376 Chapter 17
Since we are analyzing a position which includes two different futures months, wewill have to make some assumptions about the relationship between the two futurescontracts. The simplest relationship is based on a constant cost of carry. The currentspread of .45 (49.85 - 49.40) between the May and July futures months is based onthis relationship, since:
56/365 x 6% x 49.40 = .45
If the spread were based only on cost of carry, and interest rates remained constant at6%, July futures would always be .92% (56/365 x 6%) greater than May futures.Unfortunately, many markets, particularly in energy and agricultural products, aresubject to supply and demand forces which often distort the cost of carry relationshipbetween futures months. For this reason, traders often choose to assume a fixed spreadrelationship when analyzing option positions in several different futures months. In ourexample, we will make the assumption that the spread between May and July futuresis always .45.
We have also added the assumption that July futures will be slightly more volatile(16%) than May futures (15%). Even though these volatility assumptions seem to beslightly higher than the implied volatilities in the marketplace, the trader’s volatilityrelationship between May and July is consistent with the marketplace’s opinion wherethe July implied volatility seems to be slightly higher than the May implied volatility.
Figures 17-6a through | 7-6f show the current characteristics of the position, as wellas the characteristics under different time and volatility assumptions. At the currentunderlying price (May futures = 49.40, July futures = 49.85), the passage of time or anydecline in volatility will hurt the position (negative theta, positive vega). But if theunderlying market should either decline or rise several points, the passage of time willbegin to help the position (Figure 17-6a). Even though the theta may become positivewith a move of several points, the vega of this position always remains positive (Figure17-6b). Regardless of underlying price, any increase (decline) in volatility will help (hurt)the position. The position acts like a long straddle with the underlying close to 50, butit acts like a long time spread with the underlying close to 46 or 53. This is typical ofhow difficult it can sometimes be to classify complex positions. If the position is largeenough, even small changes in market conditions can cause the position to change itscharacteristics radically.
Note that if one were to look only at the delta and gamma, it might seem that thebest thing which could happen to this trader would be a swift decline in the underlyingmarket {negative delta, positive gamma). If, however, the decline is too violent, withthe market falling five or six points, the trader will have to consider his net contractposition. All calls will collapse to zero, but the net total of 46 puts which he is short willbegin to act like long futures. These will be partially offset by the 22 futures which heis actually short. But the result will be a position which is net long 24 contracts, leavingthe trader with unlimited downside risk. Of course, a violent upside move will neverbe a problem since the trader has an upside contract position which is net long 29contracts (long 51 calls, short 22 futures).Position Analysis 377
 
Figure 17-6a
200-—
 
1505
5 weeks to May expiration (current)
at May expiration
 
1005oo8
=8aBss
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oOa5
505
 
 
 
 
 
40 42 44 46 48 50 52 54 56 58 60May futures price
 
Figure 17-6b
200
 
180" current volatility
volatility increases 25% (May= 15%, July=16%)(May= 18.75%, July=20%) *
   
 
theoretical profit & loss
 
volatility declines 25%(May=1 1.25%, July=12%)
 
 
 
 
T T T T
-150-+- rT T T T40 42 44 46 48 50 52 54 56 58 60
May tutures price
 378 Chapter 17
 
Figure 17-6c
150
 
at May expiration
 
100-1 1
 
delta position (hundreds)
  
4 2 weeks to May expiration
 
 
 
 
 
t00+ T T T T t T T T T4 42 44 46 48 50 52 84 56 58 60May futures priceFigure 17-6d
 
current volatility(May= 15%, July=16%)
 
 
 
delta position (hundreds)
-t07
 
volatility increases 25%(May=18,75%, July=20%)
-20- volatility declines 25%
 
 
 
