Under conditions of constantvolatility then, the fair value of an option can be expected todecline according to the geometry of Figure 5-2. In fact, with nochange in volatility, the percentage amount that an at-the-moneyoption can be expected to lose in value over the course of anyperiod of time is given very simply as follows:
Expected percentage lossdue to time decay alone
= 100x (1 — Vt,/t,)THE EMPEROR OF CHINA'S NOSE 69
(at-the-moneyoption price)
 
time til expiry t
FIGURE 5-2. Assuming the volatility of a future remains constant, the price of theat-the-money option will diminish at an accelerating rate (moving from right to leftalong the curve) as the time to expiry approaches zero, according to the formula,
v xvi40
 
ATMO, =
Since different futures on the same commodity have, in general, different volatilities,each at-the-money option will follow its own particular decay curve.
where ¢, is the number of days till expiry at the start of the periodand t, is the number of days at the end of the period. For exam-ple, between the tenth and ninth trading days till expiry,
Expected percentage
loss due to time = 100 x [I — V(9/10)] = 5.13 percent
decay alone
and between the tenth and fifth trading days till expiry, that is,during the second last week,
Ex ected percentage
oss due tg time _ yy, [1 — V(5/10)] = 29.3 percentdecay alone70 OPTION THEORY
It might be expected that the square-root time relationshipwould prove to hold true when comparing options on differentfutures of the same commodity. But, in general, this is not so. Forexample, on May 17, the implied volatility of the September1996 at-the-money coffee option (54 trading days till expiry) was54 percent, whereas the implied volatility of the July at-the-money coffee option (15 trading days till expiry) was only 44 per-cent. (See Example 4-3 and Example 4-4 in Chapter 4.) Why thediscrepancy? Why shouldn't the September and July at-the-mon-eys have the same implied volatilities and be priced in the ratioof ¥(54/15) according to the options formula’? It's the same cof-fee, after all!
Well, it is and it isn't. July coffee is old crop, and July optionsexpire in early June, before there is any frost danger to theBrazilian harvest—the world's largest. September coffee is newcrop, and its options expire in early August, well through freeze-scare season. The market understands that there is greaterpotential for price volatility during-the term of the Septemberoptions than during the term of the July options and will there-fore assign a higher relative price--or greater implied volatility—to the September options.
Frequently, it is the nearby option which exhibits the highestimplied volatility. This is especially true in commodities wheresupply can be rapidly expanded or rapidly curtailed in responseto price change. Crude oil is perhaps the best example. When oildemand exceeds supply, the nearby futures quickly go to a pre-mium over the deferreds, and when demand falls short of supplythe deferreds go to carrying charges over the nearbys.Consequently, price swings in the nearby crude oil future willalways be larger than price swings in the deferred contracts. Thischaracteristic of the crude oil futures market is reflected in theimplied volatilities of its different option maturities. A similarconfiguration prevails in the grain market; the September soy-bean option regularly exhibits greater implied volatility thanNovember soybean option. Both these options have the uncer-tain summer weather to contend with, but November encom-passes a postharvest period where the uncertainty level drops,and a lowered overall uncertainty level results in a loweredoption implied volatility.THE EMPEROR OF CHINA'S NOSE 71
Many commodities such as gold, silver, and stock indexfutures trade at carrying charges which only change when inter-est rates change, and in those markets you will find impliedvolatility to be relatively constant across different futures matu-rities. Sometimes an event with a massively uncertain outcome,but an outcome with large price implications, will distort the rel-ative values of different option maturities on the same commod-ity. This can be a periodic event such as the "Hogs and PigsReport" released quarterly by the U.S. Department ofAgriculture, or a once-in-a-generation event (like the referendumon Quebec separation from Canada which created massivevolatility in the price of the Canadian dollar).
A big surprise in a pig report can cause a sudden very largechange in the price of hog and pork belly futures. The uncer-tainty preceding this report holds hog option prices way abovewhat would be suggested by monitoring market volatility in hogfutures. The large option premiums are a reflection of the col-lective understanding that just prior to the release of the reportthe market is not looking at a normal probability distribution ofpossible prices at all. If anything, the market is preoccupied withthe likelihood of a sudden big price shift, either up or down.After a report of this type is released and the uncertainty isresolved, option prices will immediately shrink, though thedegree of shrinkage will depend on the time to expiry of theoption. As soon as uncertainty is removed from a futures market,its option prices almost always decline, regardless ¢ the magni-tude d the impact the removal d the uncertainty may have on thefutures price. The time horizon should always be examined forthe possibility of a major upcoming "event" whenever impliedvolatilities do not seem to line up in accordance with historicalpatterns.
