price between the first observation and last, thequantity AP,,, will be very close to zero. But, in a strongly trend-ing market of comparable real volatility the quantity A P,,, will notbe close to zero, since values of AP, will be cither mostly positive80 Option THEORY
 
 
 
Price Change 100 x (AP,- AP,,) 100 x |APIIDay P, (AP) (AP,- APey) Pay (Pav) (Pav)1 12102 1194 -16 -16 1202 1.331 1.3313 1160 34 34 1177 -2.689 2.8894 1188 28 28 1174 2.385 2.3855 1185 -3 -3 1186 0.253 0.2536 1170 -15 -15 1178 -1.274 1.2747 1153 -17 -17 1162 1.464 1.4648 1178 25 25 1166 2.145 2.1459 1172 aac] aac] 1175 0.511 0.51110 1195 23 23 1184 1.943 1.94311 1210 15 15 1202 1.247 1.247AP,, = 0 (MAD) = 1.544Standard deviation(day) = 1.82%Volatility (SD x 384) = 29.01%Mean absolute deviation = 1.54%Price Change 100 x (AP; - AP,,) 100 x |AP|Day P, (AP) (AP\- APa) Pay (Pax) (Pav)1 12102 1226 16 0 1218 0.000 1.3143 1260 34 18 1243 1.448 2.7354 1288 28 12 1274 0.942 2.1985 1285 -3 -19 1286 -1.477 0.2336 1300 15 -1 1292 0.077 1.1617 1317 17 1 1308 0.076 1.2998 1342 25 9 1330 0.677 1.8809 1336 + -22 1339 1.643 0.44810 1359 23 7 1348 0.519 1.707W 1374 15 -1 1366 0.073 1.098AP,, = 16 (MAD) = 1.407Standard deviation {eay) = 0.98%Volatility (SD xV254) = 15.78%Mean absolute deviatlon = 141%
FIGURE 5-5, A potentially serious error in the computation of volatility can resultwhen the standard deviation of price changes is calculated in a runaway bull or beartrend. In the two price series above, daily price changes are of the same magnitude,indicating that volatility should be approximately the same. Yet, in the upward trend-ing market, the calculated volatility is only half of what it is in the trading market.
In the limiting case, if a future were to advance by a constant amount every day,the variable (A?, - A%) would tend toward zero, as would the standard deviation ofdaily price changes and the volatility. Using the mean absolute deviation, MAD, as ameasure of volatility yields consistent results in both price series, the slightly lower valuein the second series resulting from an increase in the absolute price level (a divisor).THE EMPEROR OF CHINA'S NOSE 81
or mostly negative. If AP,, turns out to be significantly nonzero,the standard deviation calculated from the formula above will notreflect the true volatility of the market (Figure 5-5).
In a strongly trending market, a reading of market volatility —as calculated by the standard deviation in the equation above —will be much lower than the implied volatility calculated fromactual option prices, and a trader comparing these two volatilitiesmight conclude that the options are overvalued and therefore beinclined to the sell side rather than the buy side. Were enoughsellers to be drawn in for this reason, the net effect would be anartificial depression of option prices during runaway bull or bearphases in a futures market. Whether in reality this happens canonly be answered empirically, if at all. It remains an interestingconjecture, though, and like all good conjectures it is based onreasoning, rather than on accidental observation.
In estimating market volatility from historical price data onemust think clearly about what is being measured. In particular, itis crucial never to confuse absolute daily prices with daily pricechanges. Into this pothole, even the mightiest have stumbled(Figure 5-6). *
In the development of Ockham’s equation, it was shown thatthe simple average of a series of price change —taking all pricechanges to be positive—could be used instead of the standarddeviation to estimate the fair value of an option. It was alsodemonstrated that the validity of this estimate was independentof the nature of the distribution of price changes. And further-more, it has been shown in this chapter that using the meanabsolute deviation of a set of price changes rather than the stan-dard deviation of these price changes leads to market volatilityestimates that are unaffected by trend. For the remainder of thismanuscript, therefore, option fair value and futures marketvolatility will be estimated using the mean absolute deviation asthe primary empirical statistic.
