trade is explained in detail, here is an overview. Tom’sﬁrst trade occurs on Monday when he establishes the delta-neutralposition. He subsequently makes adjusting stock trades each day at theend of the day. Finally, he closes the entire position on Friday. Thereare six trades in Table 8-6B, each of which will be explained next.Transaction costs are not included for the sake of simplicity.
Long Volatility Step 1—Opening the PositionTom’s ﬁrst trade in Table 8-6B creates a delta-neutral position. Heestablishes the position some time on Monday in a two-part trade.With a stock price of $90.80, Tom buys 100 of the 90 Calls at 5.60each and simultaneously sells 5,800 shares of stock short. As indicatedin Table 8-6A, the volatility assumption is 30 percent. Tom will usethis information later. Tom calculated how many shares to sell shortusing the process presented in Tables 8-1 through 8-5. He ﬁrst ﬁguredout the share-equivalent position and then traded that number ofshares in a way to bring the net delta of the total position to zero. Inthis example, the delta of a 90 Call is 0.58, and Tom purchased 100Calls, so his option position is equivalent to long 5,800 shares (100options  100 shares/option  0.58  5,800). He therefore sells short5,800 shares at $90.80 to create a delta-neutral position.
Long Volatility Step 2—The Adjusting TradesTom makes a second trade on Monday, but this time at the end of theday just before the market closes. Between the time of Tom’s openingtrade and the end of the trading day, the stock price rises to 92.00.Because of the change in stock price, the delta of the option haschanged. The change in delta is explained by the concept of gamma,which is discussed in Chapter 4.Table 8-6A indicates that with a stock price of 92.00 and 75 days toexpiration (Monday), the delta of the 90 Call is 0.60. Long 100 ofthe 90 Calls with a delta of 0.60 each represents a share-equivalentposition of long 6,000 shares. Thus the two-part position that Tom
Delta-Neutral Trading: Theory and Reality • 251
established in his ﬁrst trade is no longer delta-neutral. To reestablishdelta-neutrality, Tom must sell short 200 more shares of stock. Thistrade is described in the second row of Table 8-6B, “Short 200 shares@ 92.00.” In column 5 the explanation is “adjusting trade to get deltaneutral.” And in column 6, Tom’s new stock position, “6,000,”appears. He now has a position of short 6,000 shares.Tom makes his third trade on Tuesday. The 90 Calls do not tradeon this day, and the stock price closes at 90.20, down 1.80 on the day.Just as in the second trade, the change in stock price changes the deltaof the 90 Call. This time the delta dropped to 0.56, and once again,Tom’s previously delta-neutral position is no longer delta-neutral.Consequently, he must make an adjusting stock trade to bring thedelta back to zero. Now, with a delta of 0.56, the long call positionis equivalent to long 5,600 shares of stock. Since Tom’s stock positionafter the second trade was short 6,000 shares, he must buy, or cover,400 shares. By buying 400 shares Tom reduces his stock position toshort 5,600 shares, or “5,600,” as indicated in column 6. The resulting position, long 100 of the 90 Calls and short 5,600 shares at 90.20,is delta-neutral. Again, the explanation in column 5 is “adjusting tradeto get delta-neutral.”The fourth and ﬁfth trades on Wednesday and Thursday, respectively, are also adjusting trades. For the fourth trade, the stock pricerises to 91.40, making the delta of the 90 Call 0.59. Tom thereforemust sell 300 shares short to increase his stock position to short 5,900shares. When the stock price falls to 89.60 and the delta of the 90 Callis 0.52, the delta-equivalent number of shares becomes 5,200. Tommust purchase 700 shares to make the total position delta-neutral.After the ﬁfth trade, Tom’s stock position is short 5,200 shares.