Strictly speaking, then, an implied volatility is specific to oneparticular futures maturity. In practice you are not likely toencounter such a refinement in its definition. In the absence ofinformation to the contrary, it is probably safe to assume that astated implied volatility has been computed from option data per-tinent to the the nearest future. It is indeed something of an odd-ity that where the implied volatility ought to be constant72 OPTION THEORY
(different strikes on the same future), it is considered to be vari-able, whereas across different futures where implied volatilityought to be variable, it is usually (by omission) thought to be con-stant. So it goes.
*
Implied volatility is the quantity obtained when an optionprice is known, the time to expiry is known, and the option equa-tion is solved for v, the unknown. Implied volatility is simply away of expressing an option price so that it may be assessed inrelation to market volatility — whatever that may be defined to be.Like implied volatility, market volatility is always expressed inannualized form, even though the data from which it is derivedare measured on a daily basis. As discussed in Chapter 3, thestandard deviation of I-day price changes can be converted to astandard deviation on any time base by multiplying the standarddeviation of daily price changes by the square root of the newtime base, expressed in trading days. By convention I year istaken as the time base for specifying volatility.
If a full year's readings of actual daily price change (about254 for the typical commodity) are assembled into a frequencydistribution and the standard deviation of that distribution cal-culated, the resulting number is still a standard deviation of dailyprice changes. To convert this “daily” number to a reading ofvolatility it must be multiplied by 1254, and the resulting prod-uct will be the average volatility observed over a I-year period.
It is obvious that market conditions vary widely over a periodas long as 1 year. Over the course of a year, all futures marketsgo through quiescent periods (where small daily price ranges arethe norm) as well as through active periods (where large dailyprice ranges are the norm). These very different types of marketsseem to come and go in more or less random fashion. Option val-ues drop in unison during quiescent periods and rise in unison inactive periods, but gradually rather than suddenly. The optionpricing structure can change suddenly, but for other reasons.
The reason that option prices change gradually with time isstraightforward enough; option traders are always wondering ifTHE EMPEROR OF CHINA'S NOSE 73
an apparent change in the trading pattern of the future will besustained, or if the apparent change is a temporary conditionwhich will quickly revert to some longer-term norm. Because thisquestion can be answered only after the fact, there will always besome option traders who vote in favor of a sustained change, andother option traders who vote in favor of regression towards themean. Forecasting the market volatility of a commodity futurefrom its recent or historic volatility is very much like forecastingthe weather a few days in advance, say, without the benefit of anymeteorological information.
Suppose you are in New York City in mid-July, in the middleof a heat wave, and that you are still able to breathe and think.Imagine yourself isolated in an apartment, with no access to anynews whatsoever. The only information reaching you comes froma giant temperature indicator you can see out of the window, anindicator that has registered over 98° at noon every day for thepast week. You know from experience that temperatures are wayabove normal (about 86°) and will eventually come down. Butyou are also aware that the heat wave has already lasted for aweek and may well last for another week. Someone holds a gunto your head and asks you for your best estimate for the noontime temperature three days hence. Chances are you will opt fora temperature around 90°. And this will be a good estimate, forit makes maximum use of the information at your disposal —inthis case an observed current high temperature, and prior expe-rience of two opposing forces; the force of regression to the meanopposing the inertia of an established trend.
In the options market, exactly the same intuitions are at work,but the intuitions of thousands and thousands of individuals,each contributing a little bit of his own particular experience ofhow the future is linked to the past. Intuition is not instinct; ithas to be learned. And some traders learn a great deal more fromtheir experiences than others.
*
Market volatility exists in the eye of the beholder, and thereare as many estimates of what market volatility really is as there74 OPTION THEORY
are option traders playing the market. The trader best able to pro-ject upcoming market volatility from historical precedents—overthe long haul—is the trader who will have the greatest tradingedge in the market. The same historical data are available to alltraders, but option data are notoriously hard to analyze becauseof the way prices are reported and records are kept. (Chapter 6is devoted entirely to structuring historical option data in such away that option prices can be related to futures volatility in a sta-tistically meaningful way.)
Current market volatility may be estimated from historicalvolatility in two very different ways, each with its own set ofadvantages and disadvantages. First, let's be clear on the waysvolatility is defined:
Implied volatility The consensus of opinion on what theupcoming volatility of a future is going to be, as expressedthrough actual option prices.
Market volatility. What the volatility of a future has been inthe recent or distant past, as expressed through a statisticalanalysis of actual futures price changes.
