y Spread in Three Trades: Calculation of Costand Theoretical Value
Section 1: Option TradingCol 1
Col 2
Col 3
Qty
Price
Col 4Debit/Credit
121
80 Call7.1485 Call4.0390 Call2.02SpreadCosts†0.04Gross cost per spread
(7.14)8.06(2.02)(1.10)(0.04)(1.14)
Col 5Theor.Price*
Col 6Debit/Credit
6.723.982.12
(6.72)7.96(2.12)(0.88)(0.04)(0.92)
T.V. per spread
Section 2: Stock Trading1,750  (84.60  84.00)750  (83.60  84.00)Net stock proﬁtStock proﬁt per spreadStock proﬁt per share per spread
 1,050  300  750  30 (750 proﬁt  25 spreads)  0.30
Section 3: Net Cost per Butterﬂy SpreadGross cost per spread  stock proﬁt per spread  0.84 (1.14  0.30)Conclusion: With a bid-ask spread of 0.5% volatility, the butterﬂy spread in thisexample is purchased for 84 cents, or 8 cents below theoretical value.* Theoretical prices assume a stock price of 84.00. Note that the theoretical value of the butterﬂyspread is between 0.88 and 0.92 when the stock price is between 83.60 and 85.00, the stock price rangein Table 9-7.† Costs are 1 cent per share or $1 per option.
302 • Trading Options As a Professional
Columns 1 through 4 of section 1 in Table 9-9C show that thegross cost of each butterﬂy spread is 1.14, including 4 cents for transaction costs. Ross purchased each 80 Call for 7.14. He sold twiceas many 85 Calls for 4.03 each and purchased each 90 Call for 2.02.Columns 5 and 6 conclude that the theoretical price of this butterﬂy spread is 92 cents, including 4 cents for transactions costs.Section 2 of Table 9-9C calculates that Ross’s three stock tradesresulted in a net proﬁt of $750, or $30 per spread, or 30 cents per shareper spread. In the ﬁrst stock trade, from step 2 of Table 9-9B, Ross purchased 2,500 shares at 84.00. Subsequently, in steps 4 and 6, he sold1,750 and 750 shares at 84.60 and 83.60, respectively. The net stocktrading proﬁt of $750 is divided by 25 (the number of Ross’s long callbutterﬂy spreads) to get $30 per spread. This per-spread proﬁt isdivided by $100 to get 30 cents per share per spread.Section 3 of Table 9-9C shows that the net cost per butterﬂy spreadis 84, which is 8 cents below the theoretical value of 92 cents. The netcost of 84 cents is the difference between the gross cost of 1.14 andthe stock proﬁt of 30 cents per share. The conclusion, therefore, is thatwith a bid-ask spread of one-half percent volatility, a long butterﬂyspread can be established for 8 cents below its theoretical value. Thespeciﬁcs of this particular example will not apply to all situations inthe real world, of course, but the concept is valid. Buying on the bidprice and selling at the ask price and staying delta-neutral make itpossible to establish positions at advantageous prices.
Exercise 3: Creating a Reverse Conversionin Two TradesTables 9-10A, 9-10B, and 9-10C show how two trades of buyingoptions on the bid, selling at the ask, and trading delta-neutral can create a proﬁtable reverse-conversion position. Reverse conversions arediscussed in detail in Chapter 6. You may want to review that chapterbefore reading this section.
Setting Bid-Ask Prices • 303
As in the preceding two exercises, Table 9-10A contains a list ofinstructions to hypothetical trader Ross and an overview of the eventsthat set bid and ask prices, make trades, and create a position. Thenext two tables show how Ross follows the instructions and how toevaluate his position.Steps 1 through 4 in Table 9-10B detail how Ross’s actions carry outthe instructions. In step 1, Ross sets bid and ask prices for the 80 Callat implied volatility levels of 32.0 and 33.0 percent, respectively, withthe stock price at 84.60. Given a theoretical value of 7.12 and assuming 32.0 percent volatility and a vega of 0.08, the bid price is 7.12 (32.0percent), and the ask price is 7.20 (33.0 percent).
