The resulting position is:Long 194 March 100 callsShort 333 March 110 callsLong 28 March 110 putsHence, this position is very similar to the one from the earlier examples. It is slightly larger because we wanted vega risk to be $1,000 perpoint, whereas it was $900 per point in the other example.
Now that the theory for determining an appropriate position hasbeen set forth, we can look at the other strategies for trading volatil-
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ity. The preceding example is a little simplistic in that we have limitedour choices of calls to use in the spread to just the two shown—theMarch 100 and the March 110. In reality, when we are looking atways to sell volatility, we have many options to choose from. While itis reasonable to expect that we would use the at-the-money call asthe long side of the spread, there are still several choices as to whichout-of-the-money call to sell. Usually, the strategist can narrow theselection process down to very few choices; but as we shall see in thenext example, there are times when the selection of the “wrong”options changes the position to something the strategist doesn’treally want.For selling volatility, the one strategy that interests most people isthe short naked combo. What happens to this position when youneutralize gamma? The following example illustrates the situation.We start out by showing you a gamma–delta neutral position, withspecific vega risk that is based on a sell combo.Example: Assume, as in the previous examples, that a trader wants tosell volatility that is currently at the 30 percent level but normally tradesin a range of 20 percent to 30 percent; and he wants $1,000 of volatility risk per point of volatility increase. However, in this example, hewants to use a sell combo to approach the problem. Recall that we prefer to use a combo (two striking prices) rather than a straddle as the shortposition in these cases.With the stock at 98, the trader is considering selling the March 95 putand March 105 call as a sell combo, and then he is looking to hedge thatsale by buying out-of-the-money options. In this example, he is going to usethe Feb 90 puts and Feb 110 calls as the out-of-the-money combo to purchase as a hedge. When he buys an out-of-the-money combo such as this toprotect the short combo, he is said to be buying the “wings.”The following data exists for XYZ: 98, on January 2:Option
Delta
Gamma
Vega
Feb 90 putMarch 95 putMarch 105 callFeb 110 call
–0.18–0.350.370.18
0.0250.0270.0280.024
0.0960.1670.1700.095
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Everything is expressed as a positive number, except for the deltas ofthe puts. The trader will have two equations as before—one for gamma andone for vega. But it appears as if he has four unknowns, since there are fouroptions listed before. In reality, though, he can consider this as twounknowns: one being the March sell combo and the other being the Febcombo to be purchased as a hedge. With this simplification, he substitutesthe sum of the components in the equation.So, let x = the number of March combos to sell, and y = the number ofFeb combos to buy. The two equations then become:Gamma neutral: (0.027 + 0.028)x + (0.025 + 0.024)y = 0Vega risk:(0.167 + 0.170)x + (0.096 + 0.095)y = –10Solving this equation, we get x = –81 and y = 91. That is, if the tradersells 81 of the March combos, he should buy 91 of the Feb combos as ahedge. If he wants, he can calculate the position gamma and position vegato ensure that they satisfy his criteria— gamma neutral and vega risk of –10.The delta of this position can be computed as:Position
Delta
ESP (Position Delta)
Long 91 Feb 90 putsShort 81 March 95 putsShort 81 March 105 callsLong 91 Feb 110 calls
–0.18–0.350.370.18
–1,638+2,835–2,9971,638
Total Position Delta:
–162
Thus, the position is essentially delta neutral. If the trader wanted to bereally picky, he could buy 100 or 200 shares of the underlying to neutralizethat delta; but as a practical matter, it is not necessary.
