Table 9-4, for example, column 1 is blank because1.75 in column 2 and 1.80 in column 3 are the initial bid and askprices. Column 4 contains the trade description, “Sell 1 at 1.80,” andcolumn 5 contains the average sale price of 1.80. Since only one contract has been sold at 1.80 at this point, this is the average price. Column 6 indicates that up to this point there is one short contract in thetotal position.In the second row of Table 9-4, there is a 1 in column 1 becausethis is the ﬁrst time that the bid and ask prices are raised. The bid priceis now 1.76 (column 2), and the ask price is now 1.81 (column 3). Thetrade description in column 4 is, “Sell 1 at 1.81.” Therefore, the average sale price in column 5 is 1.805 because this is the average of 1.80and 1.81, the sale prices of the contracts sold in the ﬁrst and secondrows, respectively. Column 6 indicates that there are now a total oftwo short contracts in the position.The process is repeated in the next seven rows, in which the bidand ask prices are raised by 1 cent after another contract is sold.The last row of Table 9-4 is the ninth time that the bid and askprices have been raised (column 1), and as indicated in column 2, thebid price of 1.84 equals the average sale price in column 5 of the previous row. The trade in the last row (column 4) is buying nine at thebid price of 1.84. Nine is the total number of short contracts in theposition, as indicated in the previous row in column 6, and buyingthem at 1.84 results in zero net proﬁt or loss for all contracts, notincluding transaction costs.The conclusion drawn from the exercise in Table 9-4 is that in thisexample, a market maker can raise the bid and ask prices nine timesand still break even if all the contracts are purchased and closed onthe ninth, or last, time. There are two constant factors in this example. First, the bid-ask spread is 5 cents, and second, the bid and askprices increase by 1 cent after each sale. Although these factors could
290 • Trading Options As a Professional
be different in real trading situations, a general conclusion can bedrawn from this speciﬁc example.
Drawing a General ConclusionThe concept presented in Table 9-4 can be applied to other tradingsituations. Generally, the number of times that a bid-ask spread canbe adjusted equals two times the bid-ask spread minus one divided bythe increment of change. The increment of change is the amount thatbid and ask prices are raised or lowered in each adjustment.In Table 9-4, the bid-ask spread is 5 cents, and the increment ofchange is 1 cent. Therefore, two times the bid-ask spread minus oneequals nine (2  5  1  9). This number divided by the change isalso nine (9  1  9). This simple formula tells a market maker howmany times a delta-neutral position can be increased before there isa risk of incurring a loss.The exercises in Tables 9-1 through 9-4 omit the real-world factorsof changing stock prices and changing implied volatility. When thestock price is static, it is easy to see that an option price of 1.81 is higherthan a price of 1.80. It gets more difﬁcult, however, when the stockprice changes. When the stock price is 53.75, is a 55 Call price of 2.20relatively higher, lower, or the same as a price of 2.80 when the stockprice was 54.60? How can a similar judgment be made when stockprices ﬂuctuate? Market makers need is a simple method of evaluatingoption prices as market conditions change. The method requires twoskills that are shown in Tables 9-5 and 9-6 and are discussed next.
Estimating Option Prices as Volatility ChangesTable 9-5 demonstrates the ﬁrst skill, which is the ability to quicklyestimate a new option price when the implied volatility changes. Theformula that accomplishes this estimation is stated at the top of thetable. It starts with an initial theoretical value of the option, the volatility assumption of which is known. The option’s vega or a fraction
Setting Bid-Ask Prices • 291
Table 9-5
From Option Price to Implied Volatility
Theoretical option value at known implied volatility + fraction of vega theoretical option value at new implied volatility
Theoretical valueVolatilityVega
Stock Price  81.5080 Call85 Call
90 Call
4.0030%0.10
0.6530%0.06
1.7530%0.08
Change Volatility AssumptionNew volatilityEstimated price80 Call:
85 Call:
90 Call:
31%31.5%32%4.101.870.77Theor value at 30% imp. vol.plus 1.0  vega (0.10) 80 Call value at 31% imp. vol.Theor. value at 30% imp. vol.plus 1.5  vega (0.08) 85 Call value at 31.5% imp. vol.Theor. value at 30% imp. vol.plus 2.0  vega (0.06) 90 Call value at 32% imp. vol.