To estimate the fair value of an at-the-money option fromempirical data, Eq. 3-1 from Chapter 3 is simply expanded toinclude ¢, the time till expiry of the option:82 Option THEORY
Some very peculiar advice.. .
*The computation of volatility is always a difficult problem for mathematicalapplication. In the Black-Scholes model, volatility is defined as the annualstandard deviation of the stock price. This is the regular statistical definitionof the standard deviation.
fX (Pi P)?s° = i=1(n-1)vePpwhereP = average stock price of ail P'sP, = daily stock pricen= number of days obseivedv = volatility
* From Options as a Strategic Investment, by Lawrence McMillan (NYIF Corp.,1993), p. 462.
FIGURE 5-6. The method of estimating historical volatility described above, liftedverbatim from a popular text on option trading, will produce very misleading results.At face value, the formula is plausible in that it seems to make use of all the availabledata, namely each price in the price series.
The error in using absolute prices rather than daily price changes arises when theorder in which absolute prices occur is lost, as when using the formula above. In astrongly trending market, the quantity Z(P; ~ P)? will be a very large number, while ina trading market with the same magnitude of daily price swings, the quantity £(P,; — P)*will be relatively small. Same volatility, two very different answers. Something is clearlyamiss, Daily price changes must be used when estimating volatility.
Fair valueofan = 9.5 xix MAD (Eq. 5-1)at-the-money option
In Eq. 5-1, fair value and MAD may be expressed either inunits of absolute price or as percentages of the base future price.To estimate market volatility (which you will recall is defined asannualized standard deviation) from a series of price changes,THE EMPEROR OF CHINA'S NOSE 83
and to make this estimate consistent with the implied volatility inOckham’s equation, daily standard deviation is first related tomean absolute deviation, as before, by the formula:
SDaaily = 1.25 x MAD
Next, annualized standard deviation is related to daily standarddeviation, as before by the formula:
SDannual = V254 x SDaaily
So that:
SD V254 x 1.25 x MAD
annual —
Or, to a very good approximation:Market volatility = 20 x MAD (Eq. 5-2)
Ii: Eg. 5-2, market volatility should be expressed as a per-centage, if it is to be directly compared with an implied volatilitycalculated from an options formula. That is to say, MAD shouldbe expressed as a percentage of futures price. Strictly speaking,the validity of Eq. 5-2 depends on the special relationship thatexists between the standard deviation and the mean absolutedeviation of a normal distribution. This equation may have to bemodified later after empirical testing of actual market data.
*
This completes the theoretical discussion on options pricing.Some of the analysis may seem unnecessarily detailed— and itundoubtedly is—but it has been included so that the reader maycorrelate what I have to say with what has already been publishedby others in the field.
Options are not obliged to price themselves to conform to anymathematical theory, mine or anyone else's; the reality of themarketplace is what really counts. At this point, it will be appro-84 OPTION THEORY
priate to switch from theory to observation, for it can only bethrough empirical analysis, through an extensive investigation ofwhat has happened in the past, that a systematically profitableapproach to options trading—if it exists at all—is likely to beuncovered. Historical data on options are hard to get at and hardto structure for analysis. But the information is certainly there.PARTTHREE
OPTIONREALITYCHAPTERSIX
PHANTOMOF THE OPTION
he difficulty in analyzing historical option data is that so
many of the parameters seem to be changing at the sametime. With a commodity future, the only variable that changesday to day is its price, so that any sequence of prices can be log-ically compared with any other sequence of prices. Not so withan option. Each day, the difference between a specific option'sstrike price and the price of its associated future changes, caus-ing the option price itself to change in a rather complicated wayGf chis were not complicated there would be no need for the mil-lion dollar formula or any of its surrogates). Furthermore, thetime to expiry of the option diminishes by 1 day, every day, alsocausing the price of the option to change in a nonlinear fashion.