Table 9-10AStep 1Step 2Step 3Step 4
Creating a Reverse Conversion in Two Trades: Instructions
Stock price 84.60. Make a market for the 80 Call at volatility levelsof 32.0% bid and 33.0% ask.Buy 10 80 Calls on the bid delta-neutral.Stock price 84.00. Make a market for the 80 Put at volatility levelsof 32.0% bid and 33.0% ask.Sell 10 80 Puts at the ask delta-neutral.
Table 9-10B
Creating a Reverse Conversion in Two Trades: Step-by-StepExplanation of Trades
Step 1:
80 CallPriceImplied vol.
Bid7.1232.0%
Step 2:
Buy 10 80 CallsSell short 700 shares
7.1284.60
Step 3:
80 PutPriceImplied vol.
Bid2.2432.0%
Step 4:
Sell 10 80 PutsSell short 300 shares
2.3384.00
Ask7.2033.0%
Stock price  84.6080 Call  7.12 (32.0%)Delta  0.70; vega  0.08(Implied vol.  32.0%)
Ask2.3333.0%
Stock price  84.0080 Put  2.24 (32.0%)Delta  0.30; vega  0.09(Implied vol.  33.0%)
304 • Trading Options As a Professional
In step 2, Ross buys 10 of the 80 Calls on the bid price of 7.12 andsells stock short to hedge the options delta-neutral. Since the 80 Callhas a delta of 0.70 with the stock at 84.60, buying 10 of these callsrequires Ross to sell 700 shares short.In step 3, Ross sets bid and ask prices for the 80 Put after the stockprice has declined to 84.00. The level of volatility is not adjusted in thisstep because the purchase of 10 Calls was not large enough, in theRoss’s opinion, to warrant a change. Given the new stock price of 84.00,the put’s theoretical value of 2.24, assuming 32.0 percent volatility anda vega of 0.09, the market for the 80 Put is 2.24 bid (32.0 percent) and2.33 ask (33.0 percent).In step 4, Ross sells 10 of the 80 Puts at the ask price and sells stockshort to hedge delta-neutral. With the stock at 84.00, the 80 Put hasa delta of 0.30. Selling 10 of these puts therefore requires that Rosssell 300 shares short. Since Ross shorted 700 shares previously, thistrade increases his stock position to short 1,000 shares.After step 4 in Table 9-10B, Ross’s total position consists of threeparts: long 10 of the 80 Calls, short 10 of the 80 Puts, and short 1,000shares of stock. This position is a reverse conversion, as described inChapter 6. The question now is, “Did Ross establish this reverse conversion at a good price?” Table 9-10C answers this question.Section 1 of Table 9-10C lists the option trades, and section 2 showsthat Ross sold 1,000 shares of stock short at an average price of 84.42.Section 3 calculates the net credit required for a reverse conversion ata strike price of 80, with an interest rate of 4 percent, 60 days to expiration, and total costs of 4 cents per share. As stated in Chapter 6, thenet credit required is the amount that makes the reverse-conversionposition proﬁtable. Step 1 calculates the DPV of the strike price,which is 79.51 in this example, and step 2 adds the costs of 4 cents pershare and the target proﬁt of 5 cents per share. The net credit requiredtherefore is 79.60 per share.Table 9-10C concludes with section 4, which calculates the actualnet credit per share by the trades in this exercise. The stock sold shortbrought in 84.42 per share. The purchased 80 Calls each cost 7.14,
Setting Bid-Ask Prices • 305
Table 9-10C
Creating a Reverse Conversion in Two Trades Calculation of Costand Theoretical Value
Section 1: Option TradingQty
Option
Price
1010
80 Call80 Puts
7.142.33
Section 2: Stock TradingPrice

WeightedPrice
70084.6030084.00Average weighted price

59.2225.2084.42
Qty

Section 3: Calculating the Net Credit Required (NC)Step 1: Calculate discounted present value (DPV) of strike 80  (1  0.04  56/365)  79.51Step 2: Calculate the net credit DPV of strike  costs  proﬁt  NC 79.51  0.04  0.05  79.60 NCSection 4: Calculation of Actual Net Credit (per Share)Stock sold short80 Calls purchased80 Puts soldNet
84.42(7.14)2.3379.61
CreditDebitCreditCredit (79.60 NC)
Conclusion: With a bid-ask spread of 1.0% volatility in this example, a proﬁtablereverse-conversion position can be established.