4.000.104.101.750.121.870.650.120.77
thereof is then added to or subtracted from the initial theoretical value.The result is a new theoretical value with a new volatility assumption.The vega, remember, is the change in option theoretical value for aone percentage point change in volatility. For a refresher on vega, referto Chapter 4.After the formula, Table 9-5 lists the theoretical values of threecalls, their vegas, and the volatility assumptions. The 80 Call, the 85Call, and the 90 Call have theoretical values of 4.00, 1.75, and 0.65,respectively. The volatility assumption is 30 percent, and the stockprice is 81.50.The row below the vega lists a new level of volatility. For the 80Call, the new level of volatility is 31 percent. For the 85 Call, it is 31.5percent, and for the 90 call, it is 32 percent. The next row containsthe estimated new option prices under the new volatility assumptions.
292 • Trading Options As a Professional
The 80 Call, for example, increases from 4.00 to 4.10. The 85 Callincreases from 1.75 to 1.87, and the 90 Call increases to 0.77.The bottom section of Table 9-5 shows that the estimated prices arecalculated in two steps. First, the percentage change in volatility isrelated to the vega. If volatility changes by 1 percent, for example, theoption theoretical value changes by one vega. This is what happenedto the 80 Call. The increase in volatility of one percentage point from30 to 31 percent caused the option theoretical value to rise by onevega from 4.00 to 4.10.The increase in volatility for the 85 Call, however, is one and ahalf percentage points. Its value therefore rises by one and a halfvegas. The vega of the 85 Call is 0.08, so the increase in volatilityfrom 30 to 31.5 percent causes the theoretical value to increase by0.12 (0.08 vega  1.5) from 1.75 to 1.87. Finally, the two percentagepoint increase in volatility for the 90 Call causes its theoretical valueto rise by 0.12 from 0.65 to 0.77 (twice its vega of 0.06).
Expressing Bid and Ask Prices in Volatility TermsThe second skill needed to quickly evaluate option prices as stockprices change is the ability to state the market for an option in volatility terms, which is shown in Table 9-6.Table 9-6
The Market in Volatility Terms
80 CallTheoretical valueVolatilityVega
4.0030%0.10
Stock Price  81.5085 Call1.7530%0.08
90 Call0.6530%0.06
Market QuotesBid-askMarket stated involatility terms:
3.90–4.10
1.75–1.83
0.68–0.77
29–31%
30–31%
30.5–32%
Setting Bid-Ask Prices • 293
Table 9-6 starts with the same three calls and their theoreticalvalues and vegas from Table 9-5. The next line in Table 9-6 statesbid and ask prices for each of the calls. For the 80 Call, for example, the bid price is 3.90, and the ask price is 4.10. For the 85 and90 Calls, the bid-ask prices are 1.75 and 1.83 and 0.68 and 0.77,respectively.The last row in Table 9-6 states the bid and ask prices in volatilityterms. For the 80 Call, this is 29 percent bid and 31 percent ask. Thesepercentages are calculated the same way that prices with a new volatility assumption were calculated in Table 9-5. The vega or a fraction ofit is added to or subtracted from the initial theoretical value with theknown volatility.Consider the bid and ask prices for the 80 Call. Given the theoretical value of 4.00, the volatility of 30 percent, and the vega of 0.10,a price of 3.90 is one vega less than 4.00, which is an implied volatility level of 29 percent. Similarly, a price of 4.10 for the 80 Call is onevega greater than a price of 4.00, so its implied volatility is 31 percent. Consequently, bid-ask prices of 3.90 and 4.10, respectively, forthe 80 Call can be stated in volatility terms as 29 percent bid and31 percent ask.The 85 Call has a bid price of 1.75 and an ask price of 1.83. Thetop of Table 9-6 indicates that 1.75 is the theoretical value, assuming30 percent volatility. Adding the vega of 0.08 to this price yields a newprice of 1.83, which is 1 percent higher in volatility terms, or 31 percent. The bid and ask for the 85 Call therefore can be stated involatility terms as 30 percent bid and 31 percent ask.Finally, assuming 30 percent volatility for the 90 Call, its theoretical value of 0.65 and its vega of 0.06 mean that its bid and ask pricesof 0.68 and 0.77 can be stated in volatility terms as 30.5 percent bidand 32 percent ask. The price of 0.68 is one-half a vega greater than0.65, and 0.77 is two vegas greater.Using vega to estimate new option prices if volatility changes andstating bid and ask prices in volatility terms are essential skills for professional traders to master because trading decisions often must be
294 • Trading Options As a Professional
made quickly. The following exercises in this chapter and those in thenext chapter illustrate that these skills are also valuable in creating andclosing positions, in managing positions, and in managing risk.