For reasons that have been explained in previous chapters,the only option that merits empirical investigation is the at-the-money option— specifically, the put and the call whose strikeprice exactly equals the current futures price, whatever that pricemay turn out to be at the close of trading on any given day. WhatI propose developing, and comparing on a day-to-day basis, is asequence of option prices, each related to a different strike price!
On first consideration, such a comparison might seemimprobable. Prices of at-the-money options are not quoted assuch, since the only time an at-the-money option can be mea-sured is when a future closes exactly on an option strike price—a rather rare event, occurring, perhaps, no more than one timeout of a hundred. Even the closest-to-the-money option is hardto pin down. One day the closest-to-the-money option may be—
8788 OPTION REALITY
in the case of the September S&P series, say—the September950; the next day it may be the September 960. Apples andoranges, so to speak.
Now, it is true that during most trading sessions a future willtrade at the strike price of one of its options, and theoreticallytherefore, if one were nimble enough and had ten sets of eyes, itwould be possible to get an instantaneous fix on the relationshipbetween an at-the-money option and its future on a more or lessdaily basis. Fine in theory, but hardly a practical proposition, andeven then what would be established is a price relationship exist-ing at one particular instant of time and specific to one specificoption. What's more, the at-the-money option would most likelybe one with a different strike price every day. All of which helps toexplain why empirical research into option pricing remains virginterritory. Therefore, in exploring this territory for answers, what Ido ask of the reader is a temporary suspension of disbelief.
Every day, closing prices are posted for the closest-to-the-money puts and calls. These options may not be identified explic-itly as being the closest-to-the money options, but there is alwaysa closest strike price by which to identify them as such. Imaginenow that these closest-to-the-money option prices can somehowbe corrected for the amount by which they are out-of-the moneyso that they become surrogate at-the-money options-phantomoptions, if you will. These phantom options will now be directlycomparable on a day-to-day basis. For, although the strike priceof the at-the-moncy option is certainly going to be changingalmost every day, this parameter will now have been effectivelyremoved as a variable. The key problem now reduces to whetherclose-to-the-money option prices be effectively corrected so thatthey express what true at-the-money option prices would haveclosed at.
*
As discussed in Chapter 3, the price of an at-the-moneycall must equal the price of the corresponding at-the-money put.If a put and a call having the same strike price are bought orwritten as a pair, the combination is called a straddle, and if thePHANTOM OF THE OPTION 89
transaction is completed when the options are trading exactly atthe money, the straddle premium paid or received will be exactlydouble the premium that would be paid or received for the put orthe call separately.
Straddle prices may be calculated from option price tablespublished in the financial press simply by adding together theprices of the put and the call at any particular strike price. Withrare exceptions, for any given future at any given time, a straddlewill have its minimum value when it is trading at the money(Figure 6-1).
When straddle price is plotted against strike price, the result-ing curve is parabolic with a rather flat base extending on eitherside of the minimum value. This flatness merely reflects the obvi-ous reality that relatively small fluctuations in the price of a futureare going to have similar but opposite effects on the put and callcomponents of the straddle. For small price increases in theunderlying future, what the call gains the put will lose, and vice-versa, of course, with small price decreases. Away from the money,the slope of the curve begins to rise steeply; with a large pricechange in a future and especially with a sustained series of pricechanges in the same direction one of the component options willbegir. to appreciate more rapidly than the other depreciates.
