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NN ee -.e._OOooorOption Values and Changing Market Canditions 99
Figure 6-3: The Effect of Changing Dividends on Stock Options
If... Call values will... Put values will...The dividend is increased fall riseThe dividend is decreased rise fall
THE DELTA
We touched on the delta (A) in the last chapter as needed to determine the number ofunderlying contracts to option contracts required to establish a neutral hedge. The deltahas several other interpretations, any of which may be useful to a trader.
Rate of Change
Figure 6-4! shows what happens to the theoretical value of a call as the price of theunderlying contract changes. Under some conditions, when the call is very deeplyin-the-money, its value changes at a rate almost identical to that of the underlying. If theunderlying price moves up or down one point, the call’s value will change by an equalamount. Under other conditions, when the call is far out-of-the-money, its value maychange only slightly, even with a large change in the price of the underlying. The deltais a measure of how an option’s value changes with respect to a change in the price ofthe underlying contract.
In theory, an option can never gain or lose value more quickly than the underlying,so the delta of a call always has an upper bound of 100. (Again, we will retain thecommon practice of writing the delta without the decimal point. Hence, a delta of 1.00will be written as 100, This is sometimes referred to as the percent format.) An optionwith a delta of 100 will move up or down one full point for each full point move up ordown in the price of the underlying. It is moving at 100% of the rate of the underlying.In theory, a call also cannot move in the opposite direction of the underlying market,so the delta of a call has a lower bound of zero. A call with a delta of zero will movenegligibly, even if the underlying contract makes a relatively large move.
Most calls, of course, will have deltas somewhere between zero and 100 and willmove at a slower rate than the underlying. A call with a delta of 25 can be expected tochange its value at 25% of the rate of the underlying. If the underlying rises (falls) 1.00,the option can be expected to rise (fall) .25. A call with a delta of 75 can be expectedto change its value at 75% of the rate of the underlying. If the underlying rises (falls).60, the option can be expected to gain (lose} .45 in value. Call options which areat-the-money have delta values close to 50. They rise or fall in value at just about halfthe rate of the underlying.
Thus far our discussion of deltas has focused only on calls. Puts have characteristicssimilar to calls, except that put values move in the opposite direction of the underlyingmarket, In Figure 6-5 we can see that when the underlying rises, puts lose value; when
 
1. In order to generalize characteristics of the option sensitivities, which in most cases are similarregardless of the type of underlying instrument, we assume an interest rate of zero in Figures 6-4through 6-21.100 Chapter 6
Figure 6-4: Call Theoretical Value vs. Underlying Price
 
30Time = 60 daysVolatility = 20%257207
Theoretical Valueat
107
 
 
 
 
 
