200 options that are 10 percent out ofthe money? These are questions that traders must answer individually.
Managing Directional Risk with DeltaTable 10-2 demonstrates how a trader might use delta to manage along option position to both increase proﬁt and decrease risk. This
Managing Position Risk • 315
technique is based on the behavior of stock prices, which generally donot make large price changes in a straight line; rather, stock prices typically rise for a few days and then fall back before resuming an uptrend. The goal of this managing technique, therefore, is to beneﬁtfrom normal up-and-down stock-price action by maintaining a relatively constant delta. Since long options have positive gamma, thedelta of a long call will increase as the stock price rises and decreaseas the stock price falls. This technique therefore involves selling a portion of owned calls when a stock rallies and buying them back whenthe stock declines. In the example that follows, a hypothetical tradernamed Grace implements this trading technique.The top section of Table 10-2 lists the rules that Grace created togovern when to purchase and sell 70 Calls. Grace’s initial position ofTable 10-2
Managing Directional Risk by Delta
Managing rules:Initial position delta  1,100.When delta is at or above 1,500, sell Calls to lower position delta to 1,100.When delta is at or below 900, buy Calls to raise position delta to 1,100.Row12345678910111213
Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Stock price70.0076.00Days to exp.353270 Call2.826.88Call delta0.530.84Beg. positionNone Long 20Beg. total delta0 1,680Beg. position value013,760Action/quantityBuy 20Sell 7End positionLong 20 Long 13End total delta1,060 1,092End total value5,6408,944Cash ﬂow(5,640)4,816Final proﬁt (total of cash ﬂows)
72.00283.700.66Long 13 8584,810Buy 4Long 171,1226,290(1,480)
77.00247.490.90Long 171,53012,733Sell 5Long 121,0808,9883,745
73.00214.070.74Long 128884,884Buy 3Long 151,1106,105(1,221)
78.00198.270.95Long 151,42512,405Sell 15None0012,40512,625
Proﬁt from buy and hold: Buy 20 @ 2.82  (5,640)Sell 20 @ 8.27  16,540Net proﬁt10,900Assumptions: Volatility, 31%; interest rate, 4%.
316 • Trading Options As a Professional
long 20 of the 70 Calls has a delta of approximately 1,100 (actually,1,060). Her goal is to approximately maintain this delta as the stockprice rises and falls, and she has chosen two triggers for action, deltasof 1,500 and 900. Consequently, when the position delta risesabove 1,500, Grace will sell a sufﬁcient number of 70 Calls so thatthe position delta is reduced to approximately 1,100. Conversely,when the position delta falls below 900, Grace will buy a sufﬁcientnumber of 70 Calls so that she increases the position delta. Using1,100 and 900 as triggers for buying and selling is a subjectivedecision. Traders can use Op-Eval Pro to experiment with stock-pricescenarios and levels of delta based on the number of contracts traded.The middle section of Table 10-2 has six columns and 13 rows thatdetail how Grace implements her strategy over a 16-day period from35 days to expiration to 19 days to expiration. Rows 1 and 2 list stockprices and days to expiration. In column 1, for example, the stock priceis 70.00 at 35 days to expiration. Row 3 lists the price of the 70 Call,row 4 lists its delta, and row 5 lists the initial position (“Beg. position”).Rows 6 and 7 hold the total delta and total value of the initial position,respectively. Row 8 indicates the action, buy or sell, and the quantityof calls, and the ending position is listed in row 9. Row 10 indicates thedelta of the ending position, which should be approximately 1,100,and row 11 indicates the value of the ending position. Row 12 lists thecash ﬂow from the trade in row 8, which is the product of quantity ofcalls in row 8 and the price in row 3 and the multiplier, which is 100and assumed. After Grace makes the ﬁnal trade and closes the position, the “Final proﬁt” is listed in column 6, row 13. The ﬁnal proﬁt isthe total of positive and negative cash ﬂows in row 12.This exercise starts in column 1 of Table 10-2 when the stock priceis 70.00 (row 1), the price of the 70 Call is 2.82 (row 2), and its deltais 0.53 (row 4). There is no beginning position (row 5), so there isno beginning delta or value (rows 6 and 7). When Grace buys 20 ofthese calls (row 8), she creates a position with a total delta of 1,060(row 10) and a value of $5,640 (row 11). The purchase is a negativecash ﬂow (line 12). Parentheses indicate option purchases, which
Managing Position Risk • 317