Due to the flatness of the straddle profile, the price of a close-to-the-money straddle will be almost identical to the price of thetrue at-the-money straddle. The question is how close does afutures settlement have to be to an option strike price before it issafe to take the nearest straddle as equivalent in price to the trueat the money straddle? Whenever option strike prices are rela-tively close together—roughly speaking, when the separationsbetween strike prices approximate the daily trading range of thefuture — the price of the closest-to-the-money straddle will be vir-tually equivalent to the price of the at-the-money straddle. Theintervals between option strike prices are often sufficiently closethat this equivalency prevails. However, in certain commoditiesthe interval between strikes far exceeds the average daily tradingrange, and the true at-the-money straddle price must be esti-mated by applying a correction factor to the closest-to-the-moneystraddle.90 OpTION REALITY
STRADDLE PROFILE FOR A JAPANESE YEN FUTURE
3.00
         
Straddie premium
Future closed at 82.78
81.00 81.50 82.00 8250 83.00 8350 84.00 84.50Option strike price
FIGURE 6-1. The straddle curve (straddle premium versus option strike price) isvery flat at strike prices close to where the future is trading. Over a sizable range offutures price change, the price of straddles at different strikes will vary little, becausethe call will gain what the put loses and vice-versa. The market offers no prizes forinformation everyone knows— futures prices are bound to fluctuate.
In the example above— measured at one particular instant in time—the true at-the-money straddle implied at the futures price of 82.78 is almost identical to theactual straddle premiums registered at strikes of 82.50 and 83.00. In this particularconfiguration for the Japanese yen, it would take a fast move of about 100 points inthe futures price to cause the at-the-money straddle price to increase by 10 points. Inother words, the price of a straddle written close-to-the-money will change veryslowly —at first. Of course, if the future embarks on a sustained move in one direction,either the put or call component of the straddle will begin to appreciate faster than theother side depreciates, and the total value of the straddle premium will increase at anaccelerating rate given by the slope of the straddle curve.
The only way to tell what that correction factor ought to be isto search for instances where the true at-the-money option pricesare known and to compare these prices with the also knownprices of the nearest strike options. It is not common for a com-modity future to close exactly on the strike price of one of itsoptions, but it does happen. Here are three such instances fromthe historical record:PHANTOM OF THE OPTION 91
 
AT THE MONEY LOWER STRIKE HIGHER STRIKEPut Call Straddle = Put — Call Straddle Put Call Straddle
Swiss franc 6850 1.22 1.22 2.44 0.98 149 2.47 147 102 2.49Cocoa 1450 0.59 0.59 1.18 0.34 0.89 1.23 0.82 0.40 1.22Coffee 115 4.50 4.50 9.00 2.71 7.45 10.16 7.73 2.40 10.13
Because it has many strike prices at intervals comparablewith the daily trading range of its future, the Swiss franc strad-dle premiums at strikes of 6800 and 6900 respectively are onlyslightly higher than the true at-the-money straddle premium at astrike price of 6850. But, with cocoa and coffee, strike prices arerelatively infrequent compared with the daily ranges of theirrespective futures prices, and the true straddle prices differ sig-nificantly from the straddle prices registered at the surroundingstrikes.
Consider, in detail, the Swiss franc data above. If the twohighest value option components of each of the 6800 and 6900straddles are summed and divided by the sum of the two lowestvalue option components, the following ratio is obtained:
1.47 + 1.49omss 0.98 + 1.02
= 1.48
If, at the same time, the two equidistant and nearest-to-the-money straddle values, 2.47 and 2.49, are averaged, the correc-tion multiplier CM necessary to produce the knownat-the-money straddle price can be determined as follows:
2.44
CM wis = 72.48
= 0.98492 OPTION REALITY
Repeating the procedure for cocoa and coffee produces the fol-lowing sets of paired values:
CMywiss = 0.984 with Reyiss = 1-487Meocoa = 0.963 with Regcoa = 2-311CM coffee = 0.887 with Rooffee = 3.009
These preliminary observations suggest a possible empiricalrelationship between CM and R. An extensive search of the his-torical record uncovered about 30 instances where a commodityfuture had closed within a price tick of the strike price of one ofits options. The search covered records from all actively tradedoptions — grains, financials, metals, etcetera. When 30 or so CMand R values as defined above are calculated and plotted on achart (Figure 6-2) they indicate a linear relationship between thevariables, and a good straight-line fit to the data is given by theequation:
CM = 1.04 — (0.04 x R)
Applying this correction multiplier to published option data,from which R can always be calculated, one can estimate thetrue at-the-money straddle value for any commodity future at anyclosing price. The correction may not be exact, but it will cer-tainly be close. *
Consider the correction suggested above applied on a dailybasis to the closest-to-the-money straddle price of a given com-modity future. The result will be a series of "phantom" straddleprices, since the corrected at-the-money straddle will have a the-oretical strike price (the futures price) which does not, in gen-eral, correspond to any listed strike price. Does the phantomaspect of the price series make it any less valid as a data base forhypothetical testing?