and the sold 80 Puts each brought in 2.33. The net amount broughtin, or net credit, therefore is 79.61, an amount higher than the netcredit required to achieve the target proﬁt of 79.60.The conclusion stated at the bottom of Table 9-10C is that with abid-ask spread of 1 percent in volatility terms, a trader may establish aproﬁtable reverse conversion position, which is just what Ross accomplished. As stated earlier, the speciﬁcs of this example will not apply toall situations in the real world, but the concept is valid.
306 • Trading Options As a Professional
Exercise 4: Creating a Long Box Spreadin Two TradesTables 9-11A, 9-11B, and 9-11C show how buying a call spreaddelta-neutral and then buying a put spread delta-neutral can create aproﬁtable long box spread. Box spreads are discussed in detail inChapter 6.As in the preceding exercises, Table 9-11A contains a list of instructions for Ross, the hypothetical trader in this example. The instructionsalso give an overview of the events that set bid and ask prices, maketrades, and create a position. The next two tables show how Ross follows the instructions and how to evaluate the position.Steps 1 through 4 in Table 9-11B explain how Ross implementseach instruction. Step 1 in Table 9-11B sets bid and ask prices for the85 and 90 Calls at implied volatility levels of 32.0 and 33.5 percent,respectively, with the stock price at 84.80. Note that this is the widestbid-ask spread in volatility terms so far. This wide spread is consistentwith the real world in that different options markets have bid-askspreads of varying widths. As stated earlier, such differences might beattributed to stock-price volatility, stock or option volume, or a specialcompany-related event.Following the instruction in step 1, Ross sets the bid and ask pricesfor the 85 and 90 Calls at 4.40 and 4.55 and 2.42 and 2.54, respectively.In step 2, Ross buys 10 of the 85–90 call spreads and hedges them deltaneutral. Ross establishes the call spreads by buying the 85 Calls on theTable 9-11AStep 1Step 2Step 3Step 4
Creating a Long Box Spread in Two Trades: Instructions
Stock price 84.80. Make markets for the 85 and 90 Calls atvolatility levels of 32.0% bid and 33.5% ask.Buy 10 85–90 call spreads delta-neutral. (Buy the 85 Calls on thebid; split the bid-ask for the 90 Call.)Stock price 83.80. Make markets for the 85 and 90 Puts atvolatility levels of 32.0% bid and 33.5% ask.Buy 10 90–85 put spreads delta-neutral. (Buy the 90 Puts on thebid; split the bid-ask for the 85 Put.)
Setting Bid-Ask Prices • 307
Table 9-11B
Creating a Long Box Spread in Two Trades Step-by-StepExplanation of Trades
Step 1:85 CallPriceImplied vol.90 CallPriceImplied vol.
Bid4.4032.0%Bid2.4232.0%
Ask4.5533.5%Ask2.5433.5%
Stock price  84.8085 Call  4.40 (32.0%)Delta  0.54; vega  0.10Stock price  84.8090 Call  2.42 (32.0%)Delta  0.36; vega  0.08
1.9284.80
85 Call  4.40; 90 Call  2.48Spread net delta  0.18
Ask4.7133.5%Ask7.8633.5%
Stock price  83.8085 Put  4.56 (32.0%)Delta  0.50; vega  0.10Stock price  83.8090 put  7.74 (32.0%)Delta  0.68; vega  0.08
3.1183.80
90 put  7.74; 85 put  4.63Spread net delta  0.18
Step 2:Buy 10 85–90 call spreadsSell short 180 sharesStep 3:85 PutPriceImplied vol.90 PutPriceImplied vol.
