remaining has a delta that changes rathermoderately as the underlying moves from far out-of-the-money (lefthand side of graph) to far in-the-money (right-hand side of graph).However, for a short-term option, the change is much more dramatic, as the call delta rises to nearly 1.00 rather quickly when theoption is in-the-money (right-hand side; higher line on graph), andthe delta falls to nearly zero rather quickly when the option is out-ofthe-money (left-hand side; lower line on graph).In February 1995, just a few days before February expiration, Motorolaannounced a slowdown in sales of cellular telephones, their most importantproduct. The stock fell over six points, from 64 to 58, and was down evenmore intraday. This came one day before expiration of the February options.Holders of out-of-the-money puts registered varying degrees of success,but nothing shows it more dramatically than the comparison of the Feb 55puts and the March 55 puts. On the day before the announcement, with theFigure 6.8COMPARISON OF DELTA AND TIME REMAINING
488
TRADING VOLATILITY
stock trading at 64 (up two points that day, by the way), both of these putswere trading at 1⁄16 of a dollar—the minimum value. When the stockdropped (dramatically) the next day, the Feb 55 puts never budged; theystill traded at 1⁄16. However, the March 55 puts jumped to a dollar, risingalmost a full point.Perhaps more vividly displaying the message, the January (1997) 50LEAPS put, which had almost two years of life remaining at the time butwhich was five points farther out of the money, outdid both of these otherputs, rising by 11⁄2 points. Thus, the power of time as an effect on theamount an option can move is dramatically shown by this actual example.The big move by the stock affected the option’s deltas as well: the Feb55 put had a delta of 0.00 before the drop and still had a delta of 0.00 afterit, thereby confirming the fact that the delta of a short-term out-of-themoney option doesn’t change much when the stock moves (even this far).The March 55 put’s delta went from –0.01 before the stock dropped to–0.16 after, an increase of 0.15. However, the longer-term option, eventhough it changed more in price, had a smaller change in delta, as its deltawent from –0.13 to –0.23, a change of only 0.10. These facts are summarized in the following table:
StockFeb 55 putMarch 55 putJan (1997) 50 put
OriginalDelta
PriceChange
ResultingDelta
–0.00–0.01–0.13
–60+1+11⁄2
–0.00–0.16–0.23
We mentioned earlier that the delta of an at-the-money option isnot exactly one-half (0.50). Rather, it is something larger. Even moreinteresting is the fact that the delta of this at-the-money call changes astime passes. Figure 6.9 is a chart of the call’s delta and shows this quiteclearly. A put would show a similar change; but the put’s delta wouldbe going in the opposite direction, changing from something like–0.43 to nearly –0.50 as time passed (remember that the put’s delta isthe call’s delta minus one). Consequently, the at-the-money put becomes a little more responsive to price changes by the underlying asexpiration approaches, while the call becomes slightly less responsive.However, the magnitude of difference is small in either case.
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Figure 6.9DELTA OF AT-THE-MONEY OPTION AS TIME PASSES
We have already seen that the price movement of the underlyingsecurity makes the delta change, and it has also been shown that thepassage of time changes the delta. The third major influence on acall’s delta is volatility. Remember that there are two types of volatility, implied and historical. Historical volatility is the measure of howfast the underlying instrument has been changing in price; impliedvolatility is the volatility that is currently being displayed by theoptions—a sort of projection of future volatility. Either one can affectthe delta of an option, although it is implied volatility that usually hasthe more dramatic, short-term effect.The options on a low-volatility stock have only a small amount oftime premium. Thus, out-of-the-money options are not so responsiveto short-term stock movements, and in-the-money options are veryresponsive. This means that the delta of out-of-the-money optionson a low-volatility stock are small—perhaps near zero—and that thedelta of a corresponding in-the-money option is large. The graph inFigure 6.10 shows this fact. This is somewhat similar to the wayoptions with only a short life remaining behave, as discussed earlier.