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go 8 90 95 100 105 110 115 120Underlying PriceFigure 6-5: Put Theoretical Value vs. Underlying Price30 DTime = 60 days ¢254 . Volatility = 20%@ 207.2 . .$ *x1108 15- s. :2 .5 400= ~ 10F 104 ~90 a _—55 ms, ~Or T T T T 7 oo ae _80 85 90 95 100 105 110 115 120
Underlying PriceOption Values and Changing Market Conditions 101
the underlying fails, puts gain value. For this reason, puts always have negative deltas,ranging from zero for far out-of the money puts to —100 for deeply in-the-money puts.Just like a call delta, a put’s delta is a measure of the rate of change in the put’s valuewith respect to a change in the price of the underlying, but the negative sign indicatesthat the change will be in the opposite direction of the underlying market. A put witha delta of -10 can be expected to change its value at 10% of the rate of the underlying,but in the opposite direction. If the underlying moves up (down) .50, the put can beexpected to lose (gain) .05 in value. At-the-money puts will have deltas of approximately50, and can be expected to change their value at approximately half the rate of theunderlying, but in the opposite direction.
Hedge Ratio
As discussed in Chapter 5, if we wish to hedge an option position against the underlyingcontract, the delta tells us the proper ratio of underlying contracts to options requiredto establish a neutral hedge. An underlying contract always has a delta of 100, so theproper hedge ratio can be determined by dividing 100 by the option’s delta. Anat-the-money option has a delta of 50, so the proper hedge ratio is 100/50, or 2/1. Forevery two options purchased, we need to sell one underlying contract to establish aneutral hedge. A call with a delta of 40 requires the sale of two underlying contracts foreach five options purchased, since 100/40 = 5/2,
Since puts have negative deltas, the purchase of puts will require us to hedge theposition by purchasing underlying contracts. A put with a delta of -75 will require thepurchase of three underlying contracts for each four puts purchased, since 100/75 = 4/3.
Thus far, all the hedges we have looked at have consisted of options and underlyingcontracts. But any hedge, whether options versus underlying contracts, or optionsversus options, is delta neutral as long as all the deltas in the hedge add up to zero, Forexample, suppose we buy four calls with a delta of 50 each, and ten puts with a deltaof —20 each. Our position is delta neutral since (+4 x +50) + (+10 x -20) = 0. A positioncan be highly complex, consisting of underlying contracts, calls, and puts, with differentexercise prices and expiration dates, but as long as the deltas add up to approximatelyzero we say that the position is delta neutral.
Theoretical or Equivalent Undertying Position
Many option traders come to the option market after trading in the underlyinginstruments. Futures option traders often start their careers by trading futures contracts;stock option traders often start by trading stock. If a trader has become accustomed toevaluating his risk in terms of the number of underlying contracts bought or sold (eitherfutures contracts or shares of stock), he can use the delta to equate the directional riskof an option position with a position of similar size in the underlying market.
Since an underlying contract always has a delta of 100, each 100 deltas in an optionposition represents a theoretical position equivalent to one underlying contract. A traderwho owns an option with a delta of 50 is long, or controls, approximately 4 of anunderlying contract. If he owns ten such contracts, he is long 500 deltas or, inequivalent terms, five underlying contracts. If the underlying is a futures contract, he is102 Chapter 6
theoretically long five such contracts. If the underlying is a stock contract consisting of100 shares of stock, he is theoretically long 500 shares of stock. The trader has a similartheoretical position if he sells 20 puts with a delta of -25 each, since -20 x -25 = +500.
It is important to emphasize the theoretical aspect of the delta interpretation as anequivalent to an underlying position. An option is not simply a surrogate for anunderlying position. An actual underlying position is almost exclusively sensitive todirectional moves in the underlying market. An option position, while also sensitive todirectional moves, is also sensitive to other changes in market conditions. An optiontrader who looks only at his delta position may be ignoring other risks which could havea far greater impact on his option position. He must realize that the delta represents anequivalent underlying position only under very narrowly defined market conditions.
The reader may have already noted that our three interpretations of delta—thehedge ratio, the rate of change in the theoretical value, and the equivalent underlyingposition—are all essentially the same. How a trader interprets a delta position dependsprimarily on his trading strategy. For example, suppose a trader has a delta position of+500. If the trader intends to maintain a delta neutral position, he must sell fiveunderlying contracts (the hedge ratio interpretation). However, if he believes the marketwill rise and wants to maintain his delta position, he knows that in theory he is long fiveunderlying contracts (the equivalent underlying position interpretation). And finally, ifhe does maintain his delta position of +500, the value of his position will change at fivetimes the rate, or 500%, of the underlying market (the rate of change interpretation).Even though a trader may interpret the delta differently at different times, eachinterpretation is mathematically the same.
