position. The extent to whichhe is pursuing one of these strategies will depend on the magnitude of the gamma andvega. A trader with a gamma of —100 and vega of +200 would describe his position asalong time spread, but not nearly to the degree of a trader who has a gamma of -1500and a vega of +3000.
Figure 8-20 is an evaluation table with the theoretical value, delta, gamma, theta,and vega of several different options. Following this table are examples of volatilityspreads of the types discussed in this chapter, along with their total delta, gamma, theta,and vega. (Although the examples in Figure 8-20 are based on futures options, theVolatility Spreads 161
Figure 8-19: Characteristics of Volatility Spreads
Spread Type Delta Gamma Theta VegaPosition Position Position PositionBackspread 0 + - +Long Straddle 0 + - +Long Strangle 0 + +Short Butterfly 0 + - +Ratio Vertical Spread 0 - + ~Short Straddle 0 - + ~Short Strangle 9 - + ~Long Butterfly 0 - + -~Long Time Spread 0 ~ + +Short Time Spread 0 + - -
spread characteristics are equally valid for stock options.) The reader will see that eachspread does indeed have the positive or negative sensitivities summarized in Figure8-19. Note also that a volatility spread need not be exactly delta neutral. (Indeed, as wesaw in Chapter 6, no trader can say with any certainty whether he is delta neutral.) Apractical guide is for the delta to be close enough to zero so that the volatilityconsiderations are more important than the directional considerations.
Note that no price is given for any of the option contracts in Figure 8-20, andtherefore no theoretical edge can be calculated for any of the spreads. The prices atwhich a spread is executed may be good or bad, resulting in a positive or negativetheoretical edge. But once the spread has heen established, the market conditions whichwill help or hurt the spread are determined by its type, not by the initial prices. Like alltraders, option traders must not let their previous trading activity affect their currentjudgement. A trader’s primary concern ought never be what happened yesterday, butwhat he can do today to make the most of the current situation, whether that meansmaximizing his potential profit or minimizing his potential loss.
CHOOSING AN APPROPRIATE STRATEGY
With so many spreads available, how do we know which type of spread is best? Firstand foremost we will want to choose spreads which have a positive theoretical edge toensure that if we are right about market conditions we can be confident of showing aprofit. Ideally, we would like to construct a spread by purchasing options which areunderpriced and selling options which are overpriced. If we can do this the resultingspread, whatever its type, will always have a positive theoretical edge.
More often, however, our opinion about volatility will result in all options appear-ing either underpriced or overpriced. When this happens, it will be impossible to bothChapter 8
 
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162Volatility Spreads 163
Figure 8-20: Examples of Volatility Spreads (continued)
CALL BACKSPREAD: Long more calls than short; all calls expiring at the same time.
 
Deita Gamma Theta VegaPosition Position Position Position
Long 30 March 105 calls +80 x 24 430 x 4.6 +30 x-—.0250 430 x .107
Short 10 March 95 calls -~10 x 78 -10x43  -10x-—0221 -10x.100-60 +95.0 ~-.9710 +2,210
Long 25 Jun 110 calls +26 x 23 425x2.5 +25x -—0132 +25 .182
Short 10 Jun 100 calls -10 x51 -10x3.2 -10x~0166 -i10x.234+65 +30.5 —.1640 +2,.210
PUT BACKSPREAD: Long more puts than short; all puts expinng at the same time.
Delta Gamma Theta VegaPosition Position Position PositionLong 80 March 90 puts +80 x -6 +80 x 1.7 +480 x -.0090 +80 x .040Short 10 March 100 puts ~10 x -48 -10x58 -10 x -0313 -10 x .1340 +78.0 ~.4070 +1.860Long 45 June 95 puts +45 x -31 +45x29 445 x -0151 +45 x .209Short 30 June 100 puts ~30 x -47 ~30x3.2 -30x-.0166 -30 x 234+15 +34.5 —1815 +2.385
CALL RATIO VERTICAL SPREAD: Short more calls than long; all calls expiring at the same time.
