increased in value to 3%, Finally, if at expiration the market is still unchanged, theshort-term option, since it is still at-the-money, will lose all of its remaining value of 3points. The long-term option will continue to decay but will do so at a slower rate. Ifthe long-term option loses %, it will be worth 6 points, and the spread will be worth 6points (Figure 8-13).
What will happen if the underlying market makes a larger move? Assume as beforethat both options are at-the-money and have values of 7% and 6. As the underlyingmarket rises and options move more deeply into-the-money, they begin to lose their timevalue. If the move is large enough, it won’t matter that the long-term option has threemore months remaining to expiration. Both options will eventually lose all their timevalue (see Figures 6-15 and 6-16). If the time spread consists of calls, both of which haveexercise prices of 100, and the underlying market moves from 100 to 150, both optionsmight trade at parity (intrinsic value), or 50 points. The spread will then go to zero. Evenif the long-term option retains as much as 4 point, the spread will still have collapsed to 4.
What about a large downward move in the underlying market? The situation isalmost identical to an upward move. As an option moves further out-of-the-money itsVolatility Spreads 151
Figure 8-13: The Effect of Time Passage on Time SpreadsTime to Expiration
Long-term option 6 months 5 months 4 months 3 monthsShort-term option 3 months 2 months 4 month noneOption Value
Long-term option Tip ™% 6% 6Short-term option 6 5 3 0Spread Vaiue 1% 2M 3% 6
time value also begins to shrink. Here, however, neither option will have any intrinsicvalue, so that if the market moves down far enough both options will eventually beworthless. If that happens, the time spread will be worthless. If, as above, the long-termoption retains even 1% point in value, the spread will still collapse to 14 point.
Since a short-term at-the-money option always decays more quickly than a long-term at-the-money option, regardless of whether the options are calls or puts, both along call time spread and a long put time spread want the underlying market to sit still.Ideally, both spreads would like the short-term option to expire right at-the-money sothat the long-term option will retain as much time value as possible while the short-termoption expires worthless.
Ifa long time spread wants the market to sit still, logically a short time spread wantsthe market to move. It may therefore seem that, as with backspreads and ratio verticalspreads, the major consideration in deciding whether to initiate a long or short timespread is the likelinood of movement in the underlying market. Certainly time spreadsare sensitive to movement in the price of the underlying, but time spreads are alsosensitive to changes in implied volatility.
A long time spread always benefits from an increase in implied volatility. Lookagain at Figure 6-18, the relationship between an option’s vega (sensitivity to a changein volatility) and the time remaining to expiration. As time to expiration increases, thevega of an option increases. This means that a long-term option is always more sensitivein total points to a change in volatility than a short-term option with the same exercisePrice.
For example, assume again that a 100 call time spread is worth 114 (long- andshort-term options with values of 714 and 6, respectively). Assume also that the valueof the spread is based on a volatility of 20%. What will happen to the value of the spreadif we raise volatility to 25%? Both options will increase in value since an increase involatility increases the value of all options, But the long-term option, with more timeTemaining to expiration and therefore a higher vega, will gain more value as a result ofthe increase in volatility. If the short-term option gains 14, the long-term option maygain a full point. Now the options will be worth 814 and 614, and the spread will havewidened from 11% to 2 (Figure 8-14).152 Chapter 8
Figure 8-14: The Effect of Volatility on Time Spreads
Voiatility 15% 20% 25%Option Value
Long term option 6% 7% 8%Short term option 5% 6 6%Spread Vaiue 1 1% 2
Conversely, if we lower our volatility estimate to 15%, both options will lose value.But the long-term option, with more time remaining to expiration, will be more sensitiveto the change in volatility and will lose a greater amount. The new option values mightbe 6!4 and 514, causing the spread to narrow to | point.