Bid4.5632.0%Bid7.7432.0%
Step 4:Buy 10 90–85 put spreadsBuy 180 shares
bid and selling the 90 Calls at the midpoint of the bid-ask spread. It iscommon practice to trade one-to-one vertical spreads this way for thefollowing reason: Bid-ask spreads for vertical spreads should not bewider than bid-ask spreads for individual options because vertical spreadpositions have less risk than individual option positions. Vertical spreadsalso have lower deltas, lower gammas, lower vegas, and lower thetas(absolute value) than single-option positions. They therefore are lesssensitive to changes in stock price, volatility, and time.Following the instruction in step 2, therefore, Ross buys 10 of the85–90 call spreads at a net debit of 1.92 each because the purchaseprice of the 85 Calls is 4.40 and the selling price of the 90 Calls is
308 • Trading Options As a Professional
2.48. Ross determines the hedging stock trade as follows: The 85 Callshave a delta of 0.54, and the 90 Calls have a delta of 0.36. Thenet delta of the 85–90 call spread therefore is 0.18, so the purchaseof 10 spreads requires that Ross sell 180 shares short at the currentprice of 84.80.In step 3, Ross sets bid and ask prices for the 85 and 90 Puts atvolatility levels of 32.0 and 33.5 percent, respectively, with the stockprice at 83.80. He does not adjust implied volatility in this casebecause vertical spreads have both long and short options, whichmeans that exposure to changing volatility is very low.In step 4, Ross buys 10 of the 90–85 Put spreads by buying the 90Puts on the bid and selling the 85 Puts at the midpoint of the bid-askspread. The purchase price of the 90 Puts therefore is 7.74, and thesale price of the 85 Puts is 4.63, which is approximately halfwaybetween the bid of 4.56 and the ask of 4.71. Ross purchases the 90–85put spread, therefore, for 3.11.Ross hedges the 10 put spreads by buying 180 shares of stock. The90 Puts have a delta of 0.68, and the 85 Puts have a delta of 0.50.The net delta of the spread therefore is 0.18. The purchase of 10spreads requires Ross to purchase 180 shares at the current price of83.80. Since he sold 180 shares short as part of the previous trade,Ross’s purchase in this trade closes his stock position.After step 4, the total position consists of long 10 of the 85 Calls,short 10 of the 90 Calls, long 10 of the 90 Puts, and short 10 of the 85Puts. This position is a long box spread. The question now is, “DidRoss establish this long box spread at a good price?” Table 9-11Canswers this question.Section 1 of Table 9-11C shows the option trades on a per-sharebasis. Ross bought 10 of the 85–90 call spreads for 1.92 each and 10of the 90–85 put spreads for 3.11 each, for a total gross cost of 5.03 foreach long box spread.Section 2 of the table shows that Ross sold 180 shares short at 84.80and covered at 83.80, resulting in a proﬁt of $180, or $18 per spread
Setting Bid-Ask Prices • 309
Table 9-11C
Creating a Long Box Spread in Two Trades Calculation of Costand Theoretical Value
Section 1: Option TradingPrice10 85–90 call spreads10 90–85 put spreadsGross cost of box spread
1.923.115.03
Section 2: Stock TradingSold short 180 sharesPurchased 180 sharesProﬁt per shareProﬁt on 180 sharesProﬁt per share per spreadNet cost per share per spread
84.8083.801.00$1800.184.85
Section 3: Theoretical ValueDPV of spread minus the sum of transaction costs plus target proﬁt5.00  (1  0.05  56/365)  (0.06  0.05)  4.85where borrowing rate  5%transactions costs  0.06target proﬁt  0.05
Conclusion: With a bid-ask spread of 1.5% volatility in this example, a proﬁtablelong box spread can be established.
