take a position in long-term options ought to expect greater potentialprofit in the form of larger theoretical edge.
New traders are often advised to begin trading in liquid markets. If a new traderdoes make an error resulting in a losing trade, in a liquid market he will be able to keephis loss to a minimum because he will be able to exit the trade with relative ease. OnRisk Considerations 197
the other hand, an experienced trader, especially a market maker, will often prefer todeal in less liquid markets. There may be less trading activity in such markets, but thebid/ask spread is much wider, resulting in greater theoretical edge each time a trade ismade. Of course, any mistake can be a problem with which the trader will have to livefor a long time. However, an experienced trader is expected to keep his mistakes to aminimum.
The most liquid options in any market are those which are short-term and whichare either at- or slightly out-of-the-money. Such options always have the narrowestbid/ask spread, and there are usually traders willing to buy or sell large numbers of thesecontracts. As a trader moves to longer term options, or to options which are more deeplyin-the-money, he finds that the bid/ask spread begins to widen, and there are fewer andfewer traders interested in these contracts. While at-the-money short-term options areconstantly traded, deeply in-the-money long-term options may not trade for weeks at atime.
In addition to the liquidity of an option market, a trader should also give somethought to the liquidity of the underlying market. If a trader has an option position andwants to make an adjustment, he may find it difficult to do if the option market isilliquid. If the underlying market is liquid, he will at least he be able to make hisadjustment in that market with relative ease. The most dangerous market in which totrade is one where both the options and the underlying contract are inactively traded.Only the most experienced and knowledgeable traders should enter such markets.
End-of-day volume figures and bid/ask spreads in S&P 500 index options traded atthe Chicago Board Options Exchange on July 9, 1993 are shown in Figure 9-15. Notethe lower volume and wider bid/ask spread for back month and deeply in-the-moneyoptions, versus front month and at- and out-of-the-money options.Chapter 9
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While delta neutral strategies are perhaps the most popular among active option traders,many traders prefer to trade with a bullish or bearish perspective in the underlyinginstrument. The trader who wishes to take a directional position in the underlyinginstrument has the choice of doing so in either the instrument itself, buying or selling afutures contract or stock, or in the option market. If he chooses the option market, thetrader has the opportunity to integrate option pricing theory into a bull or bear strategyin order to take advantage of theoretically mispriced options.
NAKED POSITIONS
Since the purchase of calls or the sale of puts will create a positive delta position, andthe sale of calls or purchase of puts will create a negative delta position, we can alwaystake a directional position in a market by taking an appropriate naked position in eithercalls or puts. If all options are overpriced (high implied volatility), we might sell puts tocreate a bullish position, or sell calls to create a bearish position. If all options areunderpriced (low implied volatility], we might buy calls to create a bullish position, orbuy puts to create a bearish position.
The problem with this approach is that, as with all non-hedged positions, there isvery little margin for error. If we purchase options, we will lose money not only if themarket moves in the wrong direction, but also if the market fails to move enough in ourfavor to offset the time premium in the option. If we sell options, we face the prospectof unlimited risk if the market moves violently against us. An experienced trader willalways look for a way to improve the risk/reward characteristics of his position bylooking for positions with the greatest possible margin for error. This philosophy appliesno less to directional strategies than to volatility strategies. In directional strategies, asin volatility strategies, this can often be done by finding an appropriate spread.
BULL AND BEAR RATIO SPREADS
If a trader believes that implied volatility is too high, one sensible strategy is a ratiovertical spread. For example, with the underlying market at 100 suppose a June 100call has a delta of 50 and a June 110 call has a delta of 25. A delta neutral spreader can:
