expireunexercised, at no cost. Before the premium is paid then, theoption buyer's expectation can be expressed as follows:OckHAM’s EQUATION 35
EXPECTATION OF THE BUYER OF A 1-DAY CALLOPTION BEFORE CALL PREMIUM IS DEDUCTED
             
“10 Buyer wins
Break even
Call buyer's expectation
Mean absolute deviation (MAD)
Relative frequency
02
6 5 4 3 2 1 oO 1 2 3 4 5 6Price change expressed as a percentage of absolute price
FIGURE 3-7. The call buyer's expectation before the call premium is deducted canbe determined by summing each possible payoff multiplied by its probability of occur-rence.
A 1-day at-the-money call option, by definition, pays off to the buyer with any pos-itive price change and by the amount of that positive price change. With any negativeprice change, the payoff is zero. In terms of the frequency distribution above, thebuyer's expectation is determined by summing the products of the height of each of thedarker-shaded bars (expressed as a probability) multiplied by the price change associ-ated with that bar.
The mean absolute deviation MAD of the price change distribution is defined tobe the sum of the products of the height of each of the shaded bars multiplied by theprice change associated with that bar, taking all values as positive. Defined in this way,the call buyer's expectation is exactly half of the mean absolute deviation. This rela-tionship holds true regardless of the shape of the distribution, provided it is symmetri-cal about they axis.36 OpTiIoN THEORY
Buyer's (probability (payoffexpectation _ that price when pricebefore premium change is change isis paid negative) negative)
(probability (payoffthat price when price
+ ; x .
change is change is
positive) positive)
The first term on the right side of this equation must be zerosince the payoff is zero for an at-the-money call option when thefutures price change is negative. The second term involves sum-ming a whole series of terms, each consisting of a unique proba-bility multiplied by a unique payoff, and covering all possiblevalues of payoff when payoff is positive, Mathematicallyexpressed:
Expectation beforepremium is paid
= X(p,xX;)
where p; is the probability associated with interval X; and all X;'sare positive. The mean absolute deviation MAD of the pricechange distribution is defined as its expected value taking all val-ues of price change as positive, regardless of sign. Mathemati-cally expressed:
Mean absolute deviation = Z(p;x1X;!)where p; is the probability associated with interval X; where X;may be either negative or positive. From symmetry considera-tions:
Lp x!Xj1) = 2 x Lp, x X;)
Therefore,OCKHAM’s EQUATION 37
Expectation before - 05 x MADpremium is paid
Since we know that the true buyer's expectation after the pre-mium is paid is zero, the call premium must be that quantitywhich reduces the buyer's true expectation to zero. In otherwords, the fair value call premium must be exactly one half of thenear absolute deviation:
Fair value call premium = 0.5 x MAD (Eq. 3-1)
Note that the call premium in the above equation will beexpressed in the same units as the mean absolute deviation; ifdeviation is expressed as a percentage of futures price, so too isfair value call premium. Note also that the mean absolute devia-tion of a distribution is not the same as the mean deviation ofsigned values, which would be zero for a perfectly symmetricaldistribution like a normal distribution. Both mean absolute devi-ation and standard deviation are measures of the dispersion or"spread" of a set of numbers around its average value, and areexpressed in the same units.
Equation 3-1 relates the fair value of an option to the"spread" of its futures price change distribution in as simple andconcise way as possible, using mean absolute deviation as themeasure of spread. In traditional option pricing theory, however,the accepted statistical measure of spread is the standard devia-tion, Indeed, as we shall see in the next chapter, volatility isdefined as the standard deviation—a rather unfortunate choice ofdefinition, and a definition that has befuddled a generation ofoption traders and made books on option trading twice as thickas they ought to be.
What can be done with fewer is done in vain with more. (WilliamdOckham, thirteenth-century philosopher and iconoclast.)
*38 OPTION THEORY
Had mean absolute deviation become synonymous withvolatility, life would have been much simpler. But it did not, and,like it or not, we are stuck with the standard deviation. If any-thing I have to say is going to be reconciled with what others inthe field have already said, I am therefore compelled to expandEq. 3-1 to include this term,
Frankly, I would not know how to develop, via the standarddeviation, an option pricing formula for a normal distributionof price changes. Fortunately, I don't have to. In the particularcase of a normal distribution centred on zero, there exists adirect linear relationship between the mean absolute deviationand the standard deviation.