The effects of volatility on a time spread become especially evident when there isa change in implied volatility in the option market. When implied volatility rises, timespreads tend to widen; when implied volatility falls, time spreads tend to narrow. Thiseffect can often be great enough to offset either a favorable or unfavorable move in theunderlying market. A trader who is long a time spread expects to lose money if theunderlying market makes a swift move in either direction. He knows that time spreadsbegin to collapse as the options move into- or out-of-the-money. But if this movementis accompanied by a sufficient increase in implied volatility, the increase in the spread’sprice due to the increase in implied volatility may actually be greater than the loss dueto market movement. In that case the trader may find that the spread has actuallywidened. Conversely, if the market sits still, the trader expects the spread to widenbecatise of the short-term option’s greater time decay. But if, at the same time, there isa collapse in implied volatility, the decline in the spread’s price due to the decline inimplied volatility may more than offset any gain from the passage of time. If this happens,the trader may find that the spread has narrowed.
These two opposing forces, the decay in an option’s value due to the passage oftime and the change in an option’s value due to changes in volatility, give time spreadstheir unique characteristics. When a trader buys or sells a time spread, he is not onlyattempting to forecast movement in the underlying market. He is also trying to forecastchanges in implied volatility. The trader would like to forecast both inputs accurately,but often an error in one input can be offset by an accurate forecast for the other input.
Ideally, the trader who is long a time spread wants two apparently contradictoryconditions in the marketplace. First, he wants the underlying market to sit still so thattime decay will have a beneficial effect on the spread. Second, he wants everyone tothink the market is going to move so that implied volatility will rise. This may seem animpossible scenario, the market remaining unchanged but everyone thinking it willmove. In fact, this happens quite often because events which don’t have an immediateeffect on the underlying market could be perceived to have a future effect on the market.
Suppose an announcement is made that the finance ministers of the major indus-trialized countries plan to meet to discuss exchange rates. If no one knows what theVolatility Spreads 153
outcome of the meeting will be, there is unlikely to be any significant move in thecurrency markets when the initial announcement is made. On the other hand, alltraders will assume that major changes in exchange rates could result from the meeting.This possibility will result in an increase in implied volatility in the currency optionmarket. The lack of movement in the underlying market, together with the increase inimplied volatility, will cause time spreads to widen.
Suppose that, as a result of the meeting, the finance ministers decide to maintainthe status quo. Now expectations of a major change in exchange rates fade, impliedvolatility falls, and, as a result, time spreads narrow.
Scenarios with similar consequences also occur in the interest rate options marketwhen a policy statement is expected from the Federal Reserve in the United States, orwhen a company’s earnings are due to be reported in the stock option market. Theseevents are unlikely to have an effect on the underlying market before the fact, but theycould have significant repercussions after the fact.
The effect of implied volatility is what distinguishes time spreads from the othertypes of spreads we have discussed. Backspreads (including long straddles, long stran-gles, and short butterflies) and ratio vertical spreads (including short straddles, shortstrangles, and long butterflies) want real volatility (price movement in the underlyingmarket} and implied volatility (a reflection of expectations about future price movementin the underlying market} to either both rise or both fall. A swift move in the underlyingmarket or an increase in implied volatility will help a backspread. A quiet market or adecrease in implied volatility will help a ratio vertical spread. With time spreads,however, real and implied volatility have opposite effects. A big move in the underlyingmarket or a decrease in implied volatility will help a short time spread, while a quietmarket or an increase in implied volatility will help a long time spread. This oppositeeffect is what gives time spreads their unique characteristics.
While the foregoing characteristics of time spreads apply to all option markets,there may be other considerations, depending on the specific underlying market. Wepreviously assumed that the price of the underlying contract for both the short- and long-term option was the same. In the stock option market this will always be true becausethe underlying for every expiration is the same stock. The underlying contract for allIBM options, regardless of the expiration month, is always IBM stock. And IBM stockcan only have a single price at any one time. In contrast, the underlying for a futuresoption is a specific futures contract. Consider this situation for Eurodollar futures andoptions traded at the CME:
March Eurodollar Futures = 93.90June Eurodollar Futures = 93.75
Suppose a trader initiates a long time spread:
long 10 June 94.00 callsshort 10 March 94.00 calls
The underlying for March Eurodollar options is a March Eurodollar futures con-tract; the underlying for June Eurodollar options is a June Eurodollar futures contract.154 Chapter 8
While the March and June Etrodollar futures contracts are related, they are notidentical. It is not impossible fo' one futures contract to go up while the other goesdown. As a result, in additior to volatility considerations, a trader who buys aJune/March Eurodollar option all time spread also has to worry about the risk of theJune futures contract falling whib the March futures contract is rising. Is there any wayto offset this risk?