buy 1 June 100 call (50)sell 2 June 110 calls (25)
199200 Chapter 10
Since the spread is delta neutral, it has no particular preference for upward or downwardmovement in the underlying market.
Now suppose the same trader believes that this ratio vertical spread is a sensiblestrategy, but at the same time he is also bullish on the market. There is no law thatrequires him to do this spread in a delta neutral ratio. If he wants this spread to reflecthis bullish sentiment, he might change the ratio slightly:
buy 2 June 100 calls (50)sell 3 June 110 calls (25)
The trader has essentially the same ratio vertical spread, but with a bullish bias. This isreflected in the total delta for the position of +25.
There is, however, an important limitation using this type of ratio strategy to createa bullish or bearish position. In our example the trader is initially bullish, but the positionis still a ratio vertical spread. As such, it has a negative gamma. If the trader hasunderestimated volatility, and the underlying market moves up too quickly, the spreadcan invert from a positive to a negative delta. If the market rises far enough, to 130 or140, eventually all options will go deeply into-the-money and the deltas of both the June100 and June 110 calls will approach 100. Eventually, the trader will be left with a deltaposition of -100. Even though the trader was correct in his bullish sentiment, theposition was primarily a volatility spread, so that the volatility characteristics of theposition eventually outweighed any considerations of market direction.
The delta can also invert with a backspread. Unlike a negative gamma position,where the inversion is caused by swift price movement in the underlying contract, abackspread can invert when the market is less volatile than expected. For example,suppose that with the underlying market at 100 a trader believes that implied volatilityis too low. He might decide on a delta neutral backspread:
buy 2 June 110 calls (25)sell 1 June 100 call (50)
If, however, he is bullish on the market, he can, as in the last example, change theratio to reflect this sentiment:
buy 3 June 110 calls (25)sell 1 June 100 call (50)
His delta position of +25 reflects this bullish bias.
We know that as time passes, or as volatility declines, all deltas move away from50. If time begins to pass with no movement in the underlying contract, the delta of theJune 100 call will remain at 50, but the delta of the June 110 call will start to decline.If, after a period of time, the delta of the June 100 call declines to 10, the delta of theposition will no longer be +25, but will instead be —-20, Since this spread is a volatilityspread, the primary consideration, as before, is the volatility of the market. Onlysecondarily are we concerned with the direction of movement. If the trader overesti-mates volatility, and the market moves more slowly than expected, the spread whichwas initially delta positive can instead become delta negative.Bull and Bear Spreads 201
BULL AND BEAR BUTTERFLIES AND TIME SPREADS
Butterflies and one-to-one (non-ratioed) time spreads can also be chosen in such a wayas to reflect a trader’s bullish or bearish bias. Like ratio spreads, however, their deltacharacteristics can also invert as market conditions change.
With the underlying market at 100, a delta neutral trader might buy the June95/100/105 call butterfly (buy a 95 call, sell two 100 calls, buy a 105 call). The traderhopes that the market will sit still at 100 so that at expiration the butterfly will widento its maximum value of five points. If, however, a trader wants to buy a butterfly, butis also bullish on the market, he can choose a butterfly where the inside exercise priceis above the current price of the underlying contract. If the underlying market iscurrently at 100, he might choose to buy the June 105/110/115 call butterfly. Sincethis position wants the underlying market at 110 at expiration, and itis currently at 100,the position is a bull butterfly. This will be reflected in the position having a positivedelta.
Unfortunately, if the underlying market moves too swiftly, say to 120, the butterflycan invert from a positive to a negative delta position. Since at expiration the butterflyalways has its maximum value with the underlying contract right at the inside exerciseprice, the trader will now want the market to fall back from 120 to 110. Whenever theunderlying market is below 110, the position will be bullish; whenever the underlyingmarket is above 110, the position will be bearish.
