standard deviation for four weeks is 7.61(78.50  0.097)Standard deviation for one week:0.35  √7/365  0.048For a stock at 78.50, one standard deviation for one week is 3.77(78.50  0.048)Standard deviation for one day:0.35  √1/365  0.018For a stock at 78.50, one standard deviation for one day is 1.41 (78.50  0.018)
Table 7-5 assumes a stock price of $78.50 and a volatility of 35 percent and calculates a one-standard-deviation price range for one year, forfour weeks, for one week, and for one day. Focus on the calculation forfour weeks: One standard deviation equals 7.61. Consequently, theexpectation under these assumptions is that the stock price will closeat the end of the period between 70.89 and 86.11 approximately twoof three months (a 68 percent probability). In one of three four-weekperiods, the change in closing price is expected to be above or below thisrange but between 63.62 and 83.74 (two standard deviations). And occasionally, one of 20 four-week periods, the closing price is expected to bebetween two and three standard deviations from the current price.
Op-Eval Pro Calculates DistributionsTo make it easy to get a standard deviation, the Distribution screen inOp-Eval Pro will perform the conversion calculation. Type in a stockprice, a volatility assumption, and a number of days, and Op-Eval Prowill calculate the one-standard-deviation price range for that number
Volatility • 219
of days, two times that number of days, three times that number ofdays, and four times that number of days.Figure 7-5 conﬁrms the four-week Op-Eval Pro calculation inTable 7-5. The stock price is 78.50, the volatility assumption is 35 percent, and the time period is 28 days. Op-Eval Pro calculates the onestandard-deviation price range as 70.89 to 86.11. These prices are7.61, or one standard deviation, above or below the stock price of78.50. The difference of 1 cent between Figure 7-5 and Table 7-5is due to rounding.The additional price ranges on the Distribution screen are for twicethe time period, three times the time period, and four times the timeperiod. These time periods are helpful to traders who want a quick estimate of a longer time frame. For a trader evaluating a four-week optionand its implied stock price range, the information on the DistributionOp-Eval Pro: Distribution AnalysisCALCULATING ONE STANDARD DEVIATION MOVEUNDERLYING PARAMETERSPRICE
79
VOL%
35
Now
70.89
86.11
67.74
56 days
65.32
84 days
Figure 7-5
DECIMAL PLACES
28
78.50
28 days
112days
DAYS
63.28
The Distribution Page from Op-Eval Pro
89.26
91.68
93.72
2
220 • Trading Options As a Professional
screen makes a comparison with 8-, 12-, and 16-week options fastand easy.Consider a trader who is bullish on stock XYZ, currently trading at78.50. Also assume that the trader wants to compare the purchase of thefour-week XYZ 80 Call trading at 2.50 with the purchase of the eightweek XYZ 80 Call trading at 3.85. If the trader believes that XYZ willtrade in a range consistent with 35 percent volatility, then Figure 7-5indicates that the one-standard-deviation price range is 7.61 for fourweeks and 10.76 for eight weeks. Only the trader, individually, candecide if the extra time and the wider price range provided by the eightweek call is worth the additional cost of 1.35. The implied stock priceranges, however, provide additional and helpful information.
