vega associated with the position:
4. Theta (Figure 17-3c)—As time passes a positive theta position will becomemore valuable, and the graph of such a position will shift upward. As time passesa negative theta position will become less valuable, and the graph will shiftdownward,
5. Vega (Figure 17-3d)—A positive vega position will be helped by increasingvolatility and hurt by declining volatility. The graph of such a position will shift362
Chapter 17
Figure 17-3c: Positive and Negative Theta (Time Decay)
 
positive theta - ,\ the graph shifts upward over time /,
\ /
 
 
  
\/ negative theta - \
the graph shifts downward over time *
 
 
Figure 17-3d: Positive and Negative Vega (Volatility)
 
positive (negative) vega - the graph shifts upward with higher (lower) volatility
oo™N
 
 
 
negative (positive) vega - the graph shifts downward with higher (lower) voiatility
 
 Position Analysis 363
upward when volatility increases and downward when volatility decreases. Anegative vega position will be hurt by increasing volatility and helped by decliningvolatility. The graph of this position will shift downward when volatility increasesand upward when volatility decreases.
Note that time and volatility have similar effects on an option position. But unliketime, which can only move in one direction, volatility can either rise or fall. If a traderhas a positive (negative) gamma position, he will necessarily have a negative (positive)theta position. He need only worry about one or the other, either gamma or theta. Buta trader can have a positive or negative gamma position and still be concerned withvolatility. A positive gamma position can still be hurt by a rise in volatility (e.g., a shorttime spread), while a negative gamma position can still be hurt by a decline in volatility(e.g., a long time spread).
Let’s try to put these graphing principles together to analyze a position.
Stock price = 98.75; Time to June expiration = 7 weeksVolatility = 20%; Interest rate = 6%; Dividend = 0
Option Option Option Option
Position Price Th. Val. Deita Imp. Vol.long 20 June 105 calls 75 1.02 25 17.6short 20 June 95 puts 1.38 1120-25 22.1
short 10 stock contracts?
Even though the position is delta neutral, it does not appear to fall into any easilyrecognized category. The calls appear to be underpriced and the puts appear to beoverpriced, and the position combines trades in both these options. The total theoreticaledge is:
20 x (1.02-.75) + 20 x (1.38-1.12) = +10.60
The sensitivities of the position are:
Option Gamma Theta ' VegaJune 105 call 43 -.0270 114June 95 puts 4.4 -.0192 115position totals -2.0 -.1560 ~.020
First we ought to try to visualize what the graph of this position looks like. Sincethe position has a positive theoretical edge of 10.60, it will cross the current stock priceof 98.75 at a point 10.60 above the zero P&L line. The position is delta neutral, so thegraph will be exactly horizontal at this point. The position is also very close to gammaneutral at this point, so the graph will have no curvature. This means that it is bendingneither upward (concave) nor downward (convex).
 
2. Since the number of shares in an underlying stock contract varles from market to market, we simplyexpress the underlying in terms of contracts. In U.S. markets where one option contract typicallycontrols 100 shares of stock, this position would be short 1,000 shares.364 Chapter 17
To complete the graph we need the upside and downside contract position. If themarket makes a large upward move, all the puts will eventually collapse to zero whilethe calls will act like long underlying contracts. In our example the 20 long calls willact like 20 long stock contracts. These will be partially offset by the actual 10 shortunderlying contracts, but the total will still be an upside contract position which is long10. If the market moves up far enough the position will take on a slope of +10. For eachpoint the stock price rises, the position will gain ten points in value.
If the market makes a large downward move, all the calls will eventually collapseto zero while the puts will act like short underlying contracts. Since a short put is a longmarket position, our short 20 puts will act like 20 long stock contracts. These will bepartially offset by the actual 10 short underlying contracts, but the total will still be adownside contract position which is long 10 contracts. If the market moves down farenough the position will take on a slope of +10. For each point the stock price falls, theposition will lose ten points in value.
The initial characteristics of our spread are shown in Figure 17-4a.
We know that the characteristics of a position are likely to change as marketconditions change. How might our position change as time passes? Since deltas moveaway from 50 with the passage of time, if no movement takes place in the underlyingstock, the deltas of both the 95 put and 105 call (currently at -25 and +25) will move
Figure 17-4a: Long 20 June 105 Calls @ .75Short 20 June 95 Puts @1.38Short 10 Stock Contracts @98.75
40
 