or 18 cents per share per spread. Subtracting 0.18 from the gross costof 5.03 yields a net cost per share per box spread of 4.85, which equalsthe theoretical value as calculated in section 3.The theoretical value of the 85–90 long box spread is equal to theDPV of the $5.00 difference between the strike prices minus the sumof transaction costs plus target rpoﬁt, and this is 4.85, as shown in section 3 of Table 9-11C. The conclusion, therefore, is that with the bidask spread of 1.5 percent volatility in this example, a trader can establishproﬁtable long box spread by buying a call spread delta-neutral andthen buying a put spread delta-neutral.
310 • Trading Options As a Professional
SummaryMaking markets in options has three parts: buy on the bid, sell at theask, and trade delta-neutral. The goals are to earn the bid-ask spreadand to trade proﬁtably regardless of market direction. Accomplishingthese goals requires three essential skills.The ﬁrst skill is expressing bid and ask prices in volatility terms. Starting with an option’s theoretical value at a known level of volatility, thevega is used to calculate the volatility level of a higher or lower price.The second skill is understanding how to create arbitrage strategiesand low-risk spreads in a few steps. This chapter demonstrated buyingcalls delta-neutral, creating a butterﬂy spread in three delta-neutraltrades, creating a reverse conversion in two delta-neutral trades, andcreating a box spread in two delta-neutral trades. There are many otherlow-risk positions that can be created in just a few steps.The third skill is adjusting bid and ask prices. Deciding when toadjust prices and how much to adjust them involves judgment thatcomes from experience and varies by underlying stock and market conditions. Nevertheless, by adjusting bid and ask prices when risk limitsare reached, traders can scale into and out of positions at better average levels of volatility. There is a limit, however, on the number oftimes that bid and ask prices can be raised or lowered before a positionbecomes unproﬁtable.
Chapter 10MANAGING POSITIONRISKhere are many types of option positions. Outright long and shortoptions, one-to-one spreads, and stock-and-option spreads are perhaps most common for speculators. But there are also ratio spreads,time spreads, butterﬂy spreads, condor spreads, and complex deltaneutral strategies. Each of these positions has unique potential proﬁtsand risks. If a trader can understand the potentials—both good andbad—then the chances of earning proﬁts are increased. Traders therefore must identify and quantify the risks and know alternative strategies for reducing risk. Only then can a trader choose the appropriaterisks to monitor based on individual trading style.Managing risk requires an understanding of how the Greeks change,as discussed in Chapter 4. It is essential to have a thorough understanding of that material before delving into this chapter. This chapterfocuses on the risks associated with delta, gamma, vega, and theta. Therisk of changing interest rates, rho, is not discussed because smallchanges in short-term interest rates do not have a signiﬁcant impact onshort-term option positions.This chapter will ﬁrst illustrate how position risk is calculated. Itthen will demonstrate how delta might be used to manage directionalrisk. Next, a case study analyzes the changing risks of vertical spreads.
T
• 311 •
312 • Trading Options As a Professional
The fourth topic, neutralizing position risk, asks—and answers—thisquestion, “Which Greek is best to neutralize?” The chapter thenconcludes with a discussion of setting risk limits.
Calculating Position RisksQuantifying the delta, gamma, vega, and theta risks of option positions is a straightforward task. Table 10-1 has ﬁve columns and ﬁverows that calculate the Greeks of 20 long 70 Calls that were purchasedfor 2.82 per share each. Assumptions about the current stock price,days to expiration, volatility, interest rate, and dividends are listed atthe bottom of the table.Column 1 in Table 10-1 lists ﬁve risk factors—price and fourGreeks—and column 2 quantiﬁes them on a per-share basis. Since theunderlying is 100 shares for each option, column 3 has the number100 in every row. Similarly, every row in column 4 has the number 20because that is the number of options in the position. Column 5 contains the risk factor of the entire position, which is the product of thethree numbers in columns 2, 3, and 4.The risk factor “Price” appears in column 1 of the ﬁrst row ofTable 10-1, and the per-share price of 2.82 is listed in column 2. The
Table 10-1
Position Risks of 20 Long 70 Calls
Col 1
Col 2
Col 3
Col 4
Col 5
Row
RiskFactor
IndividualOption
 Multiplier
 Quantity

Position
12345
PriceDeltaGammaVegaTheta
$2.820.5350.0590.0870.310
100100100100100
2020202020

$5,6401,070118174620
Assumptions: Stock price, 70.00; strike price, 70; days to expiration, 35; volatility, 31%; interest rates,4%, dividends, none; 7-day theta.