 
(May=1t.25%, July=12%) \-304—- T T 7 7 T 1 1 , _—40 42 44 46 48 60 52 8A 56 58 60
May futures price
 Position Analysis 379
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
Figure 17-6e50iH40- ioeo2Zc2£Cco2oaoS£EoOa-304 ;te 1 week to May expiration° j-40-+-——___-—_——_1— T T i—r T T T TT40 42 44 46 48 50 32 4 56 68 60May futures priceFigure 17-6f20| 4 current volatility: : (May=15%, July=16%)155 i ‘105 a : ;volatility increases 25% } .(May= t8.75%, July=20%) ae +3 !2 5&4 .3<2£eco= -= .oos . f€ \E 43oa“464 ‘ : volatility declines 25%\ Q (May= 11.25%, July= 12%)-20 + T 17 TT 7 T T —40 42 44 46 48 50 52 54 56 58 60
May futures price
 380 Chapter 17
Figure 17-6 also gives the implied delta for the individual options as well as for thetotal position. This can sometimes be a useful number because a trader may want totake into consideration current market conditions when deciding how to hedge hisposition. In this example there is little difference between the trader’s presumed deltaposition (-253) and his implied delta position (-242]. But this is not always the case. Ifthe trader’s volatility assumption is dramatically different than the implied volatility inthe marketplace, the presumed delta position might be significantly different than theimplied delta position. Since no one can be sure what the right volatility is, no tradercan be sure that his delta calculations are correct. Given this uncertainty, it is notuncommon among traders to hedge a position somewhere between the trader’s owndelta and the implied delta. If, for example, a trader calculated his delta position as+600, but found that the implied delta was +200, he might treat the position as if it hada delta of +400. If he wanted to get delta neutral, the trader might sell four underlyingcontracts.
We made the assumption that the spread between the July and May futurescontracts will remain constant at .45. What will happen if the spread begins to widenor narrow? The total delta of -252 indicates that the trader would like a decline in themarket. With a July delta of -635 and a May delta of +382, the trader would prefer Julyto go down faster than May. The ideal would be for July futures to decline and at thesame time for May futures to rise, but any narrowing of the Juty/May spread shouldhelp the position. In futures equivalents, we might describe the position as being roughlyshort three July/May futures spreads, and short an additional 214 July futures.
Finally, we might want to analyze a total position which consists of contracts whichare not directly related, but which have somewhat similar characteristics. For example,suppose we have the ratio vertical call spread presented at the beginning of this chapter:
ABC stock price = 99.00; Time to June expiration = 7 weeksVolatility= 20%; Interestrate=6%; Dividend = 0
Option Option Option Option
Position Price Th. Val. Delta Imp. Vol.long 10 June 95 calls 6.25 5.82 76 23.7short 30 June 105 calls 1.63 1.08 26 24:5
At the same time, suppose we also have a long strangle in a different stock:
XYZ stock price = 62.00; Time to July expiration = 11 weeksVolatility = 23%; Interest rate = 6%; Dividend = .50 in 8 weeks
Option Option Option Option
Position Price Th. Val. Delta Imp. Vol.long 20 July 65 calls 1.38 1.52 36 21.6long 20 July 60 puts 1.50 1.57 ~34 22.3
If we combine these two positions, how might we analyze the overall risk? Onepossible method might be to look at the change in value of the two positions per pointmove in the underlying stocks. But because the two stocks are trading at different prices,Position Analysis 38!
and because they are expected to have different volatilities, a one-point move in ABC isunlikely to be matched by a one-point move in XYZ. To equate movement in ABC withmovement in XYZ, it may be more useful! to look at price changes in terms of standarddeviations. For example, we can approximate a weekly standard deviation for eachstock by dividing the annual volatility by ¥52, and multiplying this by the stock price:
ABC: 20%/7.2 x 99 = 2.75XYZ: 23%/7.2 x 62 = 1.98
From these numbers we can generate graphs showing the position value given variousstandard deviation price changes in each underlying stock. This has been done in Figure17-7a. Since we have a ratio vertical spread in ABC stock options and a long strangle inXYZ stock options, it should come as no surprise that the position will show itsmaximum theoretical profit when ABC sits still and XYZ makes a big move.
In order to complete our analysis we might want to look at the delta, gamma, theta,and vega of the position. But, again, with the stocks at different prices it may be difficultto equate the sensitivities of the ABC options with the sensitivities of the XYZ options.Since our total position is an intermarket spread, rather than looking at the delta,gamma, theta, and vega of the position, we might instead look at the $delta, $gamma,Stheta, and $vega sensitivities introduced in Chapter 16. These sensitivities are shownin Figures 17-7b through 17-7e.°
 