are negative cash ﬂows. Cash inﬂows, from option sales, are numberswithout parentheses.In column 2 of Table 10-2, the stock price has risen to 76.00 (row 1)at 32 days to expiration (row 2). The price of the 70 Call has increasedto 6.88 (row 3), and its delta is 0.84. Grace’s position of 20 long calls(row 5) therefore has a total delta of 1,680 (row 6), which exceedsGrace’s trigger limit and spurs her to act. She sells seven calls (row 8)in order to reduce the delta to approximately 1,100. Grace calculatedthis quantity by subtracting the desired delta from the beginningdelta and dividing the quotient by the delta of the call in row 4; that is,(1,680 – 1,100)  (0.84  100)  6.90 ≈ 7. The actual ending delta is1,092 (row 10). Selling seven calls resulted in a positive cash ﬂow of$4,816 (row 12).This trading exercise continues in column 3 of Table 10-2 whenthe stock price falls to 72.00 at 28 days (rows 1 and 2). As a result,the position delta falls to 858 (row 6). To raise the delta to approximately 1,100, Grace must buy four of the 70 Calls (row 8). Shecalculates this quantity by subtracting the beginning delta from thedesired delta and dividing the quotient by the delta of the call; that is,(1,100 – 858)  (0.66  100)  3.7 ≈ 4. The actual ending delta onthis day is 1,122.In columns 4 and 5 in Table 10-2, the stock price rises to 77.00at 24 days and falls to 73.00 at 21 days, respectively. To adjust the deltato the desired level, Grace sells ﬁve of the 70 Calls in column 4 andbuys three in column 5. When the stock price is 78.00 at 19 days toexpiration in column 6, Grace sells the remaining 15 calls at 8.27 eachto close the position.Row 13 in Table 10-2 indicates that the proﬁt from the trades incolumns 1 through 6 amounts to $12,625, not including trading costs.The bottom section of Table 10-2 calculates the alternative outcome of a buy-and-hold strategy. Had Grace held the original positionof 20 long 70 Calls for the entire time period and sold them allat 8.27, then her proﬁt would have been $10,900, not includingtrading costs.
318 • Trading Options As a Professional
The conclusion is that given the stock-price action in Table 10-2,the technique of managing delta increased Grace’s proﬁts by $1,725($12,625 versus $10,900). Different stock-price action, of course,would lead to a different result. It is possible that a loss could exceedthe initial investment. Had the stock price declined shortly after Gracepurchased the initial 20 calls, the delta could have declined below900. The rules then would have required Grace to purchase additional calls. If the original calls plus the additional calls then allexpired worthless, Grace’s loss would have exceeded $5,640.Despite the potential for negative outcomes, the exercise in Table10-2 demonstrates that proﬁts of long option positions potentiallycan be increased by managing delta. This technique tends to increaseproﬁts when prices are trending in a volatile manner, and it tends tounderperform the buy-and-hold approach when prices trend withbelow-average volatility.
Tracking Changes in Position RiskTable 10-3 calculates the risks of 20 long 70–75 Call vertical spreads.A long vertical call spread, also known as a bull call spread, involvesthe purchase of one call and the sale of another call with the sameunderlying and same expiration date but with a higher strike price.Figure 1-9 is a graph of a long call vertical spread. There are also shortvertical call spreads, known as bear call spreads, and long and shortvertical put spreads. A comparison of Table 10-3 with Table 10-1reveals that the risks of vertical spreads are very different from the risksof outright long options.Table 10-3 is similar to Table 10-1 in that the position risks listed inthe right-most column are the product of the individual risk component, the multiplier, and the number of contracts. Table 10-3,however, has seven columns instead of ﬁve because there are twooption positions, 20 long 70 Calls and 20 short 75 Calls. The secondoption requires an additional column, and the spread value also adds
Managing Position Risk • 319
Table 10-3
Position Risks of 20 Bull Call Spreads 1
Stock price, 70, equals strike of long call.Position: Long 20 70 Calls @ 2.82.Short 20 75 Calls @ 1.02.Col 1
Col 2
Row
RiskFactor
Long 20 Short 20 Spread70 Calls 75 Calls Value
12345
PriceDeltaGammaVegaTheta
$2.820.5350.0590.0870.310
Col 3
Col 4
 $1.02  $1.80 0.262 0.273 0.049 0.010 0.066 0.021(0.225) 0.085
Col 5
Col 6
Col 7
Multiplier Quantity Position100100100100100
2020202020
 $3,600 5462042 170
Assumptions: Stock price, 70; days to expiration, 35; volatility, 31%; interest rate, 4%, dividends, none;7-day theta.
