implied volatility.
Volatility SkewsVolatility skew is a market condition in which options with the sameunderlying and the same expiration but different strike prices trade atdifferent implied volatilities. This is a common occurrence in stockindex options and options on futures contracts but less common inoptions on individual stocks.Table 7-7 contains prices and implied volatilities of calls and putswith 13 strike prices. The underlying is the XSP Index, the Mini-SPXIndex, which is based on the Standard & Poor’s 500 Stock Index. Whenthe data were gathered, the XSP Index was 132.00, the dividend yield
Table 7-7
Volatility Skew
Call Price
Strike Price
Put Price
Implied Volatility
12.6010.759.007.305.754.303.052.251.450.950.660.440.05
120122124126128130132 A-T-M134136138140142144
0.270.440.670.951.401.952.703.905.056.558.2510.0511.95
26.25%25.24%24.49%23.52%22.89%21.75%20.83%21.77%20.80%20.90%21.70%22.40%24.10%
Note: Option prices indicated are the midpoint between the bid and ask.Assumptions: XSP Index, 132.00; days to expiration, 25; dividend yield, 1.2%.
Volatility • 235
was 1.9 percent, the interest rate was 3.50 percent, and there were25 days to expiration. As the table indicates, the implied volatility ofthe at-the-money 132 Call is 20.83 percent, and the implied volatilities of the other options increase as strike prices increase or decrease.The implied volatility of the 130 Call and the 130 Put, for example, is21.75 percent.Figure 7-9 graphs the information in Table 7-7. Note that the lineabove 132 is not symmetric with the line below 132 and that neitherline is perfectly straight. Note also that this information is only fromone time on one day. Although Table 7-7 and Figure 7-9 illustratevolatility skews in numerous index options markets, in these dynamicmarkets, the overall level of implied volatility and the slopes of impliedvolatility skew change. Option traders must be aware of this potentialmarket condition and prepare themselves accordingly.
Why Skews ExistThere is no theoretical reason for the existence of volatility skews.However, one practical explanation may be that since option prices
Volatility Skew28Implied Volatility
Center Strike26242220122 124 126 128 130 132 134 136 138 140 142 144Strike PriceFigure 7-9
Graphical Depiction of Volatility Skew
236 • Trading Options As a Professional
are determined by supply and demand, different forces of supply anddemand affect options with different strike prices in varying ways.Options are analogous to insurance policies, and strike prices are analogous to deductibles. Consequently, like the forces at work in theinsurance industry, the varying elements of supply and demand forthe different amounts of protection offered by options with differentstrike prices produce volatility skews.For example, when there is more demand for cheap insurance policies—policies with a low absolute price—sellers of low-cost insurancepolicies require a high risk premium. In options, meeting this demandwould result in high implied volatility but not a high absolute price.
