vegaOEX 425 put -81 25 -3.09 $1OEX 440 put ~196 42 -4,00 74OEX 455 put 339 43 -2.36 54NYA 250 call 221 13 -4.93 27NYA 260 call 152 21 5.49 44NYA 270 call 75 18 -4.05 39NYA 250 put -41 13 ~1.85 28NYA 260 put 117 24 -2.36 44NYA 270 put 211 26 -1.26 28The total dollar sensitivities of our OEX/NYA position is:
Position $delta $gamma  $theta $vegalong 20 OEX 440 calls 5180 720 —185.80 1500long 20 OEX 440 puts -3920 840 -80.00 1480short 39 NYA 260 calls 5928 ~-819 214.11 -1716short 39 NYA 260 puts 4563 -936 92.04 -1716Totals ~105 -195 +40.35 -452
How can we interpret these sensitivities? Although the $delta of -105 indicates apreference for the market to fall rather than rise, in practice most traders would considerthe position essentially $delta neutral since the preference is only very slight. In relativeterms the $delta of -105 represents a position which is short only about 24% of an OEXcontract (with a $delta of 440), or about 40 percent of an NYA contract (with a $deltaof 260).
But why does the position have a negative $gamma, positive $theta, and negativevega?
The total negative $gamma of -195 seems to suggest that movement in theunderlying contracts will hurt our position. If both the OEX and NYA were to move atthe same rate, we can see from Figure 16-9 that this would indeed be true. But we don’texpect the OEX and the NYA to move at the same rate; we expect the NYA to move atonly 87 percent the rate of the OEX. If we multiply our NYA $gamma position by .87,and then add it to our OEX $gamma position, it turns out that the total $gamma is veryclose to zero:°
(720 + 840} — .87 x (819 + 936} = 33
This reflects the fact that we are relatively insensitive to movement in the two indexesas long as the NYA moves at 87% the rate of the OEX.
 
4. In a perfectly balanced intermarket spread the total $delta, $gamma, $theta, and $vega would in factadd up to exactly zero.
5. Ibid.Intermarket Spreading 349
The same reasoning applies to our $vega position. The total negative $vega of -452suggests that an increase in volatility will hurt our position. If the volatility of the OEXand NYA were to increase by the same amount this would be true. But if volatility rises,we expect NYA volatility to rise by only 87% of OEX volatility. If we multiply our NYA$vega position by .87, and then add it to our OEX $vega position, the total the total$vega is approximately zero:©
(1500 + 1480) - .87 x (1716 + 1716) =-6
This reflects the fact that we are insensitive to changes in volatility as long as NYAvolatility is 87% of OEX volatility.
From these examples, we can see that the actual $gamma, $theta, and $vegapositions should reflect the relative volatilities of the underlying contracts. In ourexample we have:
total $gamma = OEX $gamma + (NYA $gamma x [NYA volatility/OEX volatility]}total $theta = OEX $theta + (NYA $theta x [NYA volatility/OEX volatility])total $vega = OEX $vega + (NYA $vega x [NYA volatiliry/OEX volatility]}
When we do this, we find that our intermarket spread is insensitive to price movement,to the passage of time, and to changes in volatility, as long as the NYA moves at 87%the rate of the OEX.
Suppose that instead of selling at-the-money NYA straddles to offset our purchasedat-the:money OEX straddles, we decided to sell NYA strangles. For example, we mightestablish the following $delta neutral position:
buy 20 OEX 440 calls sell 69 NYA 270 callsbuy 20 OEX 440 puts sell 96 NYA 250 puts
Since the OEX 440 call has a $delta of 259 and the NYA 270 call has a $delta of 75, foreach OEX 440 call we buy, we must sell 259/75, or approximately 3.45 NYA 270 calls.In the same way, since each OEX 440 put has a $delta of -196 and each NYA 250 puthas a delta of -41, for each OEX 440 put which we buy, we must sell 196/41, orapproximately 4.78 NYA 250 puts. Having purchased 20 OEX 440 straddles, toestablish a $delta neutral intermarket spread we must sell 69 NYA 270 calls and 96 NYA250 puts.The sensitivities of this position are:
Position $delta $gamma $theta $vegalong 20 OEX 440 calls 5180 720 -185.80 1500long 20 OEX 440 puts -3920 840 -80.00 1480short 69 NYA 270 calls -5175 1242 279.45 —2691short 96 NYA 250 puts 3936 -1248 177.60 2688
Taking into consideration the multiplication factor of .87 for the NYA $gamma,$theta, and $vega, the total sensitivities for our spread are:
 
