two years until expiration and a $60 strikeprice he gets a bit more than $23 in option premium (assuming a30% volatility). So far, he is down $2. But he is now in position toearn an additional $10 since he has a call on company stock at $50and the obligation to sell these shares at $60. He now has the potential to increase his gain by $8 and defer the tax liability, assumingthat the 60 call is materially different from the 50 call in the eyes ofthe tax people.If the stock’s options trade with a 45% volatility, the premiumreceived is more than $25 and the position has the potential to generate an additional $10 in gains. Obviously, this type of strategy isnot for everyone given the capital required to maintain an uncovered short option position for two years, but it does represent oneof the few arbitrage possibilities between the corporate and listedoptions markets.
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If the executive in this example also owns company shares, thismight turn out to be a good news–bad news situation. The goodnews is that these shares could be deposited with his broker to coverthe call options written. The bad news is that writing calls againsthis shares (if in-the-money options are selected) may create a taxable event. See the relevant section in Chapter 10 and consult a professional tax advisor.
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OPTIONS 901:ADVANCEDTHEORETICALNOTIONS
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hania Twain is not impressed by rocket scientists, and ourgoal is not to turn readers into options nerds. Rather, it isto demystify some of the technical terms you may haveheard or read about and to underscore their practical applications (or lack thereof) for options traders and investors.
217Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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The Options Strategist
Calculating Historical VolatilityHistorical volatilities can be found on various websites, includingcboe.com, but for those instances where it is not available, or notavailable for the required time frame, here is how it is calculated.First we must choose the period over which we want to calculatethe stock’s (or index’s) volatility. Let’s take 20 trading days for BakerHughes (BHI). Table 9.1 below is an Excel spreadsheet where wehave calculated BHI’s historical volatility.In the second column are the stock’s daily closing prices. The thirdcolumn calculates the stock’s daily continuously compounded rateof return. One could use straight daily rates of returns, and theywould be very close to the continuously compounded rates, at leastwhen the returns are relatively small. There is a slight advantage inusing continuously compounded rates in that a move from $100 to$101 and a second move from $101 to $100 will result in the positive and negative rates being of equal magnitude, whereas if simplerates are used, a move from $100 to $101 is equal to a 1.00% returnand from $101 to $100 one of ⫺0.99%. To calculate continuouslycompounded rates use the ⫽LN(. . .) formula. For example, to calculate the return from day 1 to day 2 use ⫽LN(B2/B1). LN standsfor natural logarithm.At the bottom of the third column we have calculated the standard deviation of the continuously compounded returns. We simply use the built-in formula ⫽STDEV(. . .); the result obtained is thestock’s daily volatility—in this instance .0237, or 2.37%. This can beconverted to an annualized number by multiplying the daily volatility by 256. Why 256 and not 365? We are interested in the number of trading, not calendar, days, and there are between 252 and254 trading days in a year. If one assumes that the Friday to Monday volatility is no greater than that of the Monday to Tuesdayvolatility and of the volatility between the other weekdays, then onecan ignore Saturdays and Sundays. In other words, weekends donot add to a stock’s volatility. And why do we not use 253 as anaverage? Pure laziness: 256 is equal to 16, a nice round numberand not different enough from 253 to make a difference. BHI’s historical volatility clocks in at 0.3786, or 38%, a mid-range number foran oil driller.
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Table 9.1 BHI: Historical VolatilityDay 1
38.40
Day 2
38.84
0.0114
Day 3
38.46
(0.0098)
Day 4
36.91
(0.0411)
Day 5
36.43
(0.0131)
Day 6
38.10
0.0448
Day 7
36.89
(0.0323)
Day 8
37.13
0.0065
Day 9
37.48
0.0094
Day 10
37.48
0.0000
Day 11
35.96
(0.0414)
Day 12
36.86
0.0247
Day 13
35.84
(0.0337)
Day 14
35.60
(0.0011)
Day 15
35.13
(0.0133)
Day 16
36.19
0.0297
Day 17
36.84
0.0178
Day 18
36.55
(0.0079)
Day 19
36.41
(0.0038)
Day 20
36.57
0.00440.02370.3786
What Volatility Tells Us About DirectionTable 9.2 gives the closing monthly prices of three ﬁctitious stocksover a 10-month period. Which of these is the most volatile?If you use the method outlined above, you will be able to calculate these stocks’ monthly volatilities (which you can annualize by
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The Options Strategist
Table 9.2Bull Run
Bear Cave
Yoyo Corp.