Skews Affect Trading ResultsTraders must consider the existence of volatility skews when makingforecasts. If out-of-the-money option strike O is trading at a higherimplied volatility than at-the-money option strike A, for example, thenas the underlying moves from strike A to strike O, there may be a tendency for the implied volatility of the call and put with strike O, whichis not at the money, to decrease and for the implied volatility of thecall and put with strike A, which is now out of the money, to increase.Consider the forecasting problem being addressed by Barb, an experienced XSP options trader. Assuming an XSP level of 132 and theoption prices and market conditions in Table 7-7, Barb must ﬁrst stateher three-part forecast for the XSP level, for the time period, and forthe implied volatility of the option she is considering buying.Barb is considering buying an XSP 126 Put with 25 days to expiration because she is bearish on the market and predicts that XSP willdecline from 132 to 126 or lower in 10 days. Barb also believes thatimplied volatility will remain constant. Her volatility forecast, however,raises a question.What does implied volatility will remain constant mean when thereis a volatility skew? What implied volatility level should Barb use whenestimating the value of the 126 Put? If XSP declines to 126 in 10 days,
Volatility • 237
as Barb predicts, the 126 Put will have moved from six points out ofthe money to at the money. If the level of implied volatility remainsconstant and the skew does not change, then the implied volatility ofthe 126 Put will decline from 23.52 to 20.83 percent. Table 7-8 showsthe implications of this change.Column 1 shows the initial market conditions: The index levelis 132, there are 25 days to expiration, the implied volatility is 23.52percent, and the price of the 126 Put is 0.95. Column 2 estimates aprice of 2.30 for the 126 Put, assuming an index level of 126, 15 daysto expiration, and the implied volatility of the 126 Put unchanged at23.52 percent. Column 3 estimates a price of 2.00 for the 126 Put,assuming the same conditions as column 2 but with an implied volatility decline to 20.83 percent. This difference means that had Barbbought the put for 0.95, she would make 1.05 per option instead of1.35 per option. Whether this difference is sufﬁcient to dissuade Barbfrom making this trade is a subjective decision that only she can make.Nevertheless, even if Barb is conﬁdent of her forecasts for the index
Table 7-8
Barb Analyzes the Impact of the Volatility Skew
InputsIndex levelStrike priceDividend yieldVolatilityInterest ratesDays to expirationOutputs126 Put priceEstimated proﬁt
Col 1
Col 2
Col 3
InitialInputs
Index and DaysChanged,VolatilityUnchanged
Index, Days, andVolatility Changed
132.001261.9%23.52%4%25
→ 126.00→→→ 23.52%→→ 15
0.95–
→
2.30+1.35
→ 20.83%15→
2.00+1.05
238 • Trading Options As a Professional
level and the time period, the volatility skew could have an impact onher decision.The conclusion from this example is that if other factors remainconstant, the existence of implied volatility skew tends to be a disadvantage for buyers of out-of-the-money options. Other factors, ofcourse, are rarely equal. There could be a change in the overall levelof implied volatility, or there could be a change in the slope of thevolatility skew. Changes in either or both of these market conditionscould produce favorable or unfavorable results for a particular optionstrategy. Consequently, option traders must consider the overall levelof implied volatility and the volatility skew, if any.
SummaryVolatility is a measure of price change without regard to direction. Mathematicians, option traders, and the market each view volatility somewhat differently. Mathematically, volatility is the annualized standarddeviation of daily returns. A volatility percentage, such as 35 percent, isan annual standard deviation, which can be converted to another timeperiod by multiplying it by the square root of time.There are many terms that describe volatility. Historic volatility isa measure of stock-price ﬂuctuations during some deﬁned period inthe past. Expected volatility is a trader’s prediction of what volatilitywill be in the future and is used to calculate theoretical values. Realized volatility is a measure of actual stock-price ﬂuctuations betweennow and some point in the future and is unknown. Implied volatilityjustiﬁes the current market price of an option.Implied volatility is the common denominator of option prices. Justas the price-earnings ratio makes possible comparisons of stock pricesover a range of variables such as total earnings and number of sharesoutstanding, implied volatility facilitates comparisons of options ondifferent underlying instruments and comparisons of the same optionat different times.
Volatility • 239
Theoretical value of options is a statistical concept only. Tradersshould focus on relative value, not absolute value. The terms overvalued and undervalued describe a relationship between implied volatility and expected volatility. Two traders could differ in their opinionof the relative value of the same option if they had different marketforecasts and trading styles.Volatility skew is a market condition in which options with the sameunderlying and same expiration but with different strike prices tradeat different levels of implied volatility. Since option prices, like allprices in free markets, are determined by the forces of supply anddemand, volatility skews likely exist because there are differing levelsof supply and demand for different options.