 
 
 
 
Figure 17-7a100 7XYZ moves down L XYZ moves up4st. devs. 4 st, devs.604 xyz moves up2 st. devs.{@ 405 |9 eee —_2 ee TTT oe= _ ~~= 204 —6 bo . ce we fen m=a ~~ ce ao sO SN§ of << ~3 S8 ~ SA= -204 x O™XYZ moves down me A2 st. devs. PON-40-l NO oN40 XYZ doesn’t move SONS-605-80-- T T T t T T T4 3 -2 1 0 1 2 3 4
 
ABC Price Movement (st. devs.)
3. In Figures 17-7b through 17-7e we make the assumption that each point is worth $100. 
 
      
 
 
 
 
 
       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
 
382 Chapter 17Figure 17-7b1500 rXYZ moves up4st. devs.10005EI“0 XYZ moves uptee ~ 2 st. devs.- ~ Te NN2 oF= ; S.é | .£ 500+ ~ ~ NN8 Na NNg NNe£ -1000- .XYZ moves down *-1600- 481. devs. XYZ moves down . .2 st. devs. NNXYZ doesn’t move-20007-2500 ——__—___— { — , —4 -3 -2 - 0 1 2 3 4ABC Price Movement (st. devs.)Figure 17-7c200 i{i150-4< 1004 oN XYZ doesn’t move2 ws=9a== 504 XYZ moves& up or downa 2 st. devs.gSe AEe 0XYZ movesup or down-605 4 st. devs.-100+ ——T T a —— ; +-4 3 2 -1 9 1 2 3 4
ABC Price Movement (st. devs.)
 Position Analysis
383
 
 
   
 
 
      
 
 
 
 
Figure 17-7d100:XYZ moves down4st, devs.80>60-4XYZ moves down2 st. devs.40-<7g& 205g - XYZ moves upEc 4, st. devs.a og XYZ doesn't moveEeXYZ moves up2. st. devs.“80 1 1 — —- 1 1 r4 3 2 “4 0 1 2 3 4ABC Price Movement (st. devs.)Figure 17-7e500wee oe, |4004-20
Total $Vega Pesition
 
200°)
XYZ moves down1007 2st. devs.
      
i| XYZ moves up| 2. st, devs.|
XYZ doesn't move
 
 
 
 
 