another column. Note that the assumptions about stock price, daysto expiration, volatility, etc. are the same in Tables 10-1 and 10-3.Comparison of the position risks in the two tables therefore is valid.If you compare column 7 in Table 10-3 with column 5 in Table 10-1,you can see that the 20 long call spreads have lower risks in every respectthan the outright long calls. Whereas the 20 long 70 Calls have a position price of $5,640 (Table 10-1, row 1, column 5), the 20 long 70–75 callspreads have a position price of $3,600 (Table 10-3, row 1, column 7).The position delta of the long call spreads is 546, signiﬁcantly less thanthe delta of 1,070 for the long calls. Also, the position gamma of 20indicates that the delta of the long call spreads is less sensitive to stockprice change than the delta of the long calls, which have a gammaof 118.The sensitivity to volatility, as measured by the vega, is also less for thecall spreads, 42 versus 174, and ﬁnally, a comparison of the thetasindicates that the call spread position is less sensitive to time erosion thanthe outright long calls. The call spreads will lose $170 in one week fromtime decay, whereas the long calls will lose $620 in one week.
320 • Trading Options As a Professional
The conclusion is that vertical spreads are less sensitive to allGreeks than outright long option positions. How can traders use thisinformation? Consider the trading example presented next.
Vertical Spreads versus Outright Long OptionsTable 10-4 compares two bullish strategies in two market scenarios.The ﬁrst strategy is long one 70 Call, and the second is long one 70–75call vertical spread. Rows 1, 2, and 3 contain the assumptions aboutthe stock price, the days to expiration, and the level of implied volatility, and rows 4 and 5 contain the prices of the 70 Call and the 70–75call spread. The price of the 70–75 call spread is calculated bysubtracting the price of the 75 Call from the price of the 70 Call. Toavoid confusion, the price of the 75 Call is not shown. Column 1 contains the initial market assumptions and the initial prices. The initialstock price is $70.00, there are 35 days to expiration, and the impliedvolatility is 31 percent.
Table 10-4
Strategy Comparison: Long Calls versus Long Vertical CallSpreads
Strategy 1: Long 70 Call @ 2.82.Strategy 2: Long 70–75 call spread @ 1.80.Scenario 1Row123
45
Stock priceDays to exp.Implied volatility
Price of 70 Call70–75 call spread
Col 1
Col 2
70.003531%
73.501431%
2.821.80
4.112.92
Scenario 2Col 3
Col 4
Col 5
73.501424%Proﬁt(Loss)1.291.12
3.853.05
Proﬁt(Loss)1.031.25
Assumptions: Interest rate, 4%, dividends, none.Conclusion: Vertical spreads are less susceptible to changes in implied volatility and perform better thanoutright long options in certain market scenarios.
Managing Position Risk • 321
Columns 2 and 3 in Table 10-4 show the estimated option prices andproﬁt of the ﬁrst scenario in which the stock price rises to 73.50 (row 1),three weeks pass, leaving 14 days remaining to expiration (row 2), butthe implied volatility remains unchanged at 31 percent (row 3). In thisscenario, the 70 Call rises in price to 4.11 for a proﬁt of 1.29 (row 4), andthe 70–75 call spread rises to 2.92 for a proﬁt of 1.12 (row 5).In the second scenario, the stock price (73.50) and time to expiration (14 days) are the same as in scenario 1, but the implied volatilityhas declined to 24 percent. This scenario is presented in columns 4and 5 of Table 10-4. The 70 Call rises in price to 3.85 for a proﬁt of1.03, and the 70–75 call spread rises to 3.05 for a proﬁt of 1.25. In thisscenario, the proﬁt from the 70–75 call spread increases by 0.13,whereas the proﬁt of the 70 Call declines by 0.26. Proﬁts from thesetwo strategies change because implied volatility decreases—this is theonly difference between the two scenarios. Table 10-4 demonstratesthat vertical spreads sometimes can perform better in an environmentof declining implied volatility than outright long options.
