isti-cated instrument — the ultimate derivative, perhaps. Consider thelevels of abstraction implicit, for example, in a put option on atreasury bond futures contract. The buyer of a Treasury bond putoption is betting with an unknown opponent that the value of thegovernment's obligation to an unknown lender, 30 years hence,will, within the short life of the option, decline by an amount suf-ficient to cover the price of the bet and still yield a profit!PARTTWO
OPTIONTHEORYCHAPTERTHREE
OCKHAM'SEQUATION
n the very short term, no one knows what the price of a com-
modity future will do. Everyone knows what it has done in thepast, of course, but market tacticians disagree on how much use-ful information—as far as predicting upcoming price action—isencoded in recent price patterns. Some observers, myself included,believe there is little or no information on future price direction tobe found in historical prices. Others swear by technical analysis, tothe extent of ignoring market fundamentals altogether.
Regardless of trading philosophy, few serious players woulddispute that in the very short term at least the price of a freelytraded entity like a commodity future will fluctuate in a virtuallyrandom manner, even as it is responding to supply and demandconsiderations such as weather forecasts, farmers productionintentions, the whims of consumers and economic policymakers,and the occasional mass-hysterical phenomenon sometimescalled "the madness of crowds."
Commodity prices may change abruptly, as when instantaneousand substantial news must suddenly be absorbed into the market-place. Jolts of this type arrive, by definition, in a random mannerbut create seemingly nonrandom commodity price patterns, espe-cially when these patterns are viewed in retrospect on price chartsand divorced from the news that gave rise to them in the first place.Regardless of how nonrandom a trading market may appear in retro-spect, at each instant of time that it was open and trading freely a tem-porary balance existed between the forces of supply and demand, asdid a state of very temporary price equilibrium.
1718 OpTION THEORY
Since the price of an option is a function of the price actionin its underlying instrument, be it a commodity future or a stock,the price of an option is a derivative variable rather than an inde-pendent variable. Some pundits will argue that price action in anoption presages upcoming action in the underlying instrument.Whereas this may be true in the case of stock options, where asudden huge increase in options volume might be the result ofinsider trading, it is certainly not true of commodity futureswhere inside information does not really exist. I intend to treatoptions as pure derivatives, which means that I am going to bemuch more interested in the variability of futures prices than inthe variability of options prices.
The relationship of paramount interest to option strategists isthe relationship between an option price and the variability of itsunderlying future isolated from all other variables. The variabilityof the option price itself is of secondary importance, for that isaffected by factors other than the variability of the underlyingfuture: The price of an option, for example, will vary with thetime remaining to expiry and also with the differential betweenthe current futures price and the strike price of the option. Allthese numbers are continuously changing, making interpretationof an option price profile over time a rather pointless exercise.Needless to say, option price charts of the high/low/close varietyare rarely seen.
*
There is considerable debate among market theoreticianson whether futures prices are random long-term. Fortunately,this debate is not relevant to the analysis of option prices. Anoption reacts as if the price of its underlying commodity futurewere a random variable and is not concerned with the directionof the futures market. Recent price direction in a commodityfuture, then, is irrelevant to the pricing of its options. The sizeof recent daily price fluctuations in a commodity future, how-ever, is the single most important variable in the pricing of itsoptions. The point is illustrated in Figure 3-1, a schematic rep-resentation of the familiar highMow/close daily bar chart for aOCKHAM’S EQUATION 19
 
FIGURE 3-1. Daily price variability, not price direction of a futures market, is whatgoverns the price of its options. Although market A has been trending steadily upward,while market B is stuck in a trading range, from an options valuation standpoint theyare equivalent, and options at comparable strike prices would be priced approximatelythe same in both markets.
Markets C and D have risen by the same amount over the same time interval(about 20 days). Options on market D would be priced substantially higher thanoptions on market C, because market D exhibits greater daily price variability. MarketsE and F are both stuck in trading ranges, but again, options in market F would bepriced higher than options in market E, because of the greater daily price variability.20 OPTION THEORY
variety of price patterns that might be generated by a commod-ity future.
The value that the free market places on an option is an indi-cation of the price the market expects the commodity future tobe trading at the instant the option expires. Even though themost likely outcome is always that the futures price will not havechanged at all by the time the option expires, the option marketrecognizes that there is a range of possibilities for the price of thefuture, a range of possibilities distributed more or less symmetri-cally about the unchanged level (Figure 3-2), Other things beingequal, larger expected ranges will result in larger option premi-ums.