Market volatility may be subdivided into short term—measured from the most recent price behavior of the future —and long term— measured over a period as long as | year, say. Theprincipal argument in favor of using short-term market volatilityas the primary predictor of upcoming volatility is that commod-ity price profiles do change rather dramatically from month tomonth, or even week to week, and that to ignore this demonstra-ble fact is to ignore obviously useful information. It makes sense,the argument goes, to increase one's best estimate of marketvolatility in a commodity future as soon as the daily price swingsin that future start to increase, and conversely, to decrease one'sbest estimate of market volatility as soon as daily price swingsbegin to decrease. There is, however, another argument whichfavors the longer-term view.
The argument in favor of using long-term volatility as a pre-dictor rests on the observation that commodity prices alwaysTHE EMPEROR OF CHINA'S NOSE 75
regress to typical behavior patterns, and that periods of unusuallyhigh or low activity in a futures market should therefore beviewed as temporary aberrations which ought to exert minimalinfluence on estimates of upcoming market volatility. A traderworking on a long-term volatility model would be very slow toadjust an estimate of volatility in response to changing conditionsin a futures market. Proponents of the long-term viewpoint canalso argue that they are working with statistically more signifi-cant data, in that a long-term data set will contain up to a year'sworth of readings— about 250— whereas a short-term data setmay contain only 20 or so.
The short-term and long-term approaches to estimating mar-ket volatility can lead to conflicting conclusions on option valua-tior. For example, consider a futures market that hasexperienced a number of wild trading sessions and has now set-tled back into a trading pattern characterized by rather smalldaily price changes (Figure 5-3). The Japanese yen is a marketwith a tendency to generate such price patterns. A trader work-ing with a short-term market volatility estimator will be focusedon recent futures price data (perhaps the previous month's) andwill be adjusting the volatility estimate downward, rather quickly,as the futures market quiets down. Actual option prices (impliedvolatilities) will be coming down more slowly and when com-pared with short-term volatility, may appear overvalued andtherefore candidates for writing. A trader working with a long-term estimator of market volatility will be adjusting the estimateof market volatility very slowly, so that the implied volatility of theoptions may drop below the estimated market volatility. Underthis scenario, the options may appear undervalued and candi-dates for buying.
One set of assumptions says buy, the other says sell. Which iscorrect? No one can say. The question of whether a short-termvolatility estimator is superior to a long-term volatility estimatorcannot be answered before-the-fact in any specific case. Can thequestion be answered in general terms? Possibly, but onlythrough observation and analysis of a great deal of historicaldata. Before we get into drawing statistical inferences fromempirical data, it will be helpful to look at ways in which market76 OPTION THEORY
           
mame implied volatilityvases market volatility (long-term)meme market volatility (short-term)
H,
 
Futures price
tlt
HHt np
Volatility
Time
FIGURE 5-3, Estimating volatility from historical data is highly subjective and canproduce very different interpretations of whether an option is overvalued or underval-ued. In the example above, a futures market (daily high, low, close) moves ratherquickly from a period characterized by large daily price swings to a period of muchsmaller daily price movement.
The short-term volatilityestimator adjusts rapidly to changes in the daily price pat-terns of the future, while the long-term estimator hardly varies at all. If these marketvolatility estimators are compared with the implied volatility of the at-the-moneyoption on the future, one estimator will be higher than the implied volatility while theother estimator will be lower; the option at time t appears overvalued by the short-termestimator, but undervalued by the long-term estimator.
volatility calculations are handled numerically, and in particular atsome peculiar suggestions offered by certain people who appearto trade imaginary options from the confines of ivory towers.
*THE EMPEROR OF CHINA'S NOSE 77
Any statistically grounded attempt to forecast market volatil-ity of a commodity future will involve calculating the standarddeviation (or mean absolute deviation) of daily price changes,with the period chosen entirely at the discretion of the trader. A6-week, continuously updated calculation of standard deviationwould generally be considered a short-term volatility estimator.In Figure 5-4, the standard deviation of daily price changes forcocoa has been calculated using data from a 30-trading-day timeinterval. The procedure is straightforward enough. At the close ofday 2, for example, you determine the price change from day lL,divide this number by the average of day 1 and day 2 futurescloses, then multiply by 100 for a daily price change expressed asa percentage of its futures value. After calculating the standarddeviation of these thirty observations and multiplying by ¥254,you atrive at a estimate of market volatility—in this case 33.3percent. Were a time interval of 15 days or 60 days to be usedinstead of 30, the estimate of market volatility would, of course,be different.