6. Ibid.350 Chapter 16
$delta = 5180 ~ 3920 - 5175 + 3936 = +21
$gamma = 720 + 840 — .87 x (1242 + 1248) = -606
$theta = —185.80 — 80.00 + .87 x (279.45 + 177.60) = +131.83$vega = 1500 + 1480 ~ .87 x (2691 + 2688) =-1700
Even though the $delta position is approximately neutral, the negative $gamma,positive $theta, and negative $vega suggest that the spread will act like a short straddle:any large move in the underlying market or increase in volatility will hurt the position,and the passage of time or a decline in volatility will help the position. If we do such aspread, we are not only taking a position on the relative volatilities of the two underlyingmarkets, we are also taking a position that the implied volatilities in the two marketsare too high.
What type of position would we initiate if the relative volatilities were incorrect,and at the same time the trader felt that implied volatilities were too low? In this casewe would try to create a position with a positive $gamma, negative $theta, and positive$vega. For example, we might buy the OEX 425/455 strangle and sell the NYA 260straddle:
buy 20 OEX 425 calls sell 18 NYA 260 callsbuy 20 OEX 455 puts sell 14 NYA 260 puts
Since the OEX 458 call has a $delta of 140 and the NYA 260 call has a $delta of 152,for each OEX 455 call we buy, we must sell 140/152, or approximately .92 NYA 260calls. In the same way, since each OEX 425 put has a $delta of -81 and each NYA 260put has a delta of -117, for each OEX 455 put which we buy, we must sell 81/117, orapproximately .69 NYA 260 puts. Having purchased 20 OEX 425/455 strangles, toestablish a $delta neutral intermarket spread we must sell approximately 18 NYA 260calls and 14 NYA 260 puts.The sensitivities of this position are:
Position $delta $gamma  $theta $vegalong 20 OEX 455 calls 2800 660 -137.40 1380long 20 OEX 425 puts —2736 500 -61.80 1020short 18 NYA 260 calls -1620 -378 98.82 -792short 16 NYA 260 puts 1638 336 33.04 -616
Taking into consideration the multiplication factor of .87 for the NYA $gamma,$theta, and $vega, the total sensitivities for our spread are:
$delta = 2800 — 2736 - 1620 + 1638 = +82
$gamma = 660 + 500 — .87 x (378 + 336} =-1781
$theta = -137.40 - 61.80 + .87 x (98.82 + 33.04) = +84.48$vega = 1380 + 1020-87 x (792 + 616) =-1175
The positive $gamma, negative $theta, and positive $vega suggest that the spreadwill act like a long straddle: any large move in the underlying market or increase inIntermarket Spreading 351
volatility will help the position, and the passage of time or a decline in volatility will hurtthe position. If we do such a spread, we are not only taking a position on the relativevolatilities of the two underlying markets, we are also taking a position that the impliedvolatilities in the two markets are too low.
While intermarket volatility spreads begin with a trader’s opinion about the relativevolatilities in two different markets, such spreads can be carried one step further. Whenthe implied volatilities appear to be incorrect with respect to each other, the trader cansimply buy straddles in the market which is relatively cheap and sell straddles in themarket which is relatively expensive. However, if the trader also has an opinion thatimplied volatilities in the two markets are generally too high or too low, by taking anappropriate $gamma and $vega position the trader can create volatility spreads whichalso exploit this mispricing. Depending on his opinion, he will want to create one of thefour basic types of volatility spreads:
positive $gamma / positive $vega (backspread]
nepative $gamma / negative $vega (ratio vertical spread)negative $gamma / positive $vega (long time spread)positive $gamma / negative $vega (short time spread)
For a more detailed discussion of volatility spreads and their characteristics, the readershould refer to Chapter 8.
OPTIONS ON SPREADS
There is one last type of intermarket spread worth mentioning. Even though suchspreads are not currently traded on exchanges, it is possible to create an option wherethe value of the option at expiration depends on a spread between two differentunderlying markets. For example, suppose two underlying markets, ABC and XYZ, aretrading at 175 and 150, respectively. The ABC over XYZ spread is therefore trading at25. If a trader believes this spread might widen to as much as 60 points, he might liketo buy an option on the spread with an exercise price below the projected spread of 60.If he buys a spread with an exercise price of 25, and the spread does widen to 60 atexpiration, he will receive a payoff of 35, the difference between the 60-point spreadand the 25-point exercise price.
Such a spread is not as exotic as it sounds, since there are many situations wherea trader might believe that a spread will expand or contract over time. A portfolioManager might believe that the stock market of one country will outperform the stockmarket of another country. If such an option were available, the manager might wantto buy a call option on the spread between the stock indexes of the two countries. If thespread widened to the point where it was greater than the chosen exercise price, theCall would go into-the-money and would have unlimited profit potential. If the spreadeither failed to widen or narrowed, the call would finish out-of-the-money and themanager's loss would be limited to the amount he paid for the call.”
 