If, in our example, the sprad between the March and June futures contracts,which is currently .15, begins b widen, the option spread will narrow regardless ofvolatility considerations. If, however, when the trader initiates the option spread, healso executes a futures spread bypurchasing March futures and selling June futures, hewill have a position which wiiloffset any loss to the option spread resulting from awidening in the futures spread.
How many futures spreads should the trader execute? He ought to trade thenumber of futures spreads requird to be delta neutral. If the delta of both options is 40and the trader executes the optien spread ten times, he will be long 400 deltas in Juneand short 400 deltas in March. Tierefore, he should buy 4 March futures contracts andsell 4 June futures contracts, “he entire spread will look like this (deltas are inparentheses):
long 10 June 94.00calls (40) short 4 June futures (100)short 10 March 94.)0 calls (40) long 4 March futures (100)
This type of balancing is 10t necessary, indeed not possible, in stock optionsbecause the underlying for all maths is identical. There is no such thing as March IBMstock or June IBM stock.
THE EFFEC] OF CHANGING INTERESTRATIS AND DIVIDENDS
Thus far we have considered ony the effects of changes in the price of the underlyingmarket and changes in volatility2n the value of volatility spreads. What about changesin interest rates and, in the case f stocks, dividends?
Because there is no carrying cost associated with the purchase or sale of a futurescontract, interest rates have onlya minor impact on futures options, and, consequently,a negligible effect on the value o futures option volatility spreads.4 If, however, we aretrading stock options and changethe interest rate, we change the forward price of stock(the current stock price plus th carrying cost on the stock to expiration). When alloptions expire at the same time as they do in backspreads and ratio vertical spreads,the forward stock price for all optons rernains the same, so that the effects of the changeare practically negligible. If, on te other hand, we are considering stock options withdifferent expiration dates, we mut consider two different forward prices. And these twoforward prices may not he equaly sensitive to a change in interest rates. For example,consider the following situation:
 
4, Interest rates can of course affect tie relative value of different futures months. As noted, we can offsetthis risk by trading a futures spreal along with the futures option time spread.Volatility Spreads 155
Stock price=100 Interestrate=12% Dividend =0Suppose a trader initiates a long time spread:
long 10 June 100 callsshort 10 March 100 calls
If there are three months remaining to March expiration and six months remaining toJune expiration, the forward prices for March and June stock are 103 and 106,respectively. If interest rates drop to 8 percent, the forward price for March will be 102and the forward price for June will be 104. With more time remaining, the June forwardprice is more sensitive to a change in interest rates. Assuming the deltas of both optionsare approximately equal, the June option will be more affected in total points by thedecline in interest rates than the March option, and the time spread will narrow. In thesame way, ifinterest rates increase, the time spread will widen because the June forwardprice will rise more quickly than the March forward price. Therefore, a long (short) calltime spread in the stock option market must have a positive (negative) rho.
Changes in interest rates have just the opposite effect on stock option puts. In ourexample, if interest rates fall from 12% to 8% the March forward price will fall from 103to 102, while the June forward price will fall from 106 to 104. Again, if we assume thatthe deltas of both options are approximately equal, and recalling that puts have negativedeltas, the June put will show a greater increase in value than the March put. The puttime spread. will therefore widen. In the same way, if interest rates increase, the puttime spread will narrow. Therefore, a long (short) put time spread in the stock optionmarket must have a negative (positive) rho.
The degree to which stock option time spreads are affected by changes in interestrates depends primarily on the amount of time between expiration dates. If there are sixmonths between expiration dates (e.g., March/September) the effect will be muchgreater than if there is only one month between expiration dates (e.g., March/April).
