inreturn for the partial downside protection afforded by the sale of the 100 call. Theposition is bullish, but with both limited risk and limited reward.
If we are bearish on the market we might create a position with limited risk andlimited reward by inverting the position in Figure 2-10. That is, we might sell the 90call and purchase the 100 call. Now our potential profit is limited to 6.65 if the marketfinishes below 90, and our potential loss is limited to 3.35 if the market finishes above100.
We can also create a bearish position with both limited risk and limited reward bypurchasing a put with a higher exercise price and selling a put with a lower exerciseprice. For example, we might buy a 105 put for 7.10 and sell a 100 put for 3.70, foratotal debit of 3.40 (Figure 2-11). If the underlying contract is below 100 at expiration,Elementary Strategies 23
aFigure 2-10
 
10long a 90 call at 9.358 short a 100 call at 2.70
Profit
 
Loss
 
 
 
 
85 90 95 100 105 110 115
the 105 put will be worth exactly 5 points more than the 100 put, and we will realizeour maximum profit of 1.60. If the underlying contract is above 105 at expiration, bothoptions will be worthless and we will lose our entire investment of 3.40. The positionwill do no worse than break even if the underlying contract is at or below 101.60, forthen the 105 put will be worth at least 3.40. Like the outright purchase of a put, thisposition is bearish. However, we have chosen to give up the unlimited downside profitpotential associated with the outright purchase of the 105 put in return for the partialupside protection afforded by the sale of the 100 put.
CONSTRUCTING AN EXPIRATION GRAPH
From the foregoing examples, we can formulate some simple rules for drawing expira-tion profit and loss graphs:
1. If the graph bends, it will do so at an exercise price. Therefore, we cancalculate the profit or loss at each exercise price involved and simply connect thesepoints with straight lines.
2. If the position is long and short equal numbers of calls (puts), the potentialdownside (upside) risk or reward will be equal to the total debit or credit requiredto establish the position.24 Chapter 2
 
 
Figure 2-1110 -8- long a 105 put at 7.10short a 100 put at 3.7064~ 4©o
 
 
 
 
 
100 105 -3. 40__+ 4! '648-5-10 T r T 185 90 95 100 105 110 115
3. Above the highest exercise price all calls will go into-the-money, so the entireposition will act like an underlying position which is either long or short underlyingcontracts equal to the number of net Jong or short calls. Below the lowest exerciseprice all puts will go into-the-money, so the entire position will act like anunderlying position which is either long or short underlying contracts equal to thenumber of net long or short puts.
To see how we can use these rules to construct an expiration graph, consider thefollowing position:
long one 95 call at 5.50short three 105 calls at 1.15
The first step is to determine the profit or loss at each exercise price (95 and 105).If the underlying contract finishes at 95, both the 95 and 105 calls will be worthless.Since the entire position was established for a debit of 2.05 (-5.50 + 3 x 1.15), at 95the position will show a loss of 2.05. If the underlying contract finishes at 105, the 95call will be worth 10.00 and the 105 calls will be worthless. Since we own the 95 call,the position will be worth 10 points less the initial debit of 2.05, or 7.95. We can plotand connect these two points on a graph (Figure 2-12a).Elementary Strategies 25
 
 
 
 
 
 
Figure 2-12a1084 long 1 March 95 call at 5.50short 3 March 105 calls at 1.1564Al +7.95es |!ao 495 1050 | |a0 { vo-27aSo a4-64-85-10 T T r 1 T85 80 95 100 105 110 115
Next, we note that there are no puts involved in this position, so the maximumdownside loss we can incur is the 2.05 debit required to establish the position. This losswill occur anywhere below 95 (Figure 2-12b).
Finally, above 105 both the 95 and 105 calls will go into-the-money, so that alloptions will begin to act like long underlying contracts. We will be long one underlyingcontract in the form of a 95 call, and short three underlying contracts in the form ofthree 105 calls. The net result is that above 105 we have a position which is equivalentto being short two underlying contracts. For each point increase in the price of theUnderlying contract, our position will lose two points (Figure 2-12c).
Applying this method to a more complex example, what would be the expirationgraph of the following position?
short one 90 call at 9.35
long two 100 calls at 2.70short four 95 puts at 1.55long two 100 puts at 3.7026
Figure 2-12b
10
Profit
Loss
-10
Chapter 2
 
