and rho—are frequently referred to as the Greeks, and each is an estimate of thechange in option value caused by a change in one of the inputs to anoption-pricing formula, assuming that other factors remain constant.Each of the Greeks will be discussed in three steps. First, each Greekwill be deﬁned, and an example will be given of how each affectsoption prices. Second, how each Greek changes as other factorschange will be discussed. Third, the concept of position Greeks willbe explored, and a method will be presented for measuring the Greeksof multipart option positions.The following discussion gets fairly technical, but option tradersmust master the concepts. This material is also essential preparationfor Chapter 10, which discusses position-risk management.
T
OverviewAs an underlying stock price rises or falls, the value of an option willalso rise or fall. Delta is a measure of the option value’s sensitivity tothose underlying price changes. Deltas themselves also changeas stock prices rise and fall, with gamma being a measure of a delta’s• 77 •
78 • Trading Options As a Professional
sensitivity to underlying price. Vega is a measure of an option value’ssensitivity to volatility, and theta is a measure of an option value’s sensitivity to the passage of time. Rho is a measure of an option value’ssensitivity to interest rates. Each of these Greeks will be discussed indepth, with examples and some general rules as to how they change,when their effects on option prices are biggest and smallest, and howthey change when either time or volatility changes.
DeltaAs discussed in Chapter 3, the price of the underlying instrument isan important factor in the determination of option values. Delta is anestimate of the change in option value given a one-unit change inprice of the underlying instrument, assuming that other factors remainconstant. Delta answers this question: If the underlying stock rises orfalls by one point, how much should I make or lose?The Op-Eval Pro software that accompanies this text was used tocreate the tables and ﬁgures in this chapter. Tables 4-1 through 4-6assume the following initial parameters: a stock price of 100, a strikeprice of 100, volatility of 30 percent, an interest rate of 4 percent,60 days to expiration, and no dividends. Given these inputs, four ofthe outputs on the Single Option Calculator screen of Op-Eval Proare a 100 Call value of 5.19, a 100 Call delta of 0.55, a 100 Put valueof 4.59, and a 100 Put delta of 0.46.In Table 4-1, the delta of the 100 Call estimates that if the stockrises by one point to 101 and other factors remain constant, then the100 Call value will rise by 0.55 to 5.74. The delta of the 100 Put estimates that under the same circumstances, the 100 Put value will fallby 0.46 to 4.13. The values to the right of the arrows in the table showthis. Although in this example the call and put deltas appear to stateprecisely the changes in call and put values, they only predict them.In most cases there will be slight differences between the predictionsand the actual changes in the marketplace. The next section, whichdiscusses gamma, will explain the reason for such differences. First,however, you need to make some observations about delta.
The Greeks • 79
Mathematically, the delta of an option is the ﬁrst derivative of optionvalue with respect to a change in price of the underlying. While it is notimportant to know the mathematics, it is important to understand theconcept that delta is an estimate of the change in option value given achange in price of the underlying with other factors remaining constant.As will be shown later, a one-unit change in price of the underlyingcauses relatively bigger changes in the values of in-the-money options,relatively smaller changes in the values of at-the-money options, andeven smaller changes in values of out-of-the-money options.
Call Values Have Positive DeltasThe plus sign () associated with the delta of the 100 Call in Table 4-1indicates a positive, or direct, relationship between a change in price ofthe underlying instrument and a change in theoretical value of the call.As the table illustrates, when only the underlying stock price rises, thetheoretical value of the 100 Call also rises. It should be noted at this pointthat a plus or minus sign associated with an option delta may be different
Table 4-1
Illustration of DeltaInitialInput
Inputs:Stock priceStrike priceDividendsVolatilityInterest rateDays to expiration
100100None30%4%60
Input withStock Price Up→
Initial OutputsOutputs:100 Call value100 Call delta100 Put value100 Put delta
5.190.554.590.46
101
New Outputs→
5.74
→
4.13
80 • Trading Options As a Professional
from the sign associated with an option position. The subject of positiondeltas will be discussed later in this chapter.
Put Values Have Negative DeltasThe negative sign () associated with the delta of the 100 Put indicates a negative, or inverse, relationship between a change in price ofthe underlying instrument and a change in value of the put. In otherwords, a rise in the stock price caused the put value to decline, asshown in Table 4-1.