Month 1
100
100
100
Month 2
104
96
104
Month 3
103
97
103
Month 4
107
93
107
Month 5
106
94
106
Month 6
110
90
102
Month 7
109
91
103
Month 8
113
87
99
Month 9
112
88
100
Month 10
116
84
104
multiplying by the square root of 12). The historical volatilities youshould obtain: Bull Run 8.5%, Bear Cave 9.9%, and Yoyo Corp.10.8%. The results are very close because all three stocks ﬂuctuatedin similar ways, alternating between $4 changes and $1 changes. Itis the magnitude of these changes that is important when calculating volatility, not the direction. The volatilities are not identicalbecause the monthly returns were slightly different. For example,from the end of month 2 to the end of month 3, Bull Run was down$1 but this was $1 on a base of $104. Bear Cave was up $1, but froma starting point of $96, which represents a higher percentage thanfor Bull Run.The unfortunate conclusion is that calculating a stock’s historicalvolatility tells us absolutely nothing about its direction, either pastor future.
Do the Markets Have the MondayMorning Blues?In the previous section we assumed that interday volatility betweenFridays and Mondays was the same as the volatility between the
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221
other days of the week. We decided to test this hypothesis andcrunched some numbers.We looked at the closing level of the Standard & Poor’s 500 Index(SPX) from December 30, 1994, to December 31, 2001, giving usseven years of data. We ﬁrst calculated SPX’s volatility looking onlyat the changes between Fridays’ closes and Mondays’ closes. Wherelong weekends occurred, we used the Thursday to Monday changeor the Friday to Tuesday variation. We then calculated SPX’s volatility for all the other days: Monday to Tuesday, Tuesday to Wednesday, Wednesday to Thursday, and Thursday to Friday.Finally, we had to decide how to account for the change in themarket between September 10, 2001, and September 17, 2001. As weall unfortunately remember, the markets were closed on September11 and on the three following business days. Should we include thechange from September 10 to September 17 in our Friday to Monday data or not? To let everyone see the impact, we decided to perform calculations that both included and excluded this one outlier.The results are presented in Table 9.3.Ignoring the market change between September 10 and September 17, we note that the markets are slightly more volatile over theweekends than during the week, with volatility at 18.8% versus17.6%. As a point of reference, for the complete period, the volatility of the SPX (taking all days into account) was 17.97%, 17.87% ifthe September 10/September 17 change is excluded.So it appears that the weekends do add a bit of volatility to themarkets on Mondays, but the impact is less than three days ofvolatility compressed into one. We will leave it to the academicresearchers to tell us exactly how long the weekends are from avolatility perspective.