Two variables directly affect the range of possibilities for theprice of a future at option expiry. One is the future's perceivedvolatility — determined mostly by price patterns of the recentpast. The other is time. A commodity future which has been fluc-tuating a lot in price is more likely to end up with a large cumu-lative change in price than a commodity future which has beentrading in a relatively tight range. And a future with many trad-ing days left till expiry clearly has more opportunity to arrive atan extreme price than one with just a few trading days left.
If daily commodity price changes were true random variables,normally distributed and with mean values of zero, determiningthe fair value of any commodity option, mathematically, would bepossible. Indeed, a massive amount of academic firepower hasbeen directed toward achieving this very goal, on the assumptionthat futures price changes are normally distributed. The fact thatcommodity price changes form distributions that are significantlynonnormal renders a great deal of current academic researchinto option pricing essentially useless, Nobel prizes in economicsnotwithstanding.
*
Although all commodity prices go through their own particu-lar bull and bear phases, over the long term prices do not changedramatically. Periods of high prices in a commodity inducegreater supplies along with a contraction in demand, and periodsOcKHAM’s EQUATION 21
MARKET A -low price variability
 
      
=TimeMARKET B -high price variability2 wet| 'APT
 
Time
FIGURE 3-2. A high-variability futures market will project a greater range of likelyfinal values than a low-variability futures market. Time is also a factor; the longer thetrading horizon, the greater the opportunity for large accumulated price changes todevelop.
In the two charts above, showing recent price history in both a low-variability anda high-variability futures market, probability envelopes have been projected forward intime. The limits of the envelopes define the 50 percent (arbitrarily chosen) probabilitylimits within which the final futures price is expected to fall at any time in the future.The true relationship between probability and time is not a linear one as suggested inthe schematic above. This is a refinement that will be explored in later chapters.22 OPTION THEORY
of low prices curtail supplies and stimulate demand. There is along-term secular rise in the overall commodity price level, but itis small—1 or 2 percentage points a year, perhaps. Very occa-sionally, a global power shift will cause a sudden sustainedchange in the price of a commodity, such as happened with oiland gold in the early 1970s. Neglecting these one-time shocks tothe system, even gold and oil have behaved like typical com-modities for the last 20 years. Of all the major contracts, only theStandard and Poor's stock index can be said to be something of aone-way street, and even that juggernaut may eventually regressto a more gently sustainable uptrend.
Price stability over the long term implies that daily pricechanges observed in a specific commodity are going to form adistribution that is centered very close to zero. It is accepted thatcommodity prices changes are very close to being random in theshort-term, and it is well-understood that repeated observationsof random variables often approximate normal curves, or "bell"curves, when plotted as frequency distributions. If daily pricechange is a random variable centered very close to zero—and weknow this to be substantially true— the question naturally arises:Why shouldn't daily commodity price changes be normally dis-tributed?
Before attempting to answer this question, it's worth review-ing the properties of a normal distribution — in reality, a technicalterm for a rather fancy equation which in many cases accuratelydescribes the distribution of a random variable.
, The normal distribution is known to accurately describe suchrandom variables as the heights or weights of people withinclearly defined populations. For example, the average height formales in the United States is around 5'9" with above-average andbelow-average heights reasonably symmetrically distributedaround this average value. The most widely accepted statisticdefining a normal distribution is the standard deviation, a statis-tic whose value can be estimated from a large sample drawn fromthe population in question.
Once the standard deviation of a distribution is estimated, itis possible to predict, on the assumption of normality, the proba-bility of occurrence of extreme values within that distribution. IfOCKHAM’S EQUATION 23
the observed extreme values follow the expected probabilities,one can confidently assume that the original premise of normal-ity is sound—at least, there will be no reason to suspect that thepremise is unsound. But what if extreme observed values fail toconform in a big way with values projected from a normal distri-bution based on the sample data? What would be a reasonableand logical conclusion in the light of this finding?
One might conclude that the sample is nonrepresentative ofthe population it is drawn from and that the true distributionreally is normal. Or one might infer that the population distribu-tion is not normal at all. This second choice is not popular,because, if the normal assumption is suspect, it renders invalidmuch of the mathematical analysis that fills option textbooks.
Overwhelming evidence favors the hypothesis that pricechange populations are significantly nonnormal. There are simplytoo many occurrences of wildly improbable price changes—improbable, that is, on the normal assumption — to ascribe theseaberrations to sampling error (see Figure 3-3, compiled from ayear of coffee price data).
