overvalued or undervalued together. I would not expect toidentify significant overvaluation or undervaluation by compar-ing options on the same commodity. Why? For one thing, with somany professional traders in the trading pit looking for arbitrageopportunities, it seems likely that options with different strikesand different terms till expiry will be forced into some kind ofprice balance with each other, based on pit experience alone. If Iwant to question the value of an out-ofthe-money option, I willlook to the historical relationship, seasonal or secular, that hasprevailed between that option and the corresponding at-the-money option.
I am prepared to argue with the market's assessment ofabsolute value as reflected in the at-the-money option prices. Inthe empirical studies which follow in "Option Reality" (Chapters6 through 9), I shall be concerned exclusively with at-the-moneyoptions — puts, calls, and straddles, where the strike price equalsthe futures price.
*
In Chapter 3, working from first principles, I deducedOckham's equation for calculating volatility vy, time to expiry ¢, orthe at-the-money option price ATMO,, when any two of these areknown. The million dollar formula, of course, does exactly thesame thing but in a more general way. If, in the million dollar for-mula, the strike price s is set equal to the futures price p, and theinterest rate r is set to zero, we have exactly the conditions underTHE Worp OF Gop 53
THE MILLION DOLLAR FORMULA ™ a calculation
Following the nomenclature of Figure 4-1,
 
t= © _ o1068254d= t - 0:3 Vo.1868 = 0.0666
and dp =—d, = -0.0666
And, since Id,i = I'da!, the calculated values of x, y, and 2 will be the samefor both d, and d,
1
Y= 7433261 x1dl
= 0.9783
=a?2 = 0.3989423 e 2
0.3981
and x = 1 - 2 x (.43618y — .12016y? + .93729y%) = 0.5265
leading to N(d1) = x = 0.5265 (since d, > 0)
and N(d,) = (1 ~ x) = 0.4735 (since d, < 0)
For an at-the-money call, with interest rate at zero, p= $, and r= 0.So, theoretical call price = p x [N(d,) - N(d,)] = 0.0531 xp
which, expressed as a percentage of futures = 5.31%
FIGURE 4-5, A typical calculation for pricing the at-the-money call option on acommodity future using the million dollar formula. The term till expiry is 50 tradingdays and the volatility 0.3, or 30 percent. The interest rate is taken to be zero.
The million dollar formula is substantially more complex when pricing out-of-the-money options. Even this simplified version for the at-the-money option is awkward tocalculate.
which Ockham's equation was deduced, so that the two formu-lae ought to agree in this restrictive case. And indeed they do(Figures 4-5 and 4-6). There is a big difference, however, in the54 OPTION THEORY
 
Here is an alternative solution to the problem posed in Figure 4-5. As before,P= 5, (since we are dealing with an-at-the-money call), and ris taken to bezero. Term till expiry ¢ is again 50 days and volatility v again 30%
ff ATMO, is the theoretical price of a call. option expressed as a percentageof the futures price, then, by.Ockham’s Equation,ammo, = 2X_ATMO, = ~~
 
40.
=" 5.30% (as before)
FIGURE 4-6. Ockham's equation solves the problem posed in Figure 4-5 muchmore economically than the million dollar formula, yielding an identical answer.Ockham's equation has the added feature that it can be solved for either vy, t, or p, whenany of these variables are specified. This is not possible with the million dollar formulawhich involves a polynomial function.
complexity of the calculations. What's more, in contrast toOckham's equation, the million dollar formula may not be solveddirectly for volatility v, knowing time to expiry t, and the at-the-money option price ATMO,.