Calendar Days versus Trading DaysThe “calendar days” component in the formula in Table 7-5 is oftendebated. The question is, “Should the formula use calendar days ortrading days?” Those in favor of trading days argue that volatility, thatis, stock-price change, can only happen on trading days. Otherscounter that calendar days better reﬂect the actual amount of timeuntil expiration. The answer is: In most cases, either can be used without much impact on the result.The conversion formula uses square root of time in years, so theimportant question is whether calendar days or trading days best approximates time in years. It can be argued, generally, that it does not matter. Any percent of a full year is the same regardless of the number ofdays in a year. Choosing 252 (trading days) versus 365 (calendar days)for days per year for price-range estimates using volatility becomes anissue only when the time period is short. What is a “short” time period?Two examples follow that shed light on this issue.First, consider a two-month time period in which there are 61 calendar days and 43 trading days. Also assume a stock price of 78.50 and35 percent volatility. Calendar days are used to calculate a standarddeviation for the period as follows: 61 calendar days divided by 365
Volatility • 221
calendar days in a year is 0.1671, the square root of which is 0.4087.The volatility for the period, therefore, is 14.3 percent (0.35 0.4087). And for a stock trading at 78.50, one standard deviation is11.23 (78.50  0.143).Trading days are used to calculate a standard deviation for theperiod as follows: 43 trading days divided by 252 trading days in a yearis 0.1706, the square root of which is 0.4130. The volatility for theperiod, therefore, is 14.5 percent (0.35  .4130). And for a stock trading at 78.50, one standard deviation is 11.38 (78.50  0.145).The difference between using calendar days and trading days is15 cents. For a stock price of 78.50 and a period of two months, thisdifference is not signiﬁcant.Second, consider a three-day time period, again assuming a stockprice of 78.50 and volatility of 35 percent. Using calendar days, 3 dividedby 365 calendar days in a year is 0.0082, the square root of which is0.0906. The volatility for the period, therefore, is 3.2 percent (0.35 0.0906). And for a stock trading at 78.50, one standard deviation is 2.51(78.50  0.032).Using trading days, 3 divided by 252 trading days in a year is 0.0119,the square root of which is 0.1091. The volatility for the period, therefore, is 3.8 percent (0.35  0.1091). And for a stock trading at 78.50,one standard deviation is 2.98 (78.50  0.038).The difference between using calendar days and trading days is 47cents (2.51 versus 2.98). This is approximately a 17 percent differenceand arguably signiﬁcant.How much of a concern should the difference between using calendar days and trading days be to traders? For a two-month time period,given a stock price of 78.50, most traders would not consider 15 centsto be signiﬁcant. For the three-day period, the difference of 47 centsmight be signiﬁcant depending on how often a trader uses strategiestargeted at three days. In general, the answer also partly depends onhow accessible the necessary information is. Most traders have easyaccess to the number of calendar days to expiration because brokerssupply it. In contrast, the number of trading days is more difﬁcult to
222 • Trading Options As a Professional
ﬁnd and time-consuming to calculate. Many traders therefore use calendar days when converting annual volatility to shorter time periodsbecause it is easier, and it usually does not make much difference.The focus now will shift from volatility as it relates to stock-pricemovements to volatility as it relates to option prices. Remember, fromChapter 2, that volatility is one of the six components that inﬂuenceoption prices.
Implied VolatilityImplied volatility is the volatility percentage that justiﬁes the marketprice of an option. In other words, it is the volatility percentage thatreturns the option’s market price as the theoretical value. This conceptis best explained with an example.Consider Gary, who uses Op-Eval Pro to estimate the theoreticalvalue of an XYZ March 70 Call. Figure 7-6 shows a Single OptionCalculator screen from Op-Eval Pro with Gary’s inputs: current stockprice of 68.00, strike price of 70, no dividend, interest rate of 4 percent, and 75 days to expiration. Gary chose a volatility of 26 percentbecause that percentage was the historic volatility based on the 30most recent daily closing stock prices (available from www.cboe.comand www.ivolatility.com). Given Gary’s inputs, Op-Eval Pro calculatesa value of 2.57 for the XYZ March 70 Call.Gary now turns to check the current market and discovers that theXYZ March 70 Call is trading at 3.40! What is going on? Did Garydo something wrong? Or could the market be assuming somethingdifferent from Gary? For instance, could the market be assuming adifferent stock price? No, the stock price is known to be $68. A different strike price or time to expiration? No, both of these are part ofthe contract speciﬁcations and are known. A different interest rate ordividend? A slight difference in interest rates or dividend yield mightproduce a change in option value of a few pennies, but not 82 cents.So, again, the answer is no.