*
Why do price changes refuse to respect the normal distribu-tion when so many naturally occurring random variables do so?Well, for one thing there is nothing natural about a commodityfuture; it is an abstraction by definition, and the pattern of pricesit generates is the result of a highly complex set of human inter-actions. Is it possible then for commodity prices to be random,but random in some abnormal way?
When we talk about prices following a random walk, we arereally talking about market players’ reactions in a freely tradingmarket being random. If we could isolate that part of futuresprice variability represented by players’ reactions after news is "inthe market’) from that part of price variability arising from exter-nal market shocks, then indeed we might have a normal distrib-ution of price variability.
But the reality is that all commodity markets are subjected tosudden and unpredictable infusions of information which result24 OPTION THEORY
             
30 OBSERVED VALUES FORCOFFEE FUTURES (1996)-249 readings with standard2 25 deviation = 2.15%e2 203°3° 45°°2 105z A significant departure from normality530ow 253s NORMAL DISTRIBUTION= 20 DERIVED FROM COFFEE2 DATA -249 readings with° 45 standard deviation =2.15%°°2 10€3Zz
-6 -5 -4 -3-2-1 0 1 2 3 4 5 6 7 8 9 10Daily price change expressed as a percentage
FIGURE 3-3. The upper chart is a frequency distribution of daily price changes forcoffee for the whole of 1996 (249 trading days), with price change expressed as a per-centage of absolute price level. The standard deviation was calculated to be 2.15 per-cent, The lower chart is a theoretical normal distribution with the same standarddeviation and reconstructed, for comparison purposes, to correspond to a representa-tive sample of 249 readings.
Even if the observations of the upper chart constituted a sample drawn from atrue normal distribution, it could hardly be expected to show absolute conformancewith normality, since it is just a sample. However, this chart exhibits a very significantdeparture from normality at its extreme values. Such a departure from normality canintroduce serious errors into any option pricing calculation based on the assumptionthat daily price changes do come from a normal distribution.OCKHAM’S EQUATION 25
in sudden instantaneous price adjustments: I'm talking aboutthings like crop forecast surprises, unexpected political develop-ments, weather scares, and so on. The price change distributionsresulting from "external shocks" are by definition massivelyunquantifiable. However, there is no denying their existence.
When we look at a frequency distribution of daily commodityprice changes, we are really looking at two distributions, one verynormal, one highly abnormal. A failure to recognize this reality—an almost universal failure in conventional theory—can lead tomany erroneous conclusions about how options are really pricedin the marketplace.
*
Now that I have pointed out the shortcomings of the normaldistribution assumption in quantifying price change distributions,I intend to develop an option pricing model based on this veryassumption. There is method in such an apparently contradictoryapproach. Knowing the limitations of a theoretical model inadvance may allow us to correct its deficiencies after the fact usingempirical information extracted from real price data. This prag-matic approach, I submit, is quite different from the conventionaltheoretical approach to option pricing which revolves around amathematically perfect formula not applicable in the real world.
There are other benefits from proceeding initially on the nor-mal assumption. Perhaps most important, the reader will be ableto directly compare the simplified option pricing model I'm goingto develop from first principles with the "million dollar formula"that dominates options literature. Before attemping to constructthis model, I would like to make a few observations on price dis-tributions in general and discuss ways of expressing these distri-butions as succinctly as possible.
*
Commodity prices are expressed in such diverse units ascents per pound, dollars per bushel, and yen per dollar. Since wewill be interested in price changes rather than in absolute prices,26 OPTION THEORY
and since we will be wanting to compare price change distribu-tions across a number of different commodities, it will beimmensely useful to express all price changes as percentages oftheir absolute price levels.
If every daily price change — whether the commodity be soy-beans, live cattle, sugar, or Japanese yen—is made dimensionlessby dividing that price change by the absolute price of its futureand then multiplying by one hundred, then all resulting mea-sures of “spread” will be expressed as dimensionless percentagesand will thereby be directly comparable. (If every option price isalso expressed as a percentage of its futures price, then everyoption price will also be expressed in the same units as the dailyprice changes in its future.) Figure 3-4 shows daily price changesfor coffee and silver, expressed as percentages of their absolutevalues of around $1.20 per pound and $5.00 per ounce, respec-tively, over the course of calendar year 1996. One thing is imme-diately clear from the "spread" of each of these distributionsabout its mean value: During 1996, coffee prices were muchmore variable than silver prices.