long 1 March 95 call at 5.50short 3 March 105 calls at 1.15
96 105
 
 
20s { oO | |
~~
 
 
85
Figure 2-12c
10
115
 
Profit
Loss
 
long 1 March 95 call at 5.50short 3 March 105 calls at 1.15
 
90 95 100 105
 
 
115
 Elementary Strategies 27
First, what happens at the three exercise prices involved? At 90 we have:
90 call 49.35100 calls 2 x 2.7095 puts ~4x 3.45100 puts +2 x 6.30Total +2.75At 95 we have:90 call 44.35100 calls —2 x 2.709S puts +4 x 1.55100 puts +2 x1.30Total +7.75And at 100 we have:90 call -.65100 calls -2 x 2.7095 puts 44x 1.55100 puts —2 x 3.70Total -7.25
We can plot and connect these profit and loss points at each of the exercise prices(Figure 2-138).
Next, below 90 all calls will be worthless and all puts will act like short underlyingcontracts. Since the position is net short two puts, the graph will act like a positionwhich is long two underlying contracts. For each point that the underlying contract fallsbelow 90, the position will lose two points (Figure 2-13b).
Finally, above 100 all puts will be worthless and all calls will act like longunderlying contracts. Since the position is net long one call, above 100 it will act like aPosition which is long one underlying contract. For each point rise in the underlyingcontract, the position will gain one point in value (Figure 2-13c).
Using this method we can draw the expiration profit and loss graph of any position,No Matter how complex. The position may consist of underlying contracts, and calls andputs at various exercise prices, But as long as all options expire at the same time, theValue of the position at expiration will be fully determined by the price of the underlyingcontract.
Using options and underlying contracts it is also possible to create positions whichMimic other option and underlying positions. For example, what are the characteristicsof the following position?
long one 100 call at 2.70short one 100 put at 3.7028 Chapter 2
Figure 2-13a10
 
short 1 March 90 call at 9.358+ long 2 March 100 calls at 2.70short 4 March 95 puts at 1.55long 2 March 100 puts at 3.70+775
24+275 { 90 95 100
4
6
Profit
-7.25
Loss
 
 
 
85 90 95 100 105 110 115
Figure 2-13b10
 
short 1 March 90 call at 9.35
long 2 March 100 calls at 2.70short 4 March 95 puts at 1.55long 2 March 100 puts at 3.70
85
67
44
Profit
 
Losswp[|ny
-84
 
 
-10 T T T T 785 90 95 100 105 110 115
 
 Elementary Strategies 29
 
 
   
 
 
 
 
 
 
Figure 2-13¢10short 1 March 90 call at 9.3584 long 2 March 100 calls at 2.70short 4 March 95 puts at 1.5565 long 2 March 100 puts at 3.70# 42a. 42 \ 1000 -|- --2+2 /Sa-64-8--10 T T T T T85 90 95 100 105 110 115
 
With the underlying contract above 100 at expiration, the 100 put will beworthless and the 100 call will act like a long underlying contract. With the underlyingcontract below 100 at expiration, the 100 call will be worthless and the 100 put willact like a short underlying contract. However, since the position is short the 100 put,when the 100 put is in the money it will act like a long underlying contract. In otherwords, this position will mimic a long underlying position regardless of where theunderlying contract is at expiration (Figure 2-14). The only real difference between theoption position and a long position in the underlying contract is that the option positionwill create a credit of one point.
Or consider the following position:
long one 90 put at .45short one 100 call at 2.70long one underlying contract at 99.00
The value of this position at expiration is given in Figure 2-15. Note the similaritybetween this position and the one in Figure 2-10. The only difference appears to be thatWe replaced the long 90 call with the combination of a long 90 put and along underlyingContract. Therefore, a long 90 put and a long underlying contract together must mimica long 90 call. As proof, the reader should draw the expiration profit and loss graph ofthe following two positions:30
Chapter 2
10
 
long a 100 call at2.70 © ~~~87 short a 100 put at 3.70 ~~~ --combined position
Profit
Loss
 
   
 
Figure 2-1510
 
110 115
 
long a 90 put at .45short a 100 call at 2.7064 long an underlying contract at 99.00
84
Profit
 
Loss
 
 
 