490
TRADING VOLATILITYFigure 6.10COMPARISON OF DELTA AND VOLATILITY
On the other hand, if the underlying is very volatile, then its out-ofthe-money options will be more responsive to changes in the price ofthe underlying security; and its in-the-money options, since they havetime value premium, won’t be able to move complete point for pointwith the underlying. This is also shown in Figure 6.10 and is similarin nature to the behavior of a longer-term option.Implied volatility can change very quickly; that is, the market’sperception of future volatility can change overnight. The followingexample shows how this happens.In 1994, Gensia Pharmaceuticals, a fledgling drug company, was a typicalhigh-volatility stock. Its normal volatility was about 70 percent annually,which is quite large (the market as a whole is typically about 15 percent). InJuly 1994, the Food and Drug Administration (FDA) announced that it wasgoing to hold hearings on the Gensia’s main drug. The stock slipped from aprice of about 10 to 8 and then slowly climbed back to 10 over the nextmonth; so the stock was actually not volatile, but the options were. The
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implied volatility of the options immediately jumped dramatically and wasup in the 130 percent to 140 percent area. This was because traders knewthat the FDA’s ruling would either “make or break” this company. Eventually, in October, the FDA ruled against Gensia, and the stock fell 50 percentin price overnight. However, once that was over, the implied volatilitychanged overnight from 130 percent back to 70 percent, as traders nolonger needed to discount such a volatile event as the FDA ruling in thecompany’s future.
Implied volatility can experience these sudden jumps for a varietyof reasons. One is a major corporate announcement, such as the FDAruling in this example or as in the lawsuit example that was given earlier. However, another reason for implied volatility to increase dramatically is a perceived change in the future price of the underlying.For stocks, this usually means a takeover rumor; for grain futures, itmight often mean expectations of bad weather. When these situationsoccur, there is not such a well-defined reason for the change in volatility (as an FDA ruling, for example). Instead option traders may oftenfind their options changing in price quite dramatically, even thoughthere is no apparent news to account for it and there is no particularprice movement by the underlying security either. In essence, thedelta of the option is changing dramatically. We can envision this scenario by looking at the previous graph and thinking of an option’sdelta lying along one line on the graph and suddenly moving to theother line, without the stock price changing at all.It is important for the strategist to understand these concepts, sohe will also understand that any “neutral” position is not risk-free; itis only neutral with respect to the variable(s) that have been neutralized initially. Subsequent changes in the other variables will adverselyaffect the neutrality. We saw this in examples regarding delta earlierin this chapter, where an initially delta neutral position became quitenonneutral when the underlying stock price changed.
VegaThis is the nomenclature given to the Greek term that describes howa change in volatility affects the price of an option (again assuming
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TRADING VOLATILITY
that all other variables remain the same). Since the volatility that weare speaking of here is strictly the volatility that we are plugging intoour valuation model, we are really talking about implied volatility.Vega is probably not a familiar term to most option traders, but itshould be, for nothing affects the price of an option so dramaticallyas the volatility. In fact, as has been demonstrated several times inthis book, a change in implied volatility can have a large short-termaffect on the price of an option.Since the subject of the chapter is volatility trading, we will betalking about vega a good deal in succeeding sections. For if we canmeasure the effect that a change in volatility has on a position, thenwe will know how much risk we are taking when trading volatility.Example: Suppose, as we did earlier in this chapter, that we observe thatthe volatility of a certain security trades in a range between 20 percent and34 percent. Furthermore, suppose that volatility is currently 32 percent,and we are considering selling the volatility.Assume that we establish a position that has a position vega of –5.00;that is, for each one point increase in volatility (i.e., from 32 percent to 33percent), we would lose $500. This assumes that –5.00 vega is worth$500, which it would be for any option whose unit of trading was $100 perpoint. If it were a futures option, however, with a larger unit of trading, thenthe volatility risk would be –5.00 times whatever the unit of trading was.Vega is stated as a negative number here, indicating that a falling volatility would be profitable for the position, while a rising volatility would beharmful. If we truly believe in our volatility trading range having a top of 34percent, then our risk due to volatility is two points (from 32 percent to 34percent), or $1,000.If we are looking for volatility to decrease back to 27 percent, for example, which is the center of the volatility’s past trading range, then we couldmake $2,500 ($500 times 5 points of volatility decrease, from 32 percentto 27 percent) if that happened.Overall, then, this is a position that has $1,000 of risk and $2,500 ofprofit potential, as measured by the vega. Of course, as we know, other factors could influence the position, including the fact that implied volatilitycould trade to higher levels than previously seen. Nevertheless, this exampleshows that by using vega, we can measure the volatility risk of a position.