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Chapter 8DELTA-NEUTRALTRADING: THEORYAND REALITYelta-neutral trading is a nondirectional trading technique thatproﬁts, loses, or breaks even from the relationship between priceﬂuctuations in the underlying stock and the time decay of optionprices. In the language of options, this relationship is the differencebetween implied volatility and realized volatility. As this chapter willexplain, professional market makers and professional speculators havevery different motivations for using this trading technique.This chapter begins with a deﬁnition of a delta-neutral position andsome examples. Then it explains the theory of delta-neutral tradingwith two detailed examples, one in which options are purchased andone in which options are sold. After the theory is explained, two moreexamples show how and why the reality of delta-neutral trading differsfrom the theory. Then the chapter explores the different motivationsof professional market makers and professional speculators who usedelta-neutral trading. Finally, the chapter concludes by discussing theproﬁt potential and risks assumed by market makers and speculatorsand the judgments that each must make.
D
• 241 •
242 • Trading Options As a Professional
Delta-Neutral DeﬁnedA delta-neutral position is a multiple-part position that has a combinednet delta of zero or approximately zero. The components of the multiple-part position can be any combination of long and short calls,puts, or stock. Tables 8-1 through 8-4 illustrate basic two-part deltaneutral positions. Table 8-5 contains a three-part example.Tables 8-1 through 8-4 each contain two sections. The upper section describes the individual option and stock trades that create thedelta-neutral position. The lower section has four columns and threerows that calculate the delta of the position. Column 1 contains anabbreviated description of each component of the position in the uppersection of the table. Column 2 states the number of shares representedby the component in column 1. For a stock position, the number ofshares in column 2 matches the number in column 1. For option positions, however, the number of shares in column 2 equals 100 times thenumber of option contracts in column 1. A directional sign, plus orminus, is not associated with the number of shares in column 2 becausethe directional sign in front of the delta in column 3 indicates whetherthe position is long or short. As explained in Chapter 4, long stock, longcalls, and short puts have positive deltas, and short stock, short calls,and long puts have negative deltas. The number in column 4 represents the market exposure, or delta, in shares of each component. Thenumber is the product of the numbers in columns 2 and 3.
Delta Neutral with Long Calls and Short StockTable 8-1 shows a two-part position of long calls and short stock resulting in a net delta of zero. As the upper section of the table indicates,the position is created by buying 20 XYZ 90 Calls at 2.75 each andselling short 900 shares of XYZ stock at 89.05 each.In the lower section of Table 8-1, row 1, column 1 contains only anabbreviated description of the option position, “Long 20 90 Calls,”because calculation of delta does not require the option price or thename of the underlying stock. The “2,000” in column 2 indicates that
Delta-Neutral Trading: Theory and Reality • 243
Table 8-1
A Delta-Neutral Position with Long Calls and Short Stock
Creating the position:The option trade:The stock trade:
Buy 20 XYZ 90 Calls @ 2.75Short 900 XYZ shares @ 89.05
Calculation of position delta:
Row 1Row 2Row 3
Col 1
Col 2
Col 3
Col 4
Position
Numberof SharesRepresented
DeltaperShare
MarketExposurein Shares
Long 20 90 CallsShort 900 shares
2,000900


 0.45 1.00Position net delta:
900900-0-
Assumptions: Stock price, 89.05; days to expiration, 43; interest rate, 5%; no dividend; volatility, 28%.