The degree of "spread" of a set of numbers about the averagevalue (mean) of that set of numbers is most commonly specifiedby its standard deviation, a statistic which can be calculated forany set of numbers or for any continuously variable distribution.The calculation of the standard deviation of a set of numbersinvolves taking the square root of squares of differences from themean. Another measure of spread of a distribution is its meanabsolute deviation, which, in the case of daily price changes, isthe average value of these price changes taking all readings aspositive. In classical statistical analysis, the mean absolute devi-ation is much less used than the standard deviation. This isunfortunate, since the mean absolute deviation as a measure ofvariability has many advantages, not least of which is its ease ofvisualization and its simplicity of calculation.
Be that as it may, there is no denying that the standard devi-ation is the statistic conventionally used in developing optionprice models. Realistically, therefore, and for comparison pur-poses if for nothing else, the standard deviation has to be incor-porated into any independently derived option pricing formulathat I or anyone else dares to come up with!OCKHAM’s EQUATION 27
               
a OBSERVED VALUES FOR3 COFFEE FUTURES (1996)c -249 readings of daily price30 change with a calculated8 standard deviation = 2.15%°= 15a£32-6-5-4-3-2-1 0 4 2 3 4 5 6 7 8 91060 Daily price change expressed as a percentage0o2452 OBSERVED VALUES FOR5 SILVER FUTURES (1996)3 -252 readings daily price2 30 change with a calculated° standard deviation =1.12%°a2EZ215
“6 -5 -4 -3 -2 +1 0 1 2 3 4 5 6 7 8 9 10Daily price change expressed as a percentage
FIGURE 3-4. Frequency distributions of daily price changes for coffee futures andsilver futures plotted to the same scale for direct visual comparison. The amount of dis-persion about the mean value is most commonly measured by the standard deviation,a nonintuitive statistic whose calculation involves taking the square root of a sum ofthe squares. The standard deviation is the commonly accepted measure of the vari-ability of a set of observations about its mean value, although the mean absolute cevi-ation can also serve this purpose. The standard deviation of a frequency distribution isexpressed in the same units as the variable on the x axis.
In the two charts above, daily price changes have been expressed as percentagesof absolute price to make the standard deviations directly comparable. From the dis-tributions it is clear that during 1996 coffee was much more volatile than silver, almosttwice as volatile: the standard deviation of daily price changes for coffee was 2.15 per-cent, the standard deviation for silver 1.12 percent. Price variability can change dra-matically with the passage of time. Traders who were active in the 1970s will recallwhen the situation was reversed: silver was much more volatile than coffee.28 OPTION THEORY
*
The "normalized" frequency distribution of coffee price datafor 1996, first compiled in Figure 3-3, is repeated as the upperchart of Figure 3-5. The term normalized means that theobserved standard deviation of the raw data has been used toconstruct a symmetrical normal distribution having the samestandard deviation as the observed data set. The inference, ofcourse, is that the observed values really do come from a normaldistribution. We know they do not. We know they do not from thegeneral empirical observation that there are just too manyextreme readings of futures price change to ascribe these pat-terns to chance occurrence. But let us suspend disbelief, for themoment, and proceed on the erroneous assumption of the valid-ity of the normal distribution. In following this line, I am simplyfollowing classical option pricing theory.
What do we do with this normalized frequency distribution?The reason for constructing a normal distribution from observa-tional data is that the probability distribution so created (Figure3-5) can now be used to project where a commodity future —inthis case, coffee—is likely to be trading at some time in thefuture. It is possible to construct a probability distribution ofdaily price changes from data gathered over any time period onechooses. In the coffee distribution of Figure 3-5, a full year'sworth of data was used in its compilation.
The more data one uses in constructing a probability distrib-ution, the more representative and statistically sound that distri-bution will be. However, the farther back one searches in time,the more likely it is that distant data will no longer be represen-tative of current daily price action. Commodity volatilities dochange over time, and this changing volatility is definitelyreflected in changing options prices. As far as arriving at the mostrepresentative probability distribution, there is really no way todecide which time period represents the best compromisebetween the benefits of increasing sample size and the benefitsof using more recent data. If the price variability of a commoditywere to remain constant, the problem of pricing its options wouldbe much simpler, for then the observational data would beOCKHAM’s EQUATION 29
 
    
25 NORMALIZED DISTRIBUTION
o . FROM HISTORICAL DATA
8
<
2
3
8
eo
°°
°°
3
2
E
3
z
“6 -5 -4 -3 -2 -1 0 1 2 3 4 «8 6Daily price change expressed as a percentage105 PROBABILITY DISTRIBUTION: OF FUTURE OCCURRENCES
3 08
m
a
06
®
>
= .04
s
o
vd
“6 “5-40-30 -2 0 1 2 3 4 5 6Daily price change expressed as a percentage
FIGURE 3-5.’ The upper chart shows the normalized absolute frequency distribu-tion of daily price changes for coffee during 1996, with price changes expressed as apercentage of absolute price level as the x axis (repeated from Figure 3-3). The y axisof this chart can be rescaled as relative frequency by dividing number of occurrencesat any given x-bar by the total number of occurrences.