The position that is derived in the example is a little different thanyou might expect. The “normal” part is that the sale of a combo ishedged by buying a combo that consists of further out-of-the-moneyoptions. What is unusual, though, is that the long options expirebefore the short options do. With this configuration, margin considerations can be considerably different. With futures options, the longoptions are considered to “cover” the short options until the longsexpire, so the margin is merely the spread requirements. However,
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with equity or index options, which continue to operate with arcaneand illogical margin requirements, the short options are considerednaked (unless market maker requirements are obtained, that is, thetrader is trading for an exchange member).Having to margin the short options as naked options makes themargin requirement very large—much, much larger than the riskinvolved. Obviously, no matter how much the underlying moves priorto expiration of the long options—the Februarys in this example—those long options will protect against large losses; that is, the lossesare limited. For the collateral requirements not to recognize that factis preposterous.The margin requirements don’t detract from the profit potentialof the position itself, although they certainly do lower the rate ofreturn. Figure 6.16 shows how the preceding example positionwould look at February expiration, that is, the expiration of the longoptions. The lower curved line depicts the profits if the volatilityremained unchanged at 30 percent, whereas the upper curved lineFigure 6.16WINGS PROFIT AND LOSS
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509
shows the larger profits available if volatility declines to 25 percent—in the middle of the supposed range of implied volatility. Once again,it is apparent that even though the position is initially gamma–deltaneutral, it acquires risk due to the passage of enough time and themovement of the underlying stock. Nevertheless, this position hasconsiderably less risk than the mere sale of a naked combo, which ofcourse has unlimited risk if the underlying should move too far ineither direction.You might be wondering why we used the February optionsinstead of longer-term ones as the hedge for the short combo. Thereason is this: in order to begin with the sale of a naked comboand turn it into a delta neutral and gamma neutral position bybuying the “wings,” it is necessary to have the wings expire beforethe short combo does. The following example shows how agamma–delta neutral position would look if you were to use a longerterm combo on the buy side.Example: Using the same assumptions as in the previous example as far asvolatility and wanting to create a gamma neutral position with vega risk of$1,000 per point are concerned, here are the relevant statistics for XYZ:98, on January 2:Option
Delta
Gamma
Vega
April 90 putMarch 95 putMarch 105 callApril 110 call
–0.24–0.350.370.31
0.0190.0270.0280.022
0.1700.1670.1700.192
Once again, we set up two equations in two unknowns, adding thenumbers from each combo. Let x = the number of March combos to sell,and y = the number of April combos to buy. The two equations thenbecome:Gamma neutral: (0.027 + 0.028)x + (0.019 + 0.022)y = 0Vega risk:(0.167 + 0.170)x + (0.170 + 0.192)y = –10Solving this equation, we get x = 67 and y = –90. With these data,when we set up and solve the simple set of two equations in two unknowns,
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TRADING VOLATILITY
we have a solution that gives us a completely different position. Now, x is apositive number, which means that we are supposed to buy the Marchcombo and sell the April combo. In effect, we are setting up two ratiowrites, one with calls and one with puts:Long 67 March 95 putsShort 90 April 90 putsandLong 67 March 105 callsShort 90 April 110 calls
This is a totally different position than we had planned on originally, but it is the only way to create a gamma neutral position withthe required vega from the four options that we have chosen. Figure6.17 shows how the position looks in 30 days. Once again, thelower curved line represents the profits and losses if implied volatilityremains unchanged, while the higher curved line shows the results ifimplied volatility decreases to 25 percent.Figure 6.17PROFIT AND LOSS OF DOUBLE RATIO SPREAD
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Note that we haven’t really solved the dilemma of margin. Just asin the prior example, we are still short the longer-term options (Apriloptions in this case), so the position is going to require a fairly largeamount of collateral.The reason that this position takes on a shape that is somewhatunexpected is that the vega of the April options is so large. In orderto create a position that has negative vega, which was the originalintent here, it is virtually impossible to be able to buy those Apriloptions because their vega is so large. The combined vega of theMarch options is 0.337, but the combined vega of the April optionsis higher, 0.362. This means that whenever volatility declines by apoint, the March combo will lose about 34 cents; but the Aprilcombo will lose even more—36 cents. Thus, it makes no sense to sellthe March combo and buy the April combo if we want to makemoney from declining volatility.This example points out something very interesting: it is oftendifficult to “intuit” a gamma neutral position that has the volatility riskthat you want. But by using the Greeks, you can see exactly the construct needed to accomplish your goals. In this chapter, we havespent considerable time outlining the groundwork for our strategy:we want to trade volatility and remove as much of the risk as possibledue to price changes by the underlying. By using the Greeks, weguarantee that an appropriate position is taken.I have often seen traders who want to accomplish the samethings as we have outlined in this chapter take a position that doesnot work but seems like it should. Typically, if we decide to sell acombo in order to capitalize on an outlook for declining volatility, wewill then think that buying the wings that expire at the same time orin a later month will protect us. The previous example shows the fallacy of that thinking. In reality, by buying those wings, we effectivelymore than cancel out our profit potential from declining volatility.So far we have seen how two of the more common strategies forselling volatility—the ratio spread and the naked short combo—lookwhen they are transformed into gamma–delta neutral positions. Nowwe examine how the basic strategies for buying volatility—the longstraddle, the backspread, and the calendar spread— can be treated asfar as gamma–delta neutral positions are concerned.