Table 9.3Friday to MondayVolatility
WeekdaysVolatility
Including Sep. 10/Sep. 17
19.25%
17.62%
Excluding Sep. 10/Sep. 17
18.79%
17.62%
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Calculating Implied VolatilityRemember the equation for distance traveled? It looked like this:s⫽v⫻tIn this equation s stands for distance, v for speed, and t for time.If you traveled 47 miles in 1.2 hours, what was your average speed?You would solve this problem by rearranging the terms of the equation as follows:v⫽s⫼tYou would easily ﬁnd your average speed to have been 39.2 milesper hour. This general type of problem was known as solving for x.At ﬁrst glance, calculating an option’s implied volatility wouldappear to be a simple case of solving for x. We are given the answer(the option’s value) and all but one of the variables (volatility). Wejust rearrange the option pricing equation to isolate volatility on theleft side of the equal sign, and voilà!There is only a slight problem. If you try to isolate volatility on theleft side of the equal sign in the Black-Scholes option pricing equation, you will ﬁnd it impossible. (If you decide to try this, you willeither waste a lot of time or end up winning a Nobel Prize.) So calculating implied volatility is not as straightforward as it may appear.Implied volatility must be calculated through an iterative process(read: trial and error). Assume you want to calculate the impliedvolatility of an option trading at $3.40. Start with a mid-rangevolatility assumption, say 35%, and calculate the value of this optionbased on 35% volatility. You get $2.95. Your volatility assumptionwas too low; try again with a higher number, say, 40%. You get$3.55. Too high; try again with a number somewhere between 35and 40%. And so on.Solving for implied volatility by hand would be cumbersome andtime consuming to say the least. Fortunately, you probably have asmuch computing power in your personal computer as some corporations had in 1973 when Professors Black and Scholes publishedtheir ﬁndings. The process now takes approximately 12 seconds andcan be done on various websites.Implied Volatility of Different Option SeriesAssume you are looking to purchase calls on a particular stock.Obviously you want to pay as little as possible. In terms of volatil-
Options 901: Advanced Theoretical Notions
223
ity, this means buying the call with the lowest implied volatility. Isit worth your time to calculate the implied volatility of the September 45 call and that of the October 50 call to see which is trading ata lower volatility and therefore represents the better buy? We do notthink so.If you start calculating implied volatility for various series in oneclass, you will ﬁnd very few discrepancies. This should not be surprising as professional traders earn their living looking for thesmallest anomalies in option prices and stepping in when they ﬁndone. They also have a major advantage over the trader sitting athome at his computer: they are physically present on the tradingﬂoor, giving them a few seconds’ advantage over the living roomtrader. (If you would like to know how much this few seconds’ headstart is worth, check out the prices of seats on the optionsexchanges.) Floor traders also have much lower transaction costs,meaning they can execute trades at a proﬁt, while at-home traderswould see all of their gains eaten up by commissions.If you ﬁnd major differences in options’ implied volatility thisshould raise a red ﬂag and not be taken as a signal that you havefound a great buying or writing opportunity. For instance, maybethe April options are trading near a 35% volatility and the Mayoptions at roughly 40% volatility. The reason: earnings are scheduled to be announced in early May. This means that the Apriloptions will not be impacted by the earnings news, but the Mayoptions will bear the full brunt of unexpected news, good or bad.This is a relatively common occurrence. Other events that may bereﬂected in diverging implied volatilities: special dividend, spin-offof a subsidiary, announced stock buyback. If you ﬁnd somethingthat looks too good to be true, it probably is.
The Market’s Implied VolatilityWith the options software now available, calculating implied volatility is relatively easy. For the options on the Standard & Poor’s 100Index (OEX) it is even easier because this task is performed by theCBOE and the result disseminated to quote vendors under the tickersymbol VIX.When you ask your broker for a quote on VIX, or when you pullone up from the Internet, what you are getting is the implied volatil-
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The Options Strategist
ity of eight options on the OEX, four calls and four puts, fouroptions from the next monthly expiration, and four from the secondmonth out, four options from the ﬁrst strike above the current levelof the index, and four from the strike immediately below the index.These eight options (the near-term, at-the-money options) tend tobe the most liquid, so there are numerous trades and a lot of up-tothe-minute information is available.VIX Levels
OEX
Just knowing at what level VIX is currently trading does not impartmuch information. But when the current level is compared to historical levels, one can see if it is relatively high or relatively low. Itis also interesting to compare the level of VIX to the ups and downsof the underlying index. Figure 9.1 charts the progress of VIX andOEX for the calendar year 2000.What is striking in this graph is how closely VIX mirrors the upsand downs of the OEX: a higher market leads to a lower VIX, andvice-versa. The mirror image is not a perfect one, but it is fairly consistent (the correlation factor for the year, comes in at ⫺0.71).Relatively long declines in the OEX, such as the one from thebeginning of September through mid-October, are reﬂected by asteadily higher implied volatility. Short-term drops, such as thesevere sell-off in late April, also cause sharp spikes in VIX.