Delta Gamma Theta Vega
Position Position Position Position
Long 20 March 95 calls 420 x 78 +20 x 4.3 +20 x ~.0221 +20 x .100
Short 30 March 100 calls -30 x 51 -30 x 5.8 -30 x-.0313  -30x.134+30 88.0 +.4970 ~2.020
Long 10 June $0 calls +10 x 81 +10 2.1 +10x-,0090 +10 .154
Short 20 June 105 calls —20 x 36 -20x3.0 ~-20x-.0160 -20x.221+90 -~39,0 +.2300 ~2,880
PUT RATIO VERTICAL SPREAD: Short more puts than long; all puts expiring at the same time.
 
Deita Gamma Theta VegaPosition Position Position PositionLong 20 March 95 puts +20 x-21 +20 x 4.3 +20 x -.0232 +20 x .100Short 60 March 90 puts -60 x -6 -60x1.7  -—60x-.0090 -60x.040—60 16.0 +.0760 -400Long 15 June 105 puts +15 x-62 +15 x 3.0 415 x -.0149 +15 x 221Short 20 June 100 puts -20 x-47 -20x3.2  -20x-0166 -20x.234
+10 —-19.0 +.1085 1.365
 164
Figure 8-20: Examples of Volatility Spreads (continued)
Chapter 8
LONG STRADDLE: Long calls and puts with the same expiration date and exercise price.
Long 10 March 100 callsLong 10 March 100 puts
Long 10 June 95 callsLong 25 June 95 puts
DeltaPosition+10 x +51+10 x -48
+30
+10 x +67+25 x —31-105
GammaPosition+10x 5.8
+10 x 5.8
+116.0
+10 x 2.9
+25 x 2.9
+1015
ThetaPosition+10 x —0313+10 x —.0313~.6260
+10 x —0140+25 x ~.0151-.6175
VegaPositlon
+10 x 134
+10 x 134
+2.680
+10 x 209+25 x .2094+7.315
SHORT STRADDLE: Short cails and puts with the same expiration date and exercise price.
Short 30 March 105 callsShort 10 March 105 puts
Short 20 June 100 calisShort 20 June 100 puts
DeltaPosition
~30 x +24—10 x—75+30
-20 x +81
-20 x -47
—B0
GammaPosition—30 x 4.6-10 x 46—184.0
-20 x 3.2-20 x 3.2-128.0
ThetaPosition~30 x -.0250—10 x—.0240+.9900
-20 x —.0166
20 x -.0166
+.6640
VegaPosition-30 x .107-10 x .107~4,280
-20 x .234-20 x .234+9.36
LONG STRANGLE: Long calls and puts with the same expiration date but different exercise
prices.
Long 20 March 105 callsLong 20 March 85 puts
Long 30 June 110 callsLong 15 June 100 puts
DeltaPosition+20 x +24+20 x —21+60
+30 x +23+15 x —47-16
GammaPosition+20 x 4.6+20 x 4.3+178.0
+30 x 2.5+15 x 3.2+123.0
ThetaPosition+20 x —.0250+20 x —.0232-.9640
+80 x -.0132+15 x —0166—.6450
VegaPosition+20 x 107+20 x .10044.140
+30 x 182+15 x 234+8.970
SHORT STRANGLE: Short calls and puts with the same expiration date but different exercise
Prices.
Short 30 March 110 callsShort 40 March 90 puts
Short 10 June 95 callsShort 10 June 105 puts
DeitaPosition—30 x +8-40 x -6
0
~10 x +67-10 x -62—50
 
Gamma Theta VegaPosition Position Position-30x2.3  -30x-.0124 -30x.055-40x1.7 —40x-.0090 -40x.040~-137.0 +,7320 3.25010x29 —-10x-0140 -10x.209-10x3.0 -10x-.0149 -10x.221-§9.0 +.2890 —4.300
 Volatility Spreads 165
Figure 8-20: Examples of Volatility Spreads (continued)
LONG BUTTERFLY: Short two options at the same exercise price, and long one option withan immediately higher exercise price and one option with an immediately lower exercise price;all options are of the same type and expire at the same time.