Vertical Spreads—How the Risks ChangeAny calculation of position risks is only a snapshot that catches the situation at one stock price and at one point in time. Position riskschange if stock price, time, or implied volatility change, as theyinevitably will. In Table 10-3, the stock price is 70, so the 70 Callis at the money, and the 75 Call is out of the money. The Greeks ofthe 70 Call therefore are larger, in absolute terms, than the Greeksof the 75 Call.Table 10-5 calculates position risks of the 20 long 70–75 call vertical spreads assuming a stock price of 75, at which point the 70 Call isin the money and the 75 Call is at the money.A comparison of Table 10-5 with Table 10-3 reveals how—and byhow much—the Greeks change when the stock price is 75 versus 70.With the stock price at 70, the 20 long 70–75 call vertical spreads havea position delta of 546, a positive gamma, a positive vega, and a
322 • Trading Options As a Professional
Table 10-5
Position Risks of 20 Bull Call Spreads 2
Stock price, 75, equals strike price of short call.Position: Long 20 70 Calls @ 6.16.Short 20 75 Calls @ 3.03.Col 1
Col 2
Col 3
RiskFactor
Long 20 Short 2070 Calls 75 Calls
Col 4SpreadValue
Price$6.16  $3.03  $3.13Delta0.791  0.535 0.256Gamma0.040  0.055 0.015Vega0.062  0.093 0.031Theta (7-day) 0.242 (0.332) 0.090
Col 5
Col 6
Col 7
Multiplier Quantity  Position100100100100100
2020202020
 $6,260 5123062 180
Assumptions: Stock price, 75; days to expiration, 35; volatility, 31%; interest rate, 4%, dividends, none.
negative theta. With a stock price of 75, the position delta is 512,which is lower than with the stock price at 70. The gamma is now negative, and the vega also has changed from positive to negative. Thetheta, however, is now positive rather than negative.The message of Table 10-5 is that a stock price rise of $5.00 causesthe position risks to reverse completely. The position delta now willchange in the opposite direction from the change in price of theunderlying stock. The position now will be hurt if implied volatilityrises and helped if it declines. Finally, the passing of time now willhelp this position.The difference in position Greeks caused by the rise in stock pricefrom 70 to 75 means that the strategy’s primary source of proﬁt haschanged. When the stock price is 70 (see Table 10-3), a bull call spreadis a bullish strategy that proﬁts primarily from a stock-price rise and ishurt by the passing of time. When the stock price is 75 (see Table 10-5),however, a bull call spread is more of a neutral strategy; it still has apositive delta, but now the position will proﬁt from time decay.The change in position risks from Table 10-3 to Table 10-5 is onlyone example of how position risks change. Given the interaction of
Managing Position Risk • 323
the changing Greeks of long and short options, it is not always easy toanticipate how position risks will change as market conditions change.Traders must continuously update their risk analysis of positionsbecause those risks can change in unanticipated ways.
Changing Risks GraphedThe position risks of 20 of the 70–75 bull call spreads are illustratedin Figures 10-1 through 10-5. In all these ﬁgures, the straight linegraphs risk at expiration, and the curved lines represent the risk at35 and 17 days to expiration. It can be difﬁcult determining whichcurved line is 35 days and which is 17 days because they cross, soattention to detail is important.Figure 10-1 graphs position value against stock price (underlying).The value of a bull call spread is small when the stock price is belowthe lower strike price and rises to its maximum value as the stock pricerises above the upper strike price. The curved line that is upper onthe left and lower on the right depicts the strategy value at 35 days to
1100010000
Theo Price
900080007000600050004000300020001000060 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90UnderlyingFigure 10-1
Value of 20 Long 70-75 Call Vertical Spreads
324 • Trading Options As a Professional
expiration. The curved line in the middle on both sides of the graphrepresents the strategy value at 17 days to expiration. A comparison ofthe curved lines shows how the strategy value moves to the value atexpiration—the straight line—as time passes.Figure 10-2 graphs the position delta of the bull call spreads as thestock price changes. The curved lines show that delta is highest whenthe stock price is between the strike prices and falls to zero when thestock price falls below 70 or rises above 75. The straight line showsthat delta is zero at expiration if the stock price is below 70 or above75. With the stock price between 70 and 75 at expiration, the delta is2,000 because the 20 long 70 Calls are exercised, and the positionbecomes long 2,000 shares of stock.Gamma is the focus of Figure 10-3. Gammas are biggest when anoption is at the money, so a bull call spread has a positive gamma whenthe stock price is below or near the lower strike price (long call). Asexplained in Chapter 4, positive gamma means that the delta of theposition changes in the same direction as the change in price ofthe underlying. The position gamma turns negative, however, when