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TRADING VOLATILITY
The Calendar Spread. Without going into as much detail as inthe previous examples, here is how a calendar spread position that isgamma–delta neutral, with appropriate volatility exposure (vega of+10—note the plus, now), would look for XYZ: 100, on January 2,with volatility = 20%:Long 52 June 100 callsShort 44 March 100 callsLong 15 June 100 putsSo this looks something like a normal calendar spread, except weare long a few extra calls on the long side, plus long a few puts aswell. This is a very attractive position, because it can make money ifthe underlying moves far enough in either direction, while also profiting if volatility increases (which is our basic assumption). Moreover,the short calls hedge the position if the underlying remains stagnant.The profitability of this calendar spread is shown in Figure 6.18.The curved lines show the profit at March expiration, the near-termFigure 6.18CALENDAR
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expiration. The lower line on the graph shows the results if volatilityremains unchanged at 20 percent, while the higher line on the graphshows the better results that could be expected if implied volatilityincreased to 25 percent.The Backspread and the Long Straddle. These are the othertwo basic strategies that we use when we want to buy volatility (i.e.,when we want to have a positive vega). I do not believe that if we areusing these strategies, we should be so concerned with gamma neutrality. When these two strategies acquire a delta, it means we aremaking money, for the underlying instrument is moving. The neutralizing of price movement was much more important for the volatility selling strategies because a large stock movement could bedevastating in that case. But here, a large movement will be helpful;so I generally feel that delta neutral, with the appropriate volatilityrisk, is sufficient.Example: Suppose we want to establish a backspread because we think thatvolatility, which is currently at 20 percent and at the low end of its historicalrange, can go higher. The following data describe the situation for XYZ:100, on January 2; volatility = 20 percent:Option
Delta
Gamma
Vega
March 90 callMarch 100 call
0.910.57
0.0180.043
0.0790.180
Solving these two equations gives us a delta neutral position, with avega of +10. If volatility increases, we will make $1,000 per point of volatility increase.Delta neutral: 0.91x + 0.57y = 0Vega exposure: 0.079x + 0.18y = 10Solving, we get x = –48 and y = 76. So the desired position is:Long 76 March 100 callsShort 48 March 90 calls
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TRADING VOLATILITY
This is a typical backspread position. Incidentally, this has a fairly smallgamma anyway, even though we didn’t neutralize it. We can compute thatthe position gamma of this position is +2.40. If the underlying rises onepoint, the position delta, which is neutral to begin with, will increase to2.40, which means that the position will be delta long 240 shares. On theother hand, if the underlying declines by one point, the delta will become–2.40, or the position will then be delta short 240 shares.
The farther that the underlying rises, the more the delta willincrease; the farther it declines, the more the delta will decrease.Eventually, you will have a position that has quite a large delta—eitherpositive in a rising market or negative in a falling market. However,that is a “good” problem, because you will have profits after largemoves by the underlying. I feel the same way about the long straddleposition as well; it’s not necessary to zero out the gamma.If you prefer a gamma–delta neutral strategy for trading volatilityfrom the long side, then I encourage you to use the calendar spread,as shown in a recent example.