I don't think so. For, though not explicitly stated, there isalways an implied at-the-money put, call, and straddle with astrike price equal to the futures close on that day. PhantomPHANTOM OF THE OPTION 93
1,00
CM = 1.04 -0.04R
0.98
0.96
0.94
0.92
Multiplier (GM)
0.90
0.88
 
0.861.0 15 2.0 25 3.0 3.5 4.0
Maximum-minimum ratio (R)
FIGURE 6-2. In order to establish a convincing relationship between R and M, itwas necessary to search the historical record for instances where a commodity futureclosed exactly on one of its strike prices. Thirty or so such instances have been identi-fi-d and plotted on a suitably scaled chart, the object being to approximate an empiri-cal equation expressing the relationship. Fortunately, the plotted points fall more orless along a straight line, indicating a linear relationship conveniently expressed by theequation:CM = 104- (0.04 x R)
The CM versus R relationship, derived from data where the at-the-money strad-dle price was known, may now be applied as a correcting factor in situations where theat-the-money straddle price is not known.
options generate coherent price sequences whereas real optionsdo not, and the phantom option possesses the one option statis-tic that can be compared directly with that same option statisticon the previous day or on any other day. Some numerical exam-ples will help illustrate the point.
Example 6-1. On February 22, 1996, the May sugar futurecloses at 1179, making 1200 the closest option strike price.94 OPTION REALITYThe May 1200 sugar call settles at 28, and the May 1200
sugar put at 55. There are 37 trading days till option expiry.Therefore:
55=— = 1.96428
CM = 1.04 — (0.04 x 1.964)= 0.962
The value of the phantom at-the-money straddle PS is thereforegiven by:
PSpep 22 = 0.962 X (55 + 28)= 79.8Example 6-2. One day later, February 23, 1996, the Maysugar future closes at 1160, making 1150 the closest optionstrike price. The May 1150 sugar call closes at 46, and theMay 1150 sugar put at 33. There are 36 trading days tillexpiry. As before:
46R=— = 1.39433
CM = 1.04 — (0.04 x 1.394)= 0.984The value of the phantom at-the-money straddle PS is given by:PSrep 23 = 0.984 x (46 + 33)
= 77.8PHANTOM OF THE OPTION 95
The phantom straddles of February 22 and February 23 aredirectly comparable even though they are derived from optiondata pertaining to two different strike prices. By essentially freez-ing out strike price as a variable, the possibility of constructing aworkable data base on which to test option hypotheses expandsenormously.
It is a short step from estimating a phantom straddle price tocalculating an implied volatility. Ockham's equation — the funda-mental equation relating option price, time, and implied volatil-ity—states that:
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 expirySince the at-the-money straddle is known to be exactly double
the value of either the at-the-money put or the at-the-money call,Ockham's equation applied to a straddle may be restated thus:
 
vxveATMS, =20orv = 2OXATMS, (Eq. 6-1)
Vt96 OPTION REALITY
where ATMS, = the at-the-money straddle price expressed asa percentage of the futures price
With PS now clearly synonymous with ATMS, the estimated at-the-money straddle prices for May sugar on February 22 andFebruary 23 may now be converted to implied volatilities byapplication of Eq. 6-1, using the appropriate number of tradingdays to expiry, 37 and 36, respectively:
Implied volatility for May sugar — 20 X ATMS,calculated on February 22. ~ Ni
20 79.8 x 100x a
37 1179
22.23 percent
And, by a similar calculation, . 20. X 77.8 x 100implied volatility on February 23 136 1160
22.35 percent
The calculations above are presented in this detailed way toclearly demonstrate the relationship between the three variables ofparamount importance in option price evaluation — at-the-moneystraddle price, implied volatility and time remaining to expiry.