 
0-4 _— <4eeXYZ moves down-100> Ast. devs,-2004 aeXYZ moves up4. st. devs.-300-+-————— T T T T 7 a4 - 2 -1 0 1 2 3 4
ABC Price Movement (st. devs.)
 384 Chapter 17
It may initially be difficult to grasp why the $gamma, $vega, and $theta arechanging as they are. If one recalls that the gamma, theta, and vega are always greatestfor an at-the-money option, the position should show its greatest positive $gamma,negative $theta, and positive $vega when ABC is close to 95 (the position is long ABC95 calis) and when XYZ is close to either 60 or 65 (the position is long XYZ 60 puts andXYZ 65 calls). The position should show its greatest negative $gamma, positive $theta,and negative $vega when ABC is close to 105 (the position is short ABC 105 calls) andthe when XYZ is as far away from 60 and 65 as possible. Since price movement inFigures 17-7a through 17-7e is expressed in standard deviations, in order to interpretthe graphs we need to know where the exercise prices are in terms of standarddeviations. With ABC at 99 and a one standard deviation price change of 2.75, the ABCexercise prices in terms of standard deviations are:
95 call: (95- 99} 72.75 =-1.5 st. devs.105 call: (105-99) / 2.75 = +2.2 st. devs.
With XYZ at 62 and a one standard deviation price change of 1.98, the XYZ exerciseprices in terms of standard deviations are:
60 put: (60-62) / 1.98 =-1 st. dev.65 call: (65 —62) / 1.98 = +1.5 st. devs.
With these values in mind, the reader should take some time to study the graphs andconfirm that the $delta, $gamma, $theta, and $vega of the position are in fact consistentwith the principles of option evaluation.
The foregoing example consisted of positions in only two different underlyingmarkets. If a trader had option positions in several different underlying markets, adetailed analysis of the total position would require multidimensional graphs. Even ifthey could be constructed, such graphs would be too complex for most traders tointerpret. Under such circumstances the only practical solution is to separate positionsby the underlying instrument and do an individual analysis of each market.Se 18 eo
Models and the Real World
The accuracy of values generated by a theoretical pricing model rests on two points: theaccuracy of the assumptions upon which the model is based, and the accuracy of theinputs into the model. Thus far, we have tended to focus on the second area, theaccuracy of the inputs. Through attention to the sensitivities of an option position (delta,gamma, theta, vega, and rho), a trader can identify those inputs which represent thegreatest risk to his position, and thereby prepare to take protective action if things gowrong. While all the inputs into the model representa risk to the trader, we have placedspecial emphasis on volatility, since it is the one input which cannot be directly observedin the marketplace.
However, the active option trader cannot afford to ignore the first area, thepossibility that the assumptions which form the foundation of the model are themselvesinaccurate or unrealistic. Some of these assumptions pertain to the way business istransacted in the marketplace, while others pertain to the mathematics of option pricing.
To begin, we might list the most important assumptions built into a traditionalpricing madel:!
1. Markets are frictionlessa. The underlying contract can be freely bought or sold, without restrictionb. There are no tax consequences associated with trading
c. Everyone can borrow and lend money freely, and one interest rate appliesto all transactions
d. There are no transaction casts2. Interest rates are constant over the life of an option3. Volatility is constant over the life of an option
4. Trading is continuous with no gaps in the price changes of an underlyinginstrument
5. Volatility is independent of the price of the underlying contract
6. Over short periods of time the percent price changes in an underlying contractare normally distributed, resulting in a lognormal distribution of underlyingprices at expiration
1. By “traditional pricing model” we mean those that are mast commonly used: the Black-Scholes modeland {ts variations, the Cox-Ross-Rubenstein model, or the Whaley model.
385386 Chapter 18
The reader may already have an opinion about the accuracy of these assumptions, butlet’s consider them one by one.
MARKETS ARE FRICTIONLESS
In Chapter 5 we came to the obvious conclusion that markets are not frictionless, Theunderlying contract cannot always be freely bought or sold; there are sometimes taxconsequences; a trader cannot always borrow and lend money freely, nor at the samerate; there are always transaction costs.
In the futures markets the underlying cannot always be freely bought or soldbecause exchanges sometimes set daily price limits beyond which a futures contractmay not trade. When that limit is reached, trading is halted until the market comes offits limit. If it doesn’t come off limit, trading does not resume until the next business day.
The consequences of a locked futures market are usually not severe, and there areseveral ways for a trader to circumvent the problem. Instead of trading in the futuresmarket, a trader might be able to trade in the cash market. Or the trader might be ableto trade the underlying market by trading a futures spread where one side of the spreadis not locked. For example, suppose a trader wants to buy a June futures contract whichis up limit. If a March futures contract is still trading because it is not up its limit, thetrader can buy a June/March spread (buy June, sell March], and then go back into themarket and buy the March futures contract. This leaves him long a June futurescontract, which is what he wanted. Finally, if the underlying futures market is locked,but the option market is not locked, a trader can trade synthetic long or short futurescontracts.
In the stock market, there may be prohibitions against the sale of stock not actuallyowned. Or if short sales are allowed, there may be restrictions on when such sales canbe made. Ifa trader cannot freely sell stock, put prices become inflated compared to callprices, and all the conversions and reversals appear to be mispriced with respect to theirtheoretical values. If short sales are not freely permitted, many stock option traders tryto carry some long stock so they will always be in a position to sell stock if the needarises. Also, we saw in Chapter 5 in the discussion of time spreads, and in Chapter 12in the discussion of boxes, that tender offers for a stock can distort the value of thesepositions. In the case of time spreads, traders may be forced to exercise call options withsome time value remaining because no stock can be borrowed to carry a short position.A box may be trading at greater than its theoretical value if a partial tender offer ts madeand early exercise is a possibility.
While there are occasionally tax considerations, for most market participants theseare usually minor. When considering a strategy, a trader is unlikely to ask himself, “Ifthis trade makes or loses money, what will be the tax consequences?” Nor should mosttraders ask such a question. Differences in tax consequences, or changing tax conse-quences, rarely make one strategy better than another.?
