. In row E of Table 4-7, in which thestock price is 100, the theta of the 100 Call starts at 0.05 (column 1)and then decreases to 0.06, 0.07, and 0.09 before going to zeroat expiration. The theta of the at-the-money put behaves similarly.Table 4-7 and Figures 4-11A and 4-11C show that thetas of in-themoney and out-of-the-money options behave differently than thetas ofat-the-money options. They get smaller (absolute value increases) fora while, but then they get larger (absolute value decreases) as expiration approaches. Because thetas behave differently for in-the-money,at-the-money, and out-of-the-money options, traders must be careful
110 • Trading Options As a Professional
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Figure 4-11 (A) Theta of In-the-Money 90 Call vs. Time (B) Theta of At-theMoney 100 Call vs. Time (C) Theta of Out-of-the-Money 110 Call vs. Time
The Greeks • 111
when making generalizations about the impact of time decay onoption values.
Using Theta with DeltaHow does a trader use theta? Since theta estimates how much a position will make or lose over some period of time, a trader buying optionscan use theta in conjunction with delta to estimate how much theunderlying stock price must change in price in a speciﬁc time periodfor the delta effect (price movement of the underlying) to make morethan the theta effect (time decay). Assume, for example, that an optionhas a one-day theta of 0.05 and a delta of 0.35. The buyer of thisoption therefore needs a $1.00 price rise in the stock in seven days tooffset the time erosion—the delta effect of 0.35 will offset the thetaeffect of seven times 0.05. Although market forecasting is an art, nota science, having a time period and a price target give the trader aframe of reference on which to base a subjective trading decision.
Thetas and VolatilityTable 4-8 and Figures 4-12A, 4-12B, and 4-12C show that thetas of inthe-money, at-the-money, and out-of-the-money options decrease(increase in absolute value) as volatility increases. This result is logical because an increase in volatility increases option values. And, giventhe same time to expiration, a higher option value contains more timeerosion per unit of time.
How Rho ChangesRho estimates how much an option value changes when the interestrate changes and other factors remain constant. Rho is generally ofleast concern to option traders because rhos are small in absoluteterms and because interest rates generally do not change dramatically,that is, more that 1 percent, in a short period of time. Nevertheless,traders need to learn four rules about how rhos change.
112 • Trading Options As a Professional
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Figure 4-12 (A) Theta of In-the-Money 90 Call vs. Volatility (B) Theta of At-theMoney 100 Call vs. Volatility (C) Theta of Out-of-the-Money 110 Call vs. Volatility
The Greeks • 113
The ﬁrst rule is that rhos of calls are positive and rhos of putsare negative. This is a result of the cost-of-carry concept explainedin Chapter 6, where the conversion strategy is discussed. Withrising interest rates, the cost to ﬁnance a stock position increases.As a result, the time value of a call must increase relative to thetime value of the put. Therefore, if interest rates rise while stockprices and put prices remain constant, then call prices must increaseto pay for the increased ﬁnancing costs. Rhos of calls therefore arepositive.Similarly, if interest rates rise while stock prices and call pricesremain constant, then put prices must decrease so that the put-calltime-premium differential increases enough to pay for the increasedﬁnancing costs. Rhos of puts therefore are negative. In reality, neither a call nor a put remains constant while the other changes. Whathappens is that call premiums rise a little and put premiums falla little.
Rhos and Stock PriceThe second rule governing rhos describes the relative level of rhos asthe underlying stock price changes. Table 4-7 and Figures 4-13A and4-13B show that rhos increase as the stock price rises. Column 1 inTable 4-7, for example, shows that as the stock price rises from 100 to105 to 110 at 56 days (column 1), the rho of the 100 Call rises from0.08 (row E) to 0.10 (row C) to 0.12 (row A), and the rho of the100 Put rises from 0.06 (row F) to 0.04 (row D) to 0.03 (row B).Remember to keep increases and decreases straight when minus signsare involved! The same concept—that rhos rise with a rising stockprice and fall with a falling stock price—holds true for any column inTable 4-7. Figures 4-13A and 4-13B demonstrate this concept graphically. At ﬁrst glance, the graphs may appear identical, but they arenot. The rho of the call rises from 0 to 0.17, and the rho of the putrises from approximately 0.17 to 0.
