the price of the underlyingstock changes. Deltas of both calls and puts increase as the stock pricerises and decrease as the stock price falls. Table 4-7 and Figures 4-1Aand 4-1B illustrate this concept. At ﬁrst glance, the graphs may appearidentical, but they are not. Moving up the x axis, the call delta rises from0 to 1.00, whereas the put delta rises from 1.00 to 0. Column 1 inTable 4-7 shows that as the stock price rises from 100 to 105 to 110 at56 days, the delta of the 100 Call increases from 0.55 (row E) to 0.71(row C) to 0.82 (row A), and the delta of the 100 Put rises from 0.45(row F) to 0.30 (row D) to 0.18 (row B). Remember to keep increasesand decreases straight when minus signs are involved! The same concept—deltas rising with an increasing stock price and falling with adecreasing stock price—holds true for any column in Table 4-7.
Deltas and Strike PriceThe second rule on deltas concerns the relative level of deltas of in-themoney, at-the-money, and out-of-the-money options. In-the-moneyoptions have deltas with absolute values greater than 0.50. At-themoney options have deltas with absolute values of approximately 0.50,and out-of-the-money options have deltas with absolute values less than0.50 regardless of time to expiration. Table 4-7 illustrates this rule.With a stock Price of 110, the 100 Call is in the money, and the 100Put is out of the money. Row A shows that the absolute value of the deltaof the 100 Call is always above 0.50, and row B shows that the absolutevalue of the delta of the 100 Put is always below 0.50.
The Greeks • 93
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Figure 4-1 (A) Delta of 100 Call vs. Stock Price (B) Delta of 100 Put vs.Stock Price
Deltas and Time to ExpirationThe third rule on delta concerns how deltas change as expirationapproaches. The absolute values of deltas of in-the-money optionsincrease toward 1.00 as expiration approaches. This rule is illustratedgraphically in Figure 4-2A and numerically in Table 4-7. In row Aof the table, for example, the stock price is 110, which means that
94 • Trading Options As a Professional
the 100 Call is in the money, and its delta increases from 0.82 at 56days (column 1) to 0.89 at 28 days (column 3) to 1.00 at expiration (column 5).Note that “Time to Expiry” on the x axis in Figures 4-2A, 4-2B, and4-2C decreases from right to left. While it is generally intuitive to havelower numbers on the left and higher numbers on the right (see Figures 4-1, 4-3, 4-4, and others), this is not true with time to expirationand options. Take a few moments, therefore, to accustom yourself toreading and understanding these three graphs.The absolute values of deltas of at-the-money options remain near0.50. In rows E and F of Table 4-7, where the stock price is 100,both the 100 Call and 100 Put are at the money. In these rows, theabsolute value of the deltas remain near 0.50 in all columns. Thisconcept is illustrated graphically in Figure 4-2B.The absolute values of deltas of out-of-the-money options decreasetoward zero as expiration approaches. This is illustrated graphically inFigure 4-2C and in row D of Table 4-7, where the stock price is 105,and the 100 Put is out of the money. The absolute value of the deltain this row decreases from 0.30 at 56 days (column 1) to 0.25 at28 days (column 3) to 0.00 at expiration (column 5).
Deltas of Calls and Puts with the Same StrikeThe fourth rule on deltas is that the sum of the absolute values of thecall delta and the put delta is approximately 1.00. With the stock at100 at 56 days to expiration (column 1, rows E and F), for example,the delta of the 100 Call is 0.55, and the delta of the 100 Put is0.45. The sum of the absolute values of these numbers, 0.55 and0.45, is 1.00. At any point in Table 4-7, this relationship holds trueand is another result of put-call parity.
