T T0 20 40 60 80 100 120 140 160 180 200Time to Expiration (days)Figure 6-16: Put Theoretical Value vs. Time to Expiration14Pop100 eetole eee usa2©= 8 Underlying Price = 1008 Volatility = 20%2 6é 100 ee44 ae—af uea0 - T T ~ T T T T0 20 40 60 80 100 120 140 160 180 200
Time to Expiration (days)
eyOption Values and Changing Market Conditions 113
Figure 6-17: Call or Put Theta vs. Time to Expiration
 
      
 
 
 
 
0.25Underlying Price = 100; Volatility = 20%0.2070.155s 'oxF0.107 |, 1000.08- a110, —— : —— |0.00-1 = = T T T T T T T T 10 20 40 60 80 100 120 140 160 180 200
Time to Expiration (days)
 
stances, be greater than the volatility component. If this happens, and the option isEuropean (no early exercise permitted), it will have a theoretical value less than parity(less than intrinsic value). As expiration approaches, the value of the option will slowlyrise to parity. Hence, the option will have a positive theta. A put option on stock can besimilarly affected if the put is deeply in-the-money and, again, no early exercise ispermitted. If interest rates are high enough, the put may be worth less than parity, andSo wiil rise to parity as expiration approaches. In both cases, however, the option mustbe European. American options, where early exercise is a possibility, can never be worthless than parity, and so can never have positive thetas.4 Since the conditions underwhich a European option will have a positive theta are rare, in general we can assumethat an option will lose value as time passes.
THE VEGA OR KAPPA
Just as we are concerned with the effect on an option’s theoretical value by movementin the underlying contract (delta) and the passage of time (theta), we must also be
4. One can argue that a put on a stock which {s about to pay a large dividend will also have a positivetheta. When the stock gives up the dividend, it will drop by approximately the amount of the dividend,and this drop in price will increase the value of the put. The reader can make his own decision as towhether this reflects a positive theta, or is actually the result of some change in market conditions (thestock going ex-dividend).114 Chapter 6
concerned with a change in volatility. This effect is shown in Figures 6-18 and 6-19,While the terms delta, gamma, and theta are found in most option texts, there is nogenerally accepted term for the sensitivity of an option’s theoretical value to a changein volatility. The most commonly used term in the trading community is vega, and thisis the term which will be used in this book. But this is by no means universal. Sincevega is not a Greek letter, a common alternative in academic literature, where Greekletters are preferred, is kappa (K).
The vega of an option is usually given in point change in theoretical value for eachone percentage point change in volatility.5 Since all options gain value with risingvolatility, the vega for both calls and puts is positive. If an option has a vega of .15, foreach percentage point increase (decrease) in volatility, the option will gain (lose) .15 intheoretical value. If the option has a theoretical value of 3.25 at a volatility of 20%, thenit will have a theoretical value of 3.40 at a volatility of 21%, and a theoretical value of3.10 at a volatility of 19%.
Note in Figure 6-20 that an at-the-money option always has a greater vega thaneither an in-the-money or out-of-the-money option when all options are of the same typeand have the same amount of time to expiration. This means that an at-the-moneyoption is always the most sensitive in total points to a change in volatility. As a corollary,
Figure 6-18: Call Theoretical Value vs. Volatility14
12} eeee104 co
 
2— 8 nderlyi§ || - eeee—
een -
| _
 
 
 
5 10. 15 20 25 30 35 40Volatility (%)
S$. The theta and the vega are sometimes expressed in dollar (or other appropriate currency) change in theoption’s value as time passes or volatility changes,Option Values and Changing Market Conditions 115
Figure 6-19: Put Theoretical Value vs. Volatility
 
 
 
 
14125 -90 _.. .tol ~ —- oe ,o2o> 8-g Underlying Price = 1009 Time = 60 daysoaee osBe[a| 100.anpe ce /oe ! T T T T0
5 10 15 20 25 30 35 40Volatility (%)
the out-of the-money option is always the most sensitive in percent terms to a changein volatility. For example, suppose that with a volatility assumption of 15% an at-themoney and an out-ofthemoney option have theoretical values of 2.00 and .50,respectively. If we raise our volatility assumption to 20%, the options might now havetheoretical values of 3.00 and 1.00. The at-the-money option has shown the greater totalpoint increase (1.00 vs. 50), while the out-of-the-money option has shown the greaterpercent increase (50% vs. 100%). Since many option strategies involve buying andselling unequal numbers of options, the latter characteristic wil! be important when welook at trading strategies. (The reader can confirm this by looking again at Figure 4-11.)
Note also in Figure 6-20 that the vega of all options declines as expirationapproaches. Therefore, a long-term option will always be more sensitive to a change involatility than a short-term option with otherwise identical contract specifications. Afour-month option will have a greater vega, and therefore be more sensitive to a changein volatility, than a one-month option.
This last point illustrates an important principle of option evaluation, that time andvolatility are closely interconnected. More time to expiration means more time forvolatility to take effect, while less time to expiration may mean that any change involatility will have only a minor effect on an option’s value. Moreover, changes in theamount of time remaining to expiration and changes in volatility often have similareffects on an option’s value. Decreasing volatility is similar to decreasing time toexpiration. Ifa trader cannot remember what effect the passage of time will have on his116 Chapter 6
Figure 6-20: Call or Put Vega vs. Time to Expiration0.305
 