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(A) Rho of 100 Call vs. Stock Price (B) Rho of 100 Put vs.
That rhos increase with rising stock prices is another result of thecost-of-carry concept. Higher-priced stocks are more expensive toﬁnance than lower-priced stocks, and changes in interest rates have agreater impact on the absolute cost of ﬁnancing high-priced stocksthan on low-priced stocks.
The Greeks • 115
Rhos and Time to ExpirationFigures 4-14A, 4-14B, and 4-14C show the third rule. Rhos increasein an almost linear manner with increases in time. The cost-of-carryconcept also explains this consequence. At a given interest rate, ﬁnancing costs are higher—in a linear relationship—for longer time periods than for shorter periods. And if interest rates change, the absoluteimpact will be higher—linearly—for the longer time period than forthe shorter.
Rhos and VolatilityThe fourth rule describes the impact of volatility on rho. This conceptis complicated. Table 4-8 and Figures 4-15A, 4-15B, and 4-15C showthat volatility has different effects on rhos of in the money, at themoney, and out of the money options. The difﬁcult aspect to grasp isthat volatility only affects rho indirectly through its impact on optionprices.Consider rows D and A in column 1 of Table 4-8, which show thatan increase in volatility from 25 to 50 percent increases the rho of the110 Call from 0.03 to 0.05. Note also that the call price rises from 1.02to 4.39. In addition to the change in cost of carry of the underlyingstock, a change in the interest rate also would affect the cost (foregoneinterest) of owning the call. The foregone interest on 4.39 is 400 percent that of the foregone interest on 1.02 regardless of the level of interest rates. Therefore, the effect of a change in the interest rate must begreater when volatility is higher than when it is lower. Figure 4-15Cshows that rising volatility has an exponential impact on the rho of anout-of-the-money call from approximately 10 to 50 percent volatility.Above 50 percent, the impact of rising volatility levels off, and rhoapproaches its limit, just as the option price approaches its limit at highlevels of volatility (see Figure 2-6).Figure 4-15B shows that the impact of volatility on rhos of at-themoney calls is more linear—and actually declining—than for out-ofthe-money calls, and Figure 4-15A shows that rising volatility causes
116 • Trading Options As a Professional
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Figure 4-14 (A) Rho of In-the-Money 90 Call vs. Time (B) Rho of At-theMoney 100 Call vs. Time (C) Rho of Out-of-the-Money 110 Call vs. Time
The Greeks • 117
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118 • Trading Options As a Professional
rhos of in-the-money calls to decrease. Fortunately, this complicatedinteraction of volatility and rho is of little importance to the vast majority of traders, even full-time ones.
Position GreeksThe term position refers to whether an option is purchased (i.e., long)or written (i.e., short). For example, if Adam buys 25 XYZ November100 Calls, his “position” is long 25. If Matthew buys 15 QRS April 45Puts and sells 15 QRS April 40 Puts, his position is long 15 April 45Puts and short 15 April 40 Puts.What Adam and Matthew and all traders need is a method of estimating how their position will perform if market conditions change,that is, if one or more of the inputs to the option-pricing formulachanges. Position Greeks indicate whether an entire position will experience proﬁt or loss when a particular input to the option-pricing formula is changed.You will learn how to calculate and interpret position Greeks afterthe following discussion about the use of positive and negative signs.In options trading, plus and minus signs can have three differentmeanings.