200018001600
Delta
140012001000800600400200060 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90UnderlyingFigure 10-2
Delta of 20 Long 70-75 Call Vertical Spreads
Managing Position Risk • 325
Gamma
140120100806040200–20–40–60–80–10060 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90Underlying
Figure 10-3
Gamma of 20 Long 70-75 Call Vertical Spreads
the stock price equals the strike price of the short call. Negative gammameans that the delta changes in the opposite direction from the changein price of the underlying.The graph of vega in Figure 10-4 is similar to the graph of gammain Figure 10-3 because both vegas and gammas are biggest whenoptions are at the money. The proximity of the curved lines, however,is quite different. The gamma lines in Figure 10-3 are farther apartwhen the stock price is 70 or 75 because gammas of at-the-moneyoptions increase as expiration approaches, whereas gammas of outof-the-money options decrease. The difference between the 35- and17-day gamma lines therefore changes noticeably in Figure 10-3.In Figure 10-4, however, the curved vega lines are closer togetherbecause vegas of both at-the-money options and out-of-the-moneyoptions decrease as expiration approaches. As a result, the differencebetween the 35-day line and 17-day line remains fairly constant astime passes to expiration.The graph of position theta in Figure 10-5 is nearly the mirror imageof the gamma and vega graphs because the sign of a position’s theta is
326 • Trading Options As a Professional
100806040Vega
200–20–40–60–80–10060 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90Underlying
Theta
Figure 10-4
Vega of 20 Long 70-75 Call Vertical Spreads
6005004003002001000–100–200–300–400–500–60060 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90Underlying
Figure 10-5
Theta of 20 Long 70-75 Call Vertical Spreads
opposite that of a position’s gamma and vega. Thetas are negative numbers with the largest absolute values when options are at the money.Consequently, when the stock price equals the strike price of the long
Managing Position Risk • 327
call in a bull call spread, the position theta is negative, which indicatesthat the position will lose money as time passes. When the stock priceequals the strike price of a short call, however, the position theta is positive, which indicates that the position will make money as time passes.Graphs of position Greeks such as Figures 10-1 through 10-5 are valuable tools in estimating how changing market conditions will change therisks of positions. Traders should familiarize themselves with thisfeature of Op-Eval Pro and use it regularly to analyze position risks. Keeping on top of the Greeks is key to good risk management. If you donot know how your risks have changed, you cannot react to changingmarket conditions within your preestablished limits.
Greeks Of Delta-Neutral PositionsTable 10-6 and Figure 10-6 take the position of 20 long call verticalspreads analyzed in Table 10-3 and make it delta-neutral by adding546 short shares. Three important observations can be made aboutthis new position and its risk characteristics.
Table 10-6
The Greeks of a Delta-Neutral Position
20 long call spreads delta-neutral.Position: Short 546 shares @ 70.00.Long 20 70 Calls @ 2.82.Short 20 75 Calls @ 1.02.Col 1
Col 2
Row
RiskFactor
Short409 Shares
12345
PriceDeltaGammaVegaTheta
70.00(1  546)000
Col 3

Col 4
Col 5
Spread  MultiplierTotal  Quantity
Position
($1.80  100 20)(0.273  100 20)(0.010  100 20)(0.021  100 20)(0.085  100 20)
 $34,62002042170
Assumptions: Stock price, 70; days to expiration, 35; volatility, 31%; interest rate, 4%, dividends, none;7-day theta.
Profit And Loss
328 • Trading Options As a Professional
40003500300025002000150010005000–500–1000–1500–2000–2500–3000–3500–4000–450060
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
Underlying
Figure 10-6
Value of Delta-Neutral 20 Long 70-75 Call Vertical Spreads
First, the indicated “Price” of the position is –$34,620 (column 7of Table 10-6). This ﬁgure represents the credit from the stock shortsale less the net debit paid for the options. This ﬁgure, however, isnot an accurate measure of the maximum risk of the position becausethe short shares have unlimited risk. A trader therefore must estimatepotential risk by considering other factors such as pending newsand the possibility of a sharp stock-price rise. Such considerationsare subjective.The second observation is about the beneﬁts and risks of beingdelta-neutral. Table 1