384
/
305 upside contract position of +10
255
position is gamma neutral(graph is bending neither upward nor
204 downward at current underlying price}
 
position is delta neutral~<_——— (graph is horizontal at
current underlying price}} +10.60 (theoretical edge)
theoretical profit & lossaL
 
downside contract position of +105 X
-10 “17 T T T T T —T90 92 94 96 98 100 102 104 106 108 110stock price
 
 
 
 Position Analysis 365
towards zero. This means our position wil! act more and more like a position which isnaked short 10 stock contracts. Therefore, as time passes the slope of the graph isbecoming more and more negative between the exercise prices of 95 and 105 (igure17-4b).
What about changes in the gamma with respect to stock price movement? If thestock price remains near the current price of 98.75, the gamma (curvature) of theposition remains near zero. As the price moves toward either the 95 or 105 exerciseprice, the position takes on greater gamma characteristics as time passes. As the stockapproaches 95 the position begins to act like a short straddle (negative gamma), and asit approaches 105 the position begins to act like a long straddle (positive gamma).
Since the gamma of our position at the current price of 98.75 is very close to zero,we might expect the theta to also be close to zero. In this case the theta is actually—.1560. This means the position is becoming less valuable each day. The reason is thatthe sale of stock has resulted in a cash credit, a credit on which we can earn interest.As time passes the amount of interest which we expect to earn declines, so the expectedprofit also declines. This would not be the case if the underlying instrument were afutures contract, since the sale of a futures contract does not result in a cash inflow, andconsequently no interest can be earned. The theta of such a position would thereforebe close to zero.
Figure 17-4b: Long 20 June 105 Calls @ .75Short 20 June 95 Puts @1.38Short 10 Stock Contracts @98.75
507
 
405 f
305
7 weeks to expiration
 
 
theoretical profit & loss
-404 at expiration
 
 
 
 
TT —T T T “T —_T TT90 92 94 96 98 100 102 104 106 108 110stock price366 Chapter 17
Changes in volatility will also affect the characteristics of the position. Fallingvolatility is similar to the passage of time. All deltas will move away from 50, and thetotal delta position will become shorter around the current price of 98.75. If volatilityrises all deltas will move towards 50, and the total delta position will become longer.As volatility rises (falls), the position looks more and more like a position which is nakedlong (short) 10 stock contracts (Figure 17-4c).
Regardless of changes in volatility, the gamma will stay close to zero near themidpoint price of 100. But as we move towards 95 or 105 the gamma begins to increaseas either the 95 put or 105 call becomes more and more at-the-money. At the same timethe gamma of an at-the-money option falls as volatility rises and rises as volatility falls.The result is that in a low-volatility market any move away from 100 can cause thegamma risk to increase very quickly, becoming negative as the stock price moves toward95 and positive as the stock price moves toward 105.
Finally, the contract position remains the same, regardless of the passage of time orchanges in volatility. If the market makes a big move, rising far above 105 or falling farbelow 95, we still have the same contract position of +10 (long 10 underlyingcontracts).
This last example illustrates an important aspect of spreading. Many traders, inaddition to focusing on delta neutral strategies, will try to construct positions which are
Figure 17-4c: Long 20 June 105 Calls @ .75Short 20 June 95 Puts @1.38Short 10 Stock Contracts @98.75
50
 
volatility = 25%
   
volatility = 20%
  
304
theoretical profit & loss
 
volatility = 15%
 
 
 