Trading Exercises IntroducedThe four exercises that follow demonstrate three trading techniquesthat market makers use. First, they trade delta-neutral to avoid therisk of market direction. Second they use implied volatility to setand adjust bid and ask prices. Third, market makers can be indifferent about which options they buy or sell because buying on thebid and selling at the ask can lead to proﬁtable conversions, reverseconversions, butterﬂy spreads, and box spreads. The following exercises give only a glimpse of the many trades this technique makespossible.All four trading exercises use the theoretical values, deltas, andvegas in Table 9-7. The stock price range is from 83.60 to 85.00, thevolatility assumption is 32 percent, and the days to expiration, interest rates, and dividends are as stated at the bottom of the table.Each exercise has its own assumptions about the width of the bidask spread and about the number of bid-ask price adjustments, if any.These variations are consistent with the real world in that differentoptions markets have different characteristics, one of which is thewidth of bid-ask spreads. Such differences might be the result of stockprice volatility, of volume of trading in the underlying stock or in theoptions themselves, or of a speciﬁc company event, such as a pendingearnings announcement.Each exercise uses three tables that explain the activities of hypothetical trader, Ross. The ﬁrst table, labeled “Instructions,” is anoverview of the steps of the exercise. In the ﬁrst step, Ross makes a market in one or more options, which involves stating bid and ask pricesgiven levels of volatility. In the second step, Ross makes a trade at oneof those prices. Subsequently, in the third step, the stock price changes,
Setting Bid-Ask Prices • 295
Table 9-7
Theoretical Values, Deltas and Vegas Calls and Puts: 80 Strike,85 Strike, 90 Strike
StockPrice
80 CallT.V. 32%
80 PutT.V. 32%
85 CallT.V. 32%
85 PutT.V. 32%
90 CallT.V. 32%
90 PutT.V. 32%
85.00DeltaVega84.80DeltaVega84.60DeltaVega84.40DeltaVega84.20DeltaVega84.00DeltaVega83.80DeltaVega83.60DeltaVega
7.420.720.087.280.720.087.120.700.087.000.700.086.860.700.086.720.700.096.580.680.096.440.680.09
1.940.280.082.000.280.082.060.300.082.100.300.082.180.300.082.240.300.092.300.320.092.360.320.09
4.500.540.104.400.540.104.280.520.104.180.520.104.080.500.103.980.500.103.880.500.103.780.490.10
3.980.460.104.080.460.104.160.480.104.260.480.104.360.500.104.460.500.104.560.500.104.660.510.10
2.500.380.092.420.360.082.340.340.082.280.340.082.200.340.082.120.320.082.080.320.082.020.300.08
6.960.620.097.080.640.087.200.660.087.320.660.087.460.660.087.600.680.087.740.680.087.880.700.08
Days to Expiration, 56; Interest Rates, 4%; Dividends, none
and Ross establishes new bid and ask prices and makes more trades.He closes the position in the fourth step. The second table contains astep-by-step explanation of how Ross implements the instructions inthe ﬁrst table, and the third table summarizes the exercise. Proﬁt andloss are calculated by comparing the price at which a position is established to its theoretical value. A conclusion is stated at the end of thethird table that summarizes the essential point of the exercise.
296 • Trading Options As a Professional
Exercise 1: Buying Calls Delta-NeutralTable 9-8A presents an overview of the two trades in this example. Rossis instructed ﬁrst to set bid and ask prices for the 85 Call at stated levels of volatility and second to make an opening delta-neutral trade.The third instruction is to adjust the bid and ask prices, and the fourthis to close out the whole position.Steps 1 through 4 in Table 9-8B detail how Ross follows each instruction. In step 1, he sets bid and ask prices for the 85 Call at volatilitylevels of 32.0 and 33.0 percent, respectively, with the stock price 84.60.Given a theoretical value of 4.28, and assuming 32.0 percent volatilityand a vega of 0.10, Ross sets the bid price at 4.28 (32.0 percent) and theask price at 4.38 (33.0 percent). Note that the ask price is 0.10 or onevega greater than the bid price.Table 9-8AStep 1Step 2Step 3Step 4
Buying Calls Delta-Neutral: Instructions
Stock price 84.60. Make a market for the 85 Call at volatility levels of32.0% bid and 32.5% ask.Buy 10 85 Calls on the bid delta-neutral.Stock price 83.80. Make a market for the 85 Call at volatility levels of31.8% bid and 32.3% ask.Sell (to close) the 10 85 Calls at the ask, and close the stock position.
