy steps:1. Set the initial point of the boundary (on the date the position isestablished) at the then-current stock price.2. Set the boundary’s end point, on the expiration date, at the put’sbreakeven level (equal to the put’s strike price less the premiumpaid).3. Draw the boundary not as a straight line, but following the general shape illustrated in Figure 3.3. For this last step it is important to notice that the shape of the boundary will be determinedby how far the put is out-of-the-money or if it is in- or at-themoney. The further out-of-the-money the put is, the more curvedthe proﬁtability boundary will be.
Choosing an Expiration DateWhat type of bearish scenario are you envisioning? Are you expecting a quick pullback next week after the earnings announcement ordo you think this stock is about to start a long slide reminiscent ofsome of the dot-bomb debacles? Although we will discuss optionpricing in more detail in a later chapter, the following theoretical putprices may help in choosing an expiration date. We have assumeda stock price of $100, a 40% volatility, a 3% risk-free interest rate,and no dividends.Let’s start with the at-the-money 100 puts. A trader is bearish overthe short term. Should she purchase the one-month options at $4.43or the two-month options at $6.20? You can see that a strong argument can be made for purchasing more time. Whereas the ﬁrstmonth costs $4.43, the second month can be purchased for an additional $1.77. It has the look of “buy one, get the second one at halfprice.” Even if this trader expects to be out of her position in 30 days,the two-month option may be a more conservative way of trading.If the price of the stock remains unchanged, the buyer of the onemonth put would see the option expire worthless and lose 100% of
74
The Options Strategist
Table 3.11Time to Expiration
100 Put
90 Put
1 month
$4.43
$1.03
2 months
$6.20
$2.29
3 months
$7.56
$3.43
6 months
$10.48
$5.86
1 year
$14.38
$9.60
2 years
$19.40
$14.40
the amount risked on the trade. The buyer of the two-month optionwould see the value of the option drop from $6.20 to $4.43 (as thetwo-month option becomes a one-month option), a loss of 29% ofthe amount risked. If a trader purchases a two-month option whenher time frame is 30 days, she must show enough discipline to getout of the position after one month.For readers who are looking for things to avoid, we haveobserved that some traders are reluctant to sell options they havepurchased prior to their expiration date. Their argument is that theyare selling something valuable, or they fear a conspiracy in whichthe underlying stock will start moving the minute their sale orderis executed. We think they should adopt the attitude that getting adollar today may be better than getting a nickel on expiration Friday.Repeat the one-month/two-month option exercise with threeand six-month options and with one- and two-year options. Thesame relationships hold, and once again we ﬁnd ourselves arguingin favor of buying more time.Buying Out-of-the-Money PutsWhat about the out-of-the-money 90 puts? The relationship amongthe various premiums is different from the one between the at-themoneys. The one-, two-, and three-month options are roughlypriced at a dollar per month. Which of these should a bear purchase,since there are no savings in buying the longer-dated options?As in many other scenarios, the answer will depend on how theunderlying stock behaves. If a trader buys a one-month put at $1.03
Basic Trading Strategies
75
and one month later the stock is unchanged, he could purchaseanother one-month put at $1.03. The result will be roughly the sameas if he had purchased the two-month put at $2.29. But what if thestock rallies 10% during the ﬁrst month and our trader is still bearish? If the trader purchased the one-month put, he will now havethe opportunity to buy another one-month option. If he decides toremain with options that are 10% out-of-the-money, he will be ableto raise the put’s strike price to $100. The higher the stock rises during the ﬁrst month, the higher the trader can reset the put’s exerciseprice.The third possible outcome is for the stock to fall in price. Withthe stock price anywhere below $89, the put purchase will have beenproﬁtable. But if the stock falls to, say, $95 or $91, the trader will havea difﬁcult decision to make if, after 30 days, he wants to remain longput options. The 90 strike puts will expire worthless. The 90 putsexpiring one month later will be closer to being at-the-money andwill probably be trading at a premium substantially above $1. If thetrader only wants to purchase options that are 10% out-of-themoney, he will have to select puts with an 85 or 80 strike price, setting a lower and lower breakeven point.Another practice to avoid is buying the target price—selectingoptions with a strike price equal to one’s target price on a stock. Saya trader is looking for a stock to drop to $90, so he purchases the 90puts. If the stock reaches the $90 target at expiration, the put optionwill be worth . . . not a penny. For the buy-the-target-price strategyto work, the stock will have to hit the target price very quickly sothat the put option (which will then be an at-the-money option) willbe priced with a fair amount of time premium (which will dwindledown to zero by the option’s expiration date).