 
110 115
 Elementary Strategies 31
1. longa 90 call at 9.352. longa 90 put at .45 and long an underlying contract at 99.00
Although there will be a small difference in the profit and loss to each position, bothpositions will have graphs with the same general shape.
The reader who is new to options may find it useful to sit down with the businesssection of a newspaper and create and graph a variety of option and underlyingpositions. This will enable him to become comfortable with many of the conceptsintroduced in the first two chapters, and will make the subsequent material that mucheasier to understand. Typical newspaper listings for futures options and stock optionsare shown in Figure 2-16.
While elementary strategies such as the ones discussed in this chapter are aconvenient method of introducing the new trader to basic option characteristics, inactual practice it is very unlikely that a trader will put on a position and let it go toexpiration. Even if he were to initiate a position with such an intent, he would be foolishindeed to simply walk away from the position and come back at expiration to find outwhether he had made or lost money. As market conditions change, a position whichseemed sensible yesterday may not seem quite so sensible today. Based on newconditions the trader may want to—indeed, may have to—alter his strategy. This is truefor all traders, not just for option traders. A trader who buys stock in the belief that itwill rise would be foolish not to reassess his position if the stock were to unexpectedlydrop several points. An option trader who sells 105 calls in the belief that the underlyingcontract will never rise above 105 would likewise be foolish not to reassess the situationif the underlying contract were to make a rapid upward move from 99 to 104. He mayStill believe that the underlying contract will not rise above 105, but he is unlikely tohave the same degree of confidence. Every trader reserves the right to make a moreintelligent decision today than he made yesterday.
The serious trader must be able to identify potentially profitable strategies givencurrent Market conditions. But he must also be able to adjust to changing marketconditions and to take protective measures when the market moves adversely. In thenext chapter we will begin our investigation of basic option pricing theory, and showhow this theory can be used to help a trader achieve these goals.
eV—_——_—_2. In the United States, investor’s Business Daily, the New York Times, and The Wall Street JournalallCarry extensive listings of exchange traded futures options and stock options.32
Figure 2-16
Futures and Futures OptionsFRIDAY, OCTOBER 22, 1993
Chapter 2
Futures-Lifetime-Month Open High Low Settle Change High LowCORN (CBT); 5,000 bushels; ¢ per bushel (1 = $50.00)Dec 255", 256 253'/, 253'/, -2'h, 2667, 225°,Mar94 «263 264'/, 261'/, 261'/, -2'1, 2667, 232%,May 267'/, 268'/, 265 265'/, 3 2707, 238",Jul 268'/, 270 266, 266%, -3'/, 2707, 241Sep 260 260'/, 2571, 257'/, “3's, 2617, 240",Dec 251"/, 253 249 249'/, 3, 255 236"/,EURODOLLARS (CME); $1,000,000; points of 100% (.01 = $25.00)Dec 96.50 96.52 96.48 96.49 -.02 96.61 90.22Mars4s«96..50 96.52 96.46 96.47 -.04 96.62 90.28Jun 96.30 96.31 96.24 96.25 -.05 96.75 90.40Sep 96.06 96.10 96.02 96.03 -.06 96.24 90.36Dec 95.70 95.73 95.65 95.66 -.05 95.87 90.74Mar95 —s-«95.62 95.65 95.57 95.58 -.05 95.80 95.24Jun 95.42 95.44 95.37 95.37 -.05 95.60 95.74Sep 95.25 95.27 95.20 95.20 -.05 95.43 91.31Dec 94.97 94.98 94.91 94.91 -.06 95.81 91.18GERMAN BONDS (LIFFE); DM 250,000; points of 100% (.01 = DM 25.00)Dec 100.13 100.50 100.03 100.46 +29 100.58 94.25Mar94 100.26 «= 100.55 100.21 100.56 +.30 100.66 97.30A 100.46 +.27 100.49 98.94CRUDE OIL (NYMEX); 1,000 barrels; $ per barrel (.01 = $10.00)Dec 16.37 18.43 18.02 16.07 -.28 23.00 17.14Jan94. 18.47 18.57 16.18 18.22 -.26 21.15 17.40Feb tee 18.64 18.36 16.37 -.25 20.81 17.64Mar 18.71 18.74 18.48 16.50 22 21.10 17.86Apr 18.78 18.83 18.58 16.61 -.24 20.86 16.05May 418.90 18.91 18.72 16.72 -.20 21.07 = 18.20Jun 419.041 19,02 18.80 16.82 -.19 21.35 = 18.31Jul 19.07 19.08 16.89 18.89 18 20.78 18.50AUG eaters esate 18.96 “7 20.78 = 18.72
 Elementary Strategies 33Figure 2-16 (continued)
Futures and Futures OptionsFRIDAY, OCTOBER 22, 1993
Futures Options