Table 9-8B
Buying Calls Delta-Neutral: Step-by-Step Explanation of Trades
Step 1:
85 CallPriceImplied vol.
Bid4.2832.0%
Step 2:
Buy 10 85 CallsShort 520 shares
4.2884.60
Step 3:
85 CallPriceImplied vol.
Bid3.8631.8%
Step 4:
Sell 10 85 CallsBuy 520 shares
3.9683.80
Ask4.3833.0%
Stock price  84.6085 Call  4.28 (32.0%)Delta  0.52; vega  0.10(Implied vol.  32.0%)
Ask3.9632.8%
Stock price  83.8085 Call  3.88 (32.0%)Delta  0.50; vega  0.10(Implied vol.  32.8%)
Setting Bid-Ask Prices • 297
In step 2, Ross buys 10 of the 85 Calls on the bid and sells stockshort to hedge the options delta-neutral. Since the 85 Call has a deltaof 0.52 with the stock at 84.60, buying 10 of these calls requires thatRoss sell 520 shares short.Step 3 reﬂects how Ross adjusts the bid and ask prices to volatility levels of 31.8 percent bid and 32.8 percent ask. Given the newstock price of 83.80, a theoretical value of 3.88, and a vega of 0.10,the new market for the 85 Call is 3.86 bid (31.8 percent) and 3.96ask (32.8 percent). The volatility is adjusted down for two reasons.First, if another sell order comes into the market that Ross mighthave to buy, then a lower bid price makes it possible for him toscale into a bigger position of 85 Calls at a lower average levelof volatility. Second, the lower volatility hopefully will entice buyers into the market. The speciﬁc adjustment of two-tenths of apercent volatility is the result of Ross’s personal judgment. Eachtrader makes such a decision individually based on knowledge andexperience.In step 4, Ross closes the position by selling the 10 calls at the askprice of 3.96 and purchases, or covers, the short shares at 83.80.Exercise 1 concludes with the proﬁt-and-loss calculations presentedin Table 9-8C. The 10 calls Ross purchased at 4.28 each and sold at3.96 each resulted in a loss of $320 [(4.28  3.96)  $100], notincluding commissions. The 520 shares he sold short at 84.60 and covered (bought) at 83.80 resulted in a proﬁt of $416 [($84.60  $83.80) 520], not including commissions. The net result, therefore, was aproﬁt of $96 before costs.As in the exercises in Tables 9-1 and 9-2, the conclusion from thisexercise is that buying on the bid, selling at the ask, andtrading delta-neutral can earn proﬁts. There are two differences,however, between this exercise and the one in Tables 9-1 and9-2. First, in this exercise, the stock price changed between thetwo trades. Second, the bid and ask prices were expressed involatility terms.
298 • Trading Options As a Professional
Table 9-8C
Buying Calls Delta-Neutral: Calculation of Proﬁt or Loss
Sell price of call  $100Minus buy price of call  $100P/(L) per call number of calls P/(L) from callsSell price of stock per shareMinus buy price of stock per shareP/(L) per share number of shares P/(L) from stockNet proﬁt/loss
$ 396428$ (32) 10($320)$84.6083.80$ 0.80 520$416$ 96
Conclusion: Buying options on the bid, selling at the ask, and trading deltaneutral makes it possible to trade proﬁtably regardless of the direction of stockprice change. Some complicating factors not discussed in this example are timedecay and changing volatility.
Exercise 2: Creating a Butterﬂy Spreadin Three TradesTables 9-9A, 9-9B, and 9-9C show how Ross creates a long call butterﬂy spread below its theoretical value with three trades of buyingoptions on the bid and selling at the ask and trading delta-neutral.A long call butterﬂy spread is a three-part strategy involving one longcall at the lowest strike price, two short calls at the middle strikeprice, and one long call at the highest strike price. The strike pricesare equidistant, and the calls have the same underlying and sameexpiration date. Figure 1-11 is a graph of a long call butterﬂy spread.Table 9-9A gives an overview of Ross’s three trades in this exercise.Pursuant to the ﬁrst instruction, Ross sets bid and ask prices for the 85Call at stated levels of implied volatility. He then makes an openingdelta-neutral trade. The third instruction tells Ross to adjust theimplied volatility levels and to set bid and ask prices for the 80 Call,and the fourth step requires him to make an opening delta-neutraltrade with the 80 Calls. The ﬁfth and six instructions for Ross are toadjust the implied volatility levels a second time, to set bid and ask
Setting Bid-Ask Prices • 299
Table 9-9A
Creating a Butterﬂy Spread in Three Trades: Instructions
Step 1
Stock price 84.00. Make a market for the 85 Call at volatility levelsof 32.0% bid and 32.5% ask.Sell 50 85 Calls at the ask delta-neutral.Stock price 84.60. Make a market for the 80 Call at volatility levelsof 32.2% bid and 32.7% ask.Buy 25 80 Calls on the bid delta-neutral.Stock price 83.60. Make a market for the 90 Call at volatility levelsof 32.0% bid and 32.5% ask.Buy 25 90 Calls on the bid delta-neutral.