Conversely, if the trader is bearish, he can always choose to buy a butterfly wherethe inside exercise price is below the current price of the underlying market. But again,if the market moves down too quickly and goes through the inside exercise price, theposition will invert from a negative to a positive delta.
In a similar manner, a trader can choose time spreads that are either bullish orbearish. A long time spread always wants the near-term contract to expire exactlyat-the-money. A long time spread will be initially bullish if the exercise price of the timespread is above the current price of the underlying market.! With the underlying marketat 100, the June/March 110 call time spread (buy the June 110 call/sell the March 110call} will be bullish since the trader will want the underlying market to rise to 110 byMarch expiration. The June/March 90 call time spread (buy the June 90 call/sell theMarch 90 call} will be bearish, since the trader will want the underlying market to fallto 90 by March expiration. However, like a long butterfly, a long time spread has anegative gamma, so that if the market moves through the exercise price, the delta caninvert. If the market moves from 100 to 120, the June/March 110 call time spread,which was initially bullish, will become bearish. If the market moves from 100 to 80,the June/March 90 call time spread, which was initially bearish, will become bullish.
1. In the futures market the situation may be complicated by the fact that different futures months maybe trading at different prices. Instead of choosing a traditional time spread where both options have thesame exercise price, the trader may have to choose a diagonal spread to ensure that the position {s eitherbullish (delta positive) or bearish (delta negative).202 Chapter 10
VERTICAL SPREADS
It is common for traders to take bull or bear market positions by choosing appropriateratio spreads, butterflies, or time spreads, but in each of these positions volatility is stillthe primary concern. A trader can be right about market direction, but if he is wrongabout volatility, the spread may not retain the directional characteristics that the traderintended.
If a trader focuses initially on the direction of the underlying market, he might lookfor a spread where the directional characteristics of the spread are the primary concern,and volatility is of secondary importance. He would like to be certain that if the spreadis initially bullish (delta positive) it will remain bullish under all possible marketconditions. If it is initially bearish (delta negative) it will remain bearish under allpossible market conditions.
The class of spreads which meet the above requirements are known as verticalspreads. A vertical spread always consists of one long (purchased) option and one short(sold) option, where both options are of the same type (either hoth calls or both puts)and expire at the same time. The options are distinguished only by their differentexercise prices. Typical vertical spreads might be:
buy 1 June 100 callsell 1 June 105 call
or
buy 1 March 105 putsell 1 March 95 put
Vertical spreads are not only initially bullish or bearish, but they remain bullish orbearish no matter how market conditions change. Two options which have differentexercise prices but which are identical in every other respect cannot have identicaldeltas. In the first example, where the trader is long a June 100 call and short a June105 call, the June 100 call will always have a delta greater than the June 105 call. Ifboth options go deeply into-the-money or very far out-of-the-money, the deltas may healmost identical. Even then, the June 100 call will have a delta fractionally greater thanthe June 105 call. In the second example, no matter how market conditions change,the March 105 put will always have a delta greater than the March 95 put.
At expiration, a vertical spread will have a minimum value of zero, if both optionsare out-of-the-money, and a maximum value of the amount between exercise prices, ifboth options are in-the-money. If the underlying market is below 100 at expiration, theJune 100/105 call spread will be worthless because both options will be worthless. Ifthe market is above 105, the same spread will be worth five points because the June100 call will be worth exactly five points more than the June 105 call. Similarly, theMarch 95/105 put spread will be worthless if the underlying market is above 105 atexpiration, and it will be worth 10 points if the market is below 95. The expirationvalues of typical bull and bear vertical spreads are shown in Figures 10-1 and 10-2.Bull and Bear Spreads
203
 