Finding Deltas in Op-Eval ProDelta appears in Op-Eval Pro on three screens in two different forms.On the Single Option Calculator screen (see Figure 2-2), deltasappear under the respective call and put values. On the Spread Positions and Portfolio screens (see Figures 2-3 and 2-6), deltas appear intwo places. Deltas of individual options appear in the “DELTA” row,and a “Spread delta” appears in the box labeled as such. The spreaddelta is the sum of the deltas of individual options in a position.
GammaAs discussed earlier, in most cases the delta does not exactly predictan option’s new value after a one-point change in price in the underlying. The difference occurs because the delta changes as the price ofthe underlying changes. Gamma is a measure of the change in deltafor a one-unit change in price of the underlying instrument, assuming that other factors remain constant. Mathematically, gamma is thesecond derivative of the option value with respect to change in priceof the underlying. Gamma answers this question: How much does myexposure to the market change, that is, how much does my deltachange, when the price of the underlying stock changes? Gammamakes it possible to estimate more accurately the change in optionvalue when the stock price rises or falls.
The Greeks • 81
Table 4-2 illustrates the concept of gamma. It takes the informationin Table 4-1 and then adds the new call and put deltas, as calculatedwith the options’ gammas after the change in the stock price. Whenthe stock price rises from 100 to 101, as the table shows, the delta ofthe 100 Call increases from 0.55 to 0.58, a rise of 0.03, a changeequal to the gamma.Similarly, the delta of the 100 Put increases by 0.03 from 0.46 to0.43. That’s right! This is an increase in the put delta; 0.43 is“greater” than 0.46. For readers comfortable with math, this mayseem obvious. Others should take note: When watching the Greekschange, it is important to keep track of plus and minus signs andincreases and decreases in value.While the change in call and put deltas exactly equals the gammain Table 4-2, there will frequently be small differences owing to rounding and changing gammas. In Table 4-2, the gammas of the both the100 Call and the 100 Put do not appear to change after the stock priceTable 4-2
Illustration of Gamma (1)InitialInputs
Inputs:Stock priceStrike priceDividendsVolatilityInterest rateDays to expiration
100100None30%4%60
Inputs withStock Price Up→
Initial OutputsOutputs:100 Call value100 Call delta100 Call gamma100 Put value100 Put delta100 Put gamma
5.190.550.034.590.460.03
101
New Outputs→→→→→→
5.740.580.034.130.430.03
82 • Trading Options As a Professional
does because they are only calculated to the second decimal point.Calculating the values to four decimal points reveals that the gammaof the 100 Call fell from 0.0327 to 0.0321, and the gamma of the100 Put fell from 0.0335 to 0.0328. While such small changes mayseem insigniﬁcant, this example involves only a one-point, or 1 percent, change in stock price. For stock-price changes of 5 percent ormore, these seemingly insigniﬁcant numbers can add up and becomesigniﬁcant to a trader who has a large position and ignores them.
Gammas of Option Values Are PositivePlus signs () are always associated with gammas of both calls and putsbecause change in delta is positively correlated with change in theprice of the underlying. Table 4-2 illustrates this: An increase in stockprice causes an increase in the deltas of both the 100 Call and the 100Put. The delta of the 100 Call, for example, increases from 0.55 to0.58, an amount exactly equal to the gamma of 0.03. Table 4-3, onthe other hand, shows that a decrease in the stock price causes adecrease in the deltas of both options. The delta of the 100 Calldecreases from 0.55 to 0.51, and the delta of the 100 Put decreasesfrom 0.46 to 0.50. These are examples of the change in delta notbeing exactly equal to the change in the gamma. The difference is dueto rounding. This is a positive correlation: Stock price up, delta up(Table 4-2), and stock price down, delta down (Table 4-3).Note that the gammas of calls and puts with the same underlying,same strike, and same expiration are nearly equal. In Tables 4-2 and4-3, they appear equal because they are calculated to only two decimal places. Gammas of same-strike calls and puts are nearly equalowing to put-call parity, a corollary of which is that the sum of theabsolute values of the call delta and put delta equals 1.00 (or verynearly 1.00), assuming that the call and put have the same underlying, same strike, and same expiration. This means that if the absolutevalue of the delta of the call increases, then the absolute value of theput must decrease by an equal amount. Otherwise, the sum of the
The Greeks • 83
Table 4-3
Illustration of Gamma (2)InitialInputs
Inputs:Stock priceStrike priceDividendsVolatilityInterest rateDays to expiration
100100None30%4%60
Inputs withStock Price Down→
Initial OutputsOutputs:100 Call value100 Call delta100 Call gamma100 Put value100 Put delta100 Put gamma
5.190.550.034.590.460.03
99
New Outputs→→→→→→
4.650.510.035.050.500.03
absolute values of the deltas would no longer equal 1.00. Since thedeltas of the call and put change by nearly the same amount, theirgammas also must be nearly equal because gamma is the change indelta. Put-call parity is discussed in Chapter 5.