7. Ofcourse, such an option would be complicated by the relative values of the currencies involved. Thebuyer of the option would have to devise a method of offsetting the currency risk.352 Chapter 16
Note that the underlying value of an option on a spread might take on a negativevalue. The ABC over XYZ spread, with ABC at 175 and XYZ at 150, has a value of 25,But if ABC rises to 185 while XYZ rises to 200, the spread will have a value of -15.Most traditional theoretical pricing models, with their assumption of a lognormaldistribution, do not allow for negative prices for the underlying contract. If one were tocreate a pricing model to evaluate options on spreads, a more realistic assumption mightbe a normal distribution, which does allow for negative prices. Such a variation on theBlack-Scholes model has been suggested in a paper prepared by Darrell Wilcox. Thisvariation is reproduced in Appendix B.a 17 &
Position Analysis
When a trader first enters an option market, his initial concern is usually findingstrategies with sufficient profit potential to justify making a trade. As a trader becomesmore actively involved in options, he finds that an increasingly greater portion of histime is spent analyzing the positions he has already taken. What are the risks to theposition? What are the rewards? And what action should be taken if market conditionschange either in the trader’s favor or against him? Simple volatility spreads, directionalspreads, or arbitrage strategies are relatively easy to analyze. As positions become morecomplex, the risks associated with the position may not be as easy to identify.
The first step in analyzing a position is to determine how the position will react tochanging market conditions. One way to accomplish this is by looking at the total risksensitivities of the position (delta, gamma, theta, vega, rho). Unfortunately, the risksensitivities are only reliable under narrowly defined market conditions, Since it is oftendramatic changes in market conditions about which a trader must be concerned, simplylooking at the risk sensitivities rarely enables a trader to grasp fully the risks associatedwith a position. Moreover, most traders find it easier to interpret graphic informationthan a series of numbers. For all of these reasons, it is oftenuseful to graph a position’svalue under a variety of market conditions.
SOME SIMPLE EXAMPLESLet’s start with a simple position:!
Stock price = 99,00; Time to June expiration = 7 weeksVolatility = 20%; Interest rate = 6%; Dividend = 0
Option Option Option Option
Position Price Th. Val. Delta Imp. Vol.long 10 June 95 calls 6.25 5.82 76 23.7short 30 June 105 calls 1.63 1.08 26 24.5
The position is a typical ratio vertical call spread which we might take if we feel thatimplied volatility is too high, as it appears to be here. The total theoretical edge is:
-10 x (6.25-5.82) + 30 x (1.63-1.08) = +12.20
 