the option position in column 1 represents 2,000 shares of the underlying stock. Remember, a directional sign, plus or minus, is not associatedwith the number of shares in column 2 because long or short is speciﬁed by the directional sign in front of the delta in column 3. In column3, the delta of “0.45” has two meanings. First, the “” means that theposition in column 2 represents positive exposure to the market, that is,long shares. Second, the “0.45” is the delta per share of each call in column 1. As described in Chapters 3 and 4, delta is the per-share marketexposure of an option. The “900” in column 4 is the product of theshares in column 2 and the delta in column 3 (2,000  0.45  900).The “900” means that the option position described in column 1 willbehave the same as long 900 shares of the underlying stock.Row 2 in the lower section of Table 8-1 describes the stock component of the delta-neutral position. “Short 900 shares” in column 1describes the position without the price or name. Column 2 containsthe number of shares, “900,” without indicating whether they are longor short. For a stock position, the number of shares in column 2matches the number of shares in column 1 because shares of stock donot have a multiplier, as options do. The delta in column 3, row 2,
244 • Trading Options As a Professional
“1.00,” has two meanings. First, the “” means that the position represents negative exposure to the market, that is, short shares. Second,the “1.00” is the delta of each share. Stock always has a per-share deltaof 1.00, 1.00 for long stock and 1.00 for short stock. The number“900” in column 4, row 2, is the product of the “900” in column 2and the “1.00” in column 3 and indicates that the market exposureis short 900 shares.Row 3, column 4 of Table 8-1 contains the net delta of the position, “–0–,” which is the sum of the deltas of the components incolumn 4. A position delta of zero indicates that, in fact, this is a deltaneutral position.
More Delta-Neutral Stock and Option CombinationsTables 8-2 through 8-4 illustrate that delta-neutral positions also canbe created with short calls and long stock (Table 8-2), with longputs and long stock (Table 8-3), and with short puts and short stock(Table 8-4). The explanations of these tables follow closely the explanation of Table 8-1; reviewing each table in depth, therefore, is notnecessary.Table 8-2
A Delta-Neutral Position with Short Calls and Long Stock
Creating the position:The option trade:The stock trade:
Sell 40 QRS 35 Calls @ 4.25Buy 3,000 QRS shares @ 38.00
Calculation of position delta:
Row 1Row 2Row 3
Col 1
Col 2
Col 3
Col 4
Position
Numberof SharesRepresented
DeltaperShare

MarketExposurein Shares
0.751.00Position net delta:
3,0003,000-0-
Short 40 35 CallsLong 3,000 shares
4,0003,000

Assumptions: Stock price, 38.00; days to expiration, 70; interest rate, 5%; no dividend; volatility, 35%.
Delta-Neutral Trading: Theory and Reality • 245
Table 8-3
A Delta-Neutral Position with Long Puts and Long Stock
Creating the position:The option trade:The stock trade:
Buy 40 MNO 17.50 Puts @ 1.10Buy 1,600 MNO shares @ 17.80
Calculation of position delta:
Row 1Row 2Row 3
Col 1
Col 2
Col 3
Col 4
Position
Numberof SharesRepresented

DeltaperShare
MarketExposurein Shares
4,0001,600

Long 40 17.50 PutsLong 1,600 shares

0.401.00Position net delta:
1,6001,600-0-
Assumptions: Stock price, 17.80; days to expiration, 55; interest rate, 5%; no dividend; volatility, 31%.
Table 8-4
A Delta-Neutral Position with Short Puts and Short Stock
Creating the position:The option trade:The stock trade:
Sell 50 FGH 45 Puts @ 1.20Short 1,500 FGH shares @ 47.50
Calculation of position delta:
Row 1Row 2Row 3
Col 1
Col 2
Col 3
Col 4
Position
Numberof SharesRepresented
DeltaperShare
MarketExposurein Shares
Short 50 45 PutsShort 1,500 shares
5,0001,500


 0.30 1.00Position net delta:
1,5001,500-0-
Assumptions: Stock price, 47.50: days to expiration, 28; interest rate, 5%; no dividend; volatility 45%
Consistency of UnitsWhen creating delta-neutral positions, traders must be careful to keepshares of stock and options in equivalent units. Inconsistencies willcause the delta calculation of a multiple-part position to be incorrect.
246 • Trading Options As a Professional
The relationship between stock and stock options is generally100 shares per option. However, infrequent events such as splits, mergers, and special distributions can change this relationship. Betweenfutures contracts and options on those contracts, the relationship isgenerally one futures contract per option, but sometimes that ratio alsocan vary. Consequently, keeping track of the multiplier betweenoptions and their underlying is essential for delta-neutral traders.