I have tried to make the point, using empirical evidence, thatit is only for at-the-money options that the million dollar formulaor Ockham's equation can possibly have any legitimacy. Workingback from an actual option price, both equations calculate avolatility based on the flawed normal distribution assumption,and this (implied) volatility will not necessarily correlate with amarket volatility computed from the standard deviation offutures price changes. If (implied) volatilities are restricted to theat-the-money option, however, these volatilities may still provevaluable as comparative yardsticks. Notwithstanding the equa-tions' inherent limitations, on no imaginable occasion can theTHE WorD OF GOD 55
million dollar formula provide any information not more easilyobtained from Ockham's equation, repeated here, from Chapter 3:
 
vxveATMO, =40whereATMO, = the at-the-money option price expressed as a
percentage of the futures price
v = the option volatility also expressed as a percent-age of the futures price
t = the number of days till option expiry
Ockham's equation can be solved immediately for volatility orfor option price. Alternatively, these same quantities can beobtained directly from the tables of Figure 4-7, generated from thissame equation. So beware Black-Scholes, there's a leaner, meaneroptions pricing machine about to give you a run for your money.
*
Whenever the option equation is solved for v, that is, whenthe option price is known up front and the expiry time is speci-fied, the quantity v so obtained is known universally as theimplied volatility of the option. The following examples illustratehow the tables of Figure 4-7 may be used to derive and compareimplied volatilities.
Example 4-1. On January 16, 1996, the March wheatfuture closed at $4.80 per bushel. The March 480 call optionwhich had 24 trading days till expiry closed at 11.5 cents.What is the implied volatility of this option?56 OPTION THEORY
Option price(p) Number of trading days till expiry (t) — ->2 3 4 &§ 8 7 8&8 O 0 11 12 13 14 «15 #18 17 18 20 22 24
71°58 50 45 41 38 35 33 32 30 29 26 27 26 25 24 24 22 2.1 2.0
REEEEEEEé~s©$
SP B2OnO?Y> Aye£2gog°3°*é»5a$»é*»5e$e&°8e£°g
BESESSRSEESRSSESRESESEEEES8g&gégggg&8
FIGURE 4-7. Options on different commodities and options with different terms toexpiry may be directly compared via a quantity called the implied volatility. From thetable above, derived from Ockham’s equation, and applicable only to at-the-money putsOption price (p)
Y 2 26 2806 4.9 «4.7 4506 «5.9 5.1 4.9Om 5.7 5.8 5.3075 «6.1 5.9 5.7000 86S 63 6.0oS 6.9 6.7 6.40 7.37.4 68os 7.8 7.5 7.2100 8.2 7.8 7.6108 86 82 7.9110 9.0 86 8.3159.4 9.0 8.712 9.8 9.4 91128 102 98 9.4190 106 102 9.8135 11.0 10.6 10.2140 11.4 11.0 10.6145 11.8 11.4 11.0180 12.2 11.8 11.3180 13.1 12.6 12.1170 13.9 13.3 12.9180 14.7 14.1 13.6190 18.5 14.9 14.4200 16.3 15.7 15.1210 17.1 16.8 15.922 «18.0 17.3 1662390 18.8 18.0 17.4240 © 19.6 18.8 18.1250 204 196 18.9260 21.2 20.4 19.7270 © 22.0 21.2 204200 © 22.9 22.0 21.2290 © 23.7 22.7 21.9300 24.5 23.5 22.73200 28.1 25.1 24.2$40 27.8 26.7 25.7360 29.4 28.2 27.2380 31.0 29.8 28.7400 32.7 31.4 302420 34.3 32.9 31.74M 35.9 34.5 33.3400 97.6 36.1 34.8490 39.2 37.7 38.3500 40.8 39.2 37.8520 42.5 40.8 38.3SMO 44.1 42.4 40.8660 45,7 43.9 42.3580 47.4 45.5 43.8600 49.0 47.1 45.4650 63.1 51.0 49.17.00 57.2 54.9 52.9750 58.8 56.78.00 60.5850
9.00
950
10.00
5
THE WorD OF GoD
Number of trading days till expiry (t) —-»
32 M 8
42
444s
38
40
42
44
48
56.0
41
50
55
60
65
70
B
57
and calls, implied volatility y may be read directly for any combination of trading daysremaining f, and option price expressed as a percentage of futures price p.58
OPTION THEORY
Price of option 11.5TT 0= — x 100 = 2.39 percentPrice of future 480
Entering the table of Figure 4-7 with an option price ratio of2.39 and a time remaining to expiry of 24 days, and interpo-lating, you arrive at an implied volatility of 19.5 percent.