Even More Advanced ConstructionsThe neutral positions that were described earlier were both gammaand delta neutral. In theory, you could construct positions that hadneutrality with respect to as many variables as you might want toconsider. Eventually, though, you have to assume some risk somewhere—if you’re a commission-paying trader—if you intend to makemoney; only market makers can be neutral with respect to all variables, for their intention is to buy on the bid and sell on the offer andtake no further risk.In order to create a position that is neutral with respect to threevariables, for example, you would merely have to set up three equations in three unknowns and solve that set of simultaneous equations.For example, you might want to be neutral with respect to gammaand theta (time) but still have volatility risk.Another risk measure that traders have come to use is the“gamma of the gamma.” This measures how fast the gammachanges when the underlying security moves by one point. Thus,many of the more theoretical traders feel they need even another
THE “GREEKS”
515
measure to help them keep a position neutral with respect to pricechanges by the underlying security. This shows just how importantthey consider price neutrality to be.So the three measures that you might want to use in your equation would be gamma, gamma of the gamma, and vega. You wouldstill neutralize delta at the end, as was done in the previous examples.In this manner, you arrive at a position that is very neutral withrespect to price movement but still has volatility risk, a risk we want,in the context of this chapter.The concept can be carried even further—four equations in fourunknowns, for example (gamma, time or theta, volatility or vega, andgamma of the gamma), with delta being neutralized at the end. Sucha position would be totally impossible to reason out, without usingthe arithmetic equations.Example: Without going into too much detail, one of the previous examplescan be used to construct a position that is neutral with respect to four variables and has volatility risk. We will once again begin with the thought of selling the March 95–March 105 combo and using the February 90–February100 combo as protection. As it turns out, this is the way the position looksafter neutralizing the necessary components for XYZ: 98, on January 2:Long 28 Feb 90 putsShort 145 March 95 putsShort 45 March 105 callsLong 82 Feb 110 callsShort 5,400 shares of XYZThis position has these qualities:Delta neutralGamma neutralTheta neutralGamma of gamma neutralVolatility risk: $1,000 per pointThere is risk to the upside in this position, but the 82 long calls mostlyoffset the risk of the 45 short calls and the 5,400 short stock. To the downside, there is more obvious risk, as the 5,400 short stock and the 28 longputs are less of an absolute hedge against all those short March 95 puts.
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TRADING VOLATILITY
It is not normally necessary for the small- or medium-sizedinvestor to attempt to get this neutral, but the example shows that itcan be done.
SummaryThis concludes the section on using the Greeks to neutralize positions, so that the volatility risk can more accurately be assessed andassumed. It is really not too complicated, as the examples show.Even with all of this preparation, though, you should realize that youcan still lose money in these positions, especially if naked optionsare involved.The examples that we used were theoretical, whereas most ofthe examples in this book have been real ones, attempting to demonstrate the viability of the various approaches in the context of actualmarket action. However, in this section, since the concept is so newto many traders, we felt that the theoretical examples quite accurately depict the ways in which these positions are analyzed andestablished. The fact that no actual examples were given does notmean that this approach is strictly a theoretical one. Quite the contrary. In fact, the next time that you see OEX options getting overlycheap or expensive, you may want to try this approach, for it worksespecially well with broad-based index options. It is also quite appropriate in the futures markets, where volatility sometimes jumps quitehigh on expectations of drought, flood, or other such conditions.
TRADING THE VOLATILITY SKEWUp to this point, we have examined the feasibility of trading volatilitybased on a two-pronged approach: (1) implied volatility is at the endof its range, and (2) it is at a more extreme level than historicalvolatility. The idea is that implied volatility will move at some point tothe middle of its range, and the volatility trader will profit.Despite all the careful work laid out in this chapter up to thispoint, it is possible to lose money by trading volatility in the mannerdescribed. For example, if you’re selling volatility, you may find that
TRADING THE VOLATILITY SKEW
517
it breaks out of the previous range and trades much higher. Thiswould cause losses, even though the position was essentially deltaand gamma neutral. This type of volatility increase has occurredmany times in the past. Sometimes, volatility increases occur from analready high volatility.In the summer of 1991, soybeans had broken down from a price of about575 to 525. Volatilities fell toward the lower end of their range, as is the custom for grain options; declining grain prices are usually accompanied bydeclining implied volatilities. Then in July, a drought scare developed, andbeans rallied back up toward 600. This rally brought implied volatilities to thetop of their previous range, well above the historical volatilities at that time.At this point, I established gamma–delta neutral call ratio spreads thathad excellent profit potential if volatility declined. What happened was thatthe drought scare was exacerbated for about another week, during whichtime beans rallied to nearly 650 and implied volatility skyrocketed to levelsway beyond the previous range.This move was too large even for the gamma–delta neutral position,and I was forced to make adjustments to protect the position. Within a fewdays it rained, and beans traded down the limit for two days before stabilizing in the 550-to-575 area. Volatilities declined at that point, but it was toolate—the adjustments had been so unprofitable that they made the wholeposition a loss.