There is one other interpretation of the delta which is perhaps of less practical usebut may still be worth mentioning. If we ignore the sign of the delta (positive for calls,negative for puts), the delta is approximately equal to the probability that the option willfinish in-the-money. A call with a delta of 25, or a put with a delta of -25, hasapproximately a 25% chance of finishing in-the-money. A call with a delta of 75, oraput with a delta of -75, has approximately a 75% chance of finishing in-the-money. Asan option’s delta moves closer to 100, or —100 for puts, the option becomes more andmore likely to finish in-the-money. As the delta moves closer to zero, the option becomesless and less likely to finish in-the-money. We can also see why at-the- money optionshave deltas close to 50. If we assume that price changes are random, there is half achance that the market will rise (the option goes into-the-money), and half a chance thatthe market will fall (the option goes out-of-the money}.
Of course, the delta is only an approximation of the probability since interest rateconsiderations and, in the case of stock options, dividends, may distort this interpreta-tion. Moreover, most option strategies depend not only on whether an option finishesin-the-money, but by how much. Ifa trader sells an option with a delta of 10 in the beliefthat the option will expire worthless nine times out of ten, he may indeed be correct.But if on the tenth time he loses an amount greater than the total premium he took inthe nine times the option expired worthless, the trade will result in a negative expectedreturn. In option trading the primary consideration is not always how often a strategyOption Values and Changing Market Conditions 103
wins or loses, but also how much it wins or loses. Every experienced trader is willingto accept several small losses if he can occasionally offset these with one big win.
THE GAMMA
In discussing the delta, we noted that under some circumstances, when the option isfar out-of the-money, its delta is close to zero. At other times, when the option is deeplyin-the-money, its delta is close to 100 (-100 for puts}. We can logically conclude thatas the underlying price changes, the delta of an option must also be changing. As theunderlying price rises, call deltas move towards 100 and put deltas move towards zero;as the underlying price falls, call deltas move towards zero and put deltas move towards—100. This effect is shown in Figures 6-6 and 6-7.
The gamma (I), sometimes referred to as the curvature of an option, is the rate atwhich an option’s delta changes as the price of the underlying changes. The gamma isusually expressed in deltas gained or lost per one point change in the underlying, withthe delta increasing by the amount of the gamma when the underlying rises, and fallingby the amount of the gamma when the underlying falls. If an option has a gamma of 5,2for each point rise (fall} in the price of the underlying, the option will gain (lose) 5 deltas.If the option originally had a delta of 25, and the underlying moves up (down) one fullpoint, then the new delta of the option will be 30 (20). If the underlying moves up(down) another point, the new delta will be 35 (15).
Note that Figures 6-6 and 6-7 seem to be identical, even though the formerrepresents call gammas and the latter put gammas. This can only mean that both callsand puts with the same exercise price and the same amount of time to expiration havethe same curvature.? This may at first seem odd, but it becomes logical when weremember that zero is greater than —S0 in the same way that —50 is greater than -100.Negative numbers become more positive (or less negative) as one moves towards zero.Therefore, both calls and puts must have positive gammas. This often confuses the newtrader since, through his use of the delta, he is accustomed to associating positivenumbers with calls and negative numbers with puts. But regardless of whether we areworking with calls or puts, we always add the gamma to the old delta as the underlyingrises, and subtract the gamma from the old delta as the underlying falls. When a traderis long options, whether calls or puts, he has a long gamma position. When he is shortOptions, he has a short gamma position.
For example, suppose an at-the-money call, with a delta of 50, and an at-the moneyput, with a delta of 50, both have gammaas of 5. If the underlying contract rises oneDoint, we add the gamma of 5 to the call delta of 50, to get the new delta of 55. To getthe new put delta if the underlying contract rises one point we also add the gamma of5 to the put delta of -50 to get the new delta of —45. This corresponds to our intuition
2. When the delta is expressed in the decimal format (0 to 1.00), the gamma ts expressed In the sameformat.
3. If interest rates are zero, as they are assumed to be in Figures 6-6 and 6-7, the gamma of calls and puts
with the same exercise price will be identical. If interest rates are not zero, and there is a possibility ofearly exercise, the gammas may differ slightly.104
Figure 6-6: Call Delta vs. Underlying Price
Delta
100
Chapter 6
 