Example 4-2. On June 14, 1996, the September wheatfuture closed at $5.005 per bushel. The September 500 putoption, which had 50 trading days till expiry, closed at 25.5cents. What is the implied volatility of this option?
Price of option 25.5————— x100= ——— x 100 = 5.09 percentPrice of future 500.5
From the table of Figure 4-7, implied volatility is 28.9percent.
Example 4-3. On May 17, 1996, the July coffee futureclosed at $1.2865 cents per pound. The July 130 call closed at4.8 cents and the July 130 put closed at 6.2 cents, for anaverage at-the-money premium of 5.5 cents. The options had15 trading days till expiry. What is the implied volatility ofthese options?
 
Price of optionx x 100 = 4.27 percentPrice of future
From the table of Figure 4-7, implied volatility is 44.2percent.
Example 4-4. On May 18, 1996, the September coffeefuture closed at $1.2435 cents per pound, The September 125put and call closed at an average premium of 12.7 cents. Theoptions had 54 trading days till expiry. What is the impliedvolatility of these options?THE WorD OF GoD 59
Price of option
 
12.70 = x 100 = 10.21 percentPrice of future 124.35
From the table of Figure 4-7, implied volatility is 54.4percent.
From the last two examples above, it is clear that theSeptember coffee option has a much higher implied volatilitythan the July coffee option (they are measured just 1 day apart).At-the-money options on different futures months need not implythe same volatility. The reasons for this seeming oddity areexplored at length in Chapter 5.
*
Implied volatility v is a descriptive statistic with no intrinsicmeaning; it is simply the standard deviation of the hypotheticalnormal distribution that would satisfy a particular pair of valuesof option price and time till expiry. Implied volatility is bestthought of as a comparative number that allows options on dif-ferent commodities and with different expiry times to be assessedfor relative price.
The implied volatility of an option must not be confused withthe market volatility of the underlying future, a statistic derivedempirically from price change data. There is no necessary equiv-alence between the implied volatility of an option and the mar-ket volatility of its related commodity future (although a bigdivergence here would certainly point to a potential overvalua-tion or undervaluation situation). A coffee option with an impliedvolatility of 40 percent is clearly projecting a more variablefutures price pattern for coffee than a gold option with animplied volatility of 10 percent is projecting for gold futures.There is no guarantee, however, that future market volatility willbear a close resemblance to an implied volatility projection, Inreality, the true volatility of a market is very difficult to define andmeasure and can only ever be known in retrospect. The implied60 Option THEORY
volatility of an option, however, is a calculable quantity known atevery instant of time.
Implied volatility as a comparative statistic has attained suchwidespread currency amongst option traders that proprietary ser-vices have sprung up for the express purpose of searching outoptions where the objectively defined implied volatility from theoptions formula seems to be out of whack with some subjectivelyderived estimate of what upcoming market volatility is likely tobe. Traders should be wary of using implied volatilities publishedby advisory services as absolute yardsticks for decisionmaking. Anoption with an apparently mispriced implied volatility does notnecessarily point to a trading opportunity; the subjectively esti-mated market volatility may fail to reflect some key informationthat the option market has already discounted.
Understanding the subtleties in the relationship betweenimplied and market volatility is the core problem in option eval-uation. The relationship is fraught with conceptual pitfalls and isdiscussed in considerable detail in the next chapter.CHAPTERFIVE
THE EMPEROROF CHINA'S NOSE
I tis January, say, and a quick check of the financial pages showsthat the price of the March at-the-money coffee call is 5.25cents. The implied volatility of this option (calculated either fromthe million dollar formula or Ockham's equation—or read fromthe table of Figure 4-7) turns out to be 41 percent. Does thisimplied volatility tell us anything useful? Can it be comparedagainst anything?