1. The analyses in this chapter were done using elther the Black-Scholes model for stock options or theBlack model for futures options.
353354 Chapter [7
While the position is currently delta neutral, we know that a ratio vertical spread has anegative gamma, positive theta, and negative vega. The gamma, theta, and vegasensitivities of the spread confirm this:
Option Gamma Theta VegaJune 95 call . 43 -.0344 13June 105 call 4.4 -.0278 117position totals -89.0 +.4900 ~2,380
The negative gamma is an indication that any large move in the underlying stockwill hurt the position. This is shown in Figure 17-1a. Under current conditions, if thestock price begins to fall and we want to remain delta neutral, for each point decline inthe stock price we will need to sell approximately 89 shares of stock (or take somesimilar negative delta action). If the stock begins to rise, for each point rise in the stockprice, we will need to buy approximately 89 shares of stock.
A ratio vertical spread also has a positive theta, so the passage of time will help theposition. This is confirmed in Figure 17-1b where the value of the position is shown asthe time remaining to expiration grows shorter. Note that not only does the position’spotential profit change as time passes, the risks associated with the position also change.As time passes the deltas of options tend to move away from 50, with the delta ofin-the-money options moving towards 100 and the delta of out-of-the-money options
Figure 17-1a: Long 10 June 95 Calls @ 6.25Short 30 June 105 Calls @ 1.63
207
 
 
-20-4]
-307
theoretical profit & loss
-407
-607
 
 
 
 
-70 T T y T T >>90 92 94 96 98 100 102 104 106 108 110stock pricePosition Analysis 355
Figure 17-1b: Long 10 June 95 Calis @ 6.25Short 30 June 105 Calls @ 1.63
400
 
at expiration1 week to expiration
80-5
 
 
 
 
 
 
3gos8a82aoO£-20-4 4 weeks to expiration 7\‘-404 7 weeks to expiration-604-B04— T T “Tr T T tT 1 T T90 92 94 96 98 100 102 104 106 108 110stock price
moving towards zero. The net effect is that with the stock price anywhere below 105our delta position will be getting longer as time passes. If, however, the stock price risesabove 105, the 95 call and the 105 call will be in-the-money, and the deltas of bothoptions will move towards 100. This will cause our delta position to become shorterwith the passage of time. This is shown in Figure 17-1b where the downward slope ofthe graph above 105 becomes more pronounced as time passes.
Gamma can also change as time passes. Atthe-money options always have thehighest gamma, and the gamma of an at-the-money option can increase rapidly asexpiration approaches. As time passes note how the curvature of the position becomesmore severe around the stock price of 105. If the stock should rise to 105 and remainthere as expiration approaches, the position will take on increasingly greater gammarisk.
Since a ratio vertical spread has a negative vega, any rise in volatility will hurt theposition, and any decline will help. The theoretical value of the position at variousvolatilities is shown in Figure 17-1c. Under current conditions the position has animplied volatility of approximately 25%, the volatility at which the position approxi-mately breaks even, showing neither profit nor loss. (We can also approximate theimplied volatility of the position by dividing the total theoretical edge of 12.20 by thevega of 2.38, and adding the result to the 20% volatility we are currently using.) If thestock price should change, the implied volatility of the position will also change. At a356
 
40
 
Chapter {7Figure 17-1c: Long 10 June 95 Calls @ 6.25Short 30 June 105 Calls @ 1.63volatility = 15%\‘\
theoretical profit & loss
volatility = 20%
-607
 