850830810790770750730710690670650
Jan
b
Fe
ryrMa Ap Ma Jun
Figure 9.1 2000
gvctJul Au Sep Oc No De
454035302520151050
OEXVIX
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225
OEX
This is not too difﬁcult to understand if one thinks of options asinsurance. In a falling market, the value of insurance is obvious toeveryone. Investors increase their coverage by purchasing puts, andtraders jump on the bandwagon hoping to beneﬁt from the downdraft. Since the laws of supply and demand are still in force in theoptions market, the price of insurance (i.e., the options’ impliedvolatility), increases.As markets drift up, investors become more comfortable and donot see any need to insure their holdings. Demand dries up, optionwriters settle for lower premiums as they see little risk in the market, and, as can be seen in the graph, VIX moves to a lower level.Not to belabor the point, but take a look at Figure 9.2, which tracksOEX and VIX during calendar 2001.The same general pattern emerges. Note here that VIX hadalready risen to a closing level of 33.87 on September 10 beforejumping to 44.94 on September 17 and peaking at 49.04 on September 20. For the year, the correlation coefﬁcient clocks in at ⫺0.60. Forreaders interested in looking at prior years or up-to-date data, bothVIX and OEX levels can be downloaded from the CBOE’s website.VIX is often referred to as the “fear index”: the more nervousinvestors and traders are, the more they are willing to pay foroptions. A lesser-known index, the VXN, calculates the impliedvolatility of options on the Nasdaq-100 Index, providing a secondmeasure of fear.
750
60
700
50
650
40
600
30
550
20
500
100
450
Jan
b
Fe
ryrMa Ap Ma Jun
Figure 9.2 2001
gvctJul Au Sep Oc No De
OEXVIX
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The Options Strategist
How should investors use this index? VIX is a thermometer—ittells the current temperature in terms of implied volatility. Whenvolatility is relatively high, options buyers are paying top dollar andstrategies that could be negatively impacted by a return to “normal”volatility (such as the straight purchase of puts and calls) shouldprobably be passed over in favor of those strategies that are lessaffected by changes in implied volatility (such as spreads).For those whose primary interest is equity rather than indexoptions, let us add our personal observations, which lead us to conclude that the same phenomenon happens with stock options’implied volatility. If a stock’s price is hit hard due to bad earningsor other news, implied volatility will rise more often than not. Asthe stock settles down and later begins to rally, one can expect areturn to more normal levels.VIX: Thermometer or BarometerImplied volatility is the market’s estimate of future volatility.Options are priced based on expectations, and the price at which anoption trades represents an agreement between buyer and sellerabout future price ﬂuctuations.How accurate is the forecast of future volatility embedded inimplied volatility? Are the professionals pricing options by gazinginto their crystal balls, or do they have one eye on the rearview mirror? We looked at VIX over a seven-year period, from December 30,1994, through December 31, 2001, and compared it to the index’sactual volatility over the following 10 and 20 trading days. This gaveus a way to measure how well VIX could forecast future volatility.To obtain a second point of reference we also measured how VIXcorrelated to the past 10 and 20 trading days. Our results are presented in Table 9.4.A perfect correlation would have an r2 of 1.00; if there is no correlation between VIX and past or future volatility, the r2 would be0.00. Our results show a higher correlation to past volatility than tofuture volatility, meaning VIX’s major appeal is as a thermometer(it tells us how “hot” volatility is at the present), not as a barometer(which would tell us if we should expect a “high“- or a “lowpressure” system).
Options 901: Advanced Theoretical Notions
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Table 9.4r2Correlation to Next 10 Trading Days
0.663
Correlation to Next 20 Trading Days
0.669
Correlation to Past 10 Trading Days
0.768
Correlation to Past 20 Trading Days
0.826
If this is a bit of a disappointment, remember that looking at theoutdoor thermometer can help you decide how to dress before stepping out. Taking a peak at VIX can help you tailor your optionstrategies for the prevailing volatility climate.