In the calculations of Figure 5-4, equal weighting is given toeach observation; that is, the price change 30 days back has thesame degree of influence on the calculated volatility as the mostrecent price change. A good case can be made for assigninggreater importance to recent observations, and this refinementcan be easily incorporated into the basic calculation. Let's stickwith the unweighted calculation for present, for there are somecomplicating suggestions from other writers regarding the calcu-lation of market volatility that demand examination.
At some point in the development of option theory, the ideatook hold that simple daily price changes could not be useddirectly to estimate daily volatility. This incorrect notion arosefrom the correct observation that while a price can never gobelow zero, it may double, triple, or go to any multiple on theupside. This latter observation is equivalent to postulating thatthe distribution of an absolute futures prices over the fullness oftime is a lognormal distribution rather than a normal distribu-tion. I have no argument with that.
However, the correct observation that absolute prices arenot normally distributed provides no insight into the expected78 OPTION THEORY
 
Price Change Too (AP) PLDay P, {aP)) (Pay) (Pav) "P1100 log, (Ri)1 13642 1347 -17.0 1356 -1,.254 0.9875 1.2543 1335 -12.0 1341 -0.895 0.9911 -0.8954 1321 -14.0 1328 1,054 0.9895 -1.0545 1346 25.0 1334 1.875 1.0189 1.8756 1370 24.0 1358 1.767 1.0178 1.7677 1411 41.0 1390 2.949 1.0299 2.9498 1372 -39.0 1392 2.803 0.9724 -2.8039 1386 14.0 1379 1.015 1.0102 1.01510 1427 41.0 1406 2.915 1.0296 2.915"1 1460 33.0 1444 2.286 1.0231 2.28612 1441 -19.0 1450 1.310 0.9870 -1.31013 1447 6.0 1444 0.416 1.0042 0.41614 1426 -21.0 1436 +1.462 0.9855 -1.46215 1482 56.0 1454 3.851 1.0393 3.85216 1513 31.0 1498 2.070 1.0209 2.07017 1490 -23.0 1602 -1,532 0.9848 -1.53218 1535 45.0 1512 2.975 1.0302 2.97519 1497 -38.0 1816 -2.507 0.9752 -2.50720 1522 25.0 1510 1.656 1.0167 1.65621 1575 53.0 1548 3.423 1.0348 3.42322 1540 -35.0 1558 -2.247 0.9778 -2.24723 1611 -29.0 1526 -1,901 0.9812 -1.90124 1495 -16.0 1503 -1.065 0.9894 -1.06525 1470 -25.0 1482 -1.686 0.9833 1.68626 1488 18.0 1479 1.217 1.0122 1.21727 1462 -26.0 1475 -1.763 0.9825 -1.76328 1447 -15.0 1454 +1.031 0.9897 1.03129 1471 24.0 1459 1.645 1.0166 1.64530 1432 -39.0 1452 -2.687 0.9735 -2.687at 1460 28.0 1446 1.936 1.0196 1.936
Standard deviation = 2.09 percentVolatility (SD x V254) = 33.31 percent
FIGURE 5-4, Market volatility projected for cocoa using the standard deviation cal-culation for daily price changes on a thirty-trading-day time base. Daily price changesare first divided by the average of the 'surrounding' daily closing prices (P,,, above) andthen multiplied by 100 to express them as percentages. To determine market volatility—by convention annualized—it is necessary to multiply the standard deviation of dailyprice changes by 1254.
It has become common practice in options literature to calculate market volatilityfrom "logarithmic returns." Using this method, each absolute price is divided by thepreceding absolute price, and the standard deviation of the logarithm of these ratios iscalculated. As is evident from the final two columns above, the logarithmic ratios areidentical to the price changes expressed as percentages, which means the logarithmicvolatility calculation will yield the same result as the simple price change volatility cal-culation. The logarithmic complication hardly seems worth the bother.THE EMPEROR OF CHINA'S NOSE 79
distribution of daily price changes, particularly when the latterare expressed as percentages of absolute values. Expressed as apercentage, a daily price change has a built-in compensator forradical shifts in the absolute price level. Furthermore, frompurely practical considerations, a futures price is very unlikely toapproach zero or double during the relatively short life span of anoption.
Nevertheless, the fashion is to calculate market volatility vialogarithms. (See the final two columns of Figure 5-4 for a com-parison with the basic calculation.) Computationally, the loga-rithmic method goes something like this: You take the price onday 2, divide by the price on day 1, and call this a "return." Youthen calculate the natural logarithm of this "return" and finallycompute the standard deviation of these logarithmic returns. Andyou wind up with exactly the same answer as the nonlogarithmiccalculation, but by a considerably more devious route.