2.  Thisis not to say that tax consequences are always insignificant. Tax considerations can sometimes playa role in portfolio management. They can also play arole in stock option strategies involving dividends,when the dividends are subject to different tax rules than the gains or losses from stock or options.Models and the Real World 387
The assumption that a trader can always borrow or lend money freely is a moreserious weakness in pricing models. Even if a trader has sufficient funds to initiate atrade, he may find at some later date that he is required to come up with additionalfunds in the form of increased margin requirements? If money were freely availablemargin would never be a problem. A trader could always borrow margin money anddeposit the money with the clearing house. Since the borrowing and lending rate areassumed to be the same, and since the clearing house in theory pays interest on themargin deposit, there would never be a problem obtaining margin money, nor wouldthere ever be a cost associated it.
In the real world traders do not have unlimited borrowing capacity. If a tradercannot meet a margin requirement he may be forced to liquidate a position prior toexpiration. Since all models, even those which allow for early exercise, assume that atrader will always have the choice of holding a position to expiration, the inability tomeet margin requirements, and therefore maintain the position, can make the valuesgenerated by the theoretical pricing model less reliable. Experienced traders learn toconsider the risk of a position not only from the standpoint of how much the positionmight lose if things go wrong, but how much margin might be required to maintain theposition over the life of the options. A trader should always have sufficient marginmoney available ta meet this latter requirement.
Even if a trader had unlimited borrowing capacity, the fact that for most tradersborrowing and lending rates are not the same can also cause problems with strategiesbased on model generated values. A trader who borrows margin money at one rate willalmost certainly receive a lower rate when he deposits this money with the clearinghouse. The difference between these rates, which will result in a loss to the trader, issomething of which the model is unaware. And the greater the difference betweenborrowing and lending rates, the less reliable will be the values generated by the model.
The most serious flaw in the frictionless markets hypothesis is the assumption thatthere are no transaction costs. While a strategy might or might not be affected by tax orinterest rate considerations, there are always transaction costs. These costs can come inthe form of brokerage fees, clearing fees, or an exchange membership. For many marketparticipants, transaction costs may be so great that a strategy which looks sensible basedon model-generated values may not be worth doing when transaction costs are alsotaken into consideration. Moreover, transaction costs can accrue not only when thestrategy is initiated or liquidated, but also every time an adjustment is made. Ifa strategywill require many adjustments because it has a high gamma and the trader intends toStay delta neutral, the transaction costs can have an especially great impact on model-generated values.
3. The possibility that a trader in a futures option market may also have to come up with additionalvarlation money, as opposed to margin money, subsequent to establishing an option position Isincorporated into most models. This is why a conversion or reversal in a futures option market may notbe delta neutral.388 Chapter 18
INTEREST RATES ARE CONSTANTOVER THE LIFE OF AN OPTION
The interest rate component assumed in a theoretical pricing model is the riskless rateover the life of an option. Any credits or debits associated with an option strategy willbe subject to an interest rate identical to the least risky rate corresponding to the termof the option. In most markets the least risky rate is the rate associated with governmentissued securities. A trader in a U.S. option market who sells straddles consisting ofthree-month calls and puts is assumed by the model to have invested the resulting creditin three-month government treasury bills. If both the option position and the treasurybills are held to expiration, this effectively locks in one interest rate (that associated withthree-month treasury bills) over the life of the options.
In reality few traders actually take the proceeds from an option trade and invest itin government securities. More often, a trader will put the proceeds in a tradingaccount, where it is subject to a variable interest rate. Moreover, while a trader can lendmoney to the government by purchasing government securities, he cannot do thereverse and borrow money from the government at the same rate. If he wants to borrow,he may have to go to a bank where the cost of borrowing could be subject to a variableinterest rate. Given the realities of the marketplace, most traders’ transactions are notsubject to a fixed and riskless rate, as the model assumes, but to a constantly changinginterest rate. Is this a cause for worry?
While a changing interest rate will cause the value of a trader’s option position tochange, interest rates usually do not change in a way which will have a significantimpact on an option’s value, at least in the short run. Since the effect of changing interestrates is a function of time to expiration, and since most listed options have terms of lessthan nine months, interest rates would have to change violently to have an impact onany but the most deeply in-the-money options. Changing interest rates become even lessof a concern when one considers how much more sensitive option values are to changesin the price of the underlying instrument or to changes in volatility.
This is not to say that a trader should completely ignore the possibility of changinginterest rates. As we saw in Chapter 8, changing interest rates can have an impact ontime spreads in the stock option market. But even then, interest rates would have tochange several percentage points over a short period of time to have a significant impact(see Figure 8-15).
With the introduction of long-term equity options, known as LEAPS, the conse-quences of changing interest rates may well become more of a concern. Using theBlack-Scholes model, we can see in Figure 18-1 that the values of two-year stock optionschanges significantly with only small changes in interest rates (delta values are inparentheses}. Note particularly how quickly the 120 put drops below its parity value of20.00. In a high interest rate environment, a long-term in-the-money American put willquickly become an early exercise candidate. Everyone will prefer to exercise the put inorder to earn a high rate of interest on the proceeds from the sale of the stock.389
 