Volatility • 223
Figure 7-6
Gary’s Estimate of the March 70 Call
What, then, can account for the difference between Gary’s calculation of 2.58 and the market price of 3.40? The only remaining factor is volatility. The market must be assuming a different volatility thanGary. Figure 7-7 shows the Single Option Calculator screen with theprice of 3.40 entered in the “CALL” box. When the “Enter” key ispressed, Op-Eval Pro recalculates the volatility as 32.84 percent. Thisis the volatility percentage that justiﬁes the market price of this XYZMarch 70 Call, and thus it is the implied volatility of this option.Examples later in this and future chapters will demonstrate thatimplied volatility is used in several ways to compare option prices, toestimate the outcome of strategies, to enter bid and ask prices, and tomake subjective judgments about the relative value of an option.
The Role of Supply and DemandThe forces of supply and demand determine option prices, just as theydetermine all prices in free markets. What varies from market to market, however, is the market-determined component of price that is used
224 • Trading Options As a Professional
Figure 7-7
Implied Volatility of the March 70 Call
to evaluate the instrument being priced. In the stock market, for example, the price-earnings ratio is used widely to make judgments abouta stock’s value.If the stock of Company A is at $80 per share with a price-earningsratio of 10, and if the stock of Company B trades at $35 per share witha price-earnings ratio of 15, then Company A is said to be less expensive than Company B. In this context, the term less expensive means thestock with the lowest price-earnings ratio, not the lowest absolute priceper share. Price-earnings ratios make it possible to compare companieswith different levels of sales, different number of shares outstanding, anddifferent stock prices. Book value and price-to-sales and debt-to-equityratios are other “common denominators” used by stock analysts.The price-earnings ratio, however, is market-determined because itis a function of stock price. The earnings per share reported by auditors is known and is determined independently of the stock price. Thestock price, however, is determined by supply and demand; so too,therefore, is the price-earnings ratio determined by supply and demand.In other words, the price-earnings ratio is market-determined.
Volatility • 225
In the options market, implied volatility is a market-determinedcomponent of option price that makes comparisons between optionspossible. Five of the inputs to the option-pricing formula are known:stock price, strike price, expiration date, interest rate, and dividend.Volatility between now and option expiration, however, is unknown.Nevertheless, given an option price, a trader can work the pricingformula backwards to ﬁnd the volatility percentage that would produce the market price of the option as the theoretical value. This percentage is the implied volatility of the option. In other words, thevolatility percentage that produces the option’s market price as thetheoretical value is the implied volatility. In Gary’s XYZ 70 Call, 32.84percent is the volatility that made the formula’s calculated value equalthe option’s market price.Just as the price-earnings ratio in the stock market is a commondenominator that makes comparison of stock prices possible, impliedvolatility also facilitates comparisons of option prices. If the options onthe stock of Company A are trading at an implied volatility of 38 percent,and if the options on the stock of Company B are trading at an impliedvolatility of 25 percent, then it can be said that the market believes thatthe price of Company A’s stock will be more volatile than the price ofCompany B’s stock. No one can guarantee that price action betweennow and option expiration—realized volatility—will bear this out, butit can be said with certainty that, today, this evaluation reﬂects themarket’s opinion.
Implied Volatility ChangesIn addition to making comparisons between stocks feasible, impliedvolatility also makes it possible to evaluate changing conditions on onestock by comparing its volatility at different times. It is common parlance, for example, to describe Gary’s XYZ March 70 Call as tradingat 32.84 percent volatility. This call, at some previous time, may havebeen trading at a higher or lower implied volatility. If all other factorsare equal, then an option trading at a lower implied volatility should
226 • Trading Options As a Professional
make a relatively good purchase, and an option trading at a higherimplied volatility should make a relatively better sale. Rarely, if ever,however, are all other factors equal! Therefore, the level of impliedvolatility cannot, in and of itself, dictate whether you should buy orsell an option. Implied volatility is, however, important informationthat a trader can incorporate into the subjective decision-makingprocess.