The Greeks: GammaAs was mentioned in Chapter 4, deltas change. And as was illustrated in Figure 4.4, deltas change at various rates. Gamma measures how rapidly deltas change. One could say that gamma is delta’sdelta.Most options software will give you gamma in addition to theoretical price and delta. If we go back to the market maker with thecomplex position in ABCDE options (whom we met in Chapter 4),we can calculate her overall gamma just as we calculated her overall delta (see Table 9.5). But note that gamma’s sign is determinedby whether a position is long or short, not by whether it involvesputs or calls. Buying options, puts or calls, creates positive gamma.Writing options results in negative gamma.Once again we have multiplied the results by 100 to adjust for thenumber of shares per option. Call gamma is 71.9, put gamma is⫺181.0, for a total gamma of ⫺109.1. But what does negative gammamean?Negative gamma tells us that if the price of the stock moves up,the position’s total deltas will decrease—in this case by about 110deltas for a $1 rise in the underlying. Remember that the options’position total delta (before hedging with short stock) was 1,010, theequivalent of being long 1,010 shares of stock; if ABCDE goes from
228
The Options Strategist
Table 9.5MMMMMMCallsMMMMMM
MMMMMMPutsMMMMMM
StrikePrice
Quantity
Gamma
Q⫻Gamma
Quantity
Gamma
Q⫻Gamma
45
⫹17
.039
66.3
⫺22
.039
⫺85.8
50
⫺32
.063
⫺201.6
⫺32
.063
⫺201.6
55
⫹37
.056
207.2
⫹19
.056
106.4
Total Call Gamma
71.9
Total Put Gamma ⫺181.0
$50 to $51, the position’s total delta would decrease from 1,010 to900. This is not good. Our trader is losing deltas as the underlyingstock is going up. Yes, she is still making money as the stock goesup (remember she has positive deltas), but the higher the stock rises,the less money she is making, and her positive delta position couldeventually become negative.What if the stock goes down? She has 1,010 deltas to start; if thestock goes down, her deltas would increase by 110, to 1,120 on a $1fall of ABCDE. She is now picking up deltas (also referred to as getting longer) as the price of the stock goes down. Not a good idea.Being long 1,010 deltas means she loses $1,010 as the stock goesfrom $50 to $49. At $49 she is now long 1,120 deltas, which meansshe would lose $1,120 on a further drop from $49 to $48. Being shortgamma reduces her gains as the stock rises, increases her losses asit drops.No need to be a rocket scientist to ﬁgure out what happens whena trader is long gamma. As the underlying stock goes up, the position picks up deltas and becomes more and more proﬁtable. As theunderlying stock drops, the position loses deltas, reducing losses ina falling market.The starting point for the analysis above was a position with netlong deltas. Had the initial position been net short deltas, the general results would have been similar. If one starts with negativedeltas and negative gamma, the position’s net deltas become morenegative as the stock rises and less negative as the stock drops. Starting with negative deltas and positive gamma means that the deltasbecome less negative as the stock increases in price and more negative as it declines.