*
In deriving the solutions above, time t was taken to be thenumber of trading days till option expiry. There is some debateabout whether trading days or calendar days should be used in animplied volatility calculation. The million dollar formula usescalendar days expressed as a fraction of 365. For options with along term to expiry the distinction between calendar days andtrading days is inconsequential. But, for an option with a shortterm to expiry the difference can be significant.PHANTOM OF THE OPTION 97
The theoretical argument in favor of calendar days contendsthat the forces affecting a futures price are independent ofwhether the market is open or closed for trading, and that thetwo weekend days ought therefore to be considered as opportu-nities for the futures price to vary and ought therefore to beincluded in the time to expiry. There is some merit to this argu-ment in the case of agricultural commodities but less merit whenit comes to financial instruments. For pragmatic reasons alone,one standard has to be selected for use in all markets.
On balance, there are good practical reasons for preferringnumber of trading days over calendar days. In the first place,market volatility calculations cannot distinguish between week-days and weekends. Therefore, neither, logically, should impliedvolatilities do so. Furthermore, the available empirical evidencestrongly suggests that number of trading days more accuratelyreflects true variability.
Whether a weekend is equivalent to two trading days (implicitin using calendar days in an option pricing formula) or no trad-ing days (implicit in using number of trading days in a formula)is a question that can be tested empirically, without reference tooption prices at all. What is needed is a large database of futuresprice changes, and futures price change is a major component ofthe data base put together for this empirical research.
Of the almost 4000 futures price changes recorded in thisdata base, 80 percent occur weekday to weekday, with theremaining 20 percent occurring over a weekend (Figure 6-3).The ratio of average daily change occurring over a weekend toaverage daily change occurring between weekdays is measuredat 1.08. If Saturday and Sundays are truly equivalent to week-days —as far as opportunity for price variation to occur — then, inaccordance with the square-root time relationship linking inde-pendent price variations over different time periods, the ratio ofaverage daily changes (weekend versus weekday) ought to be V3or approximately 1.714. If Saturdays and Sundays are not equiv-alent to weekdays, this ratio ought to be 1. The observed ratio of1.08 implies a time multiplier of 1.17, or less than an hour'sworth of open market trading.98 OPTION REALITY
AVERAGE DAILY PRICE CHANGE (%)
Weekdays Over weekend RatioFINANCIALSS&P 500 0.622 0.672 1.080T-Bonds 0.485 0.394 0.812Swiss frane 0.409 0.395 0.966Yen 0.367 0.381 1.038RESOURCEGold 0.262 0.338 1.290Silver 0.805 0.834 1.036Crude oll 1.434 1.615 1.126Cotton 0.919 0.738 0.803FOODSoybeans 0.922 1.103 1.196Wheat 1.345 1.307 0.972Com 1.099 1.381 1.257Cattle 0.724 0.767 1.059Cocoa 0.903 0.917 1.016Coffee 1.476 2.140 1.450Sugar 0.932 1.085 1,164
Average —P 1.084
FIGURE 6-3. The question of whether Saturdays and Sundays should be accordedequal weightings with weekdays can be answered by checking if the average pricechange over weekends is substantially higher from the average price change betweenregular weekdays.
A ratio of 1.714 (the square root of 3) would indicate that equal weighting begiven to all days. A ratio of 1 would indicate that weekends should be ignored in theoptions formula. The observed average ratio from 3781 observations was 1.084.
In all the tables and calculations that follow, the time tillexpiry is taken as the number of trading days. By way of compro-mise, I do count a 3- or 4-day weekend or a midweek holiday asone additional trading day.
*
Using the techniques described in this chapter, it is now pos-sible to determine, on a daily basis, the true value