Long Volatility Step 3—Closing the Remaining PositionTom’s last trade in Table 8-6B, the sixth trade, occurs during the dayon Friday when the stock price reaches 90.80, and Tom makes a twopart trade to close his entire remaining position. After the ﬁfth trade,
252 • Trading Options As a Professional
Tom’s position consisted of long 100 Calls and short 5,200 shares.Therefore, in his closing trade, Tom sells all 100 of the calls at 5.45and buys 5,200 shares at 90.80.There are two noteworthy aspects of Tom’s ﬁnal trade. First, theimplied volatility of the 90 Calls is 30 percent, the same level as whenthe original position was opened on Monday. Second, the delta of theoption was not used in determining the number of shares to tradebecause this trade closed the entire position.Tom’s six trades comprise a complete delta-neutral trade. First, heopened a delta-neutral position. Second, he made adjusting stocktrades on each day, and third, he closed the remaining position. It isnow reasonable to ask, “Did Tom make or lose money?” The proﬁtand-loss calculation is presented next.
Long Volatility Step 4—Calculation of Proﬁt and LossTable 8-6C contains two parts that calculate proﬁt and loss. Part oneaccounts for the option trades, and part two accounts for the stocktrades. Column 1 indicates the day on which a transaction is made.Column 2 states the action—buy, sell, or short—and the quantity ofshares or options traded. Columns 3 and 4 state the purchase priceand sale price, respectively. Column 5 indicates the proﬁt or loss pershare, and column 6 indicates the total dollar proﬁt or loss.In part 1 of Table 8-6C, Tom calculates his loss from the option trades.In column 1, he lists both Monday and Friday because those were thedays when he bought and sold options. Similarly, in column 2, he indicates “buy and sell.” In columns 3 and 4, Tom writes down that the purchase price and sale price were 5.60 and 5.45, respectively. Note thatthese are per-share prices. Column 5 indicates a per-share loss of 0.15,or 15 cents. Tom reaches this the result by subtracting the purchase priceof 5.60 in column 3 from the sale price of 5.45 in column 4 (sale price purchase price  proﬁt or loss). Tom’s total dollar loss of $1,500 incolumn 6 is the product of three numbers: the per-share loss in column5, the number of options in column 2, and the option multiplier of 100($0.15 per share  100 options  100 shares/option  $1,500).
Delta-Neutral Trading: Theory and Reality • 253
Table 8-6C
Col 1
Part 1:Option TradeMon & FriPart 2:Stock TradesRow 1 MonRow 2 MonRow 3 TueRow 4Row 5 WedRow 6 ThuRow 7Row 8 FriRow 9Row 10
Delta-Neutral Trading—Long Volatility: Calculation of Proﬁtand LossCol 2
Col 3
Col 4
Col 5
Col 6
Action &Quantity
PurchasePrice
SalePrice
P/(L)per Share
Total P/(L)in Dollars
Buy/sell 100
5.60
5.45
(0.15)
($1,500)
Short 5,800Short 200Buy 400
——90.20——89.60—90.80
90.80—92.00—92.00 (200) 1.8090.80 (200) 0.6091.40—91.40 (300) 1.8090.80 (400) 1.2090.800.00Proﬁt fromstock tradingCombined P/(L)stock andoptions
——$ 360$ 120—$ 540$ 4800
Short 300Buy 700Buy 5,200
$1,500
-0-
In part 2 of Table 8-6C, Tom calculates the proﬁt or loss from hisstock trades and concludes with the net dollar proﬁt from all thetrades. Row 1 shows Tom’s ﬁrst stock trade, which was part of his initial delta-neutral position. He made this trade on Monday, as indicatedin column 1, by selling short 5,800 shares, as indicated in column 2.Tom reﬂects the short sale price of 90.80 in column 4. Column 3 hasno price because this transaction did not involve a purchase. Tom alsoleaves columns 5 and 6 blank in row 1 because an opening trade willnot involve a proﬁt or loss.Starting with row 2, Tom enters his adjusting stock trades. He madethe ﬁrst before the close on Monday by selling 200 shares short at aprice of 92.00 per share. Like the columns left blank for the openingtrade in row 1, Tom leaves columns 3, 5, and 6 of row 2 blank because
254 • Trading Options As a Professional