Figure 10-1
 
 
 
 
 
 
 
 
 
sell a call (put)at a higherexercise pricea od€=oo“no—~«buy a cali (put)at a lowerexercise priceFigure 10-2sell a call (put)at a lowerexercise price—___--____#=2Qagal$ibuy a calf (put)at a higherexercise price
 
 
 204 Chapter 10
Since a vertical spread at expiration will always have a value hetween zero and theamount between exercise prices, a trader can expect the price of such a spread to besomewhere within this range. A 100/105 call vertical spread will trade for someamount between zero and five points; a 95/105 put vertical spread will trade for someamount between zero and 10 points. The exact value will depend on the likelihood ofthe underlying market finishing below the lower exercise price, above the higherexercise price, or somewhere in between. If the market is currently at 80 and gives littleindication of rising, the price of the 100/105 call vertical will be close to zero, whilethe price of the 95/105 puts vertical will be close to 10. If the market is currently at120 with little likelihood that it will fall, the price of the 100/105 call vertical will beclose to five points, while the price of the 95/105 put vertical will be close to zero.
If a trader wants to do a vertical spread, he has essentially four choices. If he isbullish he can choose a bull vertical call spread or a bull vertical put spread; if he isbearish he can choose a bear vertical call spread or a bear vertical put spread. Forexample:
bull call spread: buy a June 100 callsell a June 105 callbull put spread: buy a June 100 putsell a June 105 putbear call spread: sell a June 100 callbuy a June 105 callbear put spread: sell a June 100 put
buy a June 105 put
Note that a trader who is bullish can buy a 100 call and sell a 105 call, or buy a100 put and sell a 105 put. And a trader who is bearish can buy a 105 call and sell a100 call, or buy a 105 put and sell 100 put. This may seem counter-intuitive, since oneexpects spreads which consist of puts to have characteristics which are the opposite ofthose which consist of calls. Regardless of whether a vertical spread consists of calls orputs, whenever a trader buys the lower exercise price and sells the higher exerciseprice, the position is bullish; whenever a trader buys the higher exercise price and sellsthe lower exercise price, the position is bearish. Call and put vertical spreads whichexpire at the same time and which consist of the same exercise prices have approxi-mately the same delta, so they have approximately the same bullish or bearish charac-teristics.
Given the many different exercise prices and expiration months available, how cana trader choose the vertical spread which best reflects his directional expectations andwhich gives him the best chance to profit from those expectations?
Since options have a limited life with a fixed expiration date, a trader who wantsto use options to take advantage of an expected market move must first determine histime horizon. Is the movement likely to occur in the next month, in the next threemonths, in the next nine months? If it is currently May and the trader foresees upward
2. Weare assuming for the moment that all options are European, with no possibility of early exercise.Bull and Bear Spreads 205
movement, but helieves the movement is unlikely to occur within the next two months,it does not make much sense to take a position in June options. If his expectations arelong-term, he may have to take his position in September, or even December, options.Of course, as he moves further and further out in time, market liquidity may becomemore of a problem. This is a factor that will have to taken into consideration.
Next, a trader will have to decide just how bullish or bearish he is. Is he veryconfident, and therefore willing to take a very large directional position? Or is he lesscertain and willing to take only a limited position? Two factors determine the totaldirectional characteristics of a vertical spread:
1, The delta of the specific vertical spread2. The size in which the spread is executed
For example, a trader who wants to take a position which is 500 deltas long(equivalent to purchasing five underlying contracts) can either choose a vertical spreadwhich is 50 deltas Jong and execute it 10 times, or choose a vertical spread which is 25deltas long and execute it 20 times. Both positions will leave him long 500 deltas.
The delta value of a vertical spread is determined by various factors: time toexpiration, volatility, and distance hetween exercise prices. Since a trader will berequired to choose an expiration date which covers the period of expected directionalmovement, and since a trader will always make his best estimate of volatility over agiven period, in practice the delta will be a function of the exercise prices which hechooses. The greater the distance between exercise prices, the greater the delta valueassociated with the spread. A 95/110 bull spread will be more bullish than a 100/110bull spread, which will, in turn, be more bullish than a 100/105 bull spread. This isshown in Figure 10-3.
Once a trader decides on an expiration month in which to take his directionalposition, he must decide which specific spread is best. That is, he must decide whichexercise prices to use. A common approach is to focus on the at-the-money options. Ifa trader does this, he will have the following choices:
Bull call spread: buyanin-the-money call or buy an atthe-money call
sell an at-the-money cal! sell an out-of-the-moneyBear call spread: buy anat-the-money call or buy an out-of-the-money call
sell an in-the-money call sell an at-the-money callBull put spread: buyanat-the-moneyput or buy an out-of-the-money put
sell an in-the-money put sell an at-the-money putBear put spread: buy anin-the-money put or buy an at-the-money put
sell an at-the-money put sell an out-of-the-money put
With four different bull spreads and four different bear spreads available, how canthe trader choose the spread which represents the hest value? One way to do this is touse a theoretical pricing model to evaluate several different vertical spreads. This hasbeen done in Figure 10-4 for six different options on an underlying futures contract: anin-the-money, at-the-money, and out-of-the-money call, and an in-the-money, at-the-206
 
 
 
 
 
Chapter 10Figure 10-3As the exercise prices become furtherapart, a vertical spread takes ongreater bullish or bearish characteristics. oeen >7 f _é4+ _ f{——~ va —slightly bullish éZmoderately bullish /~—<_—- SE EEMEEEEEEEiamnAvery bullish~<_- een tent ont ee eee cen+
 
 
 
money, and out-of-the-money put. The assumptions include an underlying price of 100,12 weeks to expiration, a volatility estimate of 20%, and an interest rate of 8%.
Suppose we are interested in doing a bull call spread. Two possibilities are to buya 95 call and sell a 100 call, or buy a 100 call and sell a 105 call. The spreads and theirtheoretical values are:
 