MAD = V(2/n) x SD
The quantity v(2/m) simplifies to 0.7979, which is a numbervery close to 0.8000. Since any option pricing model is going todepend ultimately on sampled data, the degree of error in using0.8000 instead of 0.7979 will be of a lower order of magnitudethan any sampling error and therefore insignificant. I aim to keepthis book as practical as possible. Therefore, henceforth, for easeof calculation, it will be convenient to use the slightly simplifiedrelationship:
MAD = 0.8xSD (Eq. 3-2)Combining Eq. 3-1 and Eq. 3-2 yields:
Fair value call premium = 0.50.8 xSD
= 0.4xSD (Eq. 3-3)
It is worth noting that Eq, 3-1 is not limited by the shape ofthe distribution of price changes— provided the distribution issymmetrical. Equation 3-3 incorporates the normal distributionassumption and is more restrictive for that reason.
*OcKHAM’s EQUATION 39
The distribution considered in Figure 3-7 is a distribution of1-day price changes, and the standard deviation of Eq. 3-3,therefore, is the standard deviation of I-day price changes. Let'ssee what happens when the trading time interval is expandedfrom | day to 2 days. The longer a random walk continues, thefurther the random variable may travel, so that the probabilitydistribution of aceumulated futures price change after 2 days oftrading will not be the same as the probability distribution after1 day of trading. After 2 days, there is opportunity for pricechanges in the same direction to accumulate into larger netchanges than the changes possible after just 1 day's trading. Thedistribution of 2-day price changes will still be centred on zerosince there is no directional bias, and it will still be symmetricalabout zero, but the distribution will have longer tails and be"stretched" horizontally if plotted on the same scale as the 1-daydistribution. Its standard deviation will have increased. Thequestion is by how much.
It is a statistical fact that the distribution formed by summingtwo independent drawings from the same normal distributionwill also be a normal distribution and that the standard deviationof this second distribution will increase by the square root of 2.It is similarly true that the distribution formed by summing ¢independent drawings will also be normal and that the standarddeviation of this distribution will increase by the square root oft. That is:
4
SD, = V2 x SD,
SD, = vi x SD, (Eq. 3-4)where the subscripts 1,2 and , refer to 1, 2, and ¢ days, respec-tively. There are approximately 254 trading days in a calendaryear, and the statistic SD554, the standard deviation of daily pricechanges annualized, has a special significance in the lexicon ofoptions, where it is synonymous with the term volatility underthat word's technical definition. Volatility as a descriptive termhas entered the popular vocabulary due to the extremely largeprice swings witnessed in the stock market in 1997 and 1998, In40 OPTION THEORY
the field of options valuation, volatility has a restricted and defi-nite meaning, namely the annualized standard deviation of dailyprice changes. It is usually given the symbol v (by definition,therefore, v= SDy54 = 254 x SDj).
Equation 3-4 can now be expanded as follows:
vxrvi
4254
Equation 3-3 established a relationship between the fair value ofa I-day at-the-money call option and the standard deviation ofdaily price change in its underlying future. It has now beenestablished that if daily price changes are normally distributed,so too are accumulated price changes covering any period oftime. By analogy, then Eq. 3-3 can be generalized for t, the timeto expiry of the option, as follows:
(Fair value call premium), = 0.4 x SD, (Eq. 3-6)Combining Eq, 3-5 and Eq. 3-6,
0.4 xvxveV254
(Fair value call premium), =
The number of trading days in a year is an approximation; it
is not the same for all commodities and varies slightly from yearto year, If 16 is taken as an approximation to V254 (true value of15.93), no significant error will be introduced into the equation.With this simplification incorporated, it is now possible to write:
vxvi40
 
(Fair value call premium), = (Eq. 3-7)
The fundamental option equation above was derived for a calloption. By exactly the same reasoning an identical formula couldOCKHAM’S EQUATION 41
Suppose a gold future is trading at $350 per ounce, its 350 call at $6 and its350 putat $4. A trader who sells the 350 call, buys the 350 put, and buys thefutures contract is guaranteed a profit regardless of the price of the futurescontract at expiry.