there was no purchase transaction in this trade and therefore no proﬁtor loss.Tom lists his ﬁrst closing stock trade in rows 3 and 4 and, as a result,makes his ﬁrst proﬁt/loss calculation. In this example, proﬁt-and-losscalculations are made on a last-in, ﬁrst-out (LIFO) basis. Tom indicates in column 1, row 3 that he made this trade on Tuesday by purchasing 400 shares at 90.20 per share (columns 2 and 3). Column 4,rows 3 and 4 contain the prices and numbers of short shares from rows1 and 2. Since Tom purchased 400 shares in row 3, he must matchthem with shares previously sold short in order to calculate proﬁt orloss. Only 200 shares were sold short in row 2, so Tom must take a second 200-share block from row 1 in order to reach a total of 400 shares.He reﬂects these matches in column 4. Row 3 indicates that 200 sharessold short at 92.00 are matched with the ﬁrst 200 shares purchased at90.20. This 200-share purchase and sale results in 1.80 proﬁt per shareand $360 total proﬁt, as shown in row 3 in columns 5 and 6. Tom calculates proﬁt/loss for the second 200-share block in row 4. Column 4,row 4 indicates a price of 90.80 and a quantity of 200 shares, and column 5 indicates a proﬁt per share of 0.60. Tom concludes in column6 that he made a total dollar proﬁt of $120 on these 200 shares.Tom shows in row 5 his adjusting trade of shorting 300 shares at aprice of 91.40 per share on Wednesday. Since this is another openingshort trade, he leaves columns 3, 5, and 6 blank because he has noinformation about a purchase price, proﬁt per share, or total proﬁt.Rows 6 and 7 show Tom’s proﬁt/loss calculation for the adjustingtrade he made on Thursday. In this trade, he purchased 700 shares(row 6, column 2), at 89.60 per share (column 3). To calculate hisproﬁt or loss, Tom must match this purchase with two different shortsales. In row 6, Tom applies LIFO and matches the 300 shares he soldshort on Wednesday at 91.40 per share with his purchase, resulting ina per-share proﬁt and total proﬁt of 1.80 and $540, respectively.Tom then matches 400 shares (row 7, column 4) that were part of hisinitial trade made on Monday (row 1) at 90.80 per share with his
Delta-Neutral Trading: Theory and Reality • 255
purchase. His per-share proﬁt and total proﬁt from these shares equal1.20 and $480, respectively.Tom accounts for his ﬁnal stock trade in row 8. The 5,200 shareshe purchased on Friday are part of the two-part trade that closed theentire position. He matches these shares with the remaining shortshares from Monday. Since the purchase price of these shares of 90.80(row 8, column 3) matches the sale price on Monday, there is no proﬁtor loss, and “–0–” appears in column 6, row 8.Row 9, column 6 contains the sum of all the proﬁts from Tom’sstock trading, which is $1,500.Tom now completes part 2 of Table 8-6C with row 10, which combines the proﬁt or loss from option trading and from stock trading.Since the option loss of $1,500 exactly offsets the proﬁt of $1,500 fromstock trading, Tom records a “–0–” in column 6, row 10.
Long Volatility Step 5—Recapping the TradesTom established a delta-neutral position by buying calls—at 30 percent implied volatility—and shorting shares of the underlying stock.He made adjusting stock trades daily until he closed the remainingposition by selling calls—at 30 percent implied volatility—and buying shares. His accounting for proﬁt and loss revealed that the ownedoptions incurred a loss from time decay and that the proﬁts from trading stock exactly offset that loss.It may seem like Tom went to a lot of trouble to just break even,but this exercise illustrated the concept of delta-neutral trading. In theory, when buying options delta-neutral, implied volatility stays constant, and the adjusting stock trades offset the time decay of options.The theory, therefore, is that implied volatility equals realized volatility and that delta-neutral trading breaks even.What is different in reality? First, implied volatility can—anddoes—change. Second, realized volatility can—and does—differ fromimplied volatility. New information is constantly hitting the market,and investor psychology changes. Both these factors cause unforeseen
256 • Trading Options As a Professional
changes in the relative level of option prices—implied volatility—andin the ﬂuctuation of stock prices—realized volatility.The next exercise illustrates the theory of delta-neutral trading withshort volatility.
