Model, the 110 call will always have agreater value than the 90 put. In absolute terms, the lognormal distribution assumptionallows for greater upside price movement than downside price movement. Conse-quently, the 110 call will have a greater possibility of price appreciation than the 90put.
Finally, the lognormal assumption built into the Black-Scholes Model overcomesthe logical problem we initially posed. If we were to allow for the possibility of unlimitedupside price movement of an underlying instrument, a normal distribution assumptionwould force us to allow for unlimited downside movement. This would require us toaccept the possibility of negative prices for the underlying instrument, clearly not apossibility for most optionable instruments. A lognormal distribution, however, doesallow for open ended upside prices (the logarithm of +00 is +00], while boundingdownside prices by zero (the logarithm of —<o is zero). This is a more realistic repre-sentation of how prices are actually distributed in the real world.
A more complete discussion of logarithmic price changes and probability calcula-tions can be found in Appendix B.
We can now summarize the most important assumptions governing price move-ment in the Black-Scholes Model:
1. Changes in the price of an underlying instrument are random and cannot beartificially manipulated, nor is it possible to predict beforehand the directionin which prices will move.
2. The percent changes in the price of an underlying instrument are normallydistributed.
3. Because the percent changes in the price of the underlying instrument areassumed to be continuously compounded, the prices of the underlying instru-ment at expiration will be lognormally distributed.
4, The mean of the lognormal distribution will be located at the forward price ofthe underlying contract.
The first of these assumptions may meet with resistance from some traders.Technical analysts believe that by looking at past price activity it is possible to predictthe future direction of prices. One can chart support and resistance points, double topsand bottoms, head and shoulders, and many similar formations which are believed topredict future price trends. We leave debate on this question to others. The importantpoint here is that the Black-Scholes Model makes the assumption that price changes arerandom and that their direction cannot be predicted. This does not mean that there isno predictive requirement in using the Black-Scholes Model. However, price prediction
 
1. Ofcourse, this is only theory. There is no law that says the price of the 90 put in the marketplace cannotbe greater than the price of the 110 call.Volatility 65
will focus on the magnitude of the price changes, rather than on the direction ofchanges.
As we shall see later, there is also good reason to question the third assumption,that prices are lognormally distributed at expiration. This may be a reasonable assump-tion for some markets, but a very poor assumption for other markets. Again, theimportant point here is for the trader who uses a theoretical pricing model to understandthe assumptions on which the theoretical values are based. He can then make his owndecision, based on his knowledge of a particular market, as to whether these assump-tions, and hence the theoretical values generated by the model, are likely to be accurate.
DAILY AND WEEKLY STANDARD DEVIATIONS
As an annual standard deviation, we know what the volatility tells us about the likelyprice movement of a contract over a one-year period. However, this is a period of timelonger than the life of most listed options. We might want to know what a volatility tellsus about price changes over a shorter period of time, for example over a month, or aweek, ora day.
An important characteristic of volatility is that it is proportional to the square rootof time. As a result of this, we can approximate a volatility over some period of timeshorter than a year by dividing the annual volatility by the square root of the number oftrading periods in a year.
Suppose we are interested in a daily volatility. While it would take a logarithmiccalculation to give us an exact daily volatility, if we ignore the relatively minor effect ofcontinuous compounding over such a short period of time, it is possible to make anestimate of daily volatility. First we must determine the number of daily trading periodsin a year. That is, if we look at prices at the end of every day, how many times a yearcan prices change? If we restrict ourselves to exchange traded options, even thoughthere are 365 days in a year, prices cannot really change on weekends or holidays, Thisleaves us with about 256 trading days during the year. Since the square root of 256 is16, to approximate a daily volatility we can divide the annual volatility by 16.
Going back to our futures contract trading at 100 with a volatility of 20%, what isa one standard deviation price change over a day’s time? 20%/16 = 114%, so a oneStandard deviation daily price change is 114% x 100 = 1.25. We expect to see a pricechange of 1.25 or less approximately two trading days out of every three, and a pricechange of 2.50 or less approximately 19 trading days out of every 20. Only one day in20 would we expect to see a price change of more than 2.50.