Both Historic and Implied Volatility ChangeStock prices go through periods of high and low historic volatility. Thechanges may be driven by events speciﬁc to the company, such asnew-product development, earnings announcements, or managementturmoil, or perhaps the changes may be driven by events in the general market. Nevertheless, option traders need to be aware of thestock’s current level of volatility in order to make a realistic forecast.If a stock’s price has not risen or fallen $10 in any month in thelast two years, such a price change will not likely occur this month.However, if a monthly $10 price change happens frequently, thenforecasting such a change this month would be reasonable.Implied volatility also rises and falls both because of companyevents and because of changes in the general market. And just astraders need to be aware of stock-price volatility, so too do they needto follow implied volatility.Figure 7-8 shows how historic and implied volatility changedover a 12-month period for a well-known large-capitalization stock.The information in this ﬁgure was supplied by I-Volatility and isavailable for all exchange-listed stocks at www.ivolatility.com and atwww.cboe.com.The upper portion of Figure 7-8 is a line graph of the stock pricefrom April 2007 to March 2008, and the lower portion contains linegraphs for historic volatility and implied volatility. Conventionalwisdom states that volatility rises when stock prices fall. Figure 7-8,however, shows that conventional wisdom is not always correct.
Volatility • 227
IVolatility.com115110105100
StockPrice
9590Apr,07
Jun,07
Aug,07
Oct,07
Dec,07
Feb,08
IVolatility.com35%30%
Impl.Vol.
25%20%Hist. Vol.15%
Apr. 07
Figure 7-8
Jun.07
Aug.07
Oct.07
30D HV
IV Index Mean
Dec.07
Feb.08
10%
Stock Prices and Changing Volatilities (Source: ivolatility.com)
From early April through late July of 2007, the implied volatility ofthe options in Figure 7-8 rose from less than 20 percent to approximately25 percent, while the stock price also rose from approximately 95 toabove 115. Implied volatility did spike higher in early August when thestock price fell from approximately 117 to below 110, and it spiked higheragain in October–November when the stock price declined from 115 toapproximately 100. Nevertheless, traders must be careful to analyze howimplied volatility is changing and not just rely on conventional wisdom.
228 • Trading Options As a Professional
Revisiting the Insurance AnalogyOne explanation of why implied volatility changes is based on the analogy between options and insurance presented in Chapter 3. In thatanalogy, volatility is comparable with the risk factor in insurance. Thelevel of risk is one component in determining the level of insurancepremiums.If an insurance company has a record of claims showing that oneof 100 homes is destroyed by ﬁre, for example, then, in theory, ﬁreinsurance would cost 1 percent of the value of a home plus a proﬁtmargin. If, however, the insurance company forecasts that ﬁre willdestroy a greater percentage of homes in the future, then it will raiseits premiums. Similarly, if the company perceives that ﬁre will causeless damage, perhaps owing to ﬁre and smoke alarms and improvedbuilding practices, then premiums are lowered.Insurance companies, however, live in a competitive environment,and some premiums are set to meet the competition. In some marketenvironments, the competitive level of insurance premiums will behigher than the theoretical level calculated by an insurance company.In such an environment, the market expects more risk than the history of risk as calculated by the insurance company. In other marketenvironments, the competitive level of insurance premiums is belowthe theoretical level calculated by the insurance company. In thoseenvironments, the market expects less risk than history indicates.Historic volatility is like the insurance company’s records of actualclaims experience. Expected volatility is like a particular insurancecompany’s forecast of future claims.While it might seem desirable to sell insurance at higher premiumswhen the market expectation for risk is above the historic level ofclaims, one has to remember that markets are generally very efﬁcient.Frequently, prices rise before most people understand why becausethe market perceives something that many individuals do not see. Anexample of the market perceiving something was the historic rise inoil prices from 2006 to 2008 to over $140 per barrel. In the early stages,
Volatility • 229
when oil was hitting new all-time highs of $40 and $50 per barrel,there were several oil market analysts who said, “Oil above $40 is atemporary phenomenon,” and then, “Oil above $50 is unwarrantedby the fundamentals,” and then, “Oil above $70 simply cannot be sustained.” In retrospect, the market clearly foresaw that supply-demandconditions had changed, whereas many individuals did not.In options, the level of implied volatility is the market’s consensusestimate of future volatility. In many cases, the market perceives a rise orfall in stock price volatility that many individuals actively involved in themarket do not see. Professional traders must never forget this. They mustconstantly ask: What is the market—through implied volatility—sayingabout future volatility? What might the market be seeing that I do notsee? And how can I protect myself if what the market is saying turns outto be right and what I am thinking turns out to be wrong?