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Models and the Real Worid390 Chapter 18
VOLATILITY IS CONSTANTOVER THE LIFE OF THE OPTION
When a trader feeds a volatility into a theoretical pricing model, he is telling the modelthe magnitude of price changes which will occur over the life of the option. Given thecharacteristics of a normal distribution, the model uses the volatility to extrapolate acertain number of one, two, three, or more standard deviation price changes. Moreover,the model assumes that occurrences of each particular magnitude will be evenlydistributed over the life of the option. Two standard deviation price changes will beevenly distributed among the one standard deviation price changes; three standarddeviation price changes will be evenly distributed among the one and two standarddeviation price changes; and so on.
Look at Figures 18-2a and 18-2b, daily bar charts of an underlying contract overthe life of a certain option. Both bar charts exhibit exactly the same volatility over theperiod in question, 16.7%. Yet it is obvious that the order in which the volatility occursis completely different. In Figure 18-2a all the large price changes occur during the earlypart of the option’s life, while in Figure 18-2b all the large price changes occur duringthe latter part of the option’s life. This is often how price changes actually occur in thereal world, rather than the even distribution which the model assumes. A tradertypically encounters periods of high volatility where all the two and three standarddeviation price changes seem to be bunched together. Or he encounters periods of lowvolatility where all the one standard deviation price changes seem to be bunchedtogether. A graph of historical volatility is never a straight line. (For a good example of