We can do the same type of calculation for a weekly standard deviation. Now weMust ask how many times per year prices can change if we look at prices once a week.Unlike trading days, we don’t have “holiday” weeks, so we must make our calculationsusing all 52 trading weeks in a year. Dividing our annual volatility of 20% by the squareroot of 52, or approximately 7.2, we get 20%/7.2 = 234. For our futures contract
——
2. Depending on holidays, the number of trading days Is usually somewhere between 250 and 255. Weuse 256 as a reasonable approximation since its square root is a whole number and therefore easier towork with.66 Chapter 4
trading at 100, we would expect to see a price change of 2.75 or less two weeks out ofevery three, a price change of 5.50 or less 19 weeks out of every 20, and only one weekin twenty would we expect to see a price change of more than 5.50.
Since we assume that the price of a stock will appreciate by the carrying cost, itmay seem that we cannot use the same method (divide by 16 for daily volatility; divideby 7.2 for weekly volatility] to approximate expected movement in an underlying stock.However, over a short period of time, the carrying cost component, like the effect ofcontinuous compounding of volatility, will be relatively small. Therefore, we can usethe same method as a reasonable estimate of daily and weekly volatility. For example,suppose a stock is trading at $45 per share and has an annual volatility of 28%. What isan approximate one standard deviation price change over a day’s time and over a week’stime?
For a daily volatility we calculate:
28%/16 x $45 = 1.75% x $45 = $.79For a weekly volatility we calculate:28%/7.2 x $45 = 3.89% x $45 = $1.75
We expect to see a price change of approximately 34 point or less two days out ofevery three, 114 points or less 19 days out of every 20, and only one day in 20 wouldwe expect to see a price change of more than 114 points. On a weekly basis, we wouldexpect to see a price change of 154 points or less two weeks out of every three, a pricechange of 3! points or less 19 weeks out of every 20, and only one week in 20 wouldwe expect to see a price change of more than 314 points.
We have used the phrase “price change” in conjunction with our volatilityestimates. Exactly what do we mean by this? Do we mean the high/low during someperiod? Do we mean open to close price changes? Or is there another interpretation?While various methods have been suggested to estimate volatility,> the traditionalmethod has been to calculate volatility based on settlement-to-settlement price changes.Using this approach, when we say a one standard deviation daily price change is 7%point, we mean a 34 point price change from one day’s settlement price to the nextday’s settlement price. The high/low or open/close price change may have been eithermore or less than 44 point, but it is the settlement-to-settlement price change on whichwe focus.
3. See:Parkinson, Michael, “The Extreme Value Method of Estimating the Variance of the Rate of Return,”Journal of Business, 1980, vol. 53, no. 1, pp. 61-64.Garman, Mark B. and Klass, Michael J., “On the Estimation of Security Price Volatilities from HistoricalData,” Journal of Business, 1980, vol. 53, no. 1, pp. 67-78.Beckers, Stan, “Variance of Security Price Returns Based on High, Low, and Closing Prices,” Journal ofBusiness, 1983, vol. 56, no. 1, pp. 97~112.Volatility 67
VOLATILITY AND OBSERVED PRICE CHANGES
Why is it important for a trader to be able to estimate daily or weekly price changes froman annual volatility? Volatility is the one input into a theoretical pricing model whichcannot be directly observed. Yet many option strategies, if they are to be successful,require an accurate assessment of volatility. Therefore, an option trader needs somemethod of determining whether his expectations about volatility are indeed beingrealized in the marketplace. Unlike directional strategies, whose success or failure canpe immediately observed from posted prices, there is no posting of volatilities. A tradermust determine for himself whether he is using a reasonable volatility input into thetheoretical pricing model.
For example, suppose a certain underlying contract is trading at 40 and a trader isusing a 30% volatility for theoretical evaluation. A one standard deviation daily pricechange is approximately 30%/16 x 40 = .75. Over five days of trading a trader notesthe following five settlement-to-settlement price changes: ,
+.43, —06, -.61, +.50, -.28
Are these five price changes consistent with a 30% volatility?
