5 in Table 8-7C reﬂects Tom’s next adjusting trade on Monday. He bought 700 shares at a price of 35.10 per share. Since this wasanother opening trade, columns 4, 5, and 6 are blank because thereis no information about a sale price, proﬁt per share, or total proﬁt.Row 6 shows Tom’s proﬁt/loss calculation for the adjusting trade hemade on Tuesday, when he sold 700 shares at 34.20. This proﬁt/losscalculation is easy because 700 is the same number of shares purchased the day before and is easily matched under LIFO to that purchase. Row 6 reﬂects Tom’s Tuesday sale at 34.20 of the 700 sharespurchased on Monday at 35.10 per share for a loss of 0.90 per share(column 5) and a total loss of $630 (column 6).Row 7 accounts for Tom’s ﬁnal stock trade in Table 8-7C. Tomclosed his position on Wednesday by selling 4,700 shares. That sale ismatched with the remaining shares purchased on Thursday. The purchase price on Thursday of 34.80 (row 1, column 3) and the sale price
262 • Trading Options As a Professional
of 34.30 on Wednesday results in a loss of 50 cents per share (row 7,column 5) and a total loss for Tom of $2,350 (column 6).Row 8, column 6 contains the sum of all the losses from Tom’s stocktrading, which amount to $4,000.Row 9 completes part 2 of Table 8-7C by combining the proﬁt fromoption trading and the loss from stock trading. Since Tom’s optionproﬁt in part 1 exactly offsets the loss from stock trading in part 2, thenet result of “–0–” appears in column 6, row 9.
Short Volatility Step 5—Recapping the TradesTom established a delta-neutral position by selling calls and buyingstock. He made adjusting stock trades daily until he closed the remaining position. The accounting for proﬁt and loss revealed that he realized a proﬁt on the short options from time decay and that his lossesfrom trading stock exactly offset that proﬁt.As with the long volatility delta-neutral exercise presented ﬁrst,the implied volatility in this second exercise was unchanged at40 percent during the period involved. What changed in this example was the stock price, which was 34.80 on Thursday when Tommade his ﬁrst trade and was 34.30 on Wednesday when he made theclosing trade. Despite the change in stock price, the result was stillbreak-even.Also like the long volatility example, this short volatility examplemight seem like a lot of work to just break even, but that is the concept of delta-neutral trading. In theory, when selling options deltaneutral, implied volatility will stay constant, and the losses fromadjusting stock trades will offset the proﬁt from time decay of theoptions—even when the stock price changes over time. In the language of options, if implied volatility equals realized volatility, thendelta-neutral trading will break even. How traders use delta-neutraltrading and attempt to proﬁt from it is discussed after the next twoexamples that address some of the real-world issues of delta-neutraltrading.
Delta-Neutral Trading: Theory and Reality • 263
Simulated “Real” Delta-Neutral Trade 1This exercise uses the theoretical values presented in Table 8-8A,the trades presented in Table 8-8B, and the proﬁt-and-loss calculationspresented in Table 8-8C. It also employs a hypothetical trader namedSusan.Table 8-8A is an abbreviated version of Tables 8-6A and 8-7A.It contains only the stock prices with the option theoretical values anddeltas necessary for the trades in Table 8-8B and the proﬁt-and-losscalculations in Table 8-8C.Like Tom’s previous two delta-neutral trading exercises, Table 8-8Bcontains the essential details of Susan’s trades. Susan establishes adelta-neutral position on Tuesday when the stock price is 86.50. Shepurchases 100 of the 85 Puts at 2.66. Because the delta of the 85 Putis 0.40, she buys 4,000 shares of stock. This is a long volatility position because options are purchased, and as indicated in Table 8-8A,the volatility assumption is 35 percent.Susan’s second trade also occurs on Tuesday, but just before the market closes. With the stock price at 87.40, the delta of the 85 Put changesto 0.36. Therefore, Susan sells 400 shares of stock to reestablish deltaneutrality. This trade is described in column 4, row 2 and is explainedas “adjusting trade to get deltaneutral” in column 5. Susan’s newstock position is shown in column 6 as “3,600,” or long 3,600 shares.Table 8-8A