Delta-Neutral Trading—ShortVolatility ExampleThis example of delta-neutral trading by our hypothetical trader,Tom, uses the theoretical values presented in Table 8-7A, the tradespresented in Table 8-7B, and the proﬁt-and-loss calculations presented in Table 8-7C. Table 8-7A contains theoretical values andTable 8-7A
Delta-Neutral Trading—Short Volatility: 35 Call TheoreticalValues (Volatility 40%, Interest Rate 5%, No Dividends,43–37 Days)
Thursday
Friday
Monday
Tuesday
Wednesday
StockPrice
43 DaysT.V./Delta
42 DaysT.V./Delta
39 DaysT.V./Delta
38 DaysT.V./Delta
37 DaysT.V./Delta
35.4035.3035.2035.1035.0034.9034.8034.7034.6034.5034.4034.3034.2034.1034.00
2.24/0.592.18/0.582.12/0.572.07/0.562.01/0.551.96/0.541.90/0.531.85/0.521.80/0.511.75/0.501.70/0.491.65/0.481.60/0.471.56/0.471.51/0.46
2.21/0.592.15/0.572.10/0.572.04/0.551.99/0.541.93/0.541.88/0.531.83/0.521.78/0.511.73/0.501.68/0.491.63/0.481.58/0.471.53/0.471.48/0.46
2.13/0.582.08/0.572.02/0.561.97/0.541.91/0.541.86/0.531.80/0.531.75/0.521.70/0.511.65/0.501.60/0.491.55/0.481.51/0.471.46/0.461.41/0.45
2.11/0.582.05/0.572.00/0.561.94/0.541.89/0.531.83/0.531.78/0.531.73/0.521.68/0.511.63/0.501.58/0.491.53/0.481.48/0.471.43/0.461.39/0.45
2.08/0.582.03/0.571.97/0.551.91/0.541.86/0.531.81/0.531.75/0.521.70/0.511.65/0.511.60/0.501.55/0.491.50/0.481.46/0.471.41/0.461.36/0.45
Delta-Neutral Trading: Theory and Reality • 257
deltas of a 35 Call over ﬁve trading days, from a Thursday to the nextWednesday, in the columns and over a range of stock prices in therows. As in the long volatility exercise, six circles appear in Table8-7A for ease of identifying when trades are made. Also as in the preceding example, the daily stock-price action is created for the purposes of the exercise.
Short Volatility—OverviewThe difference between long volatility and short volatility is simply thedifference between buying and selling options. Short volatility meansthat a position has a negative vega, as deﬁned in Chapter 4. The position created by Tom’s ﬁrst trade in Table 8-7B is short volatility becausethe calls are short.Table 8-7B contains the essential details of all of Tom’s trades inthis example. On Thursday, Tom’s ﬁrst trade consists of selling 100 ofthe 35 Calls delta-neutral; that is, he sells calls and buys shares. As inthe preceding example, transaction costs are not included for the sakeTable 8-7B
Delta-Neutral Trading—Short Volatility: The Trades
Col 1 Col 2
Col 3
Col 4
Col 5
Col 6
StockPrice
OptionDelta
Trade
Explanation
StockPosition
Day
34.80 0.53 Sell 100 35 Calls @ 1.90 Opening tradeBuy 5,300 shares @ 34.80Thu 35.30 0.58 Buy 500 shares @ 35.30Adjusting trade toget delta-neutralFri34.10 0.47 Sell 1,100 shares @ 34.10 Adjusting trade toget delta-neutralMon 35.10 0.54 Buy 700 shares @ 35.10Adjusting trade toget delta-neutralTue 34.20 0.47 Sell 700 shares @ 34.20Adjusting trade toget delta-neutralWed 34.30 0.48 Buy 100 35 Calls @ 1.50 Closing tradeSell 4,700 shares @ 34.30
Thu
5,3005,8004,7005,4004,700-0-
258 • Trading Options As a Professional
of simplicity. After the ﬁrst trade, Tom makes adjusting stock tradeseach day at the end of the day. Finally, he closes the entire positionon Wednesday. Tom’s six trades in Table 8-7B will be explained next.