Multiple-Part Delta-Neutral PositionsTable 8-5 differs slightly from Tables 8-1 through 8-4 because itdescribes a three-part position. The three components are short 40MNO 22.50 Puts, long 40 MNO 25.00 Puts, and long 1,000 shares ofMNO stock. The process for calculating the net delta follows that fora simple delta-neutral trade except that there are more than two components. The message is simple: Delta-neutral positions come inmany shapes and sizes. What a trader needs to know is how to makemoney from them, and that is discussed later in this chapter.Table 8-5
A Three-Part Delta-Neutral Position
Creating the position:Option trade 1:Option trade 2:The stock trade:
Sell 40 MNO 22.50 Puts @ 0.20Buy 40 MNO 25.00 Puts @ 0.90Buy 1,000 MNO shares @ 25.75
Calculation of position delta:
Row 1Row 2Row 3Row 4
Col 1
Col 2
Col 3
Col 4
Position
Numberof SharesRepresented
DeltaperShare
MarketExposurein Shares
Short 40 22.50 PutsLong 40 25.00 PutsLong 1,000 shares
4,0004,0001,000


 0.12 0.37 1.00Position net delta:
 4801,4801,000-0-
Assumptions: Stock price, 25.75; days to expiration, 55; interest rate, 4%; no dividend; volatility, 32%.
Delta-Neutral Trading: Theory and Reality • 247
The Theory of Delta-Neutral TradingDelta-neutral trading involves three steps. First, a trader establishes adelta-neutral position. Second, as the underlying stock price changesand as the net delta of the total position changes away from zero, thetrader makes adjusting stock trades according to predetermined rules.Third, the trader closes the entire position, hopefully for a net proﬁt.An adjusting stock trade is the purchase or sale of a speciﬁc number of shares of stock that returns the net delta of the total position tozero or approximately zero. The predetermined rules dictating whenstock trades are made can be based on time or stock-price movement.For example, adjusting stock trades based on time might consist ofmaking trades every day at noon or every day shortly before the market closes. Adjusting trades being based on stock-price movementmight require making trades whenever the stock price rises or falls$2.00 or when the stock price rises or falls one standard deviation, asexplained in Chapter 7. Adjusting stock trades also can be based onthe net position delta.The theory of delta-neutral trading can be illustrated best with twoexamples, one involving purchased calls and the second involving soldcalls. In each example, a trader named Tom will practice delta-neutral trading, which will be explained in ﬁve steps. First, Tom will establish a delta-neutral position. The implied volatility of the options thatTom buys or sells will be assumed and identiﬁed. Second, Tom willmake adjusting stock trades at the close of each trading day. Closingstock prices for each day are chosen for the sake of the examples, butthe deltas and theoretical values of the options that appear in thetables are actual calculations based on those stock prices using theOp-Eval Pro software that accompanies this text. Third, on the ﬁfthday of trading, Tom will close the position. Fourth, Tom will calculatehis proﬁt or loss. Fifth, the conclusion will explain why the exampleis important, what concepts it illustrates, and what factors in the realworld might differ from the example. The reasons that professionalmarket makers and professional speculators might use this strategy willbe discussed later in this chapter.
248 • Trading Options As a Professional
Delta-Neutral Trading—Long Volatility ExampleThis ﬁrst example uses the theoretical values presented in Table 8-6A,the trades presented in Table 8-6B, and the proﬁt-and-loss calculationspresented in Table 8-6C. Table 8-6A contains theoretical values anddeltas of a 90 Call over ﬁve days in the columns and over a range ofstock prices in the rows. The left-most column contains stock prices,and the other columns contain option theoretical values and deltas.In the ﬁrst column to the right of a stock price of 91.00, for example,“5.71/0.58” appears. The “5.71” is the option’s theoretical value, andthe “0.58” is the option’s delta. Six circles appear in Table 8-6A forease of identiﬁcation. These circles indicate when trades are made. Asnoted earlier, the daily price action is created for the sake of the example. In the real world, of course, the market will determine priceaction.