Deita Gamma Theta VegaPosition Position Position PositionLong 10 March 95 calls +10 x 78 10x43 +10x-0221 +10x.100Short 20 March 100 calls —20 x 84 -20x5.8 ~-20x -.03139 -20x.134Long 10 March 105 calls +10 x 24 +10 x 4.6 +10x-.0250 +10x.107Q -27.0 +,.1850 —.610Long 30 June 90 puts +80 x-17 +30 x 2.1 +80 x~0111 +80 154Short 60 June 95 puts -60 x-31 -60x2.9 -60x-.0151 -60~x,.209Long 30 June 100 puts +30 x -47 +80%3.2 -30x-0166 +30 .234-60 ~15.0 +.0780 -.900
SHORT BUTTERFLY: Long two options at the same exercise price, and short one option withan immediately higher exercise price and one option with an immediately lower exercise price;all options are of the same type and expire at the same time.
Delta Gamma Theta VegaPosition Position Position PositionShort 20 March 100 puts -20 x -48 20x58  -20x~.0313 -20%x.134Long 40 March 105 puts +40 x ~75 +40 x 4.6 +40 x-.0240 +40 x.107Short 20 March 110 puts —20 x~91 —20 x 2.3 -20 x-.0103 ~20x,055~220* +22.0 —.1280 +.500Short 25 June 95 calls 25 x 67 -268x2.9  -25x-.0140 ~25 x .209Long 50 June 100 calls +50 x 51 +50 x 3.2 +50 x-.0166 +50 x .234Short 25 June 105 calls —25 x 36 —25 x 3.0 ~25 x~.0160 ~-25x.221-25 +125 —.0800 +.950
* Note that because the ratio of a butterfly is always 1 x 2 x 1, the butterfly can becomeunbalanced if the exercise prices are far away from the current underlying price.
LONG TIME SPREAD: Long a long-term option and short a short-term option where all op-tions have the same exercise price and are of the same type.
Delta Gamma Theta VegaPosition Position Position PositionLong 20 June 100 calls 420 x51 +20 x 3.2 420-0166 +20 x .234Short 20 March 100 calls =20 x 51 ~20 x 5.8 —20 x-.0313 ~20x.1340 $2.0 +.2940 +2.000Long 10 June 95 puts +10 x-31 +10x 2.9 +10 x -.0151 +10 x .209Short 10 March 95 puts -10 x -21 ~10x 43 -10 x-.0232 -10x.100
—100 14.0 +.0810 +1.090
 166 Chapter 8
Figure 8-20: Examples of Volatility Spreads (continued)
SHORT TIME SPREAD: Short a long-term option and long a short-term option where all op-tions have the same exercise price and are of the same type.
Delta Gamma Theta VegaPosition Position Position Position
Long 25 March 100 puts +25 x-48 +25 x 6.8 +25 x ~,0313 +25 x 134Short 25 June 100 puts -25 x47 -25x3.2  -25x-0166 -25~x.234
—25 +65.0 ~.3675 —2.500“Long 10 March 1085 calls +10 x 24 410x 4.6 +10x-.0250 +10x.107Short 10 June 110 calls -10 x 23 -10x25 -10x-.0132 —10x.182
+10 +21.0 —.1180 ~.750
* Although this is technically a diagonal spread, it will tend to act like a time spread because theoptions have deltas which are approximately equal.
buy underpriced and sell overpriced options, Such a market can be easily identified bycomparing our volatility estimate to the implied volatility in the option marketplace. Ifimplied volatility is generally lower than our volatility estimate, all options will appearunderpriced. {f implied volatility is generally higher than our estimate, all options willappear overpriced.
If options generally appear underpriced (low implied volatility), look for spreadswith a positive vega. This includes strategies in the backspread or long time spreadcategory. Jf options generally appear overpriced (high implied volatility), look forspreads with a negative vega. This includes strategies in the ratio vertical or short timespread category.
It may seem at first glance that if one encounters a market where all options areeither underpriced or overpriced, the sensible strategies are either long straddles andstrangles, or short straddles and strangles. Such strategies will enable a trader to take aposition with a positive theoretical edge on both sides of the spread. Straddles andstrangles are certainly possible strategies when all options are too cheap or too expen-sive. We will see in Chapter 9 that straddles and strangles, while often having a large,positive theoretical edge, can also be among the riskiest of all strategies. For this reason,a trader will often want to consider other spreads in the backspread or ratio verticalspread category, even if such spreads entail buying some overvalued options or sellingsome underpriced options.