From the relative frequency distribution of the lower chart, it is possible to pro-ject, on the normal distribution assumption, the probability that an upcoming dailyprice change will lie between any two limits of x. For example, the probability that aprice change will lie in the range — 1.5 to —3.0 percent is the sum of the three darker-shaded bars above. This probability turns out to be 0.042 plus 0.052 plus 0.065 whichequals 0.159. Note that in a relative frequency distribution plotted as discrete verticalbars, as in the example above, the sum of the heights of the bars must necessarily addup to one—a certainty.30 OPTION THEORY
coming from a single time-invariant distribution. Again, we knowthis isn't so.
A frequency distribution of the type shown in Figure 3-5 doesnot provide any information on the sequence of observations. Toget an idea of how price variability does change over time, oneneeds to look at daily price charts showing highs, lows, and clos-ing prices. Scan any price chart for any commodity and you willfind days of large price swings interspersed with days when theprice hardly changes at all, You will notice strings of successiveprice changes in the same direction, mixed in with strings of dayswhere advances alternate with declines. Some charts will retro-spectively exhibit strong trends, others wide trading ranges. Mostimportant, a commodity price chart will show prolonged intervalsof time where large daily price changes are the norm, and otherprolonged intervals where small daily price changes are thenorm. All of which points to the conclusion that the random vari-able which is generating these price patterns is coming from anunderlying price distribution that itself is not consistentlyvolatile.
Despite these obvious limitations to extracting useful infor-mation from the historical record, the reality is that optionstraders, acting intuitively or employing statistical methods, willbe closely watching the pattern of recent daily price changes in acommodity future for clues as to its upcoming volatility. Theyhave little else to go on. Therefore, it seems reasonable to pro-ceed on the basis that a probability distribution of price variabil-ity derived from recent price history —albeit over an arbitrarytime interval—will prove useful in constructing an options pric-ing model, provided the limitations of that model are understood.
And where do we go from here? The answer is that for themoment we continue along the same well-traveled road othertheorists have taken, working with an idealized model, but evermindful of its limitations and of the ultimate need for stringentreality checks before any theoretically derived or empiricallymodified options pricing formula can be introduced into the realworld of options trading.
*OCKHAM’S EQUATION 31
Before the task of fair pricing an option from first principlescan be undertaken, the concept of mathematical expectation hasto be clearly understood. A commodity option is in essence astraightforward wager. When two parties make a wager on theoutcome of a game of chance, both hope to win rather thanexpect to win. The truth is that in a fair wager the mathematicalexpectations of both parties are zero. One party may be morelikely to win than the other, but expectations will be the same,because the underdog will be receiving odds from the favorite.
For example, a racehorse quoted at odds of 8 to 1 against ispriced this way because the market, collectively, rightly orwrongly, believes that the horse has one chance out of nine ofwinning and eight chances out of nine of not winning. In otherwords, the market believes that the bookmaker has eight chancesout of nine of winning, while the bettor has only one chance outof nine of winning. If a horse is fairly priced, expectations ofbookmaker and bettor will be equal (neglecting the bookmaker'sbuilt-in edge), because the bookmaker will get only $1 upon win-ning whereas the bettor will collect $8.
The expected value of a random variable is the sum of each ofits possible values, or intervals of values, multiplied by the prob-ability of that value's occurrence. In the case of a bettor wager-ing on a horse at odds of 8 to 1 against, the random variable isthe bettor's payoff. In a straight win bet there are only two possi-ble values for this random variable; a positive value of 8 units ifthe horse wins, and a negative value of | unit if the horse loses.