StrikePrice Calls-Settle Puts-SettleCORN OPTIONS (CBT)Dec Mar May Dec Mar May230 23%, 314, 35 %/, "y " ‘I,240 14%, 2271, 25 11, I, 1 471,250 6, 1471, 18", 3 3's, 3%,260 2 9, 121, 8's, 7h, 7,270 * 5°, 84, 17 13, 13280 ‘tl, 3, 5", 26 te anaes290 sy 1", 3%, 36, eee eats300 1, I, 2 vee nanaEURODOLLAR OPTIONS (CME)Dec Mar Jun Dec Mar Jun95.25 1.26 1.26 1.07 cab cab 0495.50 1.01 1.01 84 cab 041 0595.75 75 77 62 cab .02 0996.00 51 54 43 .01 04 596.25 27 .30 26 02 06 2496.50 08 +13 -13 .08 AT 3796.75 01 03 05 25 32 5097.00 cab 04 01 49 500GERMAN BOND OPTIONS (LIFFE)Dec Mar Jun Dec Mar Jun98.50 1.99 2.33000 wu. 03 QT eee99.00 1.62 194 oa .06 SB99.50 1.11 168 uae 15 52 ase100.00 72 1.280 .26 2 sae100.50 44 9.04 see -48 95 7101.00 24 TO eee .78 1.23000; oe101.50 13 ST saeee 1.17 1.51102.00 07 Me 1.51 1.85CRUDE OIL OPTIONS (NYMEX)Dec Jan Feb Dec Jan Feb15.00 3.08 aa tee .04 .03 0616.00 2.09 aese ae 02 08 1317.00 1.14 4.39 ae 07 18 .2718.00 35 65 92 28 43 5519.00 .06 .24 44 99 1.02 1.0720.00 02 .09 .20 1.95 1.86 1.6221.00 04 03 .07 2.94 2.80 vane22.00 041 01 04 3.93 tose
eee34
Figure 2-16 (continued)
Equity Options
FRIDAY, OCTOBER 22, 1993
S&P 100 INDEX-OEX 424.15Strike Calls-LastPrice Nov Dec390 ee ane395 297%, 30400 28‘, 29 |,405 21 saees410 18 18415 1's, 13 "1,420 7", 9°,425 4 6%,430 4°71, 4,435 1, 2°),440 “he 1 he445 " ",450 Ie ",455 ‘Ne "IBM 44%, (CBOE)
Strike Calis-LastPrice Nov JanBh anes teens40 41, 5°),45 1, 2"
50 *he a55 ne ‘teMOTOROLA 103‘, (AMEX)Strike Calls-LastPrice Nov Dec90 13 ‘1, 1371,95 91, 10 4,100 5°), 7105 2°), 4%,110 1 2%,115 t, 1
(CBOE)
Apr10 "/,7
Puts-Last
Chapter 2eo 3 So
Introduction toTheoretical Pricing Models
Tn the last chapter we looked at some of the simple option strategies a trader mightinitiate given an opinion on an underlying contract’s likely price movement. Whateverthe basis for the opinion, it will probably be expressed with terms such as “goodchance,” “highly likely,” “possible,” “improbable,” etc. The problem with this approachis that opinions cannot easily be expressed in numerical terms. What do we really meanby “good chance”? Or by “highly unlikely”? If we want to approach option marketslogically we will need some method of quantifying our opinions about price movements.
From the elementary strategies discussed in the previous chapter it is clear that thedirection in which the underlying market moves can have a significant effect on theprofitability of an option strategy. Consequently, option traders are sensitive to thedirection in which the underlying market moves. But an option trader has an additionalproblem: the speed of the market. A commodity trader who believes a commodity willrise in price within a specified period can be reasonably certain of making a profit if heis right. He simply buys the commodity, waits for it to reach his target price, then sellsthe commodity for a profit.
The situation is not quite so simple for an option trader. Suppose a trader believesa commodity will rise in price from $100, its present price, to $120 within the next fivemonths. Suppose also that a $110 call expiring in three months is available at a price of$4, If the commodity rises to $120 by expiration, the purchase of the $110 will resultin a profit of $6 ($10 intrinsic value less the $4 cost of the option). But is this profit acertainty? What will happen if the price of the commodity remains below $110 for thenext three months and only reaches $120 after the option expires? Then the option willexpire worthless and the trader will lose his $4 investment.
Perhaps the trader would do better to purchase a $110 call which expires in sixmonths rather than three months. Now he can be certain that when the commodityTeaches $120, the call will be worth at least $10 in intrinsic value. But what if the priceof the six-month option is $12? In that case the trader might still show a loss. Eventhough the underlying commodity reaches the target price of $120, there is noGuarantee that the $110 call will ever be worth more than its $10 intrinsic value.
; A trader in an underlying market is almost exclusively interested in the directioniM which the market will move. While the option trader is also sensitive to directionalConsiderations, he must also give careful consideration to how fast the market is likely
3536 Chapter 3