If the contract expires at $360 $340Profit on 350 call ~$4 $6Profit on 350 put —$4 $6Profit on future $10 ~$10NET PROFIT $2 $2
FIGURE 3-8. If an at-the-money call were to trade at a different price from the at-the-money put, a trader would be guaranteed a profit by selling the call, buying the put,and buying the futures contract. The numerical example above illustrates the neces-sary equivalence of the price of the put and the call.
A guaranteed profit is an impossibility—certainly on a commodity exchange.
be derived for a put option. The equivalence in price of the at-the-money call and the at-the-money put—even in hugely trend-ing markets —may strike the reader as curious, but it is borne outby direct observation. It may also be demonstrated as necessarilytrue from arbitrage arguments (Figure 3-8). The reader shouldnote, however, that put and call options that are out of the moneyby the same amount do not, in general, trade at the same price.The fundamental option equation may therefore be slightly gen-eralized to include both calls and puts:
ViATMO, = ~~~" (Eq. 3-8)40
 
where ATMO, = the at-the-moncy fair value option price (putor call) expressed as a percentage of thefutures price
v = the option volatility also expressed as apercentage of the futures price
t = the number of days until the option expires42 OPTION THEORY
*
Equation 3-8, which will henceforth be referred to asOckham’s equation (in tribute to its minimalist roots), links thetheoretical fair value price of the two most actively tradedoptions on a future with the volatility of the future and with thetime till expiry of the options. Ockham's equation is theoreticallysound and based on a number of simplifying assumptions alreadydescribed, particularly (with the inclusion of the standard devia-tion term) the assumption that daily price changes come from anormal distribution. There is no requirement, of course, foractual option prices in the marketplace to conform to the valuesindicated by Ockham's equation, or any other equation for thatmatter.
If an option formula based on normal distribution assump-tions cannot be expected to accurately forecast real optionsprices in the marketplace, what is the purpose of deriving it inthe first place? The answer is that I have to confront the statusquo. Furthermore, in the next chapter, it will become apparentthat Ockham's equation is a special case of the famous Black-Scholes formula, which is used extensively in decisionmaking bya very large segment of the options trading public.CHAPTERFOUR
THE WORDOF GOD
n 1997, the Royal Swedish Academy of Sciences awarded theNobel prize in economics, plus a cash prize of $1 million, totwo theoretical economists (and to another posthumously) fortheir research into option pricing models. From the press release:
Robert C Merton and Myron Scholes have, in collaboration with thelate Fischer Black, developed a pioneering formula for the valuationof stock options. Their methodology has paved the way for economicvaluations in many areas. It has also generated new types of finan-cial instruments and facilitated more efficient risk management insociety.
This announcement was greeted with universal acclaim.Well, almost universal. It would scarcely be an exaggeration tosay that since its appearance 25 years ago, the million dollar for-mula—the culmination of the above-mentioned research — hasdominated option thinking with an authority of biblical propor-tions. Like the Word of God, everyone is expected to revere it,and no one is expected to understand it. The million dollar for-mula has been reproduced in virtually every serious book onoptions published since 1973, usually accompanied by a dis-claimer of the derivation of this formula is beyond the scope of thisbook variety.
The original papers describing the development of theformula were written in a high academic tone, strictly forthe consumption of Ph.Ds in advanced mathematics. Not onlyis the nomenclature clumsy and bizarrely complex, there are
4344 OPTION THEORY
discontinuities in the logical presentation, where the authors, aspart of their proof, cite other authors’ proofs of such-and-suchwithout bothering to verify or explain what such-and-such is orwas. A typical rehash of the million dollar formula appears inFigure 4-1: This is the simplified version for use with options oncommodity futures.
As a result of the formula’s impenetrable logic, optionsauthors by and large have been content to accept it at face valueand simply regurgitate it when necessary. Comprehension of theformula has not been helped by explanations like this (intendedfor a general audience):
Holding constant all the inputs to the options formula except theinterest rate always increases the value of an option. To get a roughidea of why this is so, note that an increase in the interest ratereduces the present value of the exercise price. Since the exerciseprice is a potential liability to the holder of an option, this increasesthe value of the option. Fischer Black (Fact and Fantasy in the Useof Options," The Financial Analyst's Journal, July-August 1975).