The trader expects to see a price change of more than .75 (one standard deviation}about one day in three, or between one and two times over a five-day period. Yet duringthis five-day period he did not see a price change of this magnitude even once. Whatconclusions can be drawn from this?4 One thing is certain: these five price changes arenot consistent with 30% volatility. The trader might explain the discrepancy in one oftwo ways. On the one hand, perhaps this was expected to be an abnormally quiet week(perhaps it was a holiday week); and next week when trading returns to normal themarket will go right back to making moves which are more consistent with a 30%Volatility. If the trader comes to this conclusion, perhaps he ought to continue to use a30% volatility for his calculations. On the other hand, perhaps there is no apparentreason for the market being less volatile than predicted by a 30% volatility. He maysimply be using the wrong volatility. If the trader comes to this conclusion, perhaps heought to consider using a new volatility input which is more consistent with theobserved price changes. If he continues to use a 30% volatility in the face of pricechanges which are significantly less than predicted by that number, he will be assigningthe wrong probabilities to the possible price outcomes for the underlying contract.Consequently, he will generate incorrect theoretical values, defeating the purpose ofusing a theoretical pricing model in the first place.
Exactly what volatility is associated with the five price changes in the foregoingexample? Without some rather involved calculations it is difficult to say. (The answeris actually 18.8%.)] However, ifa trader has some idea beforehand of what price changeshe expects, he can easily see that the changes over the five-day period are not consistentwith a 30% volatility.
 
4. — Five days is admittedly a very small sample from which to draw a meaningful conclusion aboutvolatility. The method and reasoning, however, are still valld.68 Chapter 4
Let’s look at another example. Now the underlying contract is trading at 33214and a trader notes the following five daily price changes:
-5, +24, +1, -7%, -44
Are these price changes consistent with an 18% volatility? At 18% a one standarddeviation price change is approximately 18%/16 x 33214 = 354. Over five days weexpect to see a price change of more than 334 between one and two times. Yet here wehave a price change of more than 3% three days out of five. And once the price changewas 774 (more than two standard deviations) which we expect to see only one day intwenty. Again, unless the trader believes that the five price changes occurred during anextraordinary week, then perhaps he ought to consider changing his volatility figure sothat it is more consistent with the observed price changes.
A NOTE ON INTEREST RATE PRODUCTS
Suppose Eurodollars are at 93.00 and we assume a volatility of 16 percent. We can applythe method previously described to calculate an approximate one standard deviationdaily price change: 16%/16 x 93.00 = .93. As any trader familiar with the Eurodollarmarket will attest, a daily price change of .93 is wildly unlikely. How can we accountfor this seemingly illogical answer? One might conclude that we simply have the wrongvolatility, and some lower number is more accurate. In fact, the 16% volatility is not atall unusual for Eurodollars, so the explanation must lie elsewhere.
Eurodollars, like many other interest rate contracts (U.S. Treasury Bills, ShortSterling, Euromarks, Euroyen) are indexed from 100. This means that the interest rateassociated with a Eurodollar contract is 100 less the value of the contract. It also meansthat, barring the unlikely advent of negative interest rates, the contract cannot take ona value greater than 100. In this respect, 100 acts as a limiting value for Eurodollars inthe same way that zero acts as a limiting value for traditional underlying contracts suchas stocks and commodities. We can integrate this characteristic into our calculations byassuming that the value of a Eurodollar contract is actually 100 less its listed price. Ifthe listed price is 93.00, for theoretical evaluation purposes we must use a value of 100— 93.00, or 7.00, in our pricing model. If we define the value of the contract as 7.00, aone standard deviation price change is 16%/16 x 7.00 = .07. This is certainly a morerealistic result than .93.
To be consistent, if we index Eurodollar prices from 100 we must also indexexercise prices from 100. Therefore, a 93.50 exercise price in our pricing model is reallya 6.50 (100 ~ 93.50) exercise price. This also requires us to reverse the type of option,changing calls to puts and puts to calls. To see why, consider a 93.50 call. For this callto go into-the-money, the underlying contract must rise above 93.50. But this requiresthat interest rates fall below 6.50 percent. Therefore, a 93.50 call in listed terms is thesame as a 6.50 put in interest rate terms. A model which is correctly set up to evaluateoptions on Eurodollar or other types of indexed interest rate contracts automaticallymakes this transformation. The price of the underlying contract and the exercise priceVolatility 69
are subtracted from 100, while listed calls are treated as puts and listed puts are treatedas calls.
Note that this type of transformation is not required for most bonds and notes.Depending on the coupon rate, the prices of these products may range freely withoutupper limit, often exceeding 100, They are therefore most often evaluated using atraditional pricing model, although interest rate products present other problems thatmay require specialized pricing models.