*
The classical standard deviation formula used in volatility cal-culations involves summing a series of squared terms, each ofthese terms being defined as the difference between an observedprice change and the average of all the observed price changes:
(AP, - AP,,,)2
(Standard deviation)2 = (N_1)
where AP, = daily price change on the i th dayAP,, = average of all observations of AP;N = number of observations
In a trading market, or a market which ends up virtuallyunchanged in 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 AP,, will notbe close to zero, since values of AP, will be either mostly positive78 Option THEORY
 
Price Change 1o@P) ,_ PL
Dy PAP) Pa) Pa) Ping 10010042)1 1364
2 1347 --17.0 1356 =1.254 0.9875 = -1.2543 1335 --12.0 1341 -0.895 0.9911 -0.8954 1321 -14.0 1328 “1.054 0.9895 —-1.0545 1346 25.0 1334 1.875 1.0189 1.8756 1370 24.0 1358 1.767 1.0178 1.7677 1411 41.0 1390 2.949 1.0299 2,9498 1372 -39.0 1392 -2.803 0.9724 —--2.8039 1386 «14.0 1379 1.015 1.0102 1.01510 1427. 41.0 1406 2.915 1.0296 = 2.9151" 1460 33.0 1444 2.286 1.0231 2.28612 1441 -19.0 1450 -1.310 0.9870 —_-1.31013 1447 6.0 1444 0.416 1.0042 0.41614 1426 -21.0 1436 -1.462 0.9855 —--1.46215 1482 56.0 1454 3.851 1.0393 3.85216 1613 31.0 1498 2.070 1.0209 2.07017 1490 _-23.0 1502 +1.532 0.9848 = -1.53218 1535 45.0 1512 2.975 1.0302 -2.97519 1497 _-38.0 1516 -2.507 0.9752 —--2.50720 1522 25.0 1510 1.656 1.0167 1.65621 1575 53.0 1548 3.423 1.0348 3.42322 1540 -35.0 1558 -2.247 0.9778 -2.24723 1511 -29.0 1526 -1.901 0.9812 —--1.90124 1495 -16.0 1503 -1.065 0.9894 —-1.06525 1470 25.0 1482 -1.686 0.9833 = -1.68626 1488 18.0 1479 1.217 1.0122 1.21727 1462 -26.0 1475 “1.763 0.9825 = -1.76328 1447 -15.0 1454 -1.031 0.9897 -1.03129 1471 24.0 1459 1.645 1.0166 1.64530 1432 -39.0 1452 -2.687 0.9735 —-2.68731 1460 28.0 1446 1.936 1.0196 1.936
Standard deviation = 2.09 percentVolatility (SD x ¥254) = 33.31 percent
FIGURE 5-4, | Market volatility projected for cocoa using the standard deviation cal-culation for daily price changes on a thirty-trading-day time base. Daily price changesare first divided by the average of the 'surrounding’ daily closing prices (P,, above) andthen multiplied by 100 to express them as percentages. To determine market volatility—by convention annualized— it is necessary to multiply the standard deviation of dailyprice changes by 254.
It has become common practice in options literature to calculate market volatilityfrom "logarithmic returns." Using this method, each absolute price is divided by thepreceding absolute price, and the standard deviation of the logarithm of these ratios iscalculated. As is evident from the final two columns above, the logarithmic ratios areidentical to the price changes expressed as percentages, which means the logarithmicvolatility calculation will yield the same result as the simple price change volatility cal-culation. The logarithmic complication hardly seems worth the bother.THE EMPEROR OF CHINA'S NOSE 79
distribution of daily price changes, particularly when the latterare expressed as percentages of absolute values. Expressed as apercentage, a daily price change has a built-in compensator forradical shifts in the absolute price level. Furthermore, frompurely practical considerations, a futures price is very unlikely toapproach zero or double during the relatively short life span of anoption.
Nevertheless, the fashion is to calculate market volatility vialogarithms. (See the final two columns of Figure 5-4 for a com-parison with the basic calculation.) Computationally, the loga-rithmic method goes something like this: You take the price onday 2, divide by the price on day 1, and call this a "return." Youthen calculate the natural logarithm of this "return" and finallycompute the standard deviation of these logarithmic returns. Andyou wind up with exactly the same answer as the nonlogarithmiccalculation, but by a considerably more devious route.
*
The classical standard deviation formula used in volatility cal-culations involves summing a series of squared terms, each ofthese terms being defined as the difference between an observedprice change and the average of all the observed price changes:
AP,- A 2(Standard deviation)? = ea
where AP, = daily price change on the i th dayAP,, = average of all observations of A P;N = number of observations
In a trading market, or a market which ends up virtuallyunchanged in