One must be mindful of the fundamentals when trading volatility;and when the soybean position was established, I felt that thedrought was not serious—I felt it was probably just the typical midsummer hype that the grain markets often experience. Moreover, Ialso thought that the implied volatility being at the high end of therange was sufficiently accounting for the stories that were circulating.Obviously this was wrong, as losses, albeit limited ones, were taken.In a like manner, losses can be taken when volatility is tradedfrom the long side as well. We have already given an example ofwhen gold’s volatility dropped to then-unheard-of levels during 1994and 1995, causing losses for buyers of volatility.The major problem in these situations was that the historical and theimplied volatilities never converged, at least not for a while. In fact, that isone of the major problems with trading volatility: there is no guarantee
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TRADING VOLATILITY
that implied and historical volatilities will converge or that they willconverge during the lifetime of your position. As a result, some volatility traders prefer to use another approach. They prefer to look for situations where there is volatility skewing and trade those.When different options on the same underlying instrument havesubstantially different implied volatilities, then a volatility skew issaid to exist. Certain markets have a volatility skew almost continuously—metals and grain options, for example, and OEX and Standard & Poor's (S&P) 500 options since the crash of 1987. Othershave a skew that appears occasionally. When we talk about a volatility skew, we are describing a group of options that has a pattern ofdiffering volatilities, not just a few scattered different volatilities. Infact, for options on any stock, future, or index, there will be slightdiscrepancies between the various options of different striking pricesand expiration dates. However, in a volatility skew situation, weexpect to see rather large discrepancies between the implied volatilities of individual options, especially those with the same expirationdate; and there is usually a pattern to those discrepancies. The following examples demonstrate the patterns that we expect to see.The following example shows the type of volatility skew that has existed inOEX and S&P 500 options—and many other broad-based index options—since the crash of 1987. This data is very typical of the skew that has lastedfor over eight years. For OEX: 586 in December 1995:OptionJan 560Jan 565Jan 570Jan 575Jan 580Jan 585Jan 590Jan 595Jan 600
Implied Volatility (%)15.814.914.013.212.511.811.210.710.5
Note that we have not labeled the options in this table as puts or calls.That’s because a put and a call with the same striking price and expiration
TRADING THE VOLATILITY SKEW
519
date must have the same implied volatility, or else there will be a risklessarbitrage available.
In the preceding volatility skew, note that the lower strikes havethe highest implied volatility. This is called a reverse volatility skew. Itis sometimes caused by bearish expectations for the underlying, butthat is usually a short-term event. For example, when a commodityundergoes a sharp decline, the reverse volatility skew will appear andlast until the market stabilizes.However, the fact that the reverse skew has existed for so long inbroad-based options is reflective of more fundamental factors. Afterthe crash of 1987 and the losses that traders and brokerage firmssuffered, the margin requirements for selling naked options wereraised. Some firms even refused to let customers sell naked optionsat all. This lessened the supply of sellers. In addition, as we pointedout in Chapter 3, money managers have turned to the purchase ofindex puts as a means of insuring their stock portfolios against losses.This is an increase in demand for puts, especially out-of-the-moneyputs. Thus, we have a simultaneous increase in demand and reduction in supply. This is what has caused the options with lower strikesto have increased implied volatilities.In addition, money managers also sometimes sell out-of-themoney calls as a means of financing the purchase of their put insurance. We have described this strategy in previous chapters as the“collar.” This action exerts extra selling pressure on out-of-themoney calls, and that accounts for some of the skew in the upperstrikes, where there is low implied volatility.A forward volatility skew has the opposite look from the reverseskew, as you might expect. It typically appears in various futuresoption markets, especially in the grain option markets, although it isoften prevalent in the metals option markets, too. It is less frequentin coffee, cocoa, orange juice, and sugar, but it does appear in thosemarkets with some frequency.The following data shows how a forward volatility skew looks for March soybeans: 745:
520
TRADING VOLATILITYOptionMarch 675March 700March 725March 750March 775March 800March 825March 850March 875March 900
Implied Volatility (%)15.816.217.719.821.923.824.826.928.830.7
Notice that in a forward skew, the volatilities increase at higher strikingprices.