Changes in dividends can also affect the value of stock option time spreads.Dividends, however, have the opposite effect on stock options as changes in interestrates (see Chapter 3). An increase (decrease) in dividends lowers (raises) the forwardprice of stock. If all options in a volatility spread expire at the same time (backspreads,ratio vertical spreads), the forward stock price will be identical for all options and theeffect on the spread will be negligible. But in a time spread, if a dividend payment isexpected between expiration of the short-term and long-term option, the long-termoption will be affected by the lowered forward price of the stock. Hence, an increase individends, if at least one dividend payment is expected between the expiration dates,will cause call time spreads to narrow and put time spreads to widen. A decrease individends will have the opposite effect, with call time spreads widening and put timespreads narrowing. The effect of changing interest rates and dividends on stock optiontime spreads is shown in Figure 8-15.
Even if both options are deeply in-the-money, a call time spread in the stock optionmarket should always have some value greater than zero. If volatility is very low, thespread should still be worth a minimum of the cost of carry on the stock between156 Chapter 8
 
Figure 8-15
The Effects of Changing Interest Rates on Stock Option Time Spreads
Stock Price = 100; Volatility = 20%; Dividend = 0Time to March expiration = 6 weeksTime to June expiration = 19 weeks
If Interest rates are... 0% 3% 6% 9% 12%June 100 call 4.81 §.35 §.92 6.52 7.15March 100 call 271 2.88 3.05 3.24 3.43Call Spread Value 2.10 2.47 2.87 3.28 3.72June 100 put 4.81 4.26 3.76 3.30 2.88March 100 put 271 2.53 2.37 2.21 2.06Put Spread Value 2.10 1.73 1.39 1.09 82
The Effect of Changing Dividends on Stock Option Time Spreads
Stock Price = 100; Volatility = 20%; Interest Rate = 6.00%Time to March expiration = 6 weeksTime to June expiration = 19 weeks
if the quarterly dividendIs... 0 1.00 2.00 3.00 4,00June 100 call 5.92 4.84 3.82 2.97 2.26March 100 call 3.05 2.53 2.07 1,66 1.31Call Spread Value 2.87 2.28 1,75 1.31 95June 100 put 3.76 4.62 5.62 6.75 8.01March 100 put 2.37 2.84 3.37 3.96 4.62Put Spread Value 1.39 1.78 2.25 2.79 3.39
 
expiration months, This is only true, however, ifa trader can carry a short stock positionbetween expiration months, If a situation arises where no stock can be borrowed, thetrader who owns a time spread may be forced to exercise his long-term option, therebylosing the time value associated with the option.
As an example, suppose in February a company’s stock is trading at 70, when atender offer is made to buy a portion of the outstanding stock at a price of 80. Ifa traderis long a June/March 70 call time spread, he will be assigned on the March 70 callbecause the holder of that call wants to tender his stock for sale at 80. The trader is nowshort stock and long a June 70 call. The June call should still carry some time valuebecause of the interest which can be earned on the short stock to June expiration. Butin order to carry a short stock position, the trader must deliver the stock to the exerciserof the March 70 call. To deliver the stock, he must borrow it from someone. Unfortu-nately, no one will lend stock to the trader because everyone wants to tender the stockfor sale at the tender price of 80. Unless the trader wants to go into the market and buythe stock at the tender price of 80, he will be forced to exercise his June 70 call. This isthe only method by which the trader can obtain stock for delivery to the exerciser ofVolatility Spreads 157
the March 70 call. Even though the June 70 call should in theory have some time value,the trader will be forced to discard the time value in order to fulfill his deliveryobligation.
In the foregoing situation, sometimes referred to as a short squeeze, a trader isforced to exercise calls with some time value remaining because no stock can beborrowed to carry a short stock position. Anyone who owns the June/March 70 calltime spread will be forced to exercise the June 70 call, and the price of the time spreadwill collapse to zero. Note that this situation is not the same as a buy-out, where all thestock in a company is purchased at one price. Although a tender offer was made at 80,it was for only a portion of the company’s outstanding stock. When the tender iscompleted, the remaining stock will continue to trade, most likely at its pre-tender priceof 70.
DIAGONAL SPREADS
A diagonal spread is similar to a time spread, except that the options have differentexercise prices. While many diagonal spreads are executed one-to-one (one long-termoption for each short-term option), diagonal spreads can also be ratioed, with unequalnumbers of long and short market contracts. With the large number of variations indiagonal spreads, it is almost impossible to generalize about their characteristics as wecan with backspreads, ratio vertical spreads, and long and short time spreads. Eachdiagonal spread must be analyzed separately, often using a computer, to determine therisks and rewards associated with the spread.