Short Volatility Step 1—Opening the PositionTom establishes his delta-neutral position in Table 8-7B some time onThursday when the stock price is 34.80. He sells 100 of the 35 Callsat 1.90 each and simultaneously buys 5,300 shares of stock at 34.80.As indicated in Table 8-7A, the volatility assumption is 40 percent.Tom decides on the quantity of shares to purchase, 5,300, by multiplying the option’s delta, 0.53, by the number of options, 100, timesthe multiplier, also 100 (0.53  100  100  5,300).
Short Volatility Step 2—The Adjusting TradesTom makes his second trade also on Thursday, but this time at the endof the day just before the market closes. In this example, between thetime of the opening trade and the end of the trading day, the stockprice rose to 35.30. The change in stock price causes the delta of theoption to increase to 0.58, as indicated in Table 8-7A. Consequently,Tom’s two-part position is no longer delta-neutral. To reestablish deltaneutrality, he must purchase 500 more shares of stock. This trade isdescribed in the second row of Table 8-7B, “Buy 500 shares @ 35.30.”In column 5, the explanation for Tom’s trade is “adjusting trade to getdelta-neutral.” And in column 6, his new stock position, “5,800,”appears. This indicates Tom’s position of long 5,800 shares.Tom’s third trade in Table 8-7B occurs on Friday. The 35 Calls donot trade on this day, and the stock price closes at 34.10. Once again,the delta has changed, this time to 0.47. Consequently, Tom mustmake an adjusting stock trade that brings the delta back to zero. Now,with a delta of 0.47, the short call position is equivalent to short4,700 shares of stock. Since Tom’s stock position after trade two is long5,800 shares, he must sell 1,100 shares. Selling 1,100 shares reducesthe stock position to long 4,700 shares, or “4,700,” as indicated incolumn 6.
Delta-Neutral Trading: Theory and Reality • 259
Tom also makes adjusting trades on Monday and Tuesday, respectively. On Monday, the stock price rises to 35.10, and the delta of the35 Call rises to 0.54. Therefore, Tom must purchase 700 shares toincrease the stock position to long 5,400 shares, or “5,400.” On Tuesday, the stock price falls to 34.20, and the delta of the 35 Call decreasesto 0.47, making the delta-equivalent number of shares long 4,700.To return to a delta-neutral position, Tom must sell the 700 shares hepurchased the day before. Therefore, after the ﬁfth trade, Tom’s stockposition is long 4,700 shares.
Short Volatility Step 3—Closing the Remaining PositionThe ﬁnal trade in Table 8-7B is a two-part trade in which Tom closeshis remaining position. After the ﬁfth trade, Tom’s position consists ofshort 100 Calls and long 4,700 shares. Therefore, in his closing trade,he buys all 100 of the calls at 1.50 and sells the 4,700 shares at 34.30.As with the long volatility example presented earlier, it is noteworthy that the implied volatility of the 35 Calls, 40 percent, was the samefor both the opening and closing trades. The stock price when theposition was closed on Wednesday, 34.30, however, was not the sameas when it was opened on Thursday, 34.80. Tom’s proﬁt-and-loss calculation is presented next.