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Later in this chapter, we explore ways to isolate volatility fromthe other variables in order to produce a position that is neutral tothe market with respect to everything except volatility.
ThetaTheta is the measure of time decay on an option. It is stated as a negative number, indicating that as one day passes, the value of anoption will decrease in value. Theta is useful in describing just howthe passage of time affects a portfolio of options. If you have a lot ofshort (naked) options in your portfolio, then you will have a very positive theta, because time is your ally. On the other hand, if you own alot of options, then the theta of your portfolio will be negative, accurately describing the loss that you can expect daily due to time decay.
RhoRho measures the effect that a change in interest rates would haveon the price of an option. Recall that the interest rate used to valueoptions is the short-term, risk-free interest rate, which is normallyconsidered to be the 90-day T-bill rate. Rho is a very small numberfor short-term options, which is what most traders have in their positions, and is therefore rightfully ignored by most traders. It does havemore meaning for longer-term options, such as LEAPS. In fact, thelonger the remaining life of the option, the more important interestrates are. Figure 6.11 shows three theoretical value curves for a twoyear LEAPS option. The top curve depicts the prices if interest ratesare 9 percent, the middle curve if interest rates are 6 percent, andthe lower curve if interest rates are 3 percent. You can see from thedistance between the curves that there is a significant effect frominterest rates.Of course, you wouldn’t see short-term interest rates change by 3percent instantaneously, but you could see such a change over afairly short period of time if the Fed is aggressively trying to raise orlower rates.
494
TRADING VOLATILITYFigure 6.11TWO-YEAR LEAPS CALL PRICING CURVE
During 1994, the Fed raised the discount rate several times in an attempt toslow down what was perceived to be an overheated economy. During thecourse of this raising of rates, the risk-free rate rose from 3 percent to 5.5percent in about 10 months. Typically, when rates are raised, stock pricesfalter. And 1994 was no exception, as there were three or four fairly steepdeclines in the broad averages that year, and the averages were off slightlyfor the year.What benefited from this rise in rates were LEAPS options. In fact, therise in rates over eight months actually negated a large part of the timedecay for the longer-term options. Essentially the prices moved from thebottom line on the graph in Figure 6.11 to the middle line, although eightmonths passed in the interim.
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Gamma, RevisitedWith the preceding four measures, you can determine how youroption positions or your whole option portfolio will react to changesin the marketplace. Still, some theoreticians felt that this was notenough, because two of them—delta and vega—can change sorapidly that it is difficult to neutralize them. So, they thought that ifthe change in the delta could be measured, for example, it would beuseful. This is the gamma, which was discussed earlier in this chapter. The gamma measures how much the option’s delta changeswhen the underlying changes in price by one point. Essentially, it is ameasure of how fast the delta changes. Thus, if gamma and deltacould be neutralized, then even if the underlying price changed, theoption position in question would continue to remain delta neutral.Eventually, if the price of the underlying moved far enough, bothgamma and delta would acquire some non-zero value, and the position would lose its neutrality. Still, a delta and gamma neutral position has a much better chance of remaining delta neutral than doesone that was initially only delta neutral.Gamma is related to the other measures of risk (in fact, as wesaid earlier, they are all interrelated). Figure 6.12 shows how gammais related to the amount of time left until expiration of the option inquestion. If there is a lot of time remaining, the gamma is a relativelystable number, whether the option is in- or out-of-the-money. That isshown