 
907
805
704
60-4
50-
405
307
207
10-
 
80
Time = 60 daysVolatility = 20%
90
* 100
 
85
95
T400 105Underlying Price
Figure 6-7: Put Delta vs. Underlying Price
Delta
110
 
120
 
0
 
-10+
-207
-305
-40/
-507
-605
70+
-80-4
-90-
Time = 60 daysVolatility = 20%
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100."
 
 
 
-100
80
 
T T100 105Underlying Price
110Option Values and Changing Market Conditions 105
that as the underlying price rises, at-the-money calls move into-the-money and at-the-money puts move out-of-the- money. If the underlying contract falls one full point, inboth cases we subtract the gamma, giving the call a delta of 45 (50-5), and the put adelta of -S5 (-50-5). Now the call is moving out-of-the-money and the put is movinginto-the-money.
We know that an underlying contract has a delta of 100. But what is the gammaassociated with an underlying contract? Since the gamma is the rate of change in thedelta, and the delta of an underlying contract is always 100, the gamma must be zero.
The gamma is a measure of how fast an option changes its directional charac-teristics, acting more or less like an actual underlying position. Since directional risk isalways important, the gamma is an important risk measurement. Indeed, an optionposition can change its directional risk dramatically, even if a trader takes no action inthe marketplace. A large gamma number, whether positive or negative, indicates a highdegree of risk; a low gamma number indicates a low degree of risk. Every option traderlearns to look carefully not only at current directional risk (the delta), but also at howthat directional risk will change if the underlying market begins to move (the gamma).
Consider a trader who sells ten calls with a delta of 30 each. He is short 300 deltas(-10 x 30), equivalent in theory to being short three underlying contracts. If this traderis accustomed to dealing in lots of five underlying contracts or less, he is, for themoment, within his normal risk limits. If the market rises ten points and he considersonly his original delta position of -300, he may assume that he is still, in theory, shortthe same three underlying contracts. But what if the initial gamma of each call was 6?Then, for each point rise in the underlying contract, each call will gain 6 deltas. Whenthe underlying rises ten points, each call will gain 60 deltas, for a total delta of 90 each.The trader’s directional risk has now increased by a factor of three. Instead of the initialdelta of -300, the delta position is now -900, well beyond his normal risk limits,
A large gamma can sometimes overwhelm the inexperienced trader. New tradersare well advised to avoid large gamma positions, particularly negative ones, because ofthe speed with which such positions can change. Even experienced traders occasionallytake on gamma positions which are too risky. This was dramatically demonstrated inthe collapse of the Volume Investors clearing firm on the COMEX in the spring of 1985.Several traders, all clearing their trades through the same firm, built up extremely largenegative gamma positions by selling large numbers of out-of-the-money gold options. Inthe past this had been a very profitable strategy, since the gold market had beenrelatively quiet with low volatility. But on this occasion the market made a violentupward move, leaving the traders, who had originally been delta neutral, short thou-sands of deltas in a rapidly rising market. The losses sustained by the traders not onlyled to the collapse of the clearing firm, but also to a crisis in the COMEX’s clearingassociation. The incident might have been avoided had someone, either the tradersthemselves, the clearing firm, or the clearing association, realized that the large negativegamma position represented an unacceptable risk. Most firms which trade options orclear option trades for their customers now have risk managers responsible for identify-ing such dangerous positions.106 Chapter 6
The gamma can also help a trader maintain a delta neutral position by enabling himto make a quick estimate of how the delta is changing. Suppose a trader has a deltaposition of +500 prior to the opening of the market. If he wishes to be delta neutral heknows that he will have to sell 500 deltas on the opening. He can accomplish this byselling five underlying contracts. But suppose that his gamma position is +100 and themarket is expected to open two points higher. If it does in fact open two points higherhis delta position will no longer be +500, but +700 since he will become 100 deltaslonger for each point rise in the underlying. He now knows that in order to be deltanetitral he must sell 700 deltas, or seven underlying contracts. He can make thiscalculation without any computer assistance simply by knowing his initial delta andgamma positions.
Since the delta of an option must always remain in the range of zero to 100 (zeroto ~100 for puts), we can logically conclude that the gamma must also be changing.Otherwise, continuously adding or subtracting a constant gamma would take the deltabeyond this range. The effect of changes in market conditions on the value of gammasis shown in Figures 6-8, 6-9, and 6-10. Figure 6-8 demonstrates that the gamma isgreatest for an option which is at-the-money, and becomes progressively smaller as anoption moves either into- or out-of-the-money. Figures 6-9 and 6-10 show that thegamma of an at-the-money option can increase dramatically as expiration approaches,or as we decrease our volatility assumption. A trader’s gamma position, which may haveinitially been small, can become increasingly large through the passage of time orthrough changes in volatility. An option position must always be monitored to ensurethat its risk characteristics remain within acceptable limits.
While the gamma is perhaps the most common measure of the change in the delta,the delta is also subject to change from market conditions other than movement in theprice of the underlying contract. Figures 6-11 and 6-12 show how call and put deltaschange as time passes. Figures 6-13 and 6-14 show how deltas change with changes involatility. Note that all four sets of graphs have the same shape. As we increase the timeto expiration or increase volatility, all options tend to become more at-the-money, withcall deltas approaching 50 and put deltas approaching —50. Conversely, as we decreasethe amount of time to expiration or decrease volatility, all option deltas move away from50 (—50 for puts}. An option which is in-the-money will move further into-the-money,and an option which is out-of-the-money will move further out-of-the-money. Optionswhich are at-the-money and have deltas close to 50 tend to maintain the same deltacharacteristics regardless of changes in time or volatility.
Perhaps a commentis in order here about what atthe-money means. Traders tendto think of an at-the-money option as one whose exercise price is approximately equalto the current price of the underlying contract. By this definition a trader will instinc-tively assign a delta of 50 to any option whose exercise price is currently at-the- money.However, a theoretical pricing model will interpret the at-the-money option (the onewith a delta of 50) as the option whose current exercise price is most likely to be closestto the price of the underlying contract at expiration. Suppose that interest rates are 12%annually and that a certain stock is currently trading at $100. If two five-month calls areavailable with exercise prices of 100 and 105, which call has a delta closer to 50? WeOption Values and Changing Market Conditions 107
Figure 6-8: Call or Put Gamma vs. Underlying Price6
 