Short Volatility Step 4—Calculation of Proﬁt and LossPart 1 of Table 8-7C accounts for Tom’s option trades. Column 1 listsboth Thursday and Wednesday because Tom made trades on thosedays. Similarly, column 2 notes that he executed both buy and selltransactions. Columns 3 and 4 indicate that the purchase price andsale price were 1.50 and 1.90, respectively. Column 5 indicates a pershare proﬁt of 0.40, or 40 cents. Tom’s total dollar proﬁt of $4,000in column 6 is the product of three numbers: the per-share proﬁt incolumn 5, the number of options in column 2, and the option multiplier of 100 ($0.40 per share  100 options  100 shares per option $4,000).
260 • Trading Options As a Professional
Table 8-7C
Col 1
Part 1:Option TradeThu & WedPart 2:Stock TradesRow 1 ThuRow 2 ThuRow 3 FriRow 4Row 5 MonRow 6 TueRow 7 WedRow 8Row 9
Delta-Neutral Trading—Short Volatility: Calculation of Proﬁtand LossCol 2
Col 3
Col 4
Col 5
Col 6
Action &Quantity
PurchasePrice
SalePrice
P/(L)per Share
Total P/(L)in Dollars
Buy/sell 100
1.50
1.90
0.40
$4,000
Buy 5,300Buy 500Sell 1,100
34.80——35.30——35.30 (500)34.10(1.20)34.80 (600)—(0.70)35.10——35.1034.20 (700) (0.90)34.8034.30(0.50)Loss fromstock tradingCombined P/(L)stock andoptions
——($ 600)($ 420)—($ 630)($2,350)
Buy 700Sell 700Sell 4,700
($4,000)
-0-
Part 2 of Table 8-7C has nine rows that describe each stock tradeand calculate Tom’s net dollar loss from all the stock trades. Row 1 ofshows his ﬁrst stock trade, in which he established part of his initialdelta-neutral position. On Thursday, he purchased 5,300 shares at34.80, as indicated in columns 1, 2, and 3, respectively. Column 4 isblank because this transaction did not involve a sale. Columns 5 and6 are also blank in row 1 because an opening trade produces no proﬁtor loss.Row 2 contains Tom’s adjusting stock trade made before the closeon Thursday, in which he purchased 500 additional shares at a priceof 35.30 per share. Like the columns left blank for the opening trade,columns 4, 5, and 6 of row 2 are also left blank because there was nosale transaction in this trade and therefore no proﬁt or loss.
Delta-Neutral Trading: Theory and Reality • 261
Rows 3 and 4 reﬂect Tom’s ﬁrst closing stock trade and, as a result,his ﬁrst proﬁt/loss calculation. As in the ﬁrst example, proﬁt-and-losscalculations are made on a LIFO basis. Column 1, row 3 indicatesthat Tom made this trade on Friday by selling 1,100 shares (column2) at 34.10 per share (column 4). Rows 3 and 4 of column 3 containthe prices and numbers of purchased shares from rows 1 and 2. The1,100 shares Tom sold in row 3 must be matched with shares previously purchased in order to calculate proﬁt or loss. Under LIFO principles, the 500 shares purchased in Tom’s adjusting stock trade (row2) must be matched ﬁrst, and then an additional 600-share block mustbe taken from the initial purchase (row 1) in order to get to a total of1,100 shares. Column 3, row 3 indicates that 500 shares purchased at35.30 are matched with the ﬁrst 500 sold at 34.10. Thus this 500-sharepurchase and sale result in a loss of 1.20 per share and a total loss forTom of $600, as shown in row 3 of columns 5 and 6. The proﬁt/losscalculation for the additional 600-share block is shown in row 4. Column 3, row 4 indicates a price of 34.80 for a quantity of 600 shares,resulting in a per-share loss of 0.70, noted in column 5. Column 6then shows Tom’s total dollar loss of $420 on these 600 shares.Row 