by the bottom curve on the graph.This fact can be depicted in tabular form as well. Table 6.1assumes that the option has one year of life remaining and that thestriking price is 50. It shows the theoretical value of the option andits delta. The gamma is merely the difference in delta for a one-pointmove in the underlying stock. Thus, in this table, we are showing theobserved gamma (the actual difference in deltas), as opposed to themathematically computed gamma. The two are essentially the same.You can observe that gamma is fairly stable for this longer-termoption: the delta increases rather uniformly as the stock rises (ordecreases uniformly as it falls). However, if we consider a short-termoption, then things change substantially. Figure 6.12 shows that thegamma of a one-month option behaves quite violently when the
496
TRADING VOLATILITYFigure 6.12GAMMA VERSUS TIME
stock is near the striking price. Table 6.2, which is similar to Table6.1, shows the situation for a one-month option.In the case of this short-term option, gamma is stable and nearlyzero for options that are only moderately in- or out-of-the-money.This makes logical sense. For example, a moderately out-of-themoney option that has very little life remaining has a delta of nearlyzero. Even if the underlying rises by a point, that delta is still going tobe very nearly zero. Thus, the one-point rise by the underlyingdoesn’t increase the delta much at all; this is merely another way ofsaying that gamma is almost zero as well. The same thing holds truefor the in-the-money option, because in that case the delta is verynearly 1.00, and a one-point move by the underlying won’t changethe delta much. Hence, once again, the gamma is nearly zerobecause the delta doesn’t change for a one-point move by theunderlying.However, near the striking price, things are much more interesting. The delta changes rather quickly, especially for a short-term
Table 6.1ONE-YEAR CALL WITH STRIKE PRICE OF 50Stock Price
Theoretical Value
Delta
Gamma (Observed)
404142———495051———585960
1.021.271.57
0.240.280.32
0.040.04
4.875.506.18
0.620.660.69
0.040.03
11.7212.6013.50
0.870.890.91
0.020.02
Table 6.2ONE-MONTH CALL WITH STRIKE PRICE OF 50Stock Price
Theoretical Value
Delta
Gamma (Observed)
414243———4849505152———5960
0.000.000.01
000.01
00.01
0.470.811.291.902.64
0.280.410.550.680.79
0.130.140.130.11
9.2510.25
1.001.00
00
497
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option or a low-volatility stock. This, too, is logical, for near expiration, the time value of an option disappears quickly. Thus, a smallincrease in the stock price as it moves higher from the striking pricewill result in the delta increasing from something just above 0.50 tosomething very large quite quickly. So, you can see that gamma isrelated to time and volatility.
Gamma and Delta NeutralThe next logical step in constructing a price neutral position is toestablish one that is neutral with respect to both delta and gamma. Inthis manner, the position will not acquire much risk due to pricechanges by the underlying instrument. It is a simple matter to construct such a position, although the thought of doing it is somewhatdaunting to many traders. There are really only two steps required:(1) create a gamma neutral position, and then (2) neutralize the delta.The second step can always be accomplished with (equivalent) sharesor contracts of the underlying security.Example: Suppose that a stock is trading at 98 at the beginning of the year,and a trader is interested in setting up a gamma and delta neutral positionwith the following three-month options:Option
Price
Delta
Gamma
March 100 callMarch 110 call
52
0.500.25
0.0300.020
Note that gamma is a small number, so you should use at least threedecimal places when dealing with it, especially if you are trading large quantities.Now, in order to create a gamma neutral position, we merely need todivide the two gammas in question to determine the neutral ratio. In thiscase, a neutral position would be to buy two March 100 calls for every threeMarch 110 calls that we sell, since the gammas are in the same ratio of twoto three (0.020 to 0.030). This position is going to have a nonzero delta,and in this example we are concerned with neutralizing that delta.