Deltas and VolatilityThe ﬁfth rule on deltas explains how deltas change as volatility changes.The rule is that as volatility increases, the absolute value of a delta
The Greeks • 95
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Time to ExpiryFigure 4-2 (A) Delta of In-The-Money 90 Call vs. Time (B) Delta of At-theMoney 100 Call vs. Time (C) Delta of Out-of-the-Money 110 Call vs. Time
96 • Trading Options As a Professional
changes toward 0.50. In other words, deltas of out-of-the-moneyoptions increase, and deltas of in-the-money options decrease. This ruleis illustrated in Table 4-8, which has two sections. In both sections, thestock price is 100, so the 90 Call is in the money, the 100 Call is at themoney, and the 110 Call is out of the money. The bottom section ofTable 4-8 (rows D, E, and F) assumes a volatility of 25 percent, and theupper section (rows A, B and C) assumes a volatility of 50 percent. Withvolatility of 25 percent and 56 days to expiration, the in-the-money90 Call has a delta of 0.89 (column 1, row F). Raising the volatilityto 50 percent (column 1, row C) lowers the delta to 0.75. Comparing any two corresponding deltas in rows F and C yields the same result:The increase in volatility causes the delta of the in-the-money 90 Callto decrease. This concept is illustrated graphically in Figure 4-3A.For the delta of the out-of-the-money 110 Call, the change is opposite. In Table 4-8, with volatility of 25 percent and 56 days to expiration, the delta of the 110 Call is 0.30 (column 1, row D). Raisingthe volatility to 50 percent increases the delta to 0.36 (column 1,row A). Comparing any two corresponding deltas in rows D and Ayields the same result: The increase in volatility causes the delta of theout-of-the-money 110 Call to increase. This concept is illustratedgraphically in Figure 4-3C.Deltas of at-the-money options remain near 0.50 (absolute value ofdelta) over a wide range of volatility, as illustrated in Figure 4-3B androws E and B in Table 4-8. As the ﬁgure demonstrates, deltas of at-themoney options increase to 1.00 at very low levels of volatility becausethe option value itself is very low and has a high correlation with stockprice movement. Consider a hypothetical situation in which the stockprice is 100, there are 60 days to expiration, and the volatility is 1 percent. In this example, a 60-day at-the-money 100 Call might have a valueof 0.82 and a delta of 0.98. An increase of 10 cents to 100.10 wouldcause the call value to rise to 0.92 and the delta to rise to 0.99. Althougha stock with 1 percent volatility is highly improbable, this exercise canhelp traders to think through what might happen in unusual market conditions—which actually may occur in a 20- or 30-year trading career.
The Greeks • 97
Delta
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Figure 4-3 (A) Delta of In-the-Money 90 Call vs. Volatility (B) Delta of At-theMoney 100 Call vs. Volatility (C) Delta of Out-of-the-Money 110 Call vs. Volatility
98 • Trading Options As a Professional
Deltas change toward 0.50 when volatility increases and awayfrom 0.50 when volatility decreases because a change in volatilitychanges the size of one standard deviation while the distance fromstock price to strike price remains constant. Consider a situation inwhich the stock price is 100, one standard deviation is 5 percent, orﬁve points, and the 105 Call has a delta of 0.35. Under these circumstances, the strike price of 105 is one standard deviation (5 percent) away from the current stock price of 100. If volatility were todouble to 10 percent with the stock price unchanged, then the 105Call would be only one-half of a standard deviation from the stockprice. Since the 105 Call is now closer to the stock price—in volatility terms—then it’s delta must be closer to 0.50.Similarly, if volatility were to decrease with the stock price unchanged,then the 105 Call would be further out of the money in volatility terms;that is, it would be more than one standard deviation above the stockprice. Since options that are further out of the money have deltas withlower absolute values, the delta of the 105 Call would decrease given adecrease in volatility and other factors remaining constant.