Delta-Neutral Trading—Simulated “Real” Example 1: 85 PutTheoretical Values and Deltas (Volatility 35%; Interest Rate 4%;No Dividends, 31–28 Days)Tuesday
Wednesday
Thursday
Friday
StockPrice
31 DaysT.V./Delta
30 DaysT.V./Delta
29 DaysT.V./Delta
28 DaysT.V./Delta
87.4086.5084.6583.10
2.32/0.362.66/0.40
2.21/0.363.32/0.494.23/0.56
264 • Trading Options As a Professional
Table 8-8B
Delta-Neutral Trading—Simulated “Real” Example 1: The Trades
Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Day
StockPrice
OptionDelta
Trade
Explanation
StockPosition
Tues
86.50
0.40
Opening trade
4,000
Tues
87.40
0.36
Buy 100 85 Puts @ 2.66Buy 4,000 shares @ 86.50Sell 400 shares @ 87.40
Wed
83.10
0.56
Buy 2,000 shares @ 83.10
Thu
87.40
0.36
Sell 2,000 shares @ 87.40
Fri
84.65
0.49
Sell 100 85 Puts @ 3.32Sell 3,600 shares @ 84.65
Adjusting tradeto get delta-neutralAdjusting tradeto get delta-neutralAdjusting tradeto get delta-neutralClosing trade
3,600
5,600
3,600-0-
Susan makes another adjusting trade on Wednesday after the stockprice falls to 83.10. The change in stock price moves the delta of the85 Put to 0.56. Therefore, Susan buys 2,000 shares and increasesher stock position to long 5,600 shares (column 6). She makes anothertrade on Thursday by selling 2,000 shares at 87.40.Susan’s lat last trade in Table 8-8B closes her position on Friday.She sells all 100 of the 85 Puts at 3.32 and the remaining 3,600 sharesat 84.65.Table 8-8C calculates Susan’s proﬁt and loss in two parts. In part 1,the proﬁt from Susan’s option trades is $6,600. Part 2 calculates theproﬁt from her stock trades as $1,940 for a combined proﬁt of $8,540.
Recapping Simulated “Real”Trade 1Susan established a delta-neutral position by buying puts and buyingstock. She made adjusting stock trades daily until she closed herremaining position. The proﬁt-and-loss calculation showed that both
Delta-Neutral Trading: Theory and Reality • 265
Table 8-8C
Col 1
Delta-Neutral Trading—Simulated “Real” Example 1: Calculationof Proﬁt and LossCol 2
Col 3
Col 4
Col 5
Col 6
Action &Quantity
PurchasePrice
SalePrice
P/(L)perShare
Total P/(L)in Dollars
2.66
3.32
0.66
$6,600
——87.400.90——87.404.3084.65(1.95)Proﬁt fromstock tradingCombined P/(L)stock andoptions
—$ 360—$8,600($7,020)
Part 1: OptionTradeTue & FriBuy/sell 100Part 2: StockTradesRow 1 TueRow 2 TueRow 3 WedRow 4 ThuRow 5 FriRow 7Row 8
Buy 4,000Sell 400Buy 2,000Sell 2,000Sell 3,600
86.5086.5083.1083.1086.50
$1,940
$8,540
the option trades and the stock trades earned a proﬁt. This is the idealsituation! However, if Susan was trading delta-neutral, how did shemake a proﬁt? The following three observations and the Distributionscreen in Op-Eval Pro explain what happened.The ﬁrst observation is that implied volatility did not change.According to Table 8-8A, implied volatility was 35 percent on eachof the four days. The second observation is that there was a net stockprice decline during the four days from 87.40 to 84.65. Third, thestock-price action seemed very volatile. There was a stock-price dropof 4.30 from Tuesday’s close to Wednesday’s close and then a rise ofthe same amount from Wednesday’s close to Thursday’s close. Thosetwo moves were followed by a drop of 2.85 on Friday before Susanclosed the position.