 
    
volatility = 25%
volatility = 30%
 
 
~100 T T T T T T90 92 94 96 98 100 102
Stock price
 
stock price of 104, the position has an implied volatility of approximately 20%. At astock price of 106, the position has an implied volatility of approximately 15%. As the
stock price rises, we hope for a lower and lower volatility.
The delta risk will also change as volatility changes since deltas move away from50 as volatility falls and towards 50 as volatility rises. If volatility falls, the position willbecome delta positive; if volatility rises, the position will become delta negative. We cansee this by tooking at the slope of the graph at the current stock price of 99. With avolatility below 20%, the graph has a positive slope; with a volatility above 20%, the
graph has a negative slope.Let’s look at another position:
Stock price = 100.00Time to expiration: June = 7 weeks, September = 20 weeksVolatility = 20%; Interest rate = 6%; Dividend = 0
Option OptionPosition Price Th. Val.long 20 June 105 calls 1.88 1.35short 20 September 110 calls 3.13 2.22
OptionDelta3030
OptionImp. Vol.24.024.1Position Analysis 357
This position is a diagonal spread, but with both options having equal deltas the positionwill tend to act like a short time spread. The total theoretical edge is:
—20 x (1.88-1.35) + 20 x (3.13-2.22] = +7.60
We know that a short time spread has a positive gamma, negative theta, and negativevega. The gamma, theta, and vega sensitivities of the spread are:
Option Gamma Theta VegaJune 105 call 4.3 -.0308 128September 110 call 2.8 -.0200 216position totals +30.0 -2160 -.760
The positive gamma means that movement in the underlying stock price is working inour favor, as shown in Figure 17-2a. As the stock price rises, our delta position isbecoming longer, and as the stock price falls, it is becoming shorter. If we want to staydelta neutral we will need to sell stock as the price rises, and buy stock as the price falls.
Since a positive gamma is always accompanied by a negative theta, the passage oftime will work against us, as shown in Figure 17-2b. Under current conditions, for eachday that passes with no movement in the underlying stock price, our position willdeteriorate by approximately .216. We can also see that with the stock below 105 theslope of the graph becomes negative as time passes, reflecting the fact that our delta
Figure 17-2a: Long 20 June 105 Calls @ 1.88Short 20 September 110 Calls @ 3.13
90
 
i!80- :
theoretical profit & loss
 
 
 
 
 
OF T T T T80 85 90 95 100 105 410 WS 120 125 130
stock price358 Chapter 17
Figure 17-2b: Long 20 June 105 Calls @ 1.88Short 20 September 110 Calls @ 3.13
100
 
80-4
      
7 weeks to June expiration |4054 weeks to June expiration
theoretical profit & loss
 
. * 1 week to June expiration
 
 
 
 
-204 |NA at June expiration-407 T T T | T T T T T80 85 90 95 100 105 110 115 120 125 130stock price
 
position is becoming shorter with the passage of time. If the stock price should rise above105, our delta position becomes longer with the passage of time. At June expiration wewant the stock to be as far away from 105 as possible.
As the stock price moves toward 105, the curvature of the position also becomesmore severe, reflecting the fact that the gamma ofan at-the-money option increases withthe passage of time.
In Figure 17-2c we can see how the position’s value declines (rises) with rising(falling) volatility. The implied volatility of the position is just under 25%.
While changing volatility can affect the delta and gamma of a position, we can seein Figure 17-2c that in this spread the delta and gamma are generally unaffected bychanges in volatility. Regardless of volatility, the delta position remains neutral at thecurrent stock price of 100, becomes shorter anywhere below 100, and becomes longeranywhere above 100. Moreover, the graphs all have the same general shape, reflectingthe fact that the position’s gamma is not particularly sensitive to changes in volatility.
GRAPHING A POSITION
Every trader should know what a graph of his position looks like, if not in minute detailat least the approximate characteristics of the position. Sometimes, if a computer isunavailable to do the graphing for us, we might have to mentally visualize such a graph.Position Analysis 359
Figure 17-2c: Long 20 June 105 Calls @ 1.88Short 20 September 110 Calls @ 3.13
100
 
   
volatility = 15 aNe “Y
volatility = 20%ty NeJ
605
  
|407
theoretical profit & loss
- hae volatility = 25%NL volatility = 30%
 
 
 