Implied Volatility Can Change IntradayImplied volatility not only changes over weeks and months, Table 7-6demonstrates how implied volatility can change within a trading day.Column 1 shows the time of day, whereas column 2 lists the stock price.The stock price ﬂuctuates from a low of 76.25 to a high of 77.95.Columns 3 and 4 contain the bid and ask prices of the 80 Call, andcolumns 5 and 6 state the implied volatilities of the bid and ask prices.At 9:30 a.m., when the market opens, for example, the stock price is76.25, and the 80 Call has a bid price of 2.60 and an ask price of 2.80.The implied volatility is 31.0 percent for the bid and 32.6 percentfor the ask. By itself, this information has little value. However, considerthe situation at 12:30 p.m. when the stock price is up to 77.95, and theimplied volatilities of the bid and ask prices have increased to 33.8 and35.4 percent, respectively. Then consider the situation at 4 p.m., whenthe stock price is down to 77.40, and the implied volatilities haveretreated to 31.1 and 32.7 percent, respectively.A trader who looks at implied volatility only at the beginning andend of each day would notice little change. Only full-time traders who
230 • Trading Options As a Professional
Table 7-6
Implied Volatility Intraday
Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Time
StockPrice
80 CallBid Price
80 CallAsk Price
Imp. Vol.of Bid
Imp. Vol.of Ask
09:3010:0010:3011:0011:3012:0012:3001:0001:3002:0003:0003.3004.00
76.2576.5076.4076.5076.7077.2577.9577.6077.4077.7577.6077.5577.40
2.602.702.652.702.903.203.703.403.203.403.303.203.10
2.802.902.852.903.103.403.903.603.403.603.503.403.31
31.0%31.0%31.0%31.0%31.9%32.4%33.8%32.8%31.9%32.2%32.0%31.4%31.1%
32.6%32.6%32.6%32.6%33.5%34.0%35.4%34.3%33.5%33.8%33.6%33.0%32.7%
Assumptions: Days to expiration, 63; interest rate, 4%; dividends, none.
watch this market all day see the nearly 3 percent rise and fall inimplied volatility. Of course, Table 7-6 presents only one possible pattern of implied volatility, which, like stock prices, could behave in anynumber of ways.Just as in forecasting stock prices, predicting changes in impliedvolatility is an art, not a science. Option traders must be aware of howmuch implied volatility can change, and they must gauge the potential impact on their positions. This topic will be discussed more inChapter 10.
Expected VolatilityExpected volatility is a loosely used term that describes a trader’s forecast for either realized volatility or implied volatility. After a period oflow historic volatility of stock prices, for example, a trader might predict
Volatility • 231
that realized volatility between now and expiration will be higher. Sucha forecast could lead the trader to a long volatility delta-neutral trade, asexplained in Chapter 8. In this case, the expected volatility is the forecast level of realized volatility, the stock-price ﬂuctuations in the future.Alternatively, after analyzing recent developments in the market,a trader might conclude that the level of implied volatility in optionprices is low (high) and forecast that it will rise (fall). Such a forecast could lead the trader to buy calls (or puts) rather than sell puts(or calls). As discussed in Chapter 3, option traders need a three-partforecast, which includes a stock-price target, a time period, and a levelof implied volatility. In this case, the expected volatility is the trader’sforecast for the level of implied volatility in the future. Forecast volatility is another name for expected volatility.