What's it all supposed to mean? And, this is just to get arough idea, remember! Imagine what an in-depth explanationwould be like! Now, I am not saying that the million dollar for-mula is incorrect. As a matter of fact, I know it to be correct,within the limits of its assumptions. What I do question, how-ever, is its scope—in particular, its attempt to cover all the bases,when it should have been clear to the authors, a posteriori, thatnot all the bases could possibly be covered.
*
Consider, for amoment, the million dollar formula in its mostgeneral version as applied to commodity futures (Figure 4-1).Notice, first of all, that the formula contains a constant multi-plying term e~" where r is the prevailing short-term interest rateexpressed as a fraction, and ¢ is the term to expiry of the option,expressed as a fraction of a year. The product of r and ¢ is boundto be a very small negative number, so that the exponential mul-tiplier will be a number very close to e® which itself is a numberTHE WorD OF Gop 45
THE MILLION DOLLAR FORMULA
Theoretical call option priceTheoretical put option price
etx [pN(d 7 sN(do)]etx [pN(dy— sN(do)}
HI
loa, (p/s) + (2)xt
where dy = 2 and dy = d,—-v x vtvx vtThe variables are: = __ price of the futures contracts = _ Strike price of the optiont = time remaining to expiry expressed as a
fraction of a yeartr =~ current risk-free interest ratev= volatility measuredby the standard deviationlog, = naturallogarithmN= the cumulative normal density function
The cumulative distribution N can be read from tables or approximated fromthe formula:
x= 1—2z (.43618y — .12016y? + .93728y*)
1
nd 3080423 eo———- al Z=, 21 +.3261 xidl °
where y =
Then, N(d) =x if d>0, or N(@)=1-x ifd<0
In the particular case of the at-the-money options with the interest rate takenas zero, that is, with p = s, and r= 0, the formula simplifies to:
px [N(dy) — N(dp)]px [N(-dy) — N(-d2)}
Theoretical call option priceTheoretical put option price
where y= 4 xi andd,=-d,
FIGURE 4-1. This is the million dollar formula in its simplified form for use withoptions on futures. The formula is advertised as being applicable to all options, that is,its scope extends to pricing out-of-the-money options as well as at-the-money options. Intheory, the million dollar formula is correct. In practice, it doesn't work—unless theoption is at-the-money, in which case a much simpler formula can be used.46 OPTION THEORY
very close to 1. For example, assuming an interest rate of 5 per-cent and a term to expiry of 6 calendar weeks,
r= 0.0530t=——254
and0.05 x 0.118 = 0.994]
Using this exponential multiplier in the formula, and takinginterest at 5 percent, the value of a 6-week option would be dis-counted by about one-half of 1 percent. I have no argument here,for a discounted premium makes sense given the way debits andcredits are assigned in an exchange-traded options contract. Anoption buyer must pay the option premium to the option writerat the moment the transaction is made, and the writer may theninvest the proceeds of the premium and collect interest.Common sense, therefore, suggests that in any option pricingformula the option price should be discounted by some interestrate component.
In practical terms, however, one has to question whether thisdiscounting term, particularly an exponential term involving thevariable ¢, is worth incorporating into the formula. In a low inter-est rate environment, we are looking at a discount of one-half of1 percent on a 6-week option, with the size of this discount ris-ing or falling more or less in a linear fashion as r and t vary. Aswill presently be shown, the volatility component in an optionspricing formula contains an intrinsic inaccuracy of such a mag-nitude as to make any interest rate discount inconsequential.
In addition, as I shall also presently argue, the principal andperhaps only legitimate use of an options pricing formula is forcomparison purposes (comparing options on different commodi-ties and comparing options with different periods to expiry onthe same commodity). For these reasons, and for ease of calcula-tion, there is little harm in leaving the theoretical interest rateTHE WorD OF Gop 47
multiplier term out of any options pricing formula. If rigor bedemanded, the interest rate discount may be applied as astraightforward percentage reduction to a formula-derived priceafter all other calculations have been completed.
In the development of Ockham's equation in the previouschapter, the interest rate factor was explicitly omitted. Therefore,in comparing Ockham's equation with the million dollar for-mula—an essential test of my credibility, to be sure—it will beappropriate to set r equal to zero in the latter.