It is possible to take an instrument such as a bond and calculate the current yieldbased on its price in the marketplace. If we were to take a series of prices and from thesecalculate a series of yields, we could also calculate the yield volatility, i.e., the volatilitybased on the change in yield. We might then use this number to evaluate the theoreticalvalue of an option on the bond, although to be consistent we would also have to specifythe exercise price in terms of yield. Because it is possibie to calculate the volatility of aninterest rate product using these two different methods, interest rate traders sometimesrefer to yield volatility (the volatility calculated from the current yield on the instrument)versus price volatility (the volatility calculated from the price of the instrument in themarketplace).
TYPES OF VOLATILITIES
When traders discuss volatility, even experienced traders may find that they are notalways talking about the same thing. When a trader makes the comment that thevolatility of XYZ is 25%, this statement may take on a variety of meanings. We can avoidconfusion in subsequent discussions if we first define the various ways in which tradersinterpret volatility.
Future Volatility
Future volatility is what every trader would like to know, the volatility that bestdescribes the future distribution of prices for an underlying contract. In theory it is thisnumber to which we are referring when we speak of the volatility input into atheoretical pricing model. If a trader knows the future volatility, he knows the right“odds.” When he feeds this number into a theoretical pricing model, he can generateaccurate theoretical values because he has the right probabilities. Like the casino, hemay occasionally lose because of short-term bad luck. Butin the long run, with the oddson his side, a trader can be fairly certain of making a profit.
Of course, traders rarely talk about the future volatility since it is impossible toknow what the future holds.
Historical Volatility
Even though one cannot know the future, if a trader intends to use a theoretical pricingModel he must try to make an intelligent guess about the future volatility. In optionevaluation, as in other disciplines, a good starting point is historical data. What typicallyhas been the volatility of this contract over some period in the past? If, over the past tenyears the volatility of a contract has never been less than 10% nor greater than 30%, aguess for the future volatility of either 5% or 40% hardly makes sense. This does not70 Chapter 4
mean that either of these extremes is impossible (in option trading the impossible alwaysseems to happen sooner or later), but based on past performance, and in the absence ofany extraordinary circumstances, a guess within the historical limits of 10% and 30% isprobably more realistic than a guess outside these limits. Of course, 10% to 30% is stilla huge range, but at least the historical data offers a starting point. Additional informa-tion may further narrow the estimate.
Note that there are a variety of ways to calculate historical volatility, but mostmethods depend on choosing two parameters, the historical period over which thevolatility is to be calculated, and the time interval between successive price changes.The historical period may be ten days, six months, five years, or any period the traderchooses. Longer periods tend to yield an average or characteristic volatility, whileshorter periods may reveal unusual extremes in volatility. To become fully familiar withthe volatility characteristics of a contract, a trader may have to examine a wide varietyOf historical time periods.
Next, the trader must decide what intervals to use between price changes. Shouldhe use daily price changes? Weekly changes? Monthly changes? Or perhaps he oughtto consider some unusual interval, perhaps every other day, or every week and a half.Surprisingly, the interval which is chosen does not seem to greatly affect the result.Although a contract may make large daily moves, yet finish a week unchanged, this isby far the exception. A contract which is volatile from day to day is likely to be equallyvolatile from week to week, or month to month. This is typified by the graphs in Figure4-8. The data points on the three graphs represent the volatility of the S&P 500 indexover successive 50-day periods. For the solid line daily price changes were used, for thedotted line price changes every two days were used, and for the broken line pricechanges every 5 days were used. Even though the graphs occasionally diverge, for themost part they exhibit the same general volatility levels and trends.
As a general rule, services which supply historical volatility data base their calcula-tions on daily settlement-to-settlement price changes. If this is not the case, an explana-tion of how the volatility was calculated will usually accompany the data. If, forexample, a service gave the volatility of a contract for the month of August as 21.6%, itcan be assumed that the calculations were made using the daily settlement-to-settlementprice changes for all the business days during that month.
Historical and future volatility are sometimes referred to as realized volatility.