Option Prices and VolatilityIn addition to information about the Greeks, Table 4-8 also reveals something signiﬁcant about the impact of changing volatility on option prices.When volatility increases, values of out-of-the-money options increaseexponentially, whereas values of at-the-money options remain approximately linear. The impact on in-the-money options is less than one forone. In column 1 of Table 4-8, for example, the doubling of volatilityfrom 25 to 50 percent causes the in-the-money 90 Call to increase by 24percent from 11.26 to 13.94. In comparison, the at-the-money 100 Callincreases signiﬁcantly by 90 percent from 4.30 to 8.20, but the out-of-themoney 110 Call shoots up dramatically by 430 percent from 1.02 to 4.39.As discussed in Chapter 3 in the section on planning trades, the signiﬁcance of changing implied volatility cannot be ignored. Positionsinvolving only out-of-the-money options face increased percentage riskfrom changes in implied volatility relative to positions involving only
The Greeks • 99
at-the-money or in-the-money options. For this reason, full-time tradersholding out-of-the-money options often employ strategies, such as vertical spreads, that reduce exposure to the risk of changing impliedvolatility.
How Gamma ChangesTable 4-7 and Figure 4-4 show that gammas are biggest when optionsare at the money, and they increase as expiration approaches. Thisconcept is signiﬁcant to option traders because it explains the wayoption prices behave as an underlying stock price changes and as anoption changes from being out of the money to at the money and thenin the money. Out-of-the-money options, with low deltas and smallergammas, do not respond dramatically to small price changes in theunderlying stock. However, as the stock approaches the strike price,the newly at-the-money option seems to “explode,” moving noticeablymore than its delta. Such option-price behavior brings tears of joy tooption owners and screams of horror to option writers.Consider the case of Debra, who bought a 100 Call for 5.08 whenthe stock price was 100 at 56 days to expiration (Table 4-7, column 1,0.04
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Figure 4-4
Gamma of 100 Call vs. Stock Price
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100 • Trading Options As a Professional
row E). If the stock rises to 110 at 28 days to expiration and Debra’sCall rises to 10.88 (column 3, row A), then she will have an unrealized proﬁt of 5.80, or $580 per option, on the 10-point rise in the stockin 28 days. However, if the stock then falls $5.00 in the next two weeks,her call will decline to 5.84 (column 4, row C). Thus, in only onehalf the time and one-half the stock-price change, almost all of Debra’sproﬁt is lost. It is the delta of 0.89, up from 0.55 initially, thatexplains the potential for loss. If Debra were aware of the new sensitivities to market changes, she may be inclined to close her positionand take a proﬁt more quickly if the market starts to move down.
Equality of Call and Put GammasTable 4-7 shows that gammas are the same for calls and puts with thesame strike, same days to expiration, and same underlying. This equality is a result of put-call parity, one part of which states that the sumof the absolute values of the call and put with the same strike priceand expiration date must total 1.00. Therefore, if the absolute valueof the delta of the call (or put) rises or falls, then the absolute value ofthe put (or call) must fall or rise an equal amount so that the sum ofthe two remains at 1.00.Rows A, B, C and D in Table 4-7 and rows A, C, D and F inTable 4-8 and Figures 4-5A and 4-5C show that gammas of in-themoney and out-of-the-money options increase only slightly until about30 days before expiration, and then they decrease to zero. Since gammais the change in delta, a slightly changing gamma is saying that deltaschange at a nearly constant rate until the last month, when they changeless. For an in-the-money call with four or ﬁve months to expiration, forexample, a $1.00 stock-price rise might increase the delta from 0.75to 0.77. This is a gamma of 0.02. With only one week to expiration,however, the same $1.00 stock-price rise would raise the delta only from0.75 to 0.76, which indicates a gamma of 0.01. Similarly, for outof-the-money options, a $1.00 stock-price rise at 90 days might cause adelta to rise from 0.33 to 0.35, whereas the same $1.00 price rise at10 days might cause the delta to rise only from 0.35 to 0.36.