266 • Trading Options As a Professional
To get an estimate of what the realized volatility was during thefour-day period in this example, the Distribution screen of Op-EvalPro is useful. By setting the “PRICE” to 87.40 and the “DAYS” to 1,it is possible to estimate the “Volatility %” by trial and error. The goalis to ﬁnd the volatility percentage that has a one-day standard deviation of 4.30. This percentage is found by raising the volatility until therange in “1 days” is 83.10 to 91.70, or a range of 4.30 up or down from87.40. The volatility percentage that achieves this result is 94. Thisnumber means that the price action of the stock in simulated “real”trade 1 is consistent with 94 percent volatility.Susan’s combined proﬁt of $8,540 in Table 8-8C resulted from therealized volatility of 94 percent being greater than the implied volatility of 35 percent. In other words, the actual ﬂuctuation of the stockprice (realized volatility) was much higher than the ﬂuctuation estimated by the price of the 85 Put (implied volatility). Remember, ifrealized volatility and implied volatility are equal, then delta-neutraltrading breaks even because the proﬁt (or loss) from stock trading offsets the loss (or proﬁt) from option time decay. If realized volatility isgreater than implied volatility, however, then the stock-price swingswill more than offset the option decay. If a long volatility delta-neutral position is maintained in this environment, then proﬁts will resultfrom the relatively large stock-price swings.Susan’s proﬁt from the put resulted from the price decline of thestock. From Tuesday to Friday, the stock price fell from 87.40 to 84.65,for a net decline of 2.85. Undoubtedly, there was some time decay ofthe 85 Put, but the delta component was greater than the theta component.In fact, simulated “real” trade 1 represents the ideal long volatilitysituation. Delta-neutral traders, when trading long volatility, want situations when the realized volatility is higher than implied volatility—and the higher, the better! They also want large stock-price changesin a short period of time. In this example, there were three large stockprice changes in four days. They were large because they were consistent with 94 percent volatility, which is much higher than the
Delta-Neutral Trading: Theory and Reality • 267
implied volatility of the purchased 85 Put of 35 percent. Unfortunately, as the next example shows, not all delta-neutral trades work outthis way.
Simulated “Real” Delta-Neutral Trade 2This exercise is different from the previous three because it exploresthe issue of changing implied volatility. Table 8-9A is another abbreviated version of Tables 8-6A and 8-7A. It contains only the stockprices, option values, and deltas needed to explain the activities ofhypothetical trader Susan. The implied volatility assumptions are indicated in the heading of each column. On Thursday, for example, theimplied volatility assumption is 28 percent (“I.V. 28 percent”). On Friday, the assumption is 30 percent, and it is 32, 34, and 24 percent onMonday, Tuesday, and Wednesday, respectively.As in the preceding delta-neutral trading exercises, Table 8-9B contains the essential details of all of Susan’s trades. Her initial tradeoccurs on Thursday. With a stock price of $61.00, she buys 50 of the60 Puts at 1.86 and simultaneously buys 2,000 shares of stock to establish a delta-neutral position because the delta of the 60 Put is 0.40.