to move. If a futures trader and an option trader take long market positions in theirrespective instruments, and the market does in fact move higher, the futures trader isassured of a profit while the option trader may show a loss. If the market fails to movesufficiently fast, the favorable directional move may not be enough to offset the option’sloss in time value. This is the primary reason speculators generally lose in optionmarkets. A speculator usually buys options for their seemingly favorable risk/rewardcharacteristics (limited risk/unlimited reward). But if he purchases options, not onlymust he be right about market direction, he must also be right about market speed. Onlyif he is right on both counts can he expect to make a profit. If predicting the correctmarket direction is difficult, correctly predicting direction and speed is probably beyondmost traders’ capabilities.
The concept of speed is vital in trading options. It is so important that there aremany option strategies which depend only on the speed of the underlying market andnot at all on its direction. Indeed, ifa trader is highly proficient at predicting directionalmoves in the underlying market, he is probably better advised to stick to the underlyinginstrument. It is only when he has some feel for the speed component that a trader canhope to intelligently enter the option market.
The option trader who wants to intelligently evaluate the potential profitability ofan option trade is faced with the task of analyzing several different factors. At aminimum he must consider:
1. The price of the underlying contract
2. The exercise price
3. The amount of time remaining to expiration
4. The direction in which he expects the underlying market to move5. The speed at which he expects the underlying market to move
Ideally, he would like to express each of these factors numerically, feed thenumbers into a formula, and derive a value for the option. By comparing the value toits price in the marketplace, the trader would then know whether the purchase or saleof the option was likely to be profitable. This is essentially the goal of option evaluation:to analyze an option based on the terms of the option contract, as well as current marketconditions and future expectations.
EXPECTED RETURN
Suppose we are given the opportunity to roll a six-sided die, and each time we roll wewill receive a dollar amount equal to the number which comes up. If we roll a one, weget $1; if we roll a two, we get $2; and so on up to six, in which case we get $6. If wewere to roll the die an infinite number of times, on average, how much would we expectto receive per roll?
We can calculate the answer using some simple arithmetic. There are six numberswhich can come up, each with equal probability. If we add up the six possible outcomesIntroduction to Theoretical Pricing Models 37
14+2+3+44+5+6=21 and divide this by the six faces of the die we get 21/6 = 314. Thatis, on average we can expect to get back $312 each time we roll the die. This is theaverage, or expected, retum. If someone were to charge us for the privilege of rollingthe die, what might we be willing to pay? If we purchased the chance to roll the die forless than $31, in the long run we would expect to be winners. If we paid more than$314, in the long run we would expect to be losers. And if we paid exactly $314, wewould expect to break even. Note the qualifying phrase “in the long run.” The expectedreturn of $314 isa realistic goal only if we are allowed to roll the die many, many times.If we are allowed to roll only once, we cannot count on getting back $314. Indeed, onany one roll it is impossible to get back $31 since no face of the die has exactly 314spots. Nevertheless, if we pay less than $314 for even one roll of the die, the laws ofprobability are on our side because we have paid less than the expected return.
Ina similar vein, considera roulette bet. A roulette wheel has 38 slots, numbered1 through 36, 0 and 00.! Suppose a casino allows a player to choose a number. If theplayer’s number comes up, he receives $36; if any other number comes up, he receivesnothing. What is the expected return from this proposition? There are 38 slots on theroulette wheel, each with equal probability, but only one slot will return $36 to theplayer. If we divide the one way to win