There is, however, one type of diagonal spread about which we can generalize. Ifa diagonal spread is done one-to-one, and both options are of the same type and haveapproximately the same delta, the diagonal spread will act very much like a conven-tional time spread. An example of a such diagonal spread is shown in Figure 8-12.
OTHER VARIATIONS
The spreads we have thus far defined are the primary types of volatility spreads, and arethe ones most commonly executed in the marketplace. There are, however, somevariations of which the reader ought to be aware.
A Christmas tree (also referred to as a /adder) is a term which can be applied toavariety of spreads. The spread usually consists of three different exercise prices whereall options are of the same type and expire at the same time. In a long (short) callChristmas tree, one call is purchased (sold) at the lowest exercise price, and one call issold (purchased) at each of the higher exercise prices. In a long (short) put Christmastree, one put is purchased (sold) at the highest exercise price, and one put is sold(purchased) at each of the lower exercise prices. Christmas trees are usually deltaneutral, but even with this restriction, there are many ways to execute the spread. Someexamples of Christmas trees are shown in Figure 8-16.
Long Christmas trees, when done delta neutral, can be thought of as particulartypes of ratio vertical spreads. Such spreads therefore increase in value if the underlyingmarket either sits still or moves very slowly. Short Christmas trees can be thought of as158 Chapter &
Figure 8-16Long Christmas Trees Short Christmas Treeslong 10 March 95 calls (78) short 5 June 90 calls (81)short 10 March 100 calls (61) long § June 100 calls (51)short 10 March 105 calls (24) long § June 105 calls (36)long 25 June 110 puts (—75) short 50 March 110 puts (-91)short 25 June 100 puts (47) long 50 March 105 puts (-75)short 25 June 95 puts (—31} long §0 March 95 puts (-21)
particular types of backspreads, and therefore increase in value with big moves in theunderlying market.
It is possible to construct a spread which has the same characteristics as a butterflyby purchasing a straddle (strangle) and selling a strangle (straddle) where the straddle isexecuted at an exercise price midway between the strangle’s exercise prices. All optionsmust expire at the same time. Because the position wants the same outcome as abutterfly, it is known as an iron butterfly. Like a true butterfly, at expiration an ironbutterfly has a minimum value of zero and a maximum value of the amount betweenexercise prices.
Note that if the straddle is purchased and the strangle is sold, the position will resultin a debit (a long iron butterfly}. Such a position will show its greatest profit at expirationif the underlying market finishes beyond the outside (strangle’s) exercise prices. A longiron butterfly is therefore equivalent to a short butterfly. If the straddle is sold and thestrangle is purchased, the position will result in a debit (a short iron butterfly). Such aposition will show its greatest profit at expiration if the underlying market finishes rightat the inside (straddle’s) exercise price. A short iron butterfly is therefore equivalent toa long butterfly. Examples of long and short iron butterflies are shown in Figure 8-17.
Another variation on a butterfly, known asa condor, can be constructed by splittingthe inside exercise prices. Now the position consists of four options at consecutiveexercise prices where the two outside options are purchased and the two inside optionssold (a long condor), or the two inside options are purchased and the two outsideoptions sold (a short condor). As with a butterfly, all options must be of the same typeall calls or all puts) and expire at the same time.