The Greeks • 101
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Figure 4-5 (A) Gamma of In-the-Money 90 Call vs. Time (B) Gamma of At-theMoney 100 Call vs. Time (C) Gamma of Out-of-the-Money 110 Call vs. Time
102 • Trading Options As a Professional
Rows E and F of Table 4-7 and rows B and E of Table 4-8 andFigure 4-5B show that gammas of at-the-money options behave differently from gammas of in-the-money and out-of-the-money options.Gammas of at-the-money options are very small and nearly constant,rising only slightly, until about one month before expiration. Then theyrise dramatically until immediately before expiration, at which pointthey drop to zero. You can understand why gammas respond this wayby considering option mechanics at expiration. Traders exercise in-themoney options, converting them into stock positions. Therefore, theirdeltas at expiration are 1.00. Out-of-the-money options, however,expire worthless, so their deltas are zero. Now consider the change indelta as an option moves from slightly out of the money to in the moneyimmediately at expiration. The absolute value of its delta rises instantaneously from near zero to near 1.00. This is a very large gamma,nearly inﬁnite, as a 2 cent stock price rise from 99.99 to 100.01 changesa call’s delta from 0.00 to 1.00 and a put’s delta from 1.00 to 0.00.
Gammas and VolatilityFigures 4-6A and 4-6C illustrate the impact of volatility on gamma forin-the-money and out-of-the-money options. At low levels of volatility—from roughly 10 to 20 percent—gammas rise with volatility. As volatility rises above 30 percent, however, gamma decreases as volatility rises.This happens because, as discussed earlier, as volatility rises, the absolutevalue of delta changes toward 0.50. With absolute deltas closer to0.50, the change in delta is smaller, and that is a smaller gamma.Figure 4-6B shows that gammas of at-the-money options change differently as volatility changes. Since the absolute value of deltas of atthe-money options is near 0.50 regardless of the level of volatility,rising volatility does not change the gamma. However, at very low levels of volatility, gammas of at-the-money options rise dramatically. Thisquick rise happens because low volatility means a low standard deviation. While a $1.00 price change in a high-volatility stock might equateto a one-half-standard-deviation move or less, the same price changein a low-volatility stock might equate to a two-standard-deviation move.
The Greeks • 103
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Figure 4-6 (A) Gamma of In-the-Money 90 Call vs. Volatility (B) Gamma ofAt-the-Money 100 Call vs. Volatility (C) Gamma of Out-of-the-Money 110 Callvs. Volatility
104 • Trading Options As a Professional
Since deltas are related to the distance to the mean in standard deviation terms, an option that is two standard deviations in the money willhave a delta the absolute value of which is much greater than an optionthat is only one-half of a standard deviation in the money. Such anoption will have a high gamma because that stock-price change wouldcause the absolute delta to move from 0.50 to nearly 1.00. Similarlogic applies to out-of-the-money options and their deltas.
How Vega ChangesTables 4-7 and 4-8 and Figure 4-7 show that vegas, the change inoption value from a one percentage point change in volatility, arebiggest when options are at the money. In any column, vegas arebiggest when the 100 Call and 100 Put are at the money (rows E andF in Table 4-7 and rows B and E in Table 4-8). At-the-money optionshave the largest vegas because a change in volatility has the biggestabsolute impact on the price of at-the-money options. In Table 4-8(column 1), for example, the increase in volatility from 25 to 50 percent causes the 90 Call to increase in price from 11.26 (row F) to13.94 (row C), an increase of 2.68. The at-the-money 100 Call, however, increases by the larger amount of 3.90 from 4.30 (row E) to 8.20(row B). The 110 Call also increases more than the out-of-the-moneycall but less than the at-the-money call by 3.37 from 1.02 (row D) to4.39 (row A).
Vegas and Time to ExpirationTables 4-7 and 4-8 and Figures 4-8A, 4-8B, and 4-8C show that vegasdecrease as expiration approaches for in-the-money, at-the-money, andout-of-the-money options. Looking across any row from column 1 tocolumn 5 in Table 4-7, the vegas get smaller. Not only do option pricesdecrease, but vegas also decrease as expiration approaches because thepotential for stock-price movement is less when there is less time.Figures 4-8A, 4-8B, and 4-8C show the same concept. However, for
The Greeks • 105
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Vega of 100 Call vs. Stock Price
at-the-money options (see Figure 4-8B), vegas remain higher and takelonger to get to zero than vegas of in-the-money and out-of-the-moneyoptions.