Table 8-6A
Delta-Neutral Trading—Long Volatility: 90 Call Theoretical Valuesand Deltas (Volatility 30%, Interest Rate 3%, No Dividends,75–71 Days)Monday
Tuesday
Wednesday
Thursday
Friday
StockPrice
75 DaysT.V./Delta
74 DaysT.V./Delta
73 DaysT.V./Delta
72 DaysT.V./Delta
71 DaysT.V./Delta
92.2092.0091.8091.6091.4091.2091.0090.8090.6090.4090.2090.0089.8089.60
6.42/0.616.30/0.606.18/0.606.06/0.595.94/0.595.82/0.595.71/0.585.60/0.585.48/0.575.37/0.565.26/0.565.15/0.545.04/0.544.93/0.52
6.39/0.616.26/0.606.14/0.606.02/0.595.90/0.595.79/0.595.67/0.585.55/0.585.44/0.575.33/0.565.22/0.565.11/0.545.00/0.544.89/0.52
6.35/0.616.23/0.606.11/0.606.99/0.595.87/0.595.75/0.595.63/0.585.52/0.585.40/0.575.29/0.565.18/0.565.07/0.544.96/0.544.85/.52
6.31/0.616.19/0.606.07/0.605.95/0.595.83/0.595.71/0.595.60/0.585.48/0.585.37/0.575.26/0.565.15/0.565.04/0.544.93/0.544.82/0.52
6.28/0.616.15/0.606.03/0.605.91/0.595.79/0.595.68/0.595.56/0.585.45/0.585.33/0.575.22/0.565.11/0.565.00/0.544.89/0.544.78/0.52
Delta-Neutral Trading: Theory and Reality • 249
Table 8-6B contains the essential details of all of Tom’s trades in thisﬁrst example. Column 1 indicates the day of the week. Tom makes twotrades on Monday and then one trade on each day from Tuesdaythrough Friday. Column 2 indicates the stock price when a trade ismade. Column 3 contains the delta of the 90 Call given the day in column 1 and the stock price in column 2. Note that the deltas in column 3 of Table 8-6B are the same as the deltas in Table 8-6A in thecorresponding column (day) and row (stock price). For example, onMonday, with a stock price of 90.80, the delta of the 90 Call is 0.58in both tables. Column 4 contains the necessary information abouteach trade. Column 5 explains brieﬂy the motivation for the trade, andcolumn 6 indicates the ending stock position after the trade is made.
Long Volatility—OverviewLong volatility means that a position has a positive vega, as deﬁned inChapter 4. Long calls and long puts have positive vega. The positionin the upcoming exercise is long volatility because the calls are long;that is, they are owned.Table 8-6B
Delta-Neutral Trading—Long Volatility: The Trades
Col 1 Col 2 Col 3
Col 4
Col 5
Col 6
Stock OptionPrice Delta
Trade
Explanation
StockPosition
Day
Mon 90.80 0.58 Buy 100 90 Calls @ 5.60Opening trade5,800Short 5,800 shares @ 90.80 Delta-neutralMon 92.00 0.60 Short 200 shares @ 92.00Adjusting trade to 6,000get delta-neutralTue 90.20 0.56 Buy 400 shares @ 90.20Adjusting trade to 5,600get delta-neutralWed 91.40 0.59 Short 300 shares @ 91.40Adjusting trade to 5,900get delta-neutralThu 89.60 0.52 Buy 700 shares @ 89.60Adjusting trade to 5,200get delta-neutralFri 90.80 0.58 Sell 100 90 Calls @ 5.45Closing trade–0–Buy 5,200 shares @ 90.80
250 • Trading Options As a Professional
Before each