The forward skew tends to appear in markets where expectationsof upward price movements are overly optimistic. This does not meanthat everyone is necessarily bullish, but that they are afraid that a verylarge upward move—perhaps several limit up days— could occur andseriously damage the naked option seller of out-of-the-money calls.Occasionally, you will see both types of skews at the same time,emanating from the striking price in both directions. This is ratherrare, but it has been seen in the metals markets at times.The following skew was taken from actual gold option prices for Aprilgold: 390:OptionApril 350April 360April 370April 380April 390April 400April 410April 420April 430April 440April 450
Implied Volatility (%)10.710.08.88.28.09.611.213.114.916.618.3
TRADING THE VOLATILITY SKEW
521
Notice that the at-the-money options are the least expensive, while theout-of-the-money options are more expensive, both at lower strikes and athigher strikes. This is a dual volatility skew.
The interesting thing about volatility skew situations is that youcan ideally establish positions where you sell expensive options andbuy cheap options on the same underlying instrument simultaneously. Then, if the actual price movement of the underlying adheresto normal patterns, the position should profit. In effect, you arebeing offered an excellent opportunity to trade two differing volatilities on the same underlying—it’s as close to a sure thing as you’llfind anywhere.
Price DistributionsBefore getting into the specifics of trading the volatility skew, let’sdiscuss stock price distributions for a minute. Stock and commodityprice movements are often described by mathematicians as adheringto standard statistical distributions. The most common type of statistical distribution is the normal distribution. This is familiar even tomany people who have never taken a statistics course.Figure 6.19 shows the “bell curve,” which is a graph of the normal distribution. The center of the graph is where the average member of the population resides; that is, most of the people are near theaverage, and very few are way above or way below the average. Thenormal distribution is used in many ways to describe the total population: results of IQ tests or average adult height, for example. In thenormal distribution, results can be infinitely above or below the average, also called the median. Thus, this is not useful for describingstock price movements, since stock prices can rise to infinity but canonly fall to zero.Thus, another statistical distribution is generally used to describestock price movements. It is called the lognormal distribution, and itis pictured in Figure 6.20. The height of the curve at various pointsessentially represents the probability of stock prices being at those levels. The highest point on the curve is right at the average, reflectingthe fact that most results are near that price, as they are with the nor-
522
TRADING VOLATILITYFigure 6.19BELL CURVE OR NORMAL DISTRIBUTION
Figure 6.20LOGNORMAL DISTRIBUTION
TRADING THE VOLATILITY SKEW
523
mal distribution shown in Figure 6.19. Or, in terms of stock prices, ifthe average is defined as today’s price, then most of the time, a stockwill be relatively near the average after some period of time. The lognormal distribution allows that stock prices could rise infinitely,although with great rarity, but cannot fall below zero. In fact, theyrarely fall to zero.Mathematicians have spent a great deal of time trying to accurately define the actual distribution of stock price movements, andthere is some disagreement over what that distribution really is. However, the lognormal distribution is generally accepted as a reasonableapproximation of the way that prices move. Those prices don’t haveto be just stock prices, either. They could be futures prices, indexprices, or interest rates.Normally, option prices reflect the distribution of the prices thatthe underlying is expected to follow. For example, the Black-Scholesmodel is based on a log