The theoretical values and deltas in Figure 8-20 have been reproduced in Figures8-21 and 8-22, but now prices have been included which reflect implied volatilitiesdifferent from the volatility input of 20%. The prices in Figure 8-21 reflect an impliedvolatility of 17%. In this case the reader will find that only those spreads with a positivevega will have a positive theoretical value:
call and put backspreadslong straddles and stranglesVolatility Spreads 167
Figure 8-21IMPLIED VOLATILITY = 17%
March Futures= 100.00 Time to Expiration =6 weeks _Volatiliy= 20% Interest = 6.00%
CALLS PUTSExercise Theoretical TheoreticalPrice Price Value Delta Price Value Delta90 10,00 10.10 93 07 A7 695 5.54 §.82 78 57 85 21100 2.28 2.69 $1 2.28 2.68 ~48105 65 95 24 §.62 §.92 -7§110 12 26 8 10.05 10.16 91
June Futures = 100.00 Time to Expiration =13 weeks  Volalifity= 20% Interest = 6.00%
CALLS PUTSExercise Theoretical Theoretical
Price Price Value Delta Price Value Delta
90 10.54 10,97 81 75 1.18 -17
95 6.83 745 67 1 94 2.55 31
100 4.00 4.71 51 4.00 4.71 47
105 2.11 2.77 36 7.01 7.66 -€2
110 1.00 1.52 23 10.79 11.30 -75
 
Figure 8-22IMPLIED VOLATILITY = 23%March Futures = 100.00 ‘Time to Expiration =6 weeks Volatility =20% _— Interest = 6.00%
 
CALLS PUTSExerclse Theoretical TheoreticalPrice Price Value Delta Price Value Delta90 10.23 10.10 93 30 A7 -695 6.12 §.82 78 1.16 85 21100 3.09 2.69 §1 3.09 2.68 48105 1.28 95 24 6.25 §.92 75110 44 26 8 10.37 10.16 91June Futures = 100.00 Time to Expiration = 13 weeks Volatility= 20% — Interest = 6.00%CALLS PUTSExerclse Theoretical TheoreticalPrice Price Value Delta Price Value Delta90 11.45 10.97 81 1.66 1.18 -1795 8.08 7.45 67 3.19 2.55 -31100 §.42 4.71 51 §.42 471 47105 3.44 2.77 36 8.34 7.66 62
110 2.08 1.62 23 11.87 11,30 75
 168 . Chapter 8
short butterflieslong time spreads
The prices in Figures 8-22 reflect an implied volatility of 23%. Now the reader willfind that only those spreads with a negative vega will have a positive theoretical value:
call and put ratio vertical spreadsshort straddles and strangleslong butterflies
short time spreads
An important assumption in most theoretical pricing models is that volatility isconstant over the life of an option. The volatility input into the model is assumed to bethe one volatility which best describes price fluctuations in the underlying instrumentover the life of the option. When all options expire at the same time, it is this onevolatility which will, in theory, determine the desirability of a spread. In real life,however, a trader may believe that volatility is likely to rise or fall over some period oftime. Very often implied volatility will also rise or fall. Since time spreads are particularlysensitive to changes in implied volatility, whether volatility is rising or falling can affectthe desirability of time spreads. Consequently, we can add this corollary to our spreadguidelines: Long time spreads are likely to be profitable when implied volatility is lowbut is expected to rise; short time spreads are likely to be profitable when impliedvolatility is high but is expected to fall.
These are only general guidelines, and an experienced trader may decide to violatethem if he has reason to believe that the implied volatility will not correlate with thevolatility of the underlying contract. A long time spread might still be desirable in a highimplied volatility market, but the trader must make a prediction of how implied volatilitymight change under certain conditions. If the market were to stagnate, with nomovement in the underlying contract, but the trader felt that implied volatility wouldremain high, a long time spread would be a sensible strategy. The short-term optionwould decay, while the long-term option would retain its value. In the same way, a shorttime spread might still be desirable in a low implied volatility market if the trader feltthat the underlying instrument were likely to make a large move with no commensurateincrease in implied volatility.