*
The question of whether to include or exclude the interestrate term in an options formula is of minor significance com-pared with the more fundamental question of whether the mil-lion dollar formula in its general form has validity in the firstplace. The general formula attempts to price all options, that is,its scope extends to pricing both out-of-the-money options andin-the-money options, as well as to pricing at-the-money options.The inherent error in using a normal distribution in lieu of thetrue distribution of futures price changes has been demonstratedin the previous chapter. As a result of this error, any options pric-ing formula based on a normal distribution, using a standarddeviation calculated from observed data, will most likely generateoption prices that do not truly reflect fair value. The parameterv, a measure of the variability of the futures price and a necessaryinput to any options pricing formula, can only be estimated fromempirical data. Any option price calculated from a formula canonly be as accurate of the estimate of v used in the calculation,and if v is estimated from empirical data, there is no guaranteethat it will be truly representative of the variability of futuresprices.
One might have expected that mathematically focusedresearchers would think to question the validity of the normalassumption, or the validity of some assumption at least, sinceeven a rough comparison of actual option closing prices pub-lished in the financial press against theoretically calculated val-ues reveals tremendous discrepancies.48 OPTION THEORY
First, actual option prices do not diminish in value at strikeprices progressively further out of the money at the rate pre-dicted by the million dollar formula. This can easily be shown tobe true by holding v and t and p constant in the formula, andsolving for option price at different values of strike prices (Figure4-2). The formula progressively underprices out-of-the-moneyoptions relative to the at-the-money option.
The reason for this underpricing of out-of-the money optionsis embedded in the erroneous normal distribution assumption.Since the true distribution of commodity price changes showsmany more extreme values than the normal distribution wouldindicate, the preponderance of outcomes that will cause an at-the-money option to expire with a positive value will lie in thecentral part of the true probability distribution of daily pricechanges. With an at-the-money option, the effects of unexpect-edly large price changes in the tails of the distribution are mini-mized, because of their infrequency relative to middle of thedistribution outcomes. In contrast, the outcomes that cause anout-of-the-money option to expire with a positive value are thosethat lie in one of the extreme tails of the distribution, the area inwhich outcomes most exceed normal distribution predictions.With a far out-of-the-money option then, the error introduced bythe ragged tails of the distribution will be maximized. The mar-ket understands "abnormality" from experience and consequentlyslaps a big surcharge on low-probability options. The million dol-lar formula, blind to this reality, has no means of accommodat-ing it.
If the degree of relative underpricing at different strikes couldbe corrected, after the fact, by some consistent correction factor,it might still be possible to come up with a generalized optionspricing formula that would work equally well for all strike prices.For example, in the comparison chart of Figure 4-2, the ratio ofactual to theoretical option price clearly increases as a functionof the amount by which the option strike price is out of themoney, possibly in some quantifiable way. If deriving an empiri-cal correction factor were possible via this ratio, as was estab-lishing that this factor applied to both puts and calls, and appliedin all commodity markets, then yes, it might be possible toTHE Worb oF Gop 49
A DISCREPANCY BETWEEN THEORY AND OBSERVATION100%
         
WH Actual maket price©) Price calculated byoption formula
Ratio ¢ observedto calculated values
Option price (average of put or call)
0 Amount by which option is out-of-the-money
FIGURE 4-2. Option prices observed in the market-place and option prices calcu-lated from the million dollar formula do not, in general, correspond with each other,as demonstrated in the chart above compiled from crude oil option data. To constructthis chart, it was first necessary to determine that value of v which made the price ofthe at-the-money option calculated from the million dollar formula equal to the mar-ket price of that option. Then, by holding y, t, and p constant in the formula, out-of-the-money option prices could be calculated for different values of s, the strike price.
The discrepancy in options pricing between theory and reality results from themillion dollar formula's assumption of a normal probability distribution of futuresprices at option expiry, when the market knows from experience that this is not thecase. Following the protocol above, formula-calculated option prices are always low inrelation to observed option prices, the error increasing on a percentage basis as theoption strike price moves out of the money.
modify the million dollar formula and make it generally valid.Unfortunately, nothing could be further from the truth. Relativeoption prices prevailing within different commodity marketsexhibit no mathematically quantifiable relationships.