Table 8-9A
StockPrice61.0060.5060.0059.8059.50
Delta-Neutral Trading—Simulated “Real” Example 2:60 PutTheoretical Values and Deltas (Volatility Varies, Interest Rate4%, No Dividends, 49–45 Days)
Thursday
Friday
Monday
Tuesday
Wednesday
49 DaysI.V. 28%T.V./Delta
48 DaysI.V. 30%T.V./Delta
45 DaysI.V. 32%T.V./Delta
44 DaysI.V. 34%T.V./Delta
43 DaysI.V. 24%T.V./Delta
1.86/0.40
2.24/0.422.31/0.442.44/0.46
2.56/0.482.23/0.50
268 • Trading Options As a Professional
Table 8-9B
Delta-Neutral Trading—Simulated “Real” Example 2: The Trades
Col 1
Col 2
Col 3
Day
StockPrice
OptionDelta
Thu
61.00
0.40
Thu
59.80
Fri
Col 4
Col 5
Col 6
Trade
Explanation
StockDeltaPosition
Opening trade
2,000
0.48
Buy 50 60 puts @ 1.86Buy 2,000 shares @ 61.00Buy 400 shares @ 59.80
60.00
0.46
Sell 100 shares @ 60.00
Mon
60.50
0.44
Sell 100 shares @ 60.50
Tue
61.00
0.42
Sell 100 shares @ 61.00
Wed
59.50
0.50
Sell 50 60 calls @ 2.23Sell 2,100 shares @ 59.50
Adjusting tradeto get delta-neutralAdjusting tradeto get delta-neutralAdjusting tradeto get delta-neutralAdjusting tradeto get delta-neutralClosing trade
2,400
2,300
2,200
2,100-0-
Susan makes a second trade on Thursday just before the marketcloses after the stock price falls to 59.80. She buys 400 shares becausethe delta of the 60 Put is now 0.48. On Friday, Susan sells 100 shareswhen the stock price rises to 60. In Table 8-9A, the column headingfor Friday (“48 Days”) notes that the implied volatility has risen to30 percent. The change in the volatility assumption has a slight impacton the delta, but the impact of a 2 percent change is probably toosmall to notice.On Monday, Susan sells 100 shares at 60.50. Again, the impliedvolatility has increased, this time to 32 percent. Susan’s ﬁfth trade, onTuesday, consists of selling still another 100 shares, this time at 61.00.And again, implied volatility has increased 2 percent to 34 percent.Susan closes the position on Wednesday when the stock price is 59.50and when implied volatility has dropped—sharply—to 24 percent.
Delta-Neutral Trading: Theory and Reality • 269
Perhaps this decline in implied volatility is the result of an earningsreport, other anticipated news, or simply a sudden change in investorpsychology. Nevertheless, Susan sells all 50 of the 60 Puts at 2.23 andthe remaining 2,100 shares at 59.50.Part 1 of Table 8-9C calculates Susan’s proﬁt from the option tradesat $1,850, and part 2 calculates the loss from stock trades at $2,820 fora combined loss of $970.
Recapping Simulated “Real”Trade 2Susan established a delta-neutral position by buying puts and buyingstock. She made adjusting stock trades daily until she closed the remaining position. The net result, however, was a loss. What happened?
Table 8-9C
Delta-Neutral Trading—Simulated “Real” Example 2: Outcome1—Calculation of Proﬁt and Loss
Col 1
Part 1: OptionTradeThu & WedPart 2: StockTradesRow 1 ThuRow 2 ThuRow 3 FriRow 4 MonRow 5 TueRow 6 WedRow 7Row 8Row 9
Col 2
Col 3
Col 4
Col 5
Col 6
Action andQuantity
PurchasePrice
SalePrice
P /(L)per Share
Total P/(L)in Dollars
Buy/sell 50
1.86
2.23
0.37
$1,850
Buy 2,000Buy 400Sell 100Sell 100Sell 100Sell 100Sell 2,000
61.0059.8059.8059.8059.8059.8061.00
————60.000.2060.500.7061.001.2059.50(0.30)59.50(1.50)Loss fromstock tradingCombined P/(L)stock andoptions
——$ 20$ 70$ 120($ 30)($3,000)($2,820)
($ 970)
270 • Trading Options As a Professional
The ﬁrst observation is that implied volatility dropped sharply from34 to 26 percent from Tuesday to Wednesday. The second observationis that the stock-price action seemed very calm because three of the dailystock-price changes were 50 cents or less. The price rose 20 cents fromThursday’s close to Friday’s close, then 50 cents to Monday’s close, andthen 50 cents to Tuesday’s close. The stock-price change from Tuesday’smarket close to closing the position on Wednesday was 1.50.Susan’s combined loss of $970 in Table 8-9C is made up of onepositive component and two negative ones. The sharp down move inthe stock price from Tuesday to Wednesday was a positive componentbecause it caused the price of Susan’s puts to rise, but time decay anddecreasing implied volatility were negative components. Ultimately,the negative impacts of vega and theta were greater than the positiveimpact of delta.