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Another way to look at this is that a position that is long gammabeneﬁts from volatility in the underlying, which is why a longgamma position often is referred to as being long volatility. Conversely, a position with short gamma will get hurt if the underlyingmoves and is therefore known as being short volatility.Long Straddle/Short StraddleIf this theoretical discussion about long/short gamma has yourhead spinning, take as a starting point a long at-the-money straddle. A long at-the-money straddle has one long call with a delta ofapproximately 0.50, and one long put with a delta of approximately⫺0.50. The straddle’s total delta is therefore close to zero, which tellsus that for a small move in the price of the underlying stock thevalue of the straddle is not expected to change because the gain onone option will be offset by the loss on the other.Now think what happens if the underlying stock starts going up.The delta of the call slowly increases as this option goes in-themoney; the delta of the put increases as it goes out-of-the-money(which, since it started at ⫺0.50, means that it is converging towardzero). The straddle’s total delta becomes positive as the price of thestock rises. And the more the stock advances, the more positive thestraddle’s delta becomes. Gains on the call are greater and greateras losses on the put are smaller and smaller. This is good.Now think what happens if the price of the stock starts to fall. Theput’s delta starts dropping below ⫺0.50, and the call’s delta also isdecreasing; the straddle’s delta is becoming negative as the price ofthe stock drops. Once again, this is a good thing. A long straddle isthe purest form of a long gamma (or long volatility) position.Conversely, a short straddle is a position with short gamma (shortvolatility). The straddle writer is hoping that the price of the underlying will go nowhere. The short straddle’s initial delta is close tozero, with ⫺0.50 delta from the short call and 0.50 delta from theshort put. But if the price of the stock starts to rise, the delta of theshort call will get more negative, going from ⫺0.50 to, say, ⫺0.60,and the delta of the short put will become less positive, going from0.50 to, say, 0.40. The straddle’s delta is becoming negative as theprice of the stock is rising. Not a good idea. The same holds true ifthe stock starts to fall: the put’s delta starts increasing, and so does
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The Options Strategist
the call’s. The result is that the straddle’s net delta becomes morepositive as the price of the stock heads south. The short straddle isthe simplest illustration of short gamma (or short volatility).
The Greeks: ThetaIn Chapter 4 we saw how options lose value over time, all else beingequal. Some options (in- and out-of-the-money options) decay in arather linear fashion while others (at-the-money options) start outlosing time premium at a constant rate but see this erosion accelerate as expiration nears. Theta quantiﬁes this time decay.Theta is calculated for a given time period: one-day theta, ﬁveday theta, seven-day theta, and so on. The number of days represents calendar, not trading, days. For instance, your option pricingsoftware may give the seven-day theta for a speciﬁc option as ⫺0.08.This tells you, everything else being equal, that over the next sevendays this option will decay by $0.08. Note that this theta does nottell you what would happen to your option position in the subsequent seven days. Calculating what this option’s seven-day thetawill be one week from now, we get ⫺0.12, an indication that timedecay is accelerating.Purchasing options creates a position with negative theta, as bothputs and calls will decay over time. Writing options generates positive theta, as a short option position will beneﬁt from the passageof time. A note of caution: not everyone agrees with assigning negative theta to long and positive theta to short options positions.Some authors and computer programmers inverse the signs. Makesure you are on the same wavelength as the author you are reading.As with other Greeks, the theta from various options can beadded together. You created a convoluted spread where you arelong 6 options (each with a seven-day theta of ⫺0.14) and short 10(with a seven-day theta of ⫺0.07). Will the passage of time help orhurt you? Simply calculate your total theta: (6 ⫻ ⫺0.14) ⫹ (⫺10 ⫻⫺0.07) ⫽ ⫺0.14. You are short theta, so time will hurt your overallposition, but only to the tune of $14 over the next seven days for thecomplete position.
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The Greeks: VegaVega is not a Greek letter; it just sounds like one. Vega measures theimpact a change of 1% in an option’s volatility estimate will haveon that option’s value. Table 9.6 illustrates the vega of three puts andthree calls, all calculated assuming a stock price of $40, an initialvolatility of 35%, three months until expiration, a risk-free interestrate of 3%, and no dividends.From Table 9.6 we can see that the 40 call is valued at $2.90 undera 35% volatility assumption. Since vega measures the impact of a1% increase in volatility, a 36% volatility assumption would raisethe call’s value by approximately $0.08. The value of the 40 putwould also increase by $0.08. Puts and calls with the same strikewill have similar if not equal vegas, and vega is highest for at-themoney options. Both puts and calls have positive vegas, signifyingthat option buyers beneﬁt when volatility increases. Writing optionswill create a position with negative vegas—one