VegaVolatility will be discussed in depth in Chapter 7, but vega will bedeﬁned here. Vega is the change in option value that results from a onepercentage point change in the volatility assumption, assuming thatother factors remain constant. Mathematically, vega is the ﬁrst derivative of option price with respect to change in volatility. Since ﬁrst derivatives are theoretically instantaneous rates of change, and since vegaestimates the impact of a one percentage point change, there frequentlywill be rounding errors. Vega answers this question: If volatility changesby one percentage point, how much do I make or lose?
84 • Trading Options As a Professional
Table 4-4
Illustration of VegaInitialInputs
Inputs:Stock priceStrike priceDividendsVolatilityInterest rateDays to expiration
100100None30%4%60
Inputs withVolatility Up
→
Initial OutputsOutputs:100 Call value100 Call vega100 Put value100 Put vega
5.190.164.590.16
31%
New Outputs→
5.35
→
4.75
Table 4-4 illustrates how the 100 Call and 100 Put values changefrom 5.19 to 5.35 and from 4.59 to 4.75, respectively, when the volatility assumption is increased from 30 to 31 percent.
Vegas of Option Values Are PositiveVegas of both call values and put values are always positive becausechanges in option value are positively correlated with changes involatility; that is, volatility up, option value up, and volatility down,option value down.Another result of the put-call parity concept is that vegas of callsand puts with the same underlying, strike, and expiration are equal.According to put-call parity, there is a quantiﬁable relationshipbetween the price of the underlying instrument and the prices of callsand puts with the same strike and same expiration. In order for theput-call parity relationship to be maintained when the call valueincreases, the same-strike put must rise by an identical amount with agiven change in volatility. Thus vegas of calls and puts with the sameunderlying, strike, and expiration must be equal.
The Greeks • 85
Vega Is Not GreekReaders familiar with the Greek alphabet may note that vega is not aGreek letter. The derivation of this term’s use remains murky, but onebelief postulates that option traders wanted a short word beginningwith a v (for volatility) that sounds like delta, gamma, and theta.Exactly who coined the term and when it was ﬁrst used are not known.Some mathematicians and traders use another Greek letter such askappa or lambda instead of vega. Why no uniform terminology existsto represent a concept as important as volatility is one of the manyquirks of the options business.
ThetaTheta is an estimate of the change in option value given a one-unitchange in time to expiration, assuming that other factors remain constant. Theta answers this question: If time passes, how much doI make or lose? Table 4-5 illustrates what happens to call and put values when days to expiration are reduced from 60 to 53. The call valuedecreases from 5.19 to 4.86, a change equal to the theta of 0.33.The put value decreases from 4.59 to 4.33, a change also equal to itstheta of 0.26. Although the amount of change in option valuesexactly equals the thetas in this example, slight differences may occurowing to rounding, especially when the calculation goes out to several decimal points.The deﬁnition of theta raises an important question: What is oneunit of time? Mathematically, theta is the ﬁrst derivative of optionvalue with respect to change in time to expiration. This means, theoretically, that one unit of time is instantaneous. Such a concept, however, does not help traders who need a tool they can use to estimatethe impact of time decay on their position. While many professionaltraders use a one-day theta, nonprofessional traders generally use a different time frame, perhaps one week, 10 days, or some other timeframe that is a percentage of a typical holding period. Consequently,there is no “right” answer to what one unit of time is.
86 • Trading Options As a Professional
Table 4-5
Illustration of ThetaInputs withChanged Days toExpiration
Initial InputsInputs:Stock priceStrike priceDividendsVolatilityInterest rateDays to expiration
100100None30%4%60
→
Initial OutputsOutputs:100 Call value100 Call theta (7-day)100 Put value100 Put theta (7-day)
5.190.334.590.26
53New Outputs
→
4.86
→
4.33
Op-Eval Pro allows you to set the number of days in the theta calculation. Simply double-click on “Theta” in the bottom-right sectionof the screen, and the “Applications Settings” box will open. Any number of days up to 999 can be entered. Note, however, that when the“Days to Expiry” input is equal to or less than the number of days inthe theta calculation, then Op-Eval Pro automatically calculates aone-day theta because it would make no sense to calculate a sevenday theta when there are six or fewer days until expiration.Different software programs, of course, deﬁne the term unit of timedifferently, so be sure to know how theta is quantiﬁed in a particularprogram before attempting to use it to estimate option price behavior.An important observation to make from Table 4-5 is that the thetaof the 100 Call of 0.33 does not equal the theta of the 100 Put of0.26. For options on stocks, exchange-traded funds (ETFs), andother deliverable underlying instruments, calls and puts with the sameunderlying, same strike, and same expiration have different thetas. Thethetas differ because the call and put have different time-value
The Greeks • 87
amounts. Different time values decaying to zero over the same timeperiod means different rates of decay and hence different thetas. Callsand puts on deliverable underlying instruments have different timevalues because there is an interest component in the call value that isnot found in the put value.