Considering Another Outcome for Simulated“Real”Trade 2Would Susan’s outcome change if she had closed her position one dayearlier when implied volatility was 34 percent? Table 8-9D answersthis question.Table 8-9D shows that had Susan closed her position on Tuesday,rather than on Wednesday, she would have realized a proﬁt of $2,230instead of a loss of $970. With the stock price at 61.00 on Tuesday andimplied volatility at 34 percent, the 60 Put was trading at 2.24. OnWednesday, however, with the stock price at 59.50 and implied volatility at 24 percent, the 60 Put was trading 1 cent lower at 2.23. Eventhough the stock price declined by 1.50, the price of the 60 Put, whichhad a delta of 0.42, declined by 1 cent. You can see the signiﬁcanceof this decrease in implied volatility!Table 8-9D shows that selling 2,200 shares at 61.00 on Tuesday(rows 5 and 6 in Table 8-9D) improves the stock trading result by$3,150 compared with selling at 59.50 on Wednesday (rows 6 and 7
Delta-Neutral Trading: Theory and Reality • 271
Table 8-9D
Delta-Neutral Trading—Simulated “Real” Example 2: Outcome2—Calculation of Proﬁt and Loss (if Position Had Been Closed onTuesday)
Col 1
Part 1: OptionTradeThu & WedPart 2: StockTradesRow 1 ThuRow 2 ThuRow 3 FriRow 4 MonRow 5 TueRow 6Row 7Row 8
Col 2
Col 3
Col 4
Action andQuantity
PurchasePrice
SalePrice
Buy/sell 50
1.86
2.24
Buy 2,000Buy 400Sell 100Sell 100Sell 200Sell 2,000
61.0059.8059.8059.8059.8061.00
Col 5
Col 6
P/(L)Total P/(L)per Share in Dollar
0.38
————60.000.2060.500.7061.001.2061.00-0Proﬁt fromstock tradingCombined P/(L)stock andoptions
$ 1,900
——$$$
2070240-0-
$
330
$ 2,230
in Table 8-9C), converting a loss of $2,820 to a proﬁt of $330.The 34 percent level of implied volatility on Tuesday also allowedSusan to sell the 60 Puts at 2.24 with the stock price at 61.00 andimprove her combined proﬁt to $2,230.Note that had implied volatility been 24 percent on Tuesday, theprice of the 60 Put would have been 1.59. This price is calculated withthe Op-Eval Pro software using the assumptions other than volatilityin Table 8-9C. A selling price of 1.59 for the 60 Puts would havechanged the options proﬁt of $1,900 in Table 8-9D to a loss of $1,350,a negative swing of $3,250.The conclusion is obvious: The drop in implied volatility from 34to 24 percent from Tuesday to Wednesday caused the delta-neutral
272 • Trading Options As a Professional
position in simulated “real” trade 2 to lose money. This is one of therisks facing delta-neutral traders. A trader must constantly ask, “ShouldI exit today, or should I wait until tomorrow?” The answer depends onthe forecast for implied volatility, and it is a decision that can only bemade by traders individually.