We might look back in time and check the implied volatilityof this same option on this same date in previous years. Supposethat in the previous two years the implied volatilities were 26 per-cent and 32 percent, respectively. Does the fact that currentimplied volatility is 41 percent suggest that the option is overval-ued and a candidate for writing? Maybe, but not necessarily so,for we cannot make any assessment of the market's pricing of anoption on the basis of its implied volatility alone. The impliedvolatility of an option only takes on real significance when it canbe compared to the current market volatility of its underlyingfutures contract, and market volatility will always be a subjectiveestimate to some extent, because there are as many estimates ofmarket volatility of a commodity future as there are players in themarket. Consciously or subconsciously, whenever traders takepositions in options, based on value considerations, they aremaking their own independent estimates of market volatility andcomparing these independent estimates to the implied volatilityof the options.
6162 OPTION THEORY
Option advisory services may tell you otherwise, but there isno such thing as an obviously overpriced or underpriced option;the market as a whole is much too smart to grant "freebies." Thatis not to say the market is always perfectly priced. But, when itisn't perfectly priced, it is certainly not going to advertise thatfact. Some people seem to believe that the market is always fairlypriced.
There is, in fact, a way in which the strategist can let the market com-pute the volatility for him. This is called using the implied volatility —that is, the volatility that the market itself is implying. This conceptmakes the assumption that, for options with striking prices close tothe current stock price and for options with relatively large tradingvolume, the market is fairly priced. [Lawrence McMillan, Options asa Strategic Investment (New York: NYIF Corp., 1993, p. 464)--one ofthe best-selling options books of all time.]
Now, it is true that the strategist can let the market computeimplied volatility, but the strategist cannot expect the market toindicate whether an implied volatility fairly reflects marketvolatility. To suggest that the options market is always fairlypriced is tantamount to saying there is no point in trying to inde-pendently place a value on an option; any estimate of marketvolatility would necessarily be inferior to the implied volatilityalready incorporated in the price of the option.
Perhaps McMillan’s statement is expressing a different ideaaltogether. Could he be implying that at-the-money options arefairly priced (because of the large trading volume) while out-of-the-money options may not be fairly priced? Could this be hisway of reconciling inconsistent implied volatilities at differentprices? I believe what we have here is a piece of specious rea-soning leading to a classic conundrum: There is no reason ever totrade at-the-money options. If you make the assumption that theseare always fairly priced, how could you then disagree with thatassumption, which is exactly what you would be doing by taking aposition in the market.
Let me stress that I am talking here about taking an optionposition based on perceived valuation, for there are certainlyother reasons to take an option position. A trader might buyTHE EMPEROR OF CHINA'S NOSE 63
options based on a strong fundamental feeling about the pricetrend of the underlying future, without particular concerns aboutwhether the option appears "expensive" or "cheap." A trader sodisposed would not be interested in option pricing models.Studying the relationship between option prices and futuresprices is predicated on the belief, or at least the hope, thatoptions are not always fairly priced. Now it may turn out thatoptions are always fairly priced, which would be a disappointingdiscovery, to say the least. But why assume, as LawrenceMcMillan does, that the question has been answered before theinvestigation has begun?
*
The failure to appreciate that the implied volatility of anoption is simply a derivative of its price has produced some ratherconfusing terminology in the literature on option pricing. Forexample, it is quite common to see separate implied volatilitieslisted for each of the out-of-the-money strike prices on the samecommodity future. The volatility that is being "implied" in animplied volatility calculation is, of course, the market volatility ofthe underlying future, and a commodity future would still "pos-sess" a market volatility even if there were no options. The ideathat a future can have more than one implied volatility does notreally make sense. It is, nevertheless, common enough practiceto talk about different implied volatilities on the same future, soIam compelled to do likewise —at least for the moment.