“” And “” Have Three Different MeaningsFirst, when associated with a quantity of options, plus signs mean“long,” and minus signs mean “short.” The position description “3NDX January 2200 Puts at 12.50” is read as “long 3 NDX 2200 Puts at12.50 each.” The position description “15 XSP November 145 Callsat 9.10” is read as “short 15 XSP November 145 Calls at 9.10 each.”Second, when associated with an option’s delta, vega, theta, or rho,plus and minus signs mean that the option value is positively or negatively correlated with changes in the respective input. “The call hasa delta of 0.65” means that the value of the call is positively correlated with changes in the price of the underlying stock; that is, if the
The Greeks • 119
stock price rises, the call value will rise, and if it falls, the call valuealso will fall.A second example uses the phrase, “the put has a rho of 0.08,”which means that the value of the put is negatively correlated withchanges in interest rates. If interest rates rise, then the put value willdecrease. And a third example, “the put has a vega of 0.20,” meansthat the value of the put is positively correlated with changes in volatility. If volatility rises, put value rises; if volatility decreases, put valuedecreases.Plus signs associated with gamma mean that the delta of the position is positively correlated with changes in price of the underlying;that is, stock price up, delta up; stock price down, delta down. Andnegative signs associated with gamma mean that the option’s delta isnegatively correlated with changes in price of the underlying stock:Stock price up, delta down; stock price down, delta up.Finally, when associated with the Greek of an entire option position,plus and minus signs, with one exception, indicate whether a positionwill proﬁt from or lose from an increase in the corresponding factor.Consider, for example, “the vega of Sally’s three long calls is 2.73.”The plus sign means that Sally’s position will proﬁt by 2.73 points ifvolatility rises 1 percent, and other factors remain constant. Anotherexample is “the theta of Bill’s four long puts is –3.64.” The minus signmeans that Bill’s position will lose 3.64 points if time changes by oneunit, and other factors remain constant.The three different meanings may be hard to remember withoutsome practice, so keep this in mind: (1) long or short, (2) positively ornegatively correlated, and (3) proﬁt or loss. During the following discussion of position Greeks, keep in mind that plus and minus signscan mean any of these depending on usage.
Introduction to Tables 4-9 through 4-18Position Greeks are illustrated with 10 tables that have the same format. Each table contains two examples. The ﬁrst example involves
120 • Trading Options As a Professional
four XYZ 80 Calls initially purchased or sold for 4.20 each. Thesecond example involves 10 QRS 40 Puts initially purchased or soldfor 0.81 each. Each example has several steps in numbered rows. Thesix rows state (1) the initial position, (2) the relevant input and thechange (i.e., stock price, days to expiration, volatility, etc.), (3) theGreek of an individual option, (4) the beginning position Greek, (5)the beginning and ending position values, and (6) a conclusion thatsummarizes the signiﬁcance of the example.These examples fulﬁll two purposes. The ﬁrst purpose is to showhow to calculate position Greeks. The second is to explain how to usea position Greek to estimate the change in a position value.
Position DeltaA position with a positive delta will proﬁt if the price of the underlying rises and will lose if it declines, assuming that other factors remainconstant. Long call positions and short put positions have positivedeltas, and Table 4-9 shows an example of each. Column 1, row 1 ofthe long call example describes the position as long four XYZ 80 Calls.The purchase price is 4.20 for each option, and the ending price is4.77 (column 3). Row 2 shows that the stock price rises from 80 to 81,and row 3 indicates that the delta is 0.55 when the calls are purchased. Row 4 shows that the position delta of 2.20 (column 2) isthe product of the quantity of long calls (4) and the option delta(0.55). Row 5 shows that the initial position value is the product ofthe quantity of calls (4) and the price of each call (4.20). The resulting position value, 16.80 debit, is reﬂected in column 2. Debit meansthat a trader makes a payment to establish the position and receivesmoney when the position is closed. Thus an increase in a debit position is a proﬁt and a decrease is a loss. In this case, an amount equalto 16.80 points, or $1,680, is the amount invested in the initial position. Column 3, row 5 indicates that the ending position value is“19.08 debit,” or $1,908. This is an increase and represents a proﬁt of2.28 ($228) from the beginning position value.
The Greeks • 121
The conclusion in row 6 of the long call example in Table 4-9 statesthat “the position delta of 2.20 estimates that a one-point stock-pricerise will cause the position to proﬁt by $220, but the actual proﬁt was$228 owing to the increasing delta.” In real trading situations, theactual result will not always exactly equal the estimate for a numberof reasons. First, depending on the size of the stock-price change, thegamma may change the delta signiﬁcantly enough to make the resultvary from the estimate. Second, rounding of numbers easily can leadto differences between an actual result and an estimate. Third, theassumption that all other factors, such as time and volatility, remainconstant may not hold.