~ | of winning) payoff)
Bettor's _ | (probability x (winningpayoff
+
(probability dosingof losing) ™ payoff)
1 89% B+ 5x CD032 OPTION THEORY
The buyer and the writer of an option are essentially cast inthe same roles as the bettor and the bookmaker, respectively. Theoption buyer has a low probability of winning a large amount,while the option writer has a high probability of winning a smallamount. In an efficient market, the same equivalence of expec-tations that governs a racetrack wager holds true in an optionstransaction. Expectations in an options transaction balance atzero through the pricing of the option premium, because, unlikea racetrack wager, the amount to be won or lost in an optionstransaction cannot be specified at the time the transaction ismade. The calculation of expectation, however, is basically thesame, and the fair price of an option is that premium paid by thebuyer to the writer that makes both their expectations balance atzero.
*
Tn attemping to derive an option formula from first principles,and for reasons that will become clear later, 1 am only going toconsider at-the-money options. (Recall that an at-the-moneyoption is one whose strike price is exactly equal to the currentprice of its future).
From Figure 3-6, a symmetrical frequency distribution ofdaily price changes in an idealized commodity future, it can beseen, graphically, how an at-the-money option premium must bepriced so that expectations of the buyer and the writer are bothzero. The frequency distribution covers all possible outcomes ofdaily price change, which means that numerically the sum of thevertical bars must add up to 1. And the probability that a pricechange will lie between any two values on the x axis is got bysumming the heights of all the bars enclosed by these two valuesof x.
Consider first the purchase and sale of a 1-day at-the-moneycall option for which the writer receives the amount CP, the callpremium. From a practical standpoint, option traders are inter-ested primarily in options with weeks, even months, till expiry.The 1-day option is not commonly discussed in the literature, butall options eventually pass through the stage of being 1-dayOCKHAM’s EQUATION 33
PURCHASE AND SALE OF A 1-DAY AT-THE-MONEY CALL OPTION
“10 CP (Call Price)o8. Option buyer wins. sé Option writer wins3c@aTo= 06=a2&> 04c.02
6 5 2 3 4 5 6Price change expressed as a percentage of absolute price
 
FIGURE 3-6. In order for the buyer of an at-the-money call option to win, theunderlying futures price must change by a positive amount greater than the premiumCP paid to the writer. Clearly, the option writer has a greater probability of winningthan the option buyer.
If the futures price change is negative, the writer will profit by a fixed amount —the option premium, CP, If the futures price change is positive but still less than thecall premium, the writer will also win, but by a progressively smaller amount as the sizeof the price change increases. If the futures price change exceeds the call premium, theoption buyer wins.
The option writer's greater probability of winning is balanced by a correspondinglysmaller payoff when that occurs. The buyer, of course, is hoping for a big payoff if thefutures price change should happen to fall in the low probability, but high payoff, pos-itive tail of the distribution.
For the moment, transaction costs, which are incurred by both buyer and writer,are not being considered.
options, and understanding how to fairly value a 1-day option isa major step in understanding how to value an option of any termto expiry. There are also compelling practical reasons for choos-ing 1 day as ground zero time; newspapers and quotation servicesreport closing prices on a daily basis!34 OPTION THEORY
When a commodity future closes exactly at an option strikeprice (making that call option temporarily the at-the-money calloption), any positive price change in the future after one more dayof trading will give the call option some residual value at expiry.There is clearly a 50 percent chance that this will occur. Becausehe receives the option premium CP, the option writer has a higherprobability of winning than the option buyer; any price changefalling in the light-shaded area (Figure 3-6) is net positive to thewriter. The option buyer can only win if the price change falls inthe darker-shaded region, clearly an occurrence with a probabilityof less than 50 percent. Expectations balance out, however,because the payoffs to writer and buyer are different.
It is by no means obvious how to calculate that value of callpremium which will balance the expectations at zero. By trial anderror it might be possible to come up with a solution. A mathe-matician confronted with this problem would calculate the stan-dard deviation, assume a normal distribution, and use statisticaltables which give areas under the normal curve at different inter-vals along the x axis, but this would hardly be a straightforwardprocedure. It would also limit the scope of the solution by intro-ducing the normal distribution assumption. The statistical tablesolution (or the polynomial expansion solution which is really thesame thing) is the route followed by classical option theorists.And this is the fork in the road where we part company, becausethere is a much simpler solution to this problem unencumberedby the normal distribution assumption, a solution involvinghardly any mathematics at all!
In Figure 3-7, we see the distribution of Figure 3-6 repeatedbut highlighted to illustrate the buyer's expectation before the pre-mium is paid. The buyer's expectation before the premium is paidis net positive, because there is no price change of the commod-ity future which can cause him to lose. Remember, the terms ofthe option contract give the buyer the right to buy but not theobligation to buy. If the futures price change turns out to be neg-ative, the buyer of the call option will simply let the option