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For the purposes of this example, let’s assume that our actual trade is tobuy 200 March 100 calls and sell 300 March 110 calls. This position hasthe following ESP, or position delta:Option
Quantity
Delta
ESP
March 100 callMarch 110 call
L 200S 300
0.500.25
+10,000–7,500
Position Delta:
+2,500
To neutralize this position, we could sell (short) 2,500 shares of thecommon stock against the option spread, or we could use our knowledge ofequivalent positions and instead sell 25 March 100 calls and buy 25 March100 puts. If we used the options, our resultant position would be:Long 175 March 100 callsShort 300 March 110 callsLong 25 March 100 putsNote: the use of the equivalent option position does not change thegamma at all. Neither stock nor the equivalent position has any gamma atall (that is, the gammas of the March 100 call and the March 100 put areequal and offset each other when we buy and sell an equal quantity).
In the preceding example, it was assumed that we used the gammas to establish a neutral position that was a call ratio write. In fact,we could have done the opposite. Either of these positions would begamma neutral:Buy 2 March 100 callsSell 3 March 110 callsorSell 2 March 100 callsBuy 3 March 110 callsWe would choose one or the other, depending on what we wereactually trying to accomplish. If we were trying to sell volatility, we
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TRADING VOLATILITY
would use the first position—the call ratio write. However, if we wereattempting to buy volatility, we would use the second position—thebackspread. The example did not show that position’s exposure tovolatility; but from what we have already demonstrated in this chapter, we know that the call ratio write is one of the strategies usedwhen selling volatility, while the backspread is a strategy used for buying volatility. We address the volatility exposure of a gamma– deltaneutral position shortly.First, however, let’s examine this position a little more closely sothat you can see just what the characteristics are of a position that isgamma and delta neutral. Figure 6.13 is that of the delta of this position for various stock prices. The more horizontal line is where thedelta would be after 7 days had passed. The other line, which haswider swings, is where the delta would be in 30 days. You can seethat even though the position starts out delta and gamma neutral, itdoes acquire a delta after time passes, especially when the underlyingstock changes dramatically in price. You must remember, though,that this example position was a rather large position, so the deltasFigure 6.13POSITION DELTA OF RATIO SPREAD
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that are acquired are not all that large. In fact, the deltas after 7 daysare rather insignificant—this is the advantage of the gamma–deltaneutral position.Notice that the deltas stay relatively flat over a range—up to aprice of about 100 in 7 days and up to a price of about 107 in 30days. This means that you would probably not have to adjust yourposition unless the stock rose above those prices. Above thoseprices, you begin to get rather delta short. Also, after 30 days, youcan see that a large downside move will cause the position to acquirea short delta (that’s because we are long the 25 March 100 puts). Aneutral trader might want to adjust in that case as well.Suffice it to say that the delta of this example position is muchmore stable than a typical one-by-two call ratio would be (buying 100March 100 calls and selling 200 March 110 calls, for example,which is only a delta neutral position).The actual position that we constructed with this gamma–deltaneutral position is mostly a call ratio write, but it has a few extra putsadded for downside profit potential. Also, the ratio of the long callsto short calls is less than you would have in a strictly delta neutralposition (1.75-to-3 instead of 1-to-2), so the position doesn’t have asmuch upside exposure to naked options.In case you were wondering how this gamma–delta neutral position looked, from a profit-and-loss standpoint, Figure 6.14 shows theprofitability at expiration (straight lines) and in 30 days (curved line).This graph assumes that volatility remains the same, so the profitsand losses are accruing strictly from stock price movement and fromthe passage of time. In reality, we would only take such a position ifwe were expecting volatility to decrease. We will next see how thatevent would affect the profitability.This example position initially had an exposure to rising volatility.In other words, the position is short volatility; or in Greek terms, ithas a negative vega. This is what we would want if we were sellingvolatility. In order to see how these statements are quantified, let’sexpand on the previous example by now including volatility in ourcalculations.Example: Let’s make one more assumption about this position. Let’sassume that the trader had observed that volatility typically traded in a range
502
TRADING VOLATILITYFigure 6.14PROFIT OF RATIO SPREAD
between 20 percent and 30 percent for this stock. Now, with volatility at 30percent, he wants to establish a gamma–delta neutral position in order tosell the volatility. Our example position is just the ticket.The data listed below is exactly the same as from the previous example,except that now we can see the vega of each option as well. Remember, thevega is the amount by which the option price would increase for a (instantaneous) 1 percent increase in implied volatility. The put has been added tothe table as well.Option
Price
Delta
Gamma
Vega
March 100 callMarch 110 callMarch 100 put
5257⁄8
0.500.25–0.50
0.0300.0200.030
0.1800.1500.180
Our initial position—long 175 March 100 calls, short 300 March 100calls, and long 25 March 100 puts—was delta and gamma neutral, but ithas a volatility exposure. We can easily compute that exposure by using thevega of the options.