Table 4-9
Positive Delta Positions—Long Calls and Short Puts
Long Call Example
Col 1
Col 2
Col 3
1 PositionLong 4 XYZ 80 Calls4.20 each→4.772 Stock price80.00→81.003 Option delta0.554 Position delta4  0.55 2.205 Position value4  4.20 16.80 debit→ 19.08 debit6 Conclusion: The position delta of 2.20 estimates that a one-point stock-pricerise will cause the position to proﬁt by $220, but the actual proﬁt was $228(from 16.80 debit to 19.08 debit) owing to the increasing delta.Assumptions: Days to exp., 60; volatility, 30%; interest rate, 5%; no dividends.
Short Put Example
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1 PositionShort 10 QRS 40 Puts0.81 each→0.522 Stock price41.00→42.003 Option delta0.344 Position delta 10  0.34 3.405 Position value 10  0.818.10 credit→ 5.20 credit6 Conclusion: The position delta of 3.40 estimates that a one-point stock-pricerise will cause the position to proﬁt by $340, but the actual proﬁt was $290(from 8.10 credit to 5.20 credit) owing to decreasing delta.Assumptions: Days to exp., 40; volatility, 25%; interest rate, 5%; no dividends.
122 • Trading Options As a Professional
The short put example in Table 4-9 is 10 short QRS 40 Puts at0.81 each. The position delta of 3.40 (row 4) estimates that these10 short puts (10) will make a proﬁt of 3.40, or $340, if QRS stockrises by one point or will lose this amount if the stock falls by onepoint and other factors remain constant. Row 2 shows that the stockprice rises by $1.00 from 41 to 42, and row 5 shows that the positionvalue changes from an 8.10 Credit to a 5.20 Credit for a proﬁt of2.90 ($290). Credit means that a trader receives money when a position is established and makes a payment when the position is closed.Thus a decrease in a credit position is a proﬁt and an increase is aloss. In this example, the estimated result, a proﬁt of $340, does notexactly equal the actual result, a proﬁt of $290. As with the long callexample, the difference in the actual result is caused by the changing delta.Table 4-10 shows that short call positions and long put positionshave negative deltas. A position with a negative delta will lose if theprice of the underlying rises and proﬁt if it declines, assuming thatother factors remain constant.
Position GammasGammas of positions do not indicate proﬁt or loss. Rather, they indicate how the position delta will change when the price of the underlying changes. A positive gamma indicates that the position delta willchange in the same direction as the change in price of the underlying. A negative gamma indicates that the position delta will changein the opposite direction from the change in price of the underlying.Table 4-11 shows that long call and long put positions have positivegammas. In the ﬁrst example in Table 4-11, each long call has a gammaof 0.04 (row 4), and the position gamma is 0.16 (row 5). The positive position gamma estimates that if the stock price rises one point, andother factors remain constant, then the position delta will rise by 0.16from 2.20 to 2.36. The beginning and ending position deltas presented in row 6 verify this. Positive gamma means this: Stock price up,position delta up, or stock price down, position delta down.
The Greeks • 123
Table 4-10
Negative Delta Positions—Short Calls and Long Puts
Short Call Example
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1 PositionShort 4 XYZ 80 Calls4.20 each→4.772 Stock price80.00→ 81.003 Option delta0.554 Position delta4  0.55 2.205 Position value4  4.20 16.80 credit→ 19.08 credit6 Conclusion: The position delta of 2.20 estimates that a one-point stock-pricedecline will cause the position to lose $220, but the actual loss was $228(from 16.80 credit to 19.08 credit) owing to decreasing delta.Assumptions: Days to exp., 60; volatility, 30%; interest rate, 5%; no dividends.
Long Put Example
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1 PositionLong 10 QRS 40 Puts0.81 each→0.522 Stock price41.00→ 42.003 Option delta0.344 Position delta10  0.34 3.405 Position value10  0.818.10 debit→5.20 debit6 Conclusion: The position delta of 3.40 estimates that a one-point stock-pricerise will cause the position to lose $340, but the actual loss was $290 (from8.10 debit to 5.20 debit) owing to increasing delta.Assumptions: Days to exp., 40; volatility, 25%; interest rate, 5%; no dividends.