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OPTIONS 401:OPTION PRICING,VOLATILITY, ANDA FIRST GREEKLETTER
U
nderstanding the mechanics of option pricing and thevarious meanings of volatility can help a trader shapeappropriate options strategies. This chapter takes usbeyond the theoretical notions discussed earlier andlays the foundation for the strategies presented in thenext four chapters.77Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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The Options Strategist
Option Pricing: Beyond the BasicsThe basics of option pricing were discussed in Chapter 1. Now let’sexplore the ﬁve pricing variables in more detail.Stock PriceTable 4.1 gives the theoretical prices for 50-day at-the-money calloptions for stocks at various prices. A 30% volatility has beenassumed with a 4% risk-free interest rate and no dividends.The higher the stock price, the higher the option price. The relationship is linear: the call on the $40 stock costs twice as much asthe one on the $20 stock. This should not come as a surprise. Purchasing two of the at-the-money calls on the $20 stock would cost$186 and would give the buyer the right to acquire $4,000 worth ofshares at their current market price. Purchasing one of the at-themoney calls on the $40 stock would also cost $186 and give thebuyer the right to acquire $4,000 worth of shares at their currentprice.Another way to view these prices is in terms of puts and insurance. Assuming the same volatility, insuring an $80 stock shouldcost twice as much as insuring a $40 stock. How does this relationship hold up with options that are in- or out-of-the-money? Table4.2 looks at some 70-day call prices for three stocks, all based on a30% volatility assumption, 4% risk-free rate, and no dividend.First note that the strike prices are in- and out-of-the-money bythe same percentage (10%) for all stocks. Then look at the out-of-themoney options: the relationship stands, except for some small
Table 4.1Stock Price
At-the-Money Call Price
$20
$0.93
$40
$1.86
$60
$2.79
$80
$3.72
$100
$4.66
Options 401: Option Pricing, Volatility, and a First Greek Letter
79
Table 4.2Stock Price
Option Series
Option Price
Option Series
Option Price
$50
45 call
$6.01
55 call
$1.02
$100
90 call
$12.03
110 call
$2.03
$150
135 call
$18.04
165 call
$3.05
rounding discrepancies. The same goes for the in-the-moneyoptions. Note that the 45 call, which is 10% in-the-money, has a premium of $6.01, representing $5 of intrinsic value and $1.01 of timepremium. The 90 call, also 10% in-the-money, is worth $12.03, whichrepresents $10 of intrinsic value and $2.03 of time premium. It’s logical, to within a penny.Exercise PriceSince strike prices are ﬁxed, there doesn’t appear to be much to sayexcept that calls with lower strikes will be more valuable than callswith higher strikes, and that puts with higher strikes will be morevaluable than puts with lower strikes.In fact, there are two circumstances under which the strike priceof an option will be adjusted. The ﬁrst is a stock split. Take yourplain vanilla 2-for-1 split of a stock trading at $100; the holder of 100shares will receive an additional 100 shares, the stock will trade at$50 ex-distribution, and the total value of the holding will remainunchanged at $10,000. What happens to the trader who was holding a June 100 call on this stock? That option will be adjustedtwofold: the strike price will be halved to $50, and the number ofcontracts held will be doubled. So the holder’s right to purchase 100shares at $100 will become the right to purchase 200 shares at $50.This same adjustment will be made for all x-for-1 stock splits.The treatment is somewhat different when the split is not of thex-for-1 type, but a 3-for-2 split, a 5-for-4, or some other odd combination. Take a $60 stock that is going to split 3 for 2. The holder of100 shares will end up with 150 shares trading at $40. The holder ofone October 60 call will see the strike price adjusted to $40 but willnot end up with 1.5 contracts because there is no such thing as listed
80
The Options Strategist