ADJUSTMENTS
A volatility spread may initially be delta neutral, but the delta of the position is likely tochange as the price of the underlying contract rises or falls. Moreover, changes involatility and time to expiration can also affect the delta of spread. A spread which isdelta neutral today may not be delta neutral tomorrow, even if all other conditionsremain the same. The optimum use of a theoretical pricing model requires a trader tocontinuously maintain a delta neutral position throughout the life of the spread.Continuous adjustments are impossible in real life, so a trader ought to give somethought to when he will adjust a position. Essentially, we can consider three possibili-ties:Volatility Spreads 169
1. Adjust at regular intervals—In theory, the adjustment process is assumed tobe continuous because volatility is assumed to be a continuous measure of thespeed of the market. In practice, however, volatility is measured over regular timeintervals, so a reasonable approach is to adjust a position at similar regular intervals.If a trader’s volatility estimate is based on daily price changes, the trader mightadjust daily. If the estimate is based on weekly price changes, he might adjustweekly. This is a trader’s best attempt to emulate the assumptions built into thetheoretical pricing model.
2. Adjust when the position becomes a predetermined number of deltas longor short-—Very few traders insist on being delta neutral all the time. Most tradersrealize that this is not a realistic approach, both because a continuous adjustmentprocess is physically impossible, and because no one can be certain that all theassumptions and inputs in a theoretical pricing model, from which the delta iscalculated, are correct. Even if one could be certain that all delta calculations wereaccurate, a trader might still be willing to take on some directional risk. Buta traderought to know just how much directional risk he is willing to accept. If he wantsto pursue delta neutral strategies, but believes that he can comfortably live with aposition which is up to 500 deltas long or short, then he can adjust the position anytime his delta position reaches this limit. Unlike the trader who adjusts at regularintervals, a trader who adjusts based on a fixed number of deltas cannot be surehow often he will need to adjust his position. In some cases he may have to adjustvery frequently; in other cases he may go for long periods of time without adjusting.
The number of deltas long or short a trader chooses for his adjustment pointsdepends on the size of his positions and his capitalization. A small, independenttrader may find that he is uncomfortable with a position which is even 200 deltaslong or short, while a large trading firm may consider a position which is severalthousand deltas long or short as being approximately delta neutral.
3. Adjust by fee-—This suggestion is not made facetiously. Some traders havegood market feel. They can sense when the market is about to move in onedirection or another. If a trader has this ability, there is no reason why he shouldn’tmake use of it. For example, suppose that the underlying market is at 50.00 andatrader is delta neutral with a gamma of ~200. If the market falls to 48.00, the tradercan estimate that he is approximately 400 deltas long. If 400 deltas is the limit ofrisk he is willing to accept, he might decide to adjust at this point. If, however, heis also aware that 48.00 represents strong support for the market, he might choosenot to adjust under the assumption that the market is likely to rebound from thesupport level. If he is right, he will have avoided an unprofitable adjustment. Ofcourse, if he is wrong and the market continues downward through the supportlevel, he will regret not having adjusted. If the trader is right more often than not,there is no reason why he shouldn’t take advantage of this skill.170 Chapter 8
ENTERING A SPREAD ORDER
In sophisticated option markets, spreads are traded as if they are one, single contract,This means that a spread is quoted with a single bid price and a single offer price. Forexample, suppose a trader is interested in buying a certain straddle and receives a quotefrom a market-maker of 3.45/3,55. If the trader wants to sell the straddle, he will haveto do so at a price of 3.45 (the bid price}; if he wants to buy the straddle, he will haveto pay 3.55 (the ask price}. If the trader decides he is willing to pay 3.55, neither he northe market-maker care whether the trader pays 1.75 for the call and 1.80 for the put,or 1.55 for the call and 2.00 for the put, or some other combination of call and putprices. The only consideration is that the prices of the call and put taken together addup to 3.55.
A market-maker will always endeavor to give one bid price and one ask price foran entire spread. If the spread is a common type, such as a straddle, strangle, butterfly,or time spread, a bid and ask can usually be given very quickly. Of course, marKet-mak-ers are only human. If a spread is very complex, involving several different options inunusual ratios, it may take a market-