Delta-Neutral Trading—Opportunities and Risksfor SpeculatorsSpeculators attempt to proﬁt by forecasting direction. A speculatorbuys stock, buys calls, or sells puts because the forecast is bullish. Inthe case of delta-neutral trading, speculators must forecast the direction of implied volatility, the direction of realized volatility, and therelationship between the two. If implied volatility is deemed to be low,and if realized volatility is forecast to rise, then a speculator mightattempt to proﬁt from this forecast by buying options delta-neutral, asexplained in the long volatility example discussed earlier. Alternatively,if implied volatility is deemed to be high, and if realized volatility isforecast to fall, then a speculator might attempt to proﬁt from this forecast by selling options delta-neutral, as explained in the short volatility example.In an effort to make a proﬁt from delta-neutral trading, speculatorsassume risk over a period of time that is several trading days at minimum and several weeks at maximum. Hypothetical trader Tom’s deltaneutral trading exercise (1) involved ﬁve trading days, with abreak-even result, not including transaction costs. While Tom closedhis position after ﬁve days, in a real situation, a speculator would havenearly daily decisions to make. Should the position be closed at breakeven, as it was in the example? Or should the position be kept open?The answer to this question is a subjective one that traders must makeindividually. Just as trading market direction is an art, based on one’sinstinct to enter trades and to take proﬁts and losses, so too is deltaneutral trading more of an art than a science.
Delta-Neutral Trading: Theory and Reality • 273
Speculative Risks of Long VolatilitySpeculators engaged in delta-neutral trading carry limited but substantial risk in the case of long volatility. For example, consider a deltaneutral position created by buying 50 call options and shorting 2,000shares of stock. If each option’s vega is 0.12, or 12 cents, a one percentage point change in implied volatility would cause the optionprice to rise or fall by 12 cents, or $12 for each option. Thus, if impliedvolatility were to drop by ﬁve percentage points without the underlying stock moving, then each option in this example would lose 60cents per share, or $60 per option (5 percent volatility  $0.12 pershare per 1 percent of volatility  100 shares per option  $60). Fora long 50-option position, the loss would total $3,000 ($60 per option 50 options  $3,000), not including any loss from time decay. Andthe loss would increase if implied volatility declined further. The maximum possible loss of a long volatility delta-neutral position occurs ifthe position is held to expiration and if the stock price equals the strikeprice of the options at expiration, in which case the options expireworthless.
Speculative Risks of Short VolatilityThe risk of delta-neutral trading borne by speculators is unlimited inthe case of short volatility. Delta-neutral positions with short optionscarry two risks. The ﬁrst risk stems from rising implied volatility. If aspeculator sells 100 Call options delta-neutral, and if each call has avega of 0.09, or 9 cents per share, then the speculator will suffer a lossof $900 for each one percentage point rise in implied volatility.The second risk of delta-neutral positions with short options arisesfrom a big move in the underlying stock. Table 8-10 shows how a sudden price rise from $42 to $49 in the underlying stock can cause a largeloss for a delta-neutral position involving short options. An announcement after the close of trading can cause a stock to open sharply higheror lower on the next day. Such price action at the start of tradingis known as a gap opening and occurs frequently after earnings
274 • Trading Options As a Professional
announcements. However, gaps in stock prices also can occur duringthe trading day. Traders with short option positions always must be onalert for such events.In Table 8-10, column 1 describes the initial position, and column2 contains the initial prices. The position is short 100 of the 45 Callsat 1.00 and long 3,000 shares of stock at $42.00. The name of the stockis omitted because it is unimportant. Row 3, column 2 indicates thatthe initial position is delta-neutral because the delta of each short callis 0.30.Column 3 reﬂects prices after the big move. The stock price hasrisen to $49.00, and the call price has risen to 4.90. Column 4 contains the per-share loss of 3.90 for each option (row 1) and the pershare proﬁt of $7.00 for the stock (row 2). Column 5 calculates theloss for the 100 Calls and the proﬁt for the 3,000 shares. For the twopart position, the net loss is $18,000.The message of Table 8-10 is that having a delta-neutral position isnot necessarily protection against losses. As explained earlier, deltaneutral positions with long options proﬁt from large stock-price swings,or high volatility, and lose from little or no stock-price changes, or lowvolatility. In contrast, delta-neutral positions with short options proﬁtfrom low volatility and lose from high volatility. What constitutes lowand high varies from stock to stock, from index to index, and fromfutures contract to futures contract.Table 8-10