Managing Position Risk • 313
multiplier in column 3 and the number of options in column 4 are100 and 20, respectively. The price risk of the 20-contract positiontherefore is $5,640 (2.82  100  20), as shown in column 5.The position Greeks in column 5, rows 2 through 5, are calculatedin a similar manner to the price of the position, but the risks are stateddifferently depending on the Greek. The position delta of 1,070, forexample, indicates that this position of 20 long 70 Calls will behavelike 1,070 shares of long stock over small stock-price changes. If thestock price rises by $1.00, this position will proﬁt by approximately$1,070, and if the stock price declines by $1.00, this position will loseapproximately this same amount.The position gamma of 118 in row 3 of Table 10-1 indicates thata $1.00 move in stock price will change the position delta by 118shares in the same direction as the change in stock price. If the stockprice rises by $1.00, for example, the position delta will increase by118 from 1,070 to 1,188. Similarly, if the stock price falls by $1.00,the position delta will decrease by 118 from 1,070 to 952.The position vega of 174 in row 4 of Table 10-1 indicates that aone percentage point change in volatility will change the positionvalue by $174. If volatility rises from 31 to 32 percent and other factors remain constant, the price of one option will rise by 8.7 cents(0.087 in column 2), which would raise the value of the 20-optionposition by $174 from $5,640 to $5,814. Similarly, if volatility declinesby 1 percent, the position value would decrease by $174 from $5,640to $5,466.The position theta in Table 10-1 estimates the impact of “one unit”of time decay. In this example, “one unit” is seven days. The positiontheta of –620 indicates that the passing of seven days will cause theposition value to decrease by $620 if other factors remain constant.A trader can use the information in Table 10-1 to ask—and answer—several questions about risk. First, can I withstand a $1,070 loss if thestock price declines $1.00? What about a $2.00 or $3.00 stock-pricedecline? Where will I take my loss if the stock price declines? These arequestions that traders must answer individually.
314 • Trading Options As a Professional
One comment needs to be made on volatility risk. The positionvega in Table 10-1 tells a trader that a one percentage point changein implied volatility will change the position value by $174. The vega,however, does not estimate the likelihood of implied volatility changing or by how much it might change. Historical data, such as that provided at www.cboe.com or at www.ivolatility.com (see Figure 7-7), canassist, but forecasting volatility is an art, not a science.In contrast to vega, the position theta provides a much ﬁrmer estimate of the risk of time decay. In Table 10-1, if the stock price andother factors are unchanged in seven days, this position will lose $620.A trader can use this estimate with a dollar risk limit to determine howlong a position will be held before a loss is taken.
Risks of Short OptionsAlthough the Greeks of long and short options are opposite, the risks ofshort options are not simply opposite the risks of long options. Positionswith uncovered short options have unlimited risk in the case of short callsand substantial risk in the case of short puts. An uncovered short optionis a short option that has no offsetting stock or option position that trulylimits risk. Although, in practice, stock-price changes are never reallyunlimited, they can be very large. As experienced traders know, unexpected events can cause prices to change by 30 percent, 50 percent, ormore overnight or in very short periods of time. The risk of short optionpositions, therefore, must be considered differently than the known maximum price risk of long options. Unfortunately, there is no uniformmethod of determining the suitability of short option risk. Are 50 shortoptions too many? Can I be short