Underlying Price = 1000.257 Volatility = 20%
100.0.207 en
0.15" oa
Vega
0.1074
0.054 °
 
 
 
0.00+1 — T T T T T T T TQ 20 40 60 80 100 120 140 160 180 200
Time to Expiration (days)
position, he might consider instead what effect a reduction in volatility will have. Thiscan be seen by comparing Figures 6-15 and 6-18 (theoretical values with respect tochanges in time and volatility), Figures 6-11 and 6-13 (delta values with respect tochanges in time and volatility), and Figures 6-9 and 6-10 (gamma values with respectto changes in time and volatility). In each case the similar shapes of the graphs indicatesimilar effects of time and volatility.
Finally, we can see in Figure 6-21 that the vega of an at-the-money option isrelatively constant, regardless of changes in volatility. This makes it easy to estimate thetheoretical value of an at-the-money option under a wide variety of volatility scenarios.
THE RHO
The sensitivity of an option’s theoretical value to a change in interest rates is given byits rho (P). Unlike the other sensitivities, one cannot generalize about the rho since itscharacteristics depend on the type of underlying instrument and the settlement proce-dure for the options. The general effects have already been summarized in Figure 6-2.Note that foreign currency options which require the delivery of the currency, ratherthan the delivery of a futures contract, are affected by both domestic and foreign interestrates. Hence, such options have two interest rate sensitivities, rho, (the domesticinterest tate sensitivity) and rho, (the foreign interest rate sensitivity]. The latter issometimes denoted with the Greek letter phi (®).Option Values and Changing Market Conditions 117
Figure 6-21: Call or Put Vega vs. Volatility0.18
 
100
     
O16) °°" OT
0.147 ce a]Underlying Price = 100 wo 7”0.127 Time = 60 days .
    
0.10+
Vega
0.08
0.06-
0.04-
0.02-
 
 
 
 
T T T
0,.00+0 25 30 35 40
Volatility (%)
 
Tf both the underlying contract and options are subject to futures-type settlement,the rho must be zero, since no cash flow results from either a trade in the underlyingcontract or a trade in the options. When options on futures are subject to stock-typesettlement, the rho associated with both calls and puts is negative. An increase ininterest rates will decrease the value of such options since it raises the cost of carryingthe options. In the case of stock options, calls will have positive rhos (an increase ininterest rates will make calls a more desirable alternative to buying the stock) and putswill have negative rhos (an increase in interest rates will make puts a less desirablealternative to selling the stock).
Although changes in interest rates can affect an option’s theoretical value, theinterest rate is usually the least important of the inputs into a pricing model. For thisTreason the rho usually takes a back seat to the other, more critical, sensitivities: the delta,gamma, theta, and vega. Indeed, few traders pay much attention to the rho. However,for completeness, some of the characteristics of the rho are shown in Figures 6-22 (stockoptions) and 6-23 (futures options). In all cases the options which have the highest rhoare those which are deeply in-the-money, because they require the greatest cash outlay.And the greater the amount of time to expiration, the greater the rho. Note also thatbecause an option is a substitute for a position in the underlying instrument, the rho forstock options is much greater in magnitude than the rho for futures options. Thepurchase or sale of stock results in a much greater cash flow than the purchase or saleofa futures contract. (The vertical scale in Figure 6-23 is approximately ten times greater118
Chapter 6
Figure 6-22: Futures Option Rho vs. Underlying Price
 
-0.017
-0.02 "|
+0.034 7"
 
 
 