 
-20 T T T T T T T T90 92 94 96 98 100 102 104 106 108 110
stock pricePosition Analysis 367
also gamma neutral or vega neutral. Such strategies may be sensible if a trader isconcerned with a move in the underlying contract, or with a change in volatility. Butatrader should not make the assumption that if he is delta neutral, gamma neutral, andvega neutral, there is nothing that can go wrong. Even if a position has a positivetheoretical edge and all the risk sensitivities are small, a profit is not necessarily certain.A position is only delta neutral, or gamma neutral, or vega neutral if the theoreticalpricing model is itself correct, and if all the information which the trader has fed intothe model is also correct. If any of these assumptions are wrong, the values generatedby the theoretical pricing model are also likely to be wrong. As a result, the trader mightfind that a position which initially appears to be neutral in terms of a risk sensitivity isin fact not neutral at all.
A COMPLEX POSITION
Each of our previous examples consisted of only two different options. A more complexposition may consist of options at several different exercise prices. Consider the positionshown in Figure 17-5.
We can determine many of the characteristics of this position simply by looking atthe total sensitivities. Since the position has a positive gamma, negative theta, andpositive vega, it will initially act like a backspread. In addition, we can see that the
Figure 17-5
Stock price = 202.50; Time to June expiration = 6 weeks; Volatility = 24.0%Interest rate = 6.00%; Dividend = 0
June Th. Imp.Contract Position Price Value Delta Gamma Theta Vega Vol.190 call 15.50 15.50 82 1.6 -.077 18 24.0195 call ~23 11.63 11.80 72 2.0 ~.088 23 23.3200 call +55 8.25 8.63 61 2.3 ~.094 26 22.6208 call —35 5.75 6.06 49 2.4 —.094 i2t 22.9210 call -20 4.00 4.08 37 2.3 ~.086 26 23.7215 call +44 2.63 2.63 27 2.0 ~.074 23 24.0190 put —42 1.63 1.70 -18 1.6 ~.046 18 23.6195 put +85 2.75 2.95 —-28 2.0 —.056 23 23.1200 put +8 45 4.76 -39 2.3 —.062 .26 23.0205 put -18 6.88 7.15 —51 2.4 ~.060 27 23.0210 put 10.00 10.13 -63 2.3 -.052 26 23,5215 put —30 13.75 13.65 -73 2.0 -.039 23 24.5stock -18 100
Stock Calls Puts Delta Gamma Theta Vega P&Ltotals: -18 +18 +3 -277 +50.5 ~2.244 +610 +18.82
 368 Chapter 17
downside contract position is net short 21 (short 18 stock contracts, long 3 puts) whilethe upside contract position is flat (short 18 stock contracts, long 18 calls). The positionhas unlimited downside reward and limited upside risk/reward. Based on thesecharacteristics, the position can essentially be described as a put backspread. Thetheoretical graph of the position under current conditions is shown by the solid line inFigure 17-5a.
In Figure 17-5a we also see how the position will change as time passes. Becausethe trader is long a large number of June 195 puts and June 200 calls, as expirationapproaches the position will act more and more like a long June 195/200 strangle.Although the position is now slightly delta short, with the stock at its current price of202.50, the June 200 calls will tend to dominate, causing the delta position to becomeincreasingly long as time passes. Should the stock drop below 195, the June 195 putswill tend to dominate, and the delta will become increasingly short with the passage oftime. Consequently, as time passes the gamma of the position will become very largewith the stock price between 195 and 200. The total delta and gamma of the positionas time passes is shown in Figures 17-5b and 17-5c.
Figure 17-5d shows the sensitivity of the position to a change in volatility. As wewould expect, any increase in volatility will help the position and any decline involatility will hurt. Even if we increase or decrease volatility by 20%, to 30% or 18%,the shape of the position remains essentially the same. While the delta and gamma willchange with changes in volatility, these numbers are not as sensitive to a change in
Figure 17-5a
300
 
6 weeks to expiration (current)
\2004 % ||
      
3 weeks to expiration |
1005
 
theoretical profit & losso
-1007
1 week to eptatonat expiratio 4
-2007
 
 
-300 T T T T T T T T T180 185 190 195 200 205 210 215 220 225 230stock price
 Position Analysis 369
 
Figure 17-5b
8000
 
6000-
4000+
    
6 weeks to expiration (current)
 
 
delta position
at expiration!
 
 
 
 
 
-6000-||-8000 1 1 1 ——J 1 1 T T180 185 190 195 200 205 240 215 220 225 230
stock price
 
Figure 17-5¢
400
 
1 week to expiration f\ :
3004 Ny ‘
 
3 weeks to expiration
gamma position
 
 
 
   
 
 
 
 
-100- .6 weeks to expiration (current) sO a-2004 TT T T T T T T T T180 185 190 195 200 205 210 215 220 226 230
stock price
 370 Chapter 17
Figure 17-5d
 
    
 
 
 
 
 
300N. :250-7 *20075 \volatility = 24%B 150+% volatility = 30%2a 100732@3 50&0+-50-volatility =18%~100-7 T T T T ~ T T T T T180 185 190 195 200 205 210 215 220 225 230
stock price
 