90 Time = 60 days~ Volatility = 20%100
110
GammawL
 
 
 
= T T T T q80 85 90 95 100 105 110 115 120Underlying Price
 
Figure 6-9: Call or Put Gamma vs. Time to Expiration257-
 
Underlying Price = 100Volatility = 20%
154
Gamma
to) 1
190 ee cee cee a ee ee Loewee 90
0 ee T T T T0 20 40 60 80 100 120 140 160 180 200
Time to Expiration (days)
 
 
 
 108 Chapter 6
Figure 6-10: Calf or Put Gamma vs. Volatility25
 
Underlying Price = 100Time = 60 days
207
154
Gamma
"5,100
54
 
 
  
 
 
 
 
0 5 10 15 20 25 30 35 40Volatility (%)
Figure 6-11: Call Delta vs. Time to Expiration100
 
907 90807 aUnderlying Price = 10070- Volatility = 20%
60-
Delta
40-307
A cee ee
 
 
 
= —~T T tT T TT0 20 40 60 80 100 120 140 160 180 200
Time to Expiration (days)Option Values and Changing Market Conditions
Figure 6-12: Put Delta vs. Time to Expiration
109
 
0
-104
-204
-307
-404
Delta
-607-707-804
-90-
 
-1000
Underlying Price =100Volatility = 20%
Oo
FT T T T
80 100 120 140Time to Expiration (days)
Figure 6-13: Call Delta vs. Volatility
 