 
-0.044 7 call (150 days)Q ’ ms£ a :a 0.054 put (150 days)“0.065 ra Exercise Price = 100 onf Volatility = 20% N\.“0.075 fo Interest Rate = 8.00% NN-0,08+"-0.09 T T T T T T T80 85 90 95 100 105 110 115 120
Underlying Price
than the vertical scale in Figure 6-22.) Because of its relatively minor importance, wewill generally disregard the rho in analyzing option strategies and managing risk.
SUMMARY
It is important for the active option trader to become familiar with delta, gamma, theta,and vega characteristics, since he may need to make quick decisions about tradingstrategies and risk management, decisions which might well determine his financial
fate. The following is a summary of these characteristics:
Delta—Deltas range from zero for far outof-the-money calls to 100 fordeeply in-the-money calls, and from zero for far out-of-the-money puts to
-100 for deeply in-the-money puts.
At-the-money calls have deltas of approximately 50, and at-the-money
puts approximately —50.
As time passes, or as we decrease our volatility assumption, call deltas moveaway from 50, and puts deltas away from —SO. As we increase our volatilityassumption, call deltas move towards 50, and put deltas towards -S0.
Gamma—aAt-the-money options have greater gammas than either in- orout-of-the-money options with otherwise identical contract specifications.Option Values and Changing Market Conditions
Figure 6-23: Stock Option Rho vs. Underlying Price
Rho
0.17
r-0.25
119
 
0.4 .Exercise Price = 1000.34 Volatility = 20%Interest Rate = 8.00%
0.25
0.15
we
-0.34 ee
 
   
 
 
T
80 85 90 9 100 105 110 115Underlying Price
As we increase our volatility assumption, the gamma of an in- orout-of-the-money option rises, while the gamma of an at-the-money optionfalls. As we decrease our volatility assumption, or as time to expirationgrows shorter, the gamma of an in- or out-of-the money option fails, whilethe gamma of an at-the-money option rises, sometimes dramatically.
Theta—At-the-money options have greater thetas than either in- or out-ofthe-money options with otherwise identical contract specifications.
The theta of an at-the-money option increases as expiration approaches.A short-term, at-the-money option will always decay more quickly than along-term, at-the-money option.
As we increase (decrease) our volatility assumption, the theta of anoption will rise (fall). Higher volatility means there is greater time valueassociated with the option, so that each day’s decay will also be greaterwhen no movement occurs.
Vega—aAt-the-money options have greater vegas than either in- or out-of:the-money options with otherwise identical contract specifications.
Out-of-the-money options have the greatest vega as a percent of theoreti-cal value.120 Chapter 6
The vegas of all options decrease as time to expiration grows shorter. Along-term option is always more sensitive to a change in volatility than ashort-term option with otherwise identical contract specifications.
The vega of an at-the-money option is relatively constant with respect tochanges in volatility. If we raise or lower volatility, the option’s vega isunlikely to change significantly.
In Figure 6-3 we summarized the general effects on stock option values if theexpected dividend payment for the underlying stock changes. But we did not addressthe question of magnitude. How sensitive is an option’s value to a change in dividends?When a stock goes ex-dividend, its price will normally drop by approximately theamount of the dividend. Increasing the dividend is therefore similar to dropping thestock price, and cutting the dividend is similar to increasing the stock price. If a stockhad been expected to pay a dividend of $.75 per share, and the dividend is increased to$1.00 per share, this is similar to dropping the stock price by an additional $.25. Sincean option’s sensitivity to a change in the price of the underlying is given by the delta,the sensitivity to a change in the dividend can also be approximated by the delta. If thedividend is increased by $.25, a call with a delta of 60 will lose about $.15 in value. Aput with a delta of —40 will gain about $.10. This approximation applies only when onedividend payment is expected prior to expiration. If multiple dividend payments areexpected over the life of the option, the effect on the option’s value must be increasedaccordingly. As we will see, dividend considerations play their greatest role in arbitrageStrategies and early exercise.
Two typical evaluation tables with prices, implied volatilities, and all relevantsensitivities are shown in Figures 6-24 and 6-25, Figure 6-24 is for options on SeptemberDeutschemark futures on the Chicago Mercantile Exchange as of 22 May 1992. Thevalues are based on a volatility assumption of 10.5% and an interest rate assumption of4.50%. Figure 6-25 is for options on General Electric stock on the Chicago BoardOptions Exchange as of the same date. The values are based on a volatility assumptionof 22% and an interest rate assumption of 4.50% for all three expiration months (June,September, December).o The expected dividends for General Electric stock, and theirpayment dates, are also included in the assumptions.”
We noted previously that the delta, gamma, theta, and vega can be sensitive tochanges in volatility. Under some circumstances it may be useful for a trader to ask: ifthe implied volatility of an option were the correct volatility, what would be its delta,gamma, theta, and vega? The resulting numbers are referred to as the implied delta,gamma, theta, and vega. The implied delta and theta, in particular, often appear oncomputer printouts, the former because many traders prefer to maintain delta neutral
6. — This is a simplified example with untform volatility and interest rates. Under some circumstances itmight be more reasonable to vary the volatility and interest rate assumptions for different expirationdates,
7. Due to the greater likelihood of early exercise for stock options, and the fact that the quoted optionsare American, the calculations in Figure 6-25 were done using a binomial model, which moreaccurately evaluates the early exercise value of American options. The problem of early exercise willbe discussed further in Chapter 12.12]
Iption Values and Changing Market Conditions
 