fractional options. The number of shares per contract will thereforebe increased to 150, so the trader ends up with one October 40 callthat covers 150 shares. This also means that the premium multiplieris changed to 150. If he sells this option (post-split) at $2, his accountwill be credited $300 ($2 ⫻ 150). Options that have more or fewerthan 100 shares as underlying usually trade under an adjusted tickersymbol, such IBZ for adjusted IBM options or GMZ for GM.In the rare cases of reverse stock splits, even those of the 1-for-x,variety, the strike price will be adjusted upward, and the number ofshares covered by each contract will be reduced accordingly. So theholder of a February 10 put of a stock that consolidates on a 1-for-5basis would end up with one February 50 put covering 20 shares.The second event that will trigger an adjustment in strike pricesis a corporation’s paying of a dividend equal to 10% or more of itsstock price. Take a company whose stock is trading at $80 and thatdeclares a special cash dividend of $10. When the stock goes ex-dividend, all option strike prices (puts and calls) will be reduced by thedividend amount of $10.Spin-offs do not usually result in any strike price adjustments. IfBig Brother Corp. spins off its Little Guy Subsidiary on the basis ofone Little Guy share for every seven Big Brother held, then the 50call, which used to be the right to purchase 100 shares of Big Brother,becomes the right to purchase 100 shares of Big Brother and 14shares of Little Guy for the aggregate exercise price of $5,000. Asmall cash adjustment will be made for the fractional share of Little Guy not obtained upon exercise of this call. The options covering both Big Brother and Little Guy will also trade under anadjusted ticker symbol.Interest Rates and DividendsHave you ever noticed that for a given stock the at-the-money callsusually trade at a higher premium than the at-the-money puts?Some people say that it’s because more investors are bullish thanbearish, so the demand for calls is greater, which pushes up prices.Some claim that over the longer term the market has always goneup and calls are justiﬁably more expensive. And others say that theprice of a stock can only go down 100%, while its upside is unlimited, and this asymmetry leads to higher call prices. All of these
Options 401: Option Pricing, Volatility, and a First Greek Letter
81
Table 4.3Risk-Free Interest Rate0%
2%
4%
6%
8%
10%
Call Premium
1.67
1.74
1.81
1.89
1.97
2.05
Put Premium
1.67
1.60
1.55
1.49
1.44
1.39
explanations may make intuitive sense, but they explain nothingabout the relative prices of puts and calls.To understand what is happening, let’s look at one of our theoretical stocks, Boring Boring (YAWN), currently at $60. YAWN paysno dividends, and its options are trading at a 20% implied volatility. Using our options pricing software we can calculate the theoretical value of the 45-day YAWN at-the-money puts and calls forvarious interest rate assumptions, as shown in Table 4.3.Only Japan has (nearly) tested the 0% risk-free interest rateassumption. Nevertheless, the result we obtain is interesting: underthat assumption, the calls’ and puts’ premiums are identical. Furthermore, as we use higher and higher interest rate assumptions,the call premium increases while the put premium decreases.The difference between the put and call premiums reﬂects thetime value of money. Assume you own a call option on YAWN, thestock has moved up above the option’s strike price, and your intention is to exercise this call because you want to take possession ofthe stock. Equity options are American-style, so you could exerciseyour call even though it may still have four weeks left until expiration. But why would you exercise early? Why spend the money nowwhen you can spend it later, keeping it in an interest-bearingaccount in the interim? Call options let you defer purchasing thestock, and the higher the interest rate, the more valuable this featurebecomes, as your cash earns more interest sitting on the sidelines.The calls’ premiums reﬂect this.Higher interest rates make call options more valuable. Why dothey reduce the value of puts? The fact is, higher interest rates widenthe difference between put and call prices (remember that under the0% interest rate assumption, the put and the call were valued at thesame price). We will return to this idea in Chapter 9.