 
 
-20+t T T T t T T T T T80 85 90 95 100 105 110 115 120 125 130
stock price
From the foregoing examples, perhaps we can formulate some rules which will enableus to do this.
Using, as before, a grid where the horizontal axis (x-axis) represents movement inthe underlying contract and the vertical axis (y-axis) represents profit or loss, we caninterpret the theoretical edge, delta, and gamma as follows:
1. Theoretical Edge—The graph of a position with a positive theoretical edge willcross the current underlying price at a point above the zero profit & loss line. Thisis the first thing a trader should look for. When a trader is right about marketconditions, he wants to know that his position will be profitable.
2. Delta (Figure 17-3a]—A positive delta is theoretically equivalent to a longposition in the underlying contract. The graph of such a position will cross thecurrent underlying price at an angle extending from the lower left to the upperright. A negative delta is theoretically equivalent to a short position in the underly-ing contract. The graph of this position will cross the current underlying price at anangle extending from the upper left to the lower right. The exact slope of the graphas it crosses the current underlying price is determined by the magnitude of thedelta. As the delta position becomes larger, the slope becomes more severe; as thedelta position becomes smaller, the slope becomes less severe. The graph of a360 Chapter 17
Figure 17-3a: Positive and Negative Delta (Slope)
 
   
positive delta(graph extends from lowerleft to upper right)
negative delta(graph extends from upperleft to lower right)
 
 
 
 
position which is delta neutral will be exactly horizontal as it crosses the currentprice of the underlying contract.
3. Gamma (Figure 17-3b}—A positive gamma position will begin to bend upwardas the underlying price moves away from the current price in either direction. Thisreflects the fact that a positive gamma position likes movement. The graph of sucha position will take on a generally convex shape (a smile). A negative gammaposition will begin to bend downward as the underlying price moves away fromthe current price in either direction. This reflects the fact that a negative gammaposition prefers for the market to sit still. The graph of such a position will take ona generally concave shape (a frown).
In order to complete a graph there is one other piece of information we need.Suppose the underlying contract makes such a large upward move that all calls gointo-the-money and all puts go outof-the- money, or such a large downward move thatall puts go into-the-money and all calls go out-of-the-money. In other words, what willthe graph of a position look like at the extreme right or left tail? We can determine thisby adding up our contract or lot position. This is a measure of how many contracts aposition will be naked long or short if the underlying market makes such a large movethat either all calls or all puts act like underlying contracts.
On the upside (extreme right tail) all puts will collapse to zero while all calls willeventually act like long underlying contracts. The graph of a position which is long morePosition Analysis 361
Figure 17-3b: Positive and Negative Gamma (Curvature)
 
positive gamma(a smile}
 
negative gamma(a frown)
 
 
 
calls than short will angle up and to the right infinitely far (unlimited upside profit). Thegraph of a position which is short more calls than long will angle down and to the rightinfinitely far (unlimited upside risk). If the number of long and short calls are equal, orif the position consists only of puts, the position will eventually flatten out (limitedupside risk/reward).
On the downside (extreme left tail) all calls will collapse to zero while all puts willeventually act like short underlying contracts. The graph of a position which is longmore puts than short will angle up and to the left infinitely far (unlimited downsideprofit). The graph of a position which is short more puts than long will angle down andto the left infinitely far (unlimited downside risk). If the number of long and short putsare equal, or if the position consists only of calls, the position will eventually flatten out(limited downside risk/reward).
The value of a position may also change over time or as volatility conditions change.The changes will be determined by the theta and