If the million dollar formula could accurately accommodatethe true probability distribution of possible price changes insteadof an idealized normal distribution of possible price changes, theimplied volatilities of all options on a particular future would bethe same. In the options pit, where prices are actually made, thetrue probability distribution makes itself felt through the experi-ences of traders betting with real money. The market knows fromexperience that option prices cannot possibly conform to thestrictures of any formula based on a normal distribution, and itprices options according to true probabilities — as best it can. Thefact that the million dollar formula comes up with inconsistent64 OPTION THEORY
implied volatilities for different strike prices is a glaring indict-ment of its inadequacy as an option pricing model.
To calculate the implied volatility of an option one has towork backward from an actual option price instead of forwardfrom an actual volatility toward an implied option price. If thecalculations are performed forward, as they should be, the mil-lion dollar formula comes up with out-of-the-money option pre-miums well below actual values prevailing in the free market(Figure 5-1). If the market truly believed in the normal distribu-tion, the vertical bars of Figure 5-1 would all be the same height.Volatility profiles vary by commodity and some implied volatilityprofiles are very much flatter than others. The nonlinear aspectof the implied volatility profile is sometimes referred to as thevolatility skew or volatility smile. A volatility "frown" is neverobserved.
Working, it seems, on the theory that if you average a seriesof errors you will somehow wind up with a right answer, a num-ber of authors — uncomfortable perhaps with the inconsistency inhaving more than one implied volatility —have suggested averag-ing implied volatilities to arrive at an averaged implied volatilityor a composite volatility. No one has ever suggested a practicaluse for an averaged implied volatility, but that does not stop peo-ple from wanting to average it. For a detailed analysis of impliedvolatility averaging carried to its ludicrous extreme, includingaveraging along with weighting by options trading volume, seeMcMillan above.
Nobody was permitted to see the Emperor of China, and the questionwas, What is the length of the Emperor of China's nose? To find out,you go all over the country asking people what they think the lengthof the Emperor of China's nose is, and you average it. And that wouldbe very "accurate" because you averaged so many people. [RichardFeynman in "Surely you're joking," Mr. Feynman (New York: Norton1987, p. 303)]
*
An inexperienced trader looking over a table of impliedvolatilities generated by the million dollar formula might beTHE EMPEROR OF CHINA'S NOSE 65
DISTRIBUTION OF IMPLIED VOLATILITIES
45 (August Crude Oil June 17, 1996)
Call options
   
40 Put options
Implied Volatility (%)
30
25
17.0 175 180 185 190 195 200 205 210 215 220 225 23.0
Option strike price
FIGURE 5-1. Implied volatilities calculated from option settlement prices typicallyfollow the pattern above; the lowest value occurs at the money with values progressivelyincreasing as the strike price move out d the money. Although implied volatilities areroutinely calculated for all possible strikes, the only one that is useful for comparingdifferent commodities or for comparing different futures of the same commodity is theat-the-money implied volatility.
seduced into a strategy of buying at-the-money options whilewriting out-of-the-money options, since the latter would appearto be overvalued relative to the former. This strategy might pro-duce a lot of commission for the broker but probably little else,for the market is going to be rather astute in its relative pricingof options on the same future. Experienced traders know intu-itively that there can be only one volatility associated with a com-modity future. They also know from experience that the milliondollar formula severely underestimates the fair value of out-of-the-money options. And they vote accordingly. Here's a formerfloor trader talking about the shortcomings of a generalizedoptions pricing formula:
Whatever the model says an option with an extreme exercise price isworth, it is probably worth more. How much more, nobody really66 OPTION THEORY
knows. But because of the apparent inaccuracy in the model, no expe-rienced trader is likely to sell such an option for its theoretical value.If the model says a far out-of-the-money option is worth .05, no expe-rienced trader will sell such an option for .0S5 because he knows themodel has probably undervalued it in the real world. Even a bid of .10or .15 may be insufficient. Of course, every trader has his price, andif someone bids .50 the trader may finally be willing to sell. Themodel may be wrong, but at a price of .50, the trader may decide thathe can live with that risk. [Sheldon Natenberg in Option Volatility andPricing Strategies (Chicago: Probus Pub. Co., 1988, p. 305)—highlyrecommended reading. |
To assist professional options traders in making trading deci-sions under rapidly changing futures conditions, the commodityexchanges publish "volatility sheets” on a daily basis. If you visitthe options pit of a commodity exchange you will see many of thefloor traders scanning these volatility sheets while they keep aclose watch on what is happening in the futures pits. I asked atrader on the floor of the New York Cotton Exchange how hemade use of his volatility sheet.