82
The Options Strategist
Table 4.4Time Until Expiration45 Days
90 Days
135 Days
180 Days
Call Premium
1.81
2.65
3.33
3.92
Put Premium
1.55
2.11
2.53
2.86
Difference
0.26
0.54
0.80
1.06
Table 4.4 is a further illustration of this phenomenon. The longerthe time to expiration, the more valuable the call becomes relativeto the put.What about the impact of dividends? The call holder can earninterest on the cash until the option is about to expire; but he forgoes any dividend paid. This reduces the advantage of the long callposition. The higher the stock’s yield, the lower the advantagegained by deferring the purchase of the stock. Dividends effectivelylower the stock’s cost of carry. If you purchase shares when the riskfree interest rate is 4%, the money used to purchase the stock is notearning the 4% return it could otherwise earn. This is your cost ofcarry. If the stock yields 1%, you are forgoing the 4% interest incomebut collecting the 1% dividend. Your cost of carry has dropped to3%. And there are instances where a stock’s cost of carry is negative:think high-yielding utility stocks in a low interest rate environment.The lesson to be learned from this section is not to jump to conclusions. An analysis that once appeared in a reputable ﬁnancial publication noted that put premiums had become more expensiverelative to call premiums over the past 12 months. The author concluded that investors were becoming more bearish and paying up forputs. The reality was that during this period interest rates had beenfalling, so the author was measuring not a shift in sentiment, but simply how option prices react to a changing interest rate environment.Time to ExpirationOptions are often referred to as “wasting” or “decaying” assets;although these terms carry a negative connotation that may notreﬂect the complete picture, the fact remains that tomorrow an
83
Options 401: Option Pricing, Volatility, and a First Greek Letter
Table 4.5Time to Expiration (Months)InterestRate
1
2
3
4
5
6
7
8
9
10
11
12
0%
1.70 2.40 2.96 3.41 3.81 4.18 4.51 4.82 5.12 5.39 5.65 5.90
4%
1.78 2.56 3.20 3.73 4.20 4.65 5.06 5.44 5.81 6.16 6.49 6.82
option will be worth less than it’s worth today, all else being equal.If all other things were continuously equal, options would be worthless. This section stops everything but the clock and looks at theimpact the passage of time has on options’ premiums.We all understand instinctively why a two-month option shouldbe worth more than a one-month option with identical terms. Moretime means a greater probability that the underlying will move bya substantial amount. But we must now quantify this instinctualknowledge. Table 4.5 calculates the theoretical value of a call optionassuming a 30% volatility, a $50 stock price, a $50 exercise price, andno dividends. The values have been calculated using two differentinterest rate assumptions.The 4% interest rate assumption gives us real-life premiums,while the 0% scenario permits us to focus on a more theoretical relationship. Under the latter scenario, note that the one-month call ispriced at $1.70 and the four-month is at $3.41; that’s a 1-to-2 ratioexcept for rounding. Comparing the one-month to the nine-monthcall we get a 1-to-3 ratio. And the ratio between the two- and onemonth options (2.40 ⫼ 1.70) turns out to be 1.4, which happens tobe the square root of two. The relationship, it turns out, is a function of the square root of time. Given the value of the one-monthoption and a calculator with a square root function, it is child’s playto ﬁll in the grid.This relationship is only approximate in real life, where interestrates are rarely 0%, but it remains useful. For those who are morevisual, this time decay is illustrated in Figure 4.1.Figure 4.1 is the classic illustration of an option’s time decay. Forthe option buyer it contains two pieces of bad news: options losevalue over time, and this time decay accelerates as the expiration
84
The Options Strategist
$7
$5$4$3$2
Option Price
$6
$1$0350
300
250
200
150
100
50
0