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122Option Values and Changing Market Conditions 123
positions using the implied delta, and the latter because the rate at which the price ofan option decays is usually more useful than the rate at which the theoretical valuedecays.
Towing the total delta, gamma, theta, and vega of an option position can help atrader determine beforehand how the position is likely to react to changing marketconditions. Since all these numbers are additive, the total sensitivity of a position canpe calculated by adding up the sensitivities of the individual options. For example, atrader who has purchased five options with a gamma of 2.5 each, and sold two optionswith a gamma of 4.0 each, has a total gamma position of
(+5 x 2.5) + (-2 x 4.0) = +4.5
Similarly, if he has purchased nine options with a theta of -.05 each, and sold fouroptions with a theta of -.08 each, his total theta position is
(+9 x -.05) + (-4 x ~.08) = -.13
New traders sometimes find it difficult to remember whether a delta, gamma, theta,or vega position is long or short. The various positions and their respective signs aregiven in Figure 6-26. The sign of the delta, gamma, theta, or vega, together with themagnitude of the numbers, tell the trader which changes in market conditions willeither help or hurt his position, and to what degree. The positive or negative effect ofchanging market conditions is summarized in Figure 6-27.
Even though a trader can analyze the effect of changing market conditions on hisposition through its delta, gamma, theta, and vega characteristics, his first and primaryconcern is that the position be profitable if his assumptions about market conditions arecorrect. This means that the position ought to have a positive theoretical edge. Thetheoretical edge can be calculated in the same way as the total delta, gamma, theta, andvega. One need only multiply the theoretical edge of each individual option (thedifference between its trade or settlement price and theoretical value} by the number ofcontracts traded, and add up all the contracts involved. The amount of positive ornegative theoretical edge is a reflection of the position’s potential profit or loss.
The top portion of Figure 6-28 shows a theoretical evaluation table for individualoptions on a hypothetical futures contract. Beneath the table are several possible optionpositions for which the total theoretical edge, delta, gamma, theta, and vega have beencalculated. The reader should take a moment to look at the arithmetic of each position,and to consider the risk characteristics associated with each position. Are all thePositions likely to be profitable if the assumptions used for theoretical evaluation arecorrect? What will help or hurt each position?
There is one other option sensitivity that is not generally used by traders, but whichthe reader may encounter in other option texts. An option’s elasticity, sometimesdenoted with the Greek letter omega (Q) (or less commonly the Greek letter lambda(A}), is the relative percent change in an option’s value for a given percent change in theprice of the underlying contract. For example, suppose a call has a theoretical value of2.50 with the underlying contract at 50. If the call has a delta of 25, and the underlyingcontract rises one point to 51, the call should rise to 2.75. In percent terms the call has124 Chapter 6
 
 
Figure 6-26Your delta Your gamma Your theta Your vega(hedge ratio) (curvature) (time decay) (volatility)
If you are... positionIs... positionis... positionis... positionis...long the underlying positive 0 oO 0short the underlying negative oO 0 oOlong calls positive positive negative positiveshort calls negative hegative positive negativelong puts negative positive negative positiveshort puts positive negative positive negativeFigure 6-27{f your delta position is... you want the underlying contract to...
positive rise in price
negative fall in priceIf your gamma positionis.. . you want the underlying contract to...
positive move very swiftly, regardless of direction
negative move slowly, regardless of directionIf your theta positionis..., the passage of time will generally...
positive increase the value of your position
negative decrease the value of your positionIf your vega position is... you want volatility to...
positive rise
negative fallIf your rho position is... you want interest rates to...
positive rise
negative fall
 
changed its value five times as fast as the underlying: the underlying rose 2% (1/50),while the call rose 10% (.25/2.50). We say that the call has an elasticity of 5. Theelasticity is sometimes referred to as the option’s leverage value, The greater an option’selasticity, the more highly leveraged the option.
An easy method of calculating an option’s elasticity is to divide the price of theunderlying contract by the option’s theoretical value, and multiply this by the option’sdelta (when calculating the elasticity, we use the decimal format for the delta):
_. _ underlying priceelasticity = theoretical value * delta125
Option Values and Changing Market Conditions
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