Forecast Volatility
Just as there are services which will attempt to forecast future directional moves in theprice of a contract, there are also services which will attempt to forecast the futurevolatility of a contract. Forecasts may be for any period, but most commonly coverPeriods identical to the remaining life of options on the underlying contract. For anunderlying contract with three months between expirations, a service might forecastvolatilities for the next three, six, and nine months. For an underlying with monthlyexpirations, a service might forecast volatilities for the next one, two, and three months.Due to the relatively recent introduction of options, volatility forecasting is still in itsinfancy, and must be considered an inexact science at best. Nonetheless, a trader’s guess 
71
Volatility
 
cose ten ce ceee cone vane nee ne Aep uiS fueneAep pug AeneaRep Auane
:seBueyo sod uo paseq
    
(%) Aumelon pezienuuy
 
 
 
veAnejo, jeouoysipy Aeg-og xepul 00s d#S :s-p enbly72 Chapter 4
about the future volatility of a contract might very well take into consideration anyvolatility forecast to which he has access.Implied Volatility
Generally speaking, future, historical, and forecast volatility are associated with anunderlying contract. We can talk about the future volatility of the S&P 500 index, or thehistorical volatility of U.S. Treasury Bonds, or a forecast volatility for IBM stock. In eachcase we are referring to the volatility of the underlying contract. There is, however, adifferent interpretation of volatility which is associated with an option rather than withthe underlying contract.
Suppose a certain futures contract is trading at 98.50 with interest rates at 8%.Suppose also that a 105 call with three months to expiration is available on this contract,and that our best guess about the volatility over the next three months is 16%. If wewant to know the theoretical value of the 105 call we might feed all these inpwts intoa theoretical pricing model. Using the Black-Scholes Model, we find that the option hasa theoretical value of .96, Having done this we might compare the option’s theoreticalvalue to its price in the marketplace. To our surprise, we find that the option is tradingfor 1.34. How can we account for the fact that we think the option is worth .96, whilethe marketplace seems to believe it is worth 1.34?
One way to answer the question is to assume that everyone in the marketplace isusing the same theoretical pricing model that we are, in this case the Black-ScholesModel. If we make this assumption, then the discrepancy between our value of .96 andthe marketplace’s value of 1.34 must be due to a difference of opinion concerning oneor more of the inputs into the model. We can therefore start going down the list of inputsand try to identify the culprits.
We know that it can’t be either the amount of time to expiration or the exerciseprice, since these inputs are fixed in the option contract. What about the underlyingprice of 98.50? Perhaps we think the price of the underlying is at 98.50, but it is reallytrading at some higher price, say 99.00. Indeed, in sttch circumstances it is always wiseto double check the inputs. But suppose we still find that the underlying is at 98.50.Even given that there is a spread between the bid and ask price, if the market isreasonably liquid it is unlikely that the spread would be wide enough to cause adiscrepancy of .38 in the value of the option. Perhaps our problem is the interest rate of8%. But, as we noted in the last chapter, the interest rate component is usually the leastimportant of the inputs into a theoretical pricing model. And in the case of futuresoptions, the interest rate component is often trivial. This leaves us with only one likelyculprit, and that is volatility. In some sense, the marketplace must be using a volatilityother than 16% to evaluate the 105 call.
What volatility is the marketplace using? To find out, we can ask the followingquestion: If we hold all other inputs constant (time to expiration, exercise price,underlying price, interest rates}, what volatility must we feed into our theoretical pricingmodel to yield a theoretical value identical to the price of the option in the marketplace?In our example, we want to know what volatility will yield a value of 1.34 for the 105call. Clearly the volatility has to be higher than 16%, so we might sit down with aVolatility 73
computer programmed with the Black-Scholes Model and start to raise the volatility. Ifwe do, we will find that at a volatility of 18.5% the 105 call has a theoretical value of1.34, We refer to this volatility as the implied volatility of the 105 call. It is the volatilitywe must feed into our theoretical pricing model to yield a theoretical value identical tothe price of the option in the marketplace. We can also think of it as the volatility beingimplied to the underlying contract through the pricing of the option in the marketplace.
When we solve for the implied volatility of an option we are assuming that thetheoretical value (the option’s price) is known, but that the volatility is unknown. Ineffect, we are running the theoretical pricing model backwards to solve for thisunknown, as shown in Figure 4-9. In fact, this is easier said than done since mosttheoretical pricing models cannot be reversed. However, a number of computerprograms have been written which can quickly solve for the implied volatility when allother inputs are known.