Thetas of Option Values Are NegativeThe minus sign () associated with thetas sometimes confuses newoption traders. Option values are directly correlated with changes inthe days to expiration: The more time to expiration, the higher is anoption’s value, and the less time to expiration, the lower is the value,assuming that other factors are constant. Consequently, one mightthink that thetas should be preceded by plus signs. But they are preceded by minus signs! Why?Option traders routinely place a minus sign in front of thetas becauseoptions decrease in value over time while other factors remain constant. The minus sign associated with thetas assumes that the option isowned and that it decays, or loses money, as expiration approaches.Experienced option traders may be aware of one exception to the rulethat thetas are preceded by negative signs. The theoretical value of deepin-the-money European-style options can be less than their intrinsic valuebecause these options cannot be exercised early and because of arbitragepricing relationships, as discussed in Chapter 6. When such situationsexist, the options have a positive theta that indicates that the theoreticalvalues will increase to intrinsic values as expiration approaches.
RhoRho is an estimate of the change in option value given a one percentage point change in interest rates, assuming that other factors remainconstant. Rho answers this question: If the interest rate changes by1 percent, how much do I make or lose? Table 4-6 illustrates how the100 Call and 100 Put values change when the interest rate is increasedfrom 4 to 5 percent. The 100 Call value has a direct relationship with
88 • Trading Options As a Professional
Table 4-6
Illustration of RhoInitialInputs
Inputs withInterest Rate Up
Inputs:Stock priceStrike priceDividendsVolatilityInterest rateDays to expiration
100100None30%4%60
→
Initial OutputsOutputs:100 Call value100 Call rho100 Put value100 Put rho
5.190.084.590.07
5%New Outputs
→
5.27
→
4.52
interest rates. It increases from 5.19 to 5.27, a change equal to the call’srho of 0.08. The 100 Put value, however, has an opposite relationship with interest rates. The one percentage point change in the interest rate decreased the value of the 100 Put from 4.59 to 4.52, a changeequal to the put’s rho of 0.07.Why rhos of call values are positive and rhos of put values are negative is another consequence of the put-call parity relationship. Foroptions on deliverable underlying instruments, the time value of a callexceeds the time value of a put with the same strike and expiration byan amount equal to the cost of carry. Cost of carry, as explained inChapter 6, is the expense of ﬁnancing the ownership of the underlying stock, and it consists mostly of interest adjusted for dividends, if any.When interest rates rise, the cost of carry increases. As a result, thetime value of the call must increase relative to the time value of theput. A newcomer to options might think that the call value couldincrease while the put value remained constant or that the put valuecould decrease while the call value remained constant. In reality, however, a little of both occurs. The call value rises, and the put value
The Greeks • 89
decreases. It is the net difference between the two that equals the costof carry. This combination of changes explains why rhos are positivefor calls and negative for puts.For most option traders, the impact of changes in interest rates onshort-term option values is small. As a result, rho is of little concern tononprofessional traders of short-term stock options. Professionaltraders, however, who engage in arbitrage strategies, as explained inChapter 6, must pay attention to the impact of interest rates. Rho isthe guide to judging the impact of changing interest rates.
How the Greeks ChangeTrying to measure something when the measure itself changes posesobvious problems. Consequently, estimating changes in option valuesis complicated by the fact that the Greeks change when market conditions change. For example, when the stock price, time to expiration,volatility, or any combination of these factors change, so do the delta,gamma, theta, and vega. Sometimes an individual Greek will changesigniﬁcantly and have a great impact on an option’s value, but sometimes the change will have little impact. Both graphs and tables areeffective ways to illustrate changing Greeks, so the following discussion uses both to make several points about each Greek.Changes in underlying price, time to expiration, and volatility matter most to traders, so Tables 4-7 and 4-8 focus on changes in thesethree inputs. Table 4-7 depicts a grid of 100 Call values, 100 Put values, and corresponding Greeks at three stock prices and various daysto expiration. A study of this table reveals how delta, gamma, vega,and theta change as stock price, time to expiration, or both change.Table 4-8 is a grid of 90 Call, 100 Call, and 110 Call values andcorresponding Greeks at 25 and 50 percent volatility and at variousdays to expiration. A study of this table reveals how the Greeks of inthe-money, at-the-money, and out-of-the-money options change asvolatility, time to expiration, or both change. The concepts in Tables4-7 and 4-8 help option traders to analyze the impact of changing market conditions on their positions.