"So I know how much to bid or offer for an option," hereplied. "I check the futures price on the board, check the volatil-ity sheet at that price, and get the fair value of any option at thatfutures price. If I see a bid above fair value, I might sell it. If Tsee an offer below fair value, I might be a buyer."
He showed me the volatility sheet — about 8 pages of denselypacked statistics. For every conceivable price that a future mighttrade at on that particular day, that is, for every other price tickfrom limit up to limit down, the sheet listed fair value put andcall prices. And this for every option.
“Do you know where these numbers come from?" I asked.
"The exchange puts them out,” he said. "They use a formula.”
"What formula?”
"The Black Scholes Formula.”
"What's that?
"You're writing a book on options and you don't know Black-Scholes. You got to be kidding, pal.”
I was. "I'm trying a different approach, that's all.”THE EMPEROR OF CHINA’S NOSE 67
He shook his head. "This business is built on the Black-Scholes model.”
"Do you understand it?”
"Understand what?"
"The formula.”
"[ don't have to understand it. It’s all done on a computer, It'svery complicated."
Another trader butted in.
"You see all these option prices,” he said, pointing to hissheet. "They're based on implied volatilities calculated from theprevious day.”
"And where do these implied volatilities come from?" I asked.
"From the implied volatilities of the day before that, I guess."
I borrowed the volatility sheets and studied them quickly—they were marked: "Confidential: For exclusive internal use ofexchange personnel." The data were truly comprehensive andremarkably practical. The fair value option prices they listed hadbeen calculated not from the million dollar formula but fromempirically observed relative pricing patterns prevailing in recenttrading sessions. It seemed as if actual options closing prices hadbeen converted to implied volatilities via the million dollar for-mula— since different implied volatilities were listed for differentstrikes—and these implied volatilities then converted back toguideline option prices for use in the,next trading session. Therewas even a built-in volatility correction factor, so that in the caseof a very large price change in the future, all options wouldreceive a boost in value. This was very logical, though exactlyhow it had been done I couldn't tell.
Still, the whole process had a circular feel to it. If all traderswere to follow such guidelines, option prices (corrected for timedecay and the inevitable shift in the at-the-money strike price)might never change at all, since each day's pricing would bedetermined absolutely from the previous day's pricing, andchanges in the market volatility of the underlying future wouldnot be reflected —at least through the actions of traders using thevolatility sheets. It doesn't happen that way in the real world, ofcourse, There are enough players tracking daily price swings in68 OPTION THEORY
the futures markets, and enough players with an intuitionfor value, that any real change in futures market volatilitywill quickly be reflected in a change in overall option pricingstructure.
*
While it may be argued, defensibly, that for any given futurethere can be only one true implied volatility, the same cannot besaid of different futures on the same commodity; different futureson the same commodity can and do have different impliedvolatilities. In any logically constructed option formula (includ-ing the million dollar formula), the fair price of an at-the-moneyoption will decrease in proportion as the square root of its timetill expiry decreases. This is clear from Ockham's equation, forexample, where:
vxvi40
 
ATMO, =
(This squlare-root time decay relationship is almost certainly validfor price change distributions that are net normal as well as dis-tributions that are normal.)
Ockham's equation is certainly applicable to any particularfutures maturity for the reason that if v is constant — which is tosay, the probability distribution of daily price changes isunchanging — the only variable that can affect the price of theoption is #, the time till expiry.