volatility as they are to the passage of time. The total delta and gamma of the positionas volatility changes is shown in Figures 17-5e and 17-5f.
A trader with this position must be concerned with a significant decline in volatility.He will be especially worrted about a slow decline in the market over time. Each daythat passes is costing him 2.244 (the theta) in potential profit, and this number willaccelerate as time passes. If several weeks should pass with little movement in the stockprice, he will come under increasing pressure to reduce his theta risk. If he decides totake some protective action, he will have to sell some options, either calls or puts. If heconsiders a large move in the underlying stock a possibility, he may not want to sell callssince this will leave him net short contracts on the upside. This will probably lead himto sell off some of the puts which he is long, either the 195 or 200 puts.
Up to now we have not considered the risk associated with a change in interestrates or, in the case of stock options, dividends. While these risks are usually small withrespect to the other risks, if the position is large enough we might want to consider howa change in interest rates or dividends will impact the position.
The sale of 18 stock contracts has created a cash credit for this trader, and theinterest earned on this credit will be a part of his profit. Assuming that he is subject toa floating interest rate, any increase in rates will help, since he will earn more on hiscash credit, and any decline in interest rates will hurt, since he will earn less on his cashcredit. We can estimate the impact on the position of a 1% change in interest rates bycalculating the interest earned to expiration at the current stock price:Position Analysis
 
Figure 17-5e
500.
371
 
-600-
-1000-+
delta position
-1500-1
 
-20004
 
-2500-+
volatility =18%
volatility = 24%
 
     
volatility = 30%
 
 
r180 185
—195
200
T205210 218
stock price
T220
OT225
230
 
Figure 17-5f
 
1407
1205
100+
gamma position
 
volatility = 30%
 
 
 
volatility =18%
volatility = 24%
 
-20+
 
 
 
 
 
T180 185
—T195
-200
r205 215
stock price
210
220228
230
 372 Chapter 17
1% x 42/365 x 202.50 = .233
Since the trader is short 18 stock contracts, the total impact of a 1% change in interestrates is:
233 x 18 = 4.19
For each percentage point increase in interest rates, his position will increase in valueby approximately 4.19; for each percentage point decline in interest rates, his positionwill decline in value by approximately 4.19. If interest rates were to immediately dropfrom 6% to 4%, the position’s potential profit would drop by approximately 8.38.
The underlying stock for this position currently pays no dividend, but the tradermay want to consider the possibility that the company will decide to start paying adividend. If the company decides to pay a dividend prior to June expiration, the trader,as a borrower of stock, will be required to pay this dividend amount to the stock lender.Since he is short 18 stock contracts, for each point increase in the dividend, the traderwill lose 18 times the dividend in potential profit. If the company decides to pay adividend of 2.00, the trader’s position will be reduced in value by approximately 36.00.
If the trader believes that the company might in fact decide to pay a dividend priorto June expiration, or if he feels that interest rates could fall significantly, he will wantto take some action to reduce his risk. Since the risk results from the short stock position,he can eliminate most of his risk by eliminating the 18 short stock contracts. Assumingthat the trader wants to maintain the same delta position, he can accomplish this bybuying back the short stock and replacing it with either deeply in-the-money options(buy deeply in-the-money puts or sell deeply in-the-money calls), or by selling syntheticstock contracts {sell calls and buy puts at the same exercise price). The former strategywould be the same as executing a three-way; the latter strategy would be the same asexecuting a conversion.
The sensitivity of the position to a change in interest rates and dividends is shownin Figures 17-5g and 17-5h.
FUTURES OPTION POSITIONS
The foregoing positions have all consisted of stock options, but delta, gamma, and vegacharacteristics are similar whether a trader is dealing with stock or futures options.Except for the effect of interest rates and dividends, the analyses of the positions wouldbe essentially the same for futures options. While futures options are not subject todividend considerations, nor are they as sensitive to changes in interest rates as stockoptions, there is an additional dimension to the analysis of a futures option positionbecause the relationship between futures months is often less well defined than withstock options. For example, consider the position in Figure 17-6.
This is typical of the type of position an active trader often ends up with: a seeminglydisjointed collection of options and underlying contracts which cannot be easilycategorized. Nonetheless, it is important that the trader recognize the risks which theposition currently presents, as well as those risks which might arise if conditions change.Position Analysis
373
 
Figure 17-5g
 
350
300+
250
200-|
1505
theoretical profit & loss
100-5
      
iiii
| interest rate = 12%
interest rate = 9%
interest rate = 6% (current)
 
 
 
interest rate = 3%
 
 
 
T185
T T t205 210 215
stock price
T220
r225
 
230
 
Figure 17-5h
 
300
200-
15074
100-4
505
theoretical profit & loss
      
dividend = 0 (current)
dividend = 2.00
 
-507
 
 
dividend = 4.00
 
 
 
 
-100
—180 185
180
T T195 200
T r T205 210 215
stock price
220
230
 Chapter 17
374
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