90 • Trading Options As a Professional
Table 4-7
The Greeks of In-the-Money, At-the-Money, and Out-of-the-MoneyOptions
RowA
B
Col 1
Col 2
Col 3
Col 4
Col 5
56 Days
42 Days
28 Days
14 Days
Exp.
10.880.890.020.060.040.070.500.110.020.060.030.01
10.320.950.010.020.030.040.130.050.010.020.020.00
10.001.000.000.000.000.000.000.000.000.000.000.00
6.750.750.040.090.060.061.380.250.040.090.050.02
5.840.810.040.050.070.030.650.190.040.050.060.01
5.001.000.000.000.000.000.000.000.000.000.000.00
3.520.530.050.110.070.043.160.470.050.110.050.03
2.450.520.070.080.090.022.270.480.070.080.080.02
0.000.000.000.000.000.000.000.000.000.000.000.00
Stock Price 110100 CallDeltaGammaVegaTheta (1-day)Rho100 PutDeltaGammaVegaTheta (1-day)Rho
12.060.820.020.110.040.121.310.180.020.110.030.03
11.470.850.020.090.040.090.900.150.020.090.030.02
Stock Price 105C
D
100 CallDeltaGammaVegaTheta (1-day)Rho100 PutDeltaGammaVegaTheta (1-day)Rho
8.190.710.030.140.050.102.460.300.030.140.040.04
7.520.720.030.120.050.081.960.280.030.120.040.03
Stock Price 100E
F
100 PallDeltaGammaVegaTheta (1-day)Rho100 PutDeltaGammaVegaTheta (1-day)Rho
5.080.550.030.160.050.084.380.450.030.160.040.06
4.360.540.040.140.060.063.830.460.040.140.040.05
Assumptions: Volatility, 30%; interest rate, 5%; dividends, none.
The Greeks • 91
Table 4-8
The Greeks and Changing VolatilityCol 1
Col 2
Col 3
Col 4
Col 5
56 Days
42 Days
28 Days
14 Days
Exp.
2.190.280.020.100.090.025.730.540.030.110.110.0411.950.800.020.070.070.05
0.920.180.030.050.090.014.020.530.040.080.150.0210.820.870.020.040.080.03
0.001.000.000.000.000.000.000.000.000.000.000.0010.001.000.000.000.000.00
0.320.250.030.050.020.012.970.540.060.110.060.0410.500.950.020.030.030.06
0.050.190.010.010.010.002.060.530.080.080.080.0210.190.990.010.010.020.03
0.000.000.000.000.000.000.000.000.000.000.000.0010.001.000.000.000.000.00
RowA
B
C
Volatility 50%110 CallDeltaGammaVegaTheta (1-day)Rho100 CallDeltaGammaVegaTheta (1-day)Rho90 CallDeltaGammaVegaTheta (1-day)Rho
4.390.360.020.140.070.058.200.550.020.160.080.0713.940.750.020.120.060.09
3.380.330.020.120.080.037.060.550.020.140.090.0513.000.770.020.110.070.07Volatility 25%
D
E
F
110 CallDeltaGammaVegaTheta (1-day)Rho100 CallDeltaGammaVegaTheta (1-day)Rho90 CallDeltaGammaVegaTheta (1-day)Rho
1.020.300.030.110.030.034.300.550.040.160.040.0811.260.890.020.070.030.12
0.670.280.030.080.030.023.680.540.050.140.050.0610.870.910.020.060.030.09
Assumptions: Stock price, 100; interest rate, 5%; dividends, none.
92 • Trading Options As a Professional
How Delta ChangesDelta, as stated earlier, estimates how much an option value changeswhen the underlying stock price changes and other factors remainconstant. Five general rules govern how deltas change. Because callshave positive deltas and puts have negative deltas, the ﬁve rules willbe stated using the absolute values of the deltas.
Deltas and Stock PriceThe ﬁrst rule describes how deltas change as