At expiration a condor will have its maximum value, equivalent to the amountbetween consecutive exercise prices, when the underlying contract is at or anywherebetween the two inside exercise prices. It will be worthless whenever the underlyingcontract is outside the extreme exercise prices at expiration. Note that this is very similarto a butterfly, except that a condor has a maximum value over a wider range ofunderlying prices. A butterfly will achieve its maximum value at expiration at only oneunderlying price, when the underlying contract is right at the inside exercise price. Forthis reason a condor will usually have a higher value than a butterfly with approximatelythe same exercise prices. Examples of long and short condors are shown in Figure 8-18.Volatility Spreads
Figure 8-17
Long iron Butterfiles
long 10 March 100 calls (51)
long 10 March 100 puts (—48)short 10 March 105 calls (24)short 10 March 95 puts (-21)
long 50 June 100 calls (51)
long 50 June 100 puts (-47)short 50 June 110 calls (23)short 50 June 90 puts (—17)
159
Short Iron Butterflies
short 15 June 100 calls (51)short 15 June 100 puts (-47)long 15 June 105 calls (36)long 15 June 95 puts (-31)
short 25 March 105 calls (24)short 25 March 105 puts (-75)long 25 March 110 calls (8)Jong 25 March 100 puts (-48)
SPREAD SENSITIVITIES
Just as every individual option has a unique delta, gamma, theta, vega, and rhoassociated with it, every spread position has unique sensitivities. These numbers canhelp a trader determine beforehand how changing market conditions are likely to affectthe spread. Before proceeding further the reader may want to review Figure 6-27, whichsummarizes the significance of the signs associated with the various option sensitivities,
A trader who initiates a volatility spread is concerned primarily with the magnitudeof movement in the underlying contract and only secondarily with the direction ofmovement. Therefore all volatility spreads will be approximately delta neutral (thedeltas will add up to approximately zero). It is true that some volatility spreads mayprefer movement in one direction rather than the other, but the primary considerationis whether movement of any type will occur. If a trader has a large positive or negativedelta, such that directional considerations become more important than volatilityconsiderations, then the position can no longer be considered a volatility spread.
All spreads which are helped by movement in the underlying market have a positivegamma. These include backspreads, long straddles, long strangles, short butterflies, andshort time spreads. All spreads which are hurt by movement in the underlying marketwill have a negative gamma. These include ratio vertical spreads, short straddles, short
Figure 8-18
Long Condors
long 10 March 90 calls (93)short 10 March 95 calls (78)short 10 March 100 calls (51)long 10 March 105 calls (24)
long 25 June 95 puts (-31)
short 25 June 100 puts (~47)short 25 June 105 puts (-62)long 25 June 110 puts (-75)
Short Condors
short 50 March 95 puts (-21)long 50 March 100 puts (—48)long 50 March 105 puts (~75)short 50 March 110 puts (-91)
short 5 June 90 calls (81)long 5 June 95 calls (67)long 5 June 100 cails (51)short § June 105 calls (36)160 Chapter 8
strangles, long butterflies, and long time spreads. A trader who has a positive gamma issometimes said to be Jong premiumand is hoping for a volatile market with large movesin the underlying contract. A trader who has a negative gamma is sometimes said to beshort premium and is hoping for a quiet market with only small moves in the underlyingmarket.
Since the effect of market movement and the effect of time decay always work inopposite directions, any spread with a positive gamma will necessarily have a negativetheta. Any spread with a negative gamma will necessarily have a positive theta. If marketmovement helps, the passage of time hurts. If market movement hurts, the passage oftime helps. An option trader can’t have it both ways.
Spreads which are helped by a rise in volatility have a positive vega. These includebackspreads, long straddles, long strangles, short butterflies, and long time spreads.Spreads which are helped by a decline in volatility have a negative vega. These includeratio vertical spreads, short straddles, short strangles, long butterflies, and short timespreads. In theory, the vega refers to the sensitivity of a theoretical value to a change inthe volatility of the underlying contract over the life of the option. In practice, however,traders tend to associate the vega with the sensitivity of an option’s price to a change inimplied volatility. Spreads with a positive vega will be helped (hurt) by any increase(decrease) in implied volatility. Spreads with a negative vega will be helped (hurt) byany decrease (increase) in implied volatility.
The delta, gamma, theta, and vega associated with the primary types of volatilityspreads are summarized in Figure 8-19. Since the delta of a volatility spread is assumedto be approximately zero, and since the theta is always of opposite sign to the gammabut of similar magnitude, we can place every volatility spread into one of four categoriesbased on the sign of the gamma and vega associated with the spread:
Category Gamma VegaBackspread Positive PositiveRatio Vertical Spread Negative NegativeLong Time Spread Negative PositiveShort Time Spread Positive Negative
If a trader is approximately delta neutral—no matter how complex the position,even if it includes options of different types, at many different exercise prices, and withseveral different expiration dates—he can always place the position in one of these fourgeneral categories based on the gamma and vega of the