Vegas and VolatilityTable 4-8 and Figures 4-9A, 4-9B, and 4-9C show how vegas changeas volatility changes. The message is that for at-the-money options(see Figure 4-9B) at 10 percent volatility and higher, vega is constant.Consequently, changing volatility has a linear impact on prices of atthe-money options: If volatility rises or falls by 5 percent—from anylevel to any other level—then the rise or fall in option value will beﬁve times the vega. Consider, for example, the change in price of the100 Call in column 1 of Table 4-8 from 4.30 (row E, 25 percentvolatility) to 8.20 (row B, 50 percent volatility). The vega in row E is0.16. Then 25 times 0.16 plus 4.30 equals 8.30, which is approximately 8.20, the value that appears in row B, column 1. The difference is due to rounding because the vega is rounded to twodecimal places.
106 • Trading Options As a Professional
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Figure 4-8 (A) Vega of In-the-Money 90 Call vs. Time (B) Vega of At-theMoney 100 Call vs. Time (C) Vega of Out-of-the-Money 110 Call vs. Time
The Greeks • 107
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Figure 4-9 (A) Vega of In-the-Money 90 Call vs. Volatility (B) Vega of At-theMoney 100 Call vs.Volatility (C) Vega of Out-of-the-Money 110 Call vs. Volatility
108 • Trading Options As a Professional
Vegas and Strike PriceFor in-the-money and out-of-the-money options, vega is near zero atlow levels of volatility (below 10 percent) and rises gradually untilabout 50 percent volatility, at which point it levels out even as volatility continues to rise. Changing volatility, therefore, does not have thesame linear impact on prices of in-the-money and out-of-the-moneyoptions as it does for at-the-money options. Consider the change inprice of the 110 Call in column 1 of Table 4-8 from 1.02 (row D,25 percent volatility) to 4.39 (row A, 50 percent volatility). The vegain row D is 0.11. Then 25 times 0.11 plus 1.02 equals 3.77, whichis not very close to the value of 4.39 that appears in row A, column1 of Table 4-8. The at-the-money estimation technique fails to workhere because of the different impact volatility has on in-the-moneyand out-of-the-money options.
How Theta ChangesTraders need to understand how thetas change because the impact oftime erosion on option prices will directly affect trading strategies. Surprisingly, traders frequently misunderstand or oversimplify this concept, usually with unfortunate results. A word of warning: Theta, theestimate of the impact of time on option values, is preceded by aminus sign, which can be confusing when discussing “biggest” and“smallest” values. Read this section carefully!Table 4-7 and Figure 4-10 show that thetas are smallest (the highest absolute value) when options are at the money. The differencesshow up more clearly in the ﬁgure than they do in the table because,on an absolute level, the numbers seem small, between 0.00 and0.05. Also, even though rows E and F of Table 4-7 reﬂect thetas attheir smallest (highest absolute value) when the 100 Call and 100 Putare at the money, the differences are not obvious because the numbers are rounded to two decimal places. At-the-money options havelarger time values than in-the-money or out-of-the-money options, andit is the time-value portion of an option’s price that erodes. Therefore,
The Greeks • 109
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Theta of 100 Call vs. Stock Price
given the same amount of time to expiration, at-the-money optionslose more value per unit of time than in-the-money or out-of-themoney options.
Thetas and Time to ExpirationTables 4-7 and 4-8 and Figure 4-11B show that thetas of at-the-moneyoptions decrease (increase in absolute value) as expiration approaches.Then, almost immediately before expiration, they go to zero. Thetas ofat-the-money options are smallest (largest absolute value) during thelast unit of time prior to expiration