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PositionLong 175 March 100 callsShort 300 March 100 callsLong 25 March 100 puts
Option Vega
Position Vega
0.1800.1500.180
+3,150– 4,500+450
Total Position Vega:
503
–900
Note that vega is always expressed as a positive number; thus, longoption positions (March 100 calls and March 100 puts in this example) havea positive vega, while short option positions have a negative position vega.The preceding position vega indicates that if implied volatility falls by 1percent—from 30 percent to 29 percent, for example—then the positionwill profit by $900. That is an instantaneous measure, of course, and it isaffected by stock price changes and the passage of time. Figure 6.15 showsthe profit in 30 days, with the lower curve being the same as seen on theprevious profit graph—it is the profit if volatility remains unchanged. The
Figure 6.15PROFIT AND LOSS OF RATIO SPREAD IF VOLATILITY CHANGES
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higher curve shows what would happen if volatility declined to 25 percent,about back to the middle of the 20 percent to 30 percent range that weassumed for this example.You can see how much extra profit that the decrease in volatility wouldgarner for this position, if the stock is above 98.
This rather detailed example shows how the gamma–delta neutral position can profit nicely from a decrease in implied volatility,while basically maintaining a much more price neutral outlook than amere call ratio spread would have. We can look at each of our fivepreferred delta neutral strategies for trading volatility and see howthey would look as gamma–delta neutral positions. The transformation is quite interesting. However, before doing that, we give you onemore piece of the general puzzle, and that is how to construct a position that is both gamma and delta neutral and also has the volatilityexposure that you feel comfortable with.This is easily accomplished by solving two equations in twounknowns and then adjusting for any residual delta. As soon as yousay, “two equations in two unknowns,” many people panic. Relax.There are easy ways to accomplish this. First of all, it’s only highschool algebra; so if you have a high school or college student in yourfamily, they can probably solve the problem for you. More realistically, there are simple programs that can solve two equations in twounknowns. These are available from shareware or are possibly evengiven away. If you still can’t find one, call us and we’ll send you oneif you pay for the postage.We will continue to use the same example to illustrate this point.We are going to arrive at essentially the same position as we did earlier, but in this case, we are going to approach the whole thing witha specific idea of the risk we want to take. In the earlier example, wemerely found the gamma–delta neutral position and then looked athow large the vega was.Example: Suppose that you want to establish a call ratio spread with theMarch options—as we did earlier—and you want to risk $1,000 for eachpoint that volatility rises. Of course, that means that you will also be making$1,000 for each point that volatility declines, and that is where your profit
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will come from. In Greek terms, risking $1,000 per point of rising volatilitymeans that the position has a vega of –10.00.Option
Price
Delta
Gamma
Vega
March 100 callMarch 110 call
52
0.500.25
0.0300.020
0.1800.150
Let x stand for the number of March 100 calls to buy, and y stand forthe number of March 110 calls to sell. Remember, we want to be gammaneutral and have a vega of –10. The two equations would then be:Gamma neutral: 0.030x + 0.020y = 0Vega risk:0.180x + 0.150y = –10Solving, we get x = 222 and y = –333. That means we should buy 222of the March 100 calls and sell 333 of the March 110 calls. By doing so,our position will have some delta to it. Since we want to be delta neutral, weneed to rid ourselves of this delta.Position
Delta
ESP or Position Delta
Long 222 March 100 callsShort 333 March 110 calls
0.500.25
+11,100–8,325
Total Position Delta:
+2,775
So we want to either short 2,800 shares or use the equivalent optionposition—sell 28 March 100 calls and buy 28 March 100 puts.