Spread Theoretical Value Delta95/100 call spread 2.87 +20100/105 call spread 1.88 +20Figure 10-4Futures price = 100; Time = 12 weeks; Volatility = 20%; Interest = 8.00%
TheoreticalOption Value Delta Gamma Theta Vega95 call 6.63 71 3.4 —.017 16100 call 3.76 51 4.1 —.022 .19105 call 1.88 31 3.7 —.020 1795 put 1.72 -27 3.4 -.019 -16100 put 3.76 -47 41 —.022 19
105 put , 6.79 -67 3.7 -.019 A7Buil and Bear Spreads 207
With both spreads having the same delta of +20, it may appear that the 100/105 spreadis a better value since it is likely to be less expensive. But should this be the soledeterminant in choosing a strategy? As in all spreads, the option trader’s goal is to createa position with positive theoretical edge, by either purchasing high value at a low price,or selling low value at a high price. In order to achieve this goal we need to know notonly the theoretical values of the spreads, but also the prices of the spreads in themarketplace.
From an option trader’s point of view, the relative prices of options in themarketplace is usually represented by the implied volatility. We know the value of thespreads based on our volatility input of 20%. What might be the prices of the spreads ifimplied volatility in the option market is something other than 20%? We can answerthe question by using a theoretical pricing model with the same underlying price, timeto expiration, and interest rate, but with volatilities lower and higher than 20%. Thishas been done in Figure 10-5 using volatilities of 16% and 24%, along with our estimatedvolatility of 20%.
Going back to our bull call spread, suppose implied volatility is lower than ourestimate of 20%, say 16%. We can see that the price of the 95/100 spread will beapproximately 3.01, while the price of the 100/105 spread will be approximately 1.78.We have two choices: we can pay 3.01 for a spread that is worth 2.87 (the 95/100spread), or we can pay 1.78 for a spread that is worth 1.88 (the 100/105 spread].Clearly, we will prefer the 100/105 spread since it will result in a positive theoreticaledge of .10. If we were to buy the 95/100 spread we would end up with a negativetheoretical edge of .14.
Now suppose implied volatility is higher than our estimate of 20%, say 24%. Wecan see that the price of the 95/100 spread will be approximately 2.77, while the priceof the 100/105 spread will be approximately 1.94. Again, we have two choices: we canpay 2.77 for a spread that is worth 2.87 (the 95/100 spread), or we can pay 1.94 foraspread that is worth 1.88 (the 100/105 spread). Now we will prefer the 95/100 spreadsince it will result ina positive theoretical edge of .10. The 100/105 spread would resultin a negative theoretical edge of .06.
Figure 10-595 95/100 100 100/105 105Volatility Cali Spread Call Spread Call16% 6.02 3.01 3.01 1.78 1.2320% 6.63 (71) 2.87 (20) 3.76 (51) 1.88 (20) 2.57 (31)24% 7.28 2.77 4.51 1.94 2.5795 95/100 100 100/105 105Volatility Put Spread Put Spread Put16% 1.11 1.90 3.01 3.13 6.1420% 1.72 (-27} 2.04 (20) 3.76 (47) 3.03 (20) 6.79 (-87)
24% 2.37 2.14 4.51 2.97 7.48208 Chapter 10
What’s happening here? Even though both spreads have the same delta values,under one set of circumstances we seem to prefer the 95/100 spread, while underdifferent circumstances we seem to prefer the 100/105 spread. The reason becomesClear if we recall one of the characteristics of option evaluation introduced in Chapter 6:
If we consider three options, an in-the-money, at-the-money, and out-of-the-money option which are identical except for their exercise prices, theat-the-money option is always the most sensitive in total points to a changein volatility.
This means that when all options appear overpriced because we believe impliedvolatility is too high, in total points the at-the-money option will be the most overpriced.When all options appear underpriced because we believe implied volatility is too low,in total points the at-the-money option will be the most underpriced. This characteristicleads to a very simple rule for choosing bull and bear vertical spreads:
Jfimplied volatility ts too low, vertical spreads should focus on purchasingthe at-the-money option. If implied volatility is too high, vertical spreadsshould focus on selling the at-the-money options.
Now we can see why the 100/105 call spread is a better value if implied volatilityis 16%, while the 95/100 spread is a better value if implied volatility is 24%. If impliedvolatility is too low (16%), we want to buy the at-the-money (100) call. Having donethis, we have only one choice in order to create a bull spread—sell the out-of-the-money(105) call. If implied volatility is too high (24%), we want to sell the at-the-money (100}call. Having done this, we again have only one choice in order to create a bullsp