200
 
 
100
90+
80-
705
Delta
40730720-5
105
 
90
Underlying Price = 100Time = 60 days
 
 
Volatility (%)
40110 Chapter 6
Figure 6-14: Put Delta vs. Volatility0
 
~10- 90
“207 Underlying Price = 100
Time = 60 days
~307
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-604
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Volatility (%)
saw in Chapter 4 that the mean of the distribution assumed in most theoretical pricingmodels is the forward price (the break-even price) for the underlying contract. In ourpresent case, the forward price is the current stock price of $100 plus the $5 carryingcosts over the five-month holding period. The 105 call is therefore the one which themodel interprets as being at-the-money, and hence will have a delta of SO. Dependingon the type of underlying contract, under extreme interest rate conditions or with agreat deal of time remaining to expiration, options can have deltas decidedly differentfrom what we might instinctively surmise.
Just as a delta neutral option position can become unbalanced through movementin the underlying contract, it can also become unbalanced through the passage of timeor through changes in volatility. No trader knows for certain whether he is delta neutralbecause he can’t be certain about the accuracy of the inputs into the theoretical pricingmodel. The delta value depends, among other factors, on the volatility assumption. Andvolatility is just that, an assumption. A trader who sells four calls, each with a delta of25, and buys one underlying contract, might believe that he is delta neutral. But in orderto arrive at a call delta of 25 the trader is required to feed some volatility into atheoretical pricing model. If he subsequently decides that his original volatility assump-tion was too low, and he now raises it, we can see in Figure 6-13 that the call delta willrise towards 50. Using his new volatility assumption the delta of the call might be 35,and instead of being delta neutral, the trader is now 40 deltas short. All he did tounbalance his delta position was to change his assumptions about market conditions.Option Values and Changing Market Conditions 111
Nothing in the foregoing discussion invalidates the concept of delta neutrality,which is still an important one. However, a trader must realize that a delta neutralposition depends on an estimate of market conditions, both current and future, andthere is no guarantee that this estimate will be correct. As he changes his assumptionsabout market conditions, he must constantly change his trading strategies to fit thesenew assumptions. Delta neutral trading is only one aspect of this principle.
THE THETA
The effect of a change in time to expiration on the theoretical values of calls and puts isshown in Figures 6-15 and 6-16. Note that all options, both calls and puts, lose value asexpiration approaches. The theta(@), or time decay factor, is the rate at which an optionloses value as time passes. It is usually expressed in points lost per day, when all otherconditions remain the same. An option with a theta of .05 will lose .05 in value for eachday that passes with no change in other market conditions. If the option is worth 2.75today, then tomorrow it will be worth 2.70. The day after that, it will be worth 2.65.
Time runs in only one direction and, technically, the theta is a positive number.However, as a convenient notation, and to remind the user that the theta represents aloss in the option’s value as time passes, it is sometimes written as a negative number.This is the convention which will be followed in this text. Therefore, the theta of anoption which loses .05 per day will be written as -.05. Consequently, a long optionposition will always have a negative theta, and a short option position will always havea positive theta. Note that this is just the opposite of the gamma, where a long optionposition has a positive gamma, and a short option position has a negative gamma.
As a general principle, an option position will have a gamma and theta of oppositesigns. Moreover, the relative size of the gamma and theta will correlate. A large positivegamma goes hand in hand with a large negative theta, while a large negative gammagoes hand in hand with a large positive theta. We saw that as expiration approaches,the gamma of an at-the-money option becomes increasingly large. The same is also trueof the theta. As we get closer to expiration the rate at which an option decays begins toaccelerate. If the option is exactly at-the-money, its theta at the moment of expirationbecomes infinitely large.
Every option position is a tradeoff between market movement and time decay. Ifprice movement in the underlying contract will help a trader (positive gamma), thepassage of time will hurt (negative theta). And vice versa. The trader can’t have it bothways. Either he wants the market to move, or he wants it to sit still. This opposing effectcan be seen in Figure 6-9 (gamma vs. time to expiration) and Figure 6-17 (theta vs. timeto expiration). As the gamma becomes large, so does the theta. Just as a large negativegamma represents a high degree of risk with respect to market movement, a largenegative theta represents a high degree of risk with respect to the passage of time.
Is it ever possible for an option to have a positive theta such that if nothing changesthe option will be worth more tomorrow than it is today? When futures options aresubject to stock-type settlement, as they currently are in the United States, the carryingcost on a deeply in-the-money option, either a call or a put, can, under some circum- 
 
 
 
 
   
 
 
 
 
112 Chapter 6Figure 6-15: Call Theoretical Value vs. Time to Expiration129010-$s 8 Underlying Price = 100sg Volatility = 20%reDf —_—8 $00 eopon a —F 4! ae27 a 0o+ i + - T T T T T