Step 2Step 3Step 4Step 5Step 6
prices for the 90 Call, and to make an opening delta-neutral trade withthe 90 Calls.Steps 1 through 6 in Table 9-9B show in detail how Ross completeseach instruction. Step 1 tells Ross to set bid and ask prices for the 85Call at implied volatility levels of 32.0 and 32.5 percent, respectively,Table 9-9B
Creating a Butterﬂy Spread in Three Trades: Step-by-StepExplanation of Trades
Step 1:
85 CallPriceImplied vol.
Bid3.9832.0%
Step 2:
Sell 50 85 CallsBuy 2,500 shares
4.0384.00
Step 3:
80 CallPriceImplied vol.
Bid7.1432.25%
Step 4:
Buy 25 80 CallsSell 1,750 shares
7.1484.60
Step 5:
90 CallPriceImplied vol.
Bid2.0232.0%
Step 6:
Buy 25 90 CallsSell 750 shares
2.0283.60
Ask4.0332.5%
Stock price  84.0085 Call  3.98 (32.0%)Delta  0.50; vega  0.10(Implied vol.  32.5%)
Ask7.1832.75%
Stock price  84.6080 Call  7.12 (32.0%)Delta  0.70; vega  0.08(Implied vol.  32.25%)
Ask2.0632.5%
Stock price  83.6090 Call  2.02 (32%)Delta  0.30; vega  0.08(Implied vol.  32.0%)
300 • Trading Options As a Professional
with the stock price 84.00. Note that the bid-ask spread, in volatilityterms, is narrower in this exercise than in the preceding one. This variation is consistent with different markets in the real world having different characteristics. Given the 85 Call’s theoretical value of 3.98,assuming 32.0 percent volatility and a vega of 0.10, Ross sets the bidprice at 3.98 (32.0 percent) and the ask price at 4.03 (32.5 percent).In step 2, Ross sells 50 of the 85 Calls at the ask price of 4.03 andbuys stock to hedge the options delta-neutral. Since the 85 Call has adelta of 0.50 with the stock at 84.00, selling 50 of these calls requiresthat Ross buy 2,500 shares.In step 3, Ross sets bid and ask prices for the 80 Call after adjustingvolatility to levels of 32.25 and 32.75 percent, respectively. He adjusts thevolatility up because the previous trade was a sale of options. The higherask price—in volatility terms—will make it possible for Ross to scale intoa larger position of short calls if another buy order comes into the market.Also, he hopes the higher level of volatility will entice sellers into the market. Again, the speciﬁc volatility adjustment of up one-quarter of1 percent is a result of Ross’s individual judgment.Given the new stock price of 84.60, a theoretical value of 7.12, andassuming 32.0 percent volatility and a vega of 0.08, the market for the80 Call is 7.14 bid (32.25 percent) and 7.18 ask (32.75 percent).In step 4, Ross buys 25 of the 80 Calls at the bid price of 7.14 andsells stock to hedge the options delta-neutral. Since the 80 Call hasa delta of 0.70 with the stock at 84.60, buying 25 of these callsrequires that Ross sell 1,750 shares. However, because Ross owned2,500 shares as a result of his trade in step 2, this trade reduces hislong stock position to 750 shares.Step 5 involves adjusting the bid and ask prices again, this time backdown to volatility levels of 32.0 and 32.5 percent, respectively, andthen setting bid and ask prices for the 90 Call. Ross adjusts the impliedvolatility down this time because the previous trade was a purchase ofoptions. Given the new stock price of 83.60, a theoretical value of2.02, assuming 32.0 percent volatility and a vega of 0.08, the marketfor the 90 Call is 2.02 bid (32.0 percent) and 2.06 ask (32.5 percent).
Setting Bid-Ask Prices • 301
In step 6, Ross buys 25 of the 90 Calls on the bid price of 2.02 andsells stock to hedge the options delta-neutral. Since the 90 Call has adelta of 0.30 with the stock at 83.60, buying 25 of these calls requiresRoss to sell 750 shares, which closes his stock position. His only positions left are long 25 of the 80 Calls, short 50 of the 85 Calls, and long25 of the 90 Calls. As described earlier and in Chapter 1, this threepart position is 25 long call butterﬂy spreads. The question now is,“Were these butterﬂy spreads established at a good price?” Table 9-9Canswers this question.Table 9-9C
Creating a Butterﬂ