And there is a further problem associated with the generalformula: It is clearly symmetrical with respect to the pricing ofpricing of puts and calls with strike prices equidistant from theat-the-money strike (Figure 4-3). Simple inspection of option50 OPTION THEORY
 
(close as ot Nov 3,1997) _ (close as of Mar 27,1997)Nov Nov Jul Jul
Strike calls puts - Strike calls _— puts905 —-‘-51.00 10.40 240° 41.00 2.10918 42.90 12.30 250. 3250. 3.50925 35.30 14.70 . 260 25.75: 6.50935 25.30 17.60 - 270. 20.00 10.50*945 22.00 21.30 880° 15.75 16.25955 16.30 25.60 , -290 12.50. 23.00965 11.40 30.60 300 10.25 30.75975 7.50 38.70 _ 310 8.00 38.50985 4.60 43.80 320 6.25 46.75* the at-the-money strike price
FIGURE 4-3. As the published option prices clearly show, a generalized and sym-metrical option formula cannot possibly work for out-of-the-money options. The mil-lion dollar formula yields identical theoretical prices for puts and calls which areout-of-the-money by the same amount. Yet, on a day where December S&P futuresclosed at 945.70—making 945 the closest at-the-money strike, the November 905 SDPput closed at 10.40, while the November 985 SDP call closed at 4.60.
In some markets, calls are more expensive than puts. On March 27, with July soy-bean meal closing at 279.50, the July 320 soybean meal call closed at 6.25, almostthree times as much as the equidistant July 240 put.
tables in the financial press reveals that no such symmetry existsin the actual market.
For example, during the growing season, out-of-the-moneycalls on a crop future command higher prices than equidistant out-of-the-money puts. The collective wisdom of the market, which isbased on pocketbook experience, is smarter than any formula andrecognizes that upside price surprises in something that is growinghave the potential to be much larger than downside surprises.With stock index futures, the opposite situation prevails: Out-of-the-money puts are valued more highly than equidistant out-of-the-money calls —trading at more than double the price in manycases. To understand why stock index puts are much more expen-sive than stock index calls, the reader need only recall, in pain orin joy, the astonishing events of October 1987 (Figure 4-4).THE Worp OF Gop 51
THE NOVEMBER 1987 S&P PUT OPTION
Strike Value of put option Value of put optionPrice October 9 October 19260 0.25 61.00
265 0.45 66.00
270 0.65 71.00
275 1.00 76.00
280 1.65 81.00
285 2.25 86.00
290 3.35 91.00
295 4.50 96.00
FIGURE 4-4. For once, the doomsday scenarists were right. Buyers of wildly out-of-the-money puts who bought on October 9, 1987, must have felt like lottery winnersjust 10 days later. During this period, the S&P Stock Index future fell from 320.0 tounder 200.0, a decline of unprecedented proportions. A November 260 put option, forexample, bought for $125 on October 9, was worth $30,000 on October 19. This wind-fall for the option buyers was a disaster for the option writers. Just as maritime insur-ance rates rose sharply after the Titanic went down, so too did S&P option premiumsafter the stock market crash of 1987. They have remained high ever since, and movedeven higher during 1997 and 1998 as a result of the tremendous daily price swings thatare now commonplace.
Am I suggesting, then, that an option pricing model is of novalue? Not at all. But only if its limitations are understood. Wehave to appreciate that option pricing is not nuclear physics, thatthere are no sublime relationships to be uncovered, and thatbending the problem to suit the mathematics is counterproduc-tive. It does seem to me that there are too many players in theoption trading community who are ready to pay lip service to the-oretical economists whose objectives are quite different fromthose of the average trader. A coterie of academics who seemnever to have studied the financial columns of a newspaper—much less traded an option—have been allowed to dominateoption pricing thinking, to press advanced mathematics onto thesolution of problems which can be treated with simple mathe-matics, and in general to make the whole options business seema great deal more complicated than it really is!
*52 OpriON THEORY
The subject of volatility in all its guises will be explored morefully in the next chapter. Suffice it to say at present, that becausethe parameter v cannot be objectively determined to everyone'ssatisfaction, there can be no objective test which will concludewhether an option is overvalued or undervalued—even when his-torical futures prices are representative of what is coming up.
We can nevertheless speculate on how instances ofexploitable overvaluation or undervaluation are likely to arise. Mysuspicion is that situations of overvaluation or undervaluation—assuming these occur—will be consistent across strike prices,that is to say, all options on the futures of a given commodity willbe