Note that the implied volatility depends on the theoretical pricing model beingused. For some options, a different model can yield a significantly different impliedvolatility. The accuracy of an implied volatility also depends on the accuracy of inputsinto the model. This not only includes the price of the option, but the other inputs aswell. In particular, problems can occur when an option has not traded for some time,but market conditions have changed significantly. Suppose in our example the price of1.34 for the 105 call reflected the last trade, but that trade took place two hours agowhen the underlying futures contract was actually at 99.25. If the price of theunderlying contract is 99.25, the implied volatility of the option at a price of 1.34 isactually 17.3%. This is a significant difference, and underscores the importance ofaccurate and timely inputs when calculating implied volatilities.
Services which supply theoretical analysis of options usually include impliedvolatilities. The implied volatilities may be for each option on an underlying contract,or there may be one implied volatility which is representative of all options on the same
Figure 4-9Using the Model to Solve for a Theoretical ValueKnowns UnknownsExercise Price (105) Theoretical ValueTime to Expiration (3 months) (2? = .96)
Underlying Futures Price (98.50)Interest Rates (8%)Volatility (16%)
Using the Model to Solve for an Implied Volatility
Knowns UnknownsExercise Price (105) Implied VolatilityTime to Expiration (3 months) (2? = 18.5%)
Underlying Futures Price (98.50)Interest Rates (8%)Option Price (1.34)74 Chapter 4
underlying. In the latter case, the figure will usually represent an average of all theindividual implied volatilities. The single implied volatility is weighted according tosome criteria, such as volume of options traded, open interest, or, as is most common,by assigning the greatest weight to the at-the-money options.
The implied volatility in the marketplace is constantly changing because optionprices, as well as other market conditions, are constantly changing. It is as if themarketplace were continuously polling all participants to come up with a consensusvolatility for the underlying contract. This is not a poll in the true sense, since all tradersdo not huddle together and eventually vote on the correct volatility. However, as bidsand offers are made, the trade price of an option will represent the equilibrium betweensupply and demand. This equilibrium can be translated into an implied volatility.
Even though the term premium really refers to an option’s price, it is commonamong traders to refer to the implied volatility as the premfum or premium level If thecurrent implied volatility is high by historical standards, or high relative to the recenthistorical volatility of the underlying contract, a trader might say that premium levelsare high; if implied volatility is unusually low, he might say that premium levels are low.
Assuming a trader had a reliable theoretical pricing model, if he could look into acrystal ball and determine the future volatility of an underlying contract he would beable to accurately evaluate options on that contract. He might then look at the differencebetween each option’s theoretical value and its price in the marketplace, selling anyoptions which were overpriced relative to the theoretical value, and buying any optionswhich were underpriced. If given the choice between selling one of two overpricedoptions, he might simply seli the one which was most overpriced in total dollars.However, a trader who has access to implied volatilities might use a different yardstickfor comparison. He might compare the implied volatility of an option to either a volatilityforecast, or to the implied volatility of other options on the same underlying contract.Going back to our example of the 105 call, we might say that with a theoretical valueof .96 and a price of 1.34, the 105 call is .38 overpriced. But in volatility terms it is 2.5%overpriced since its theoretical value is based on a volatility of 16% (the trader’s volatilityestimate) while its price is based on a volatility of 18.5% (the implied volatility). Due tothe unusual characteristics of options, it is often more useful for the serious trader toconsider an option’s price in terms of implied volatility rather than in terms of its totaldollar price.
For example, suppose a Treasury Bond 98 call is trading for 3-32 ($3,500) with acorresponding implied volatility of 10.5%. Suppose also that a 102 call with the sameexpiration date is trading for 1-16 ($1,250) with an implied volatility of 11.5%. In totaldollar terms the 102 call is $2,250 cheaper than the 98 call. Yet an experienced traderwill probably conclude that in theoretical terms the 98 call is actually less expensivethan the 102 call because the implied volatility of the 98 call is a full percentage pointless than the implied volatility of the 102 call. Does this mean that one ougnt to buy the98 call and sell the 102 call? Not necessarily. If the future volatility of the TreasuryBonds turns out to be 8%, then both options are overpriced; while if volatility turns outto be 14