With a positive gamma, the change in delta works to the advantageof the position. Referring back to Table 4-2, as the underlying stockincreases from 100 to 101, the delta of the 100 Call increases from0.55 to 0.58. Underlying price up, delta up! This beneﬁts the callowner because the market exposure, that is, the delta, is changing inthe call owner’s favor. Initially, the call owner’s exposure to the market was a delta of 0.55, which means that for every one-pointincrease in the underling, with other factors constant, the call ownerparticipates by approximately 55 percent. After a one-point price risein the stock, however, the call owner’s exposure has increased toapproximately 58 percent. And as the market continues to rise, the callowner makes more and more per unit of price change because thedelta of the position is increasing toward 1.00.
124 • Trading Options As a Professional
Table 4-11
Positive Gamma Positions—Long Calls and Long Puts
Long Call Example
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1 PositionLong 4 XYZ 80 Calls4.20 each2 Stock price80.00→81.003 Option delta0.55→0.594 Option gamma0.045 Position gamma 4  0.04 0.166 Position delta4  0.55 2.20→2.367 Conclusion: The position gamma of 0.16 estimates that a one-point stockprice rise will cause the position delta to rise by 0.16, and the actual rise was0.16 (from 2.20 to 2.36).Assumptions: Days to exp., 60; volatility, 30%; interest rate, 5%; no dividends.
Long Put Example
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1 PositionLong 10 QRS 40 Puts0.81 each2 Stock price41.00→42.003 Option delta0.34→0.244 Option gamma0.115 Position gamma 10  0.11 1.106 Position delta10  0.34 3.40→2.407 Conclusion: The position gamma of 1.11 estimates that a one-point stockprice rise will cause the position delta to rise by 1.10, and the actual rise was1.00 (from 3.40 to 2.40) owing to rounding.Assumptions: Days to exp., 40; volatility, 25%; interest rate, 5%; no dividends.
What about a price decline? Look back at Table 4-3. As the priceof the underlying declines and other factors remain constant, the callowner loses less than the amount estimated by the initial delta. Thisresult happens because the delta decreases. Losing less than theamount estimated by the original delta is a beneﬁt to the call owner.The second example in Table 4-11 shows that a positive positiongamma also has a beneﬁcial impact on a long put position when themarket rises. As the stock price rises from 41 to 42 (row 2), the deltaof the position in row 6 rises from 3.40 to 2.40, only slightlydifferent from the position gamma of 1.10. Consequently, less is lostthan estimated by the original delta.
The Greeks • 125
Negative GammaTable 4-12 shows that short call and short put positions have negativegammas. Negative gamma means that the delta of the position willchange in the opposite direction from the change in price of the underlying: Stock up, position delta down, or stock down, position delta up.The ﬁrst example in Table 4-12 is short four XYZ 80 Calls at 4.20each. The position gamma of 0.16 (row 5) estimates that as the stockprice rises, the delta of the position will decrease by 0.16. The actualchange in position delta, from 2.20 to 2.36 in row 6, exactly equalsthis estimate.In the short put example in Table 4-12 the position gamma of1.10 estimates that the position delta will decrease by this amountif the stock rises one point and will increase by this amount if the stockprice falls by one point, other factors remaining constant. The actualchange in delta is 1.00 as the position delta in row six changes from3.40 to 2.40. The difference between the actual and the estimateis due to rounding.When a position has a negative gamma, any change in delta worksto the disadvantage of the position. Assume, for example, that a tradersells the 100 Call in Table 4-2. The initial delta of the short call position is 0.55, and the position gamma is 0.03. This position gammaestimates that an increase in the stock price from 100 to 101 causes theposition delta to decrease by 0.03 from 0.55 to 0.58. Underlyingprice up, delta down! This stock increase hurts the short call positionbecause the loss becomes larger than the loss estimated by the initialdelta. As the market continues to rise, the position loses more and moreper unit of price rise as the exposure to the market of the short call position continues to decline toward 1.00 per short call.Table 4-3 shows that a negative gamma also works to the disadvantage of a short call position when the stock price declines. Assumingagain that a trader sells the 100 Call in Table 4-3, the initial delta is0.55. As the stock declines from 100 to 99, the short call positionproﬁts less than the amount estimated by the delta. The proﬁt is lessbecause the delta of the short call is increasing from 0.55 to 0.51.
126 • Trading Options As a Professional
Table 4-12
Negative Gamma Positions—Short Calls and Short Puts
Short Call Example
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1 PositionShort 4 XYZ 80 Calls4.20 each2 Stock price80.00→8