Many Terms for VolatilityWords in the options business are often used loosely and with contradictory meanings. Different traders sometimes use the same words differently. Here is a rough guide to some of the terms related to the manyaspects of volatility. Past volatility is the same thing as historic volatility.Option volatility, or an option’s volatility, means implied volatility. Futurevolatility is another name for realized volatility. Expected volatility is aprediction for either realized or implied volatility; forecast volatility andpredicted volatility are other names for expected volatility.Stock-price action in the past is historic volatility. When a traderenters a number as the volatility input of an option-pricing calculator,that is expected volatility. And the stock-price ﬂuctuations betweentoday and some day in the future constitute realized volatility.
Using VolatilityTraders can use the concept of volatility and the related concept of pricerange distributions to plan trades and to choose strike prices. A volatility percentage, by itself, does not predict when the small-movement and
232 • Trading Options As a Professional
big-movement days will occur, but traders who are willing to apply theirown judgment can study a stock’s price action and develop a feel for thesize and length of small- and big-movement periods. These traders thencan choose appropriate strategies for the price action they expect basedon their experience with the stock and their knowledge of other eventsaffecting the market. The risk of applying this subjective judgment, ofcourse, is that a forecast will be wrong, and a loss will result.One variation of this approach adds a trader’s knowledge of volatility. In this strategy, the trader sells options with a strike price that isat least one standard deviation away from the current stock price. Forexample, consider a stock price of 78.50 and a volatility of 35 percent,as in Table 7-5 and Figure 7-5. Given a one-month one-standard-deviation stock price range of 7.61, the strike price of 70 is more than onestandard deviation below 78.50. One justiﬁcation for selling the onemonth 70 Put is that, based on the principles of volatility, it hasa 68 percent chance of expiring worthless and earning a proﬁt forthe seller.Remember, however, that a statistical probability of earning a proﬁtdoes not guarantee that a proﬁt will be earned. Even if such an optionexpires worthless two of three months over several years, it might notexpire worthless in the speciﬁc month that a trader sells it. Also, evenif the option ultimately is out of the money and expires worthless atexpiration in one month, stock-price action could cause it to be in themoney at some time prior to expiration. Such stock-price action couldcause the trader to cover the short option at a loss, even though theoption eventually expired worthless. This frustrating series of eventshappens occasionally to every experienced option trader.
“Overvalued” and “Undervalued”A discussion of volatility naturally leads to an exploration of the relationship of option prices to theoretical value. Many traders strive to buyoptions when they are trading below their theoretical value, so-calledundervalued options. Likewise, they try to sell overvalued options.
Volatility • 233
Like the old adage, “Buy low, sell high,” the goal of buying undervalued options and selling overvalued ones may prove illusive. A complete understanding of what overvalued and undervalued mean raisesquestions about how the goal can be achieved.As discussed in Chapter 3, the value of an option depends on sixinputs, one of which is the volatility between the current date andexpiration, that is, realized volatility. Realized volatility, though, isunknown. The true theoretical value therefore is unknown. Theoretical values are actually only estimates of that value and are, in fact,based on the volatility assumption that a trader uses as an input, thatis, expected volatility.Now consider what makes an option overvalued. An overvaluedoption is an option the market price of which is higher than its theoretical value. The difference between these two prices, however, isgenerally the volatility assumption. The volatility component ofan option’s market price is implied volatility, but the volatility component of an option’s theoretical value is expected volatility. The difference between market price and theoretical value, therefore, isthe difference between implied volatility and expected volatility. Inthe case of an overvalued option, implied volatility is higher thanexpected volatility.The logic for undervalued options is similar. An undervalued optionis an option the market price of which is lower than its theoreticalvalue. In the case of an undervalued option, implied volatility is lowerthan expected volatility.The calculation of implied volatility is objective; it uses knownvariables, stock price, strike price, days to expiration, interest rate,dividends, and the market price of an option. Theoretical value, however, is subjective because it uses expected volatility, an unknownvariable. The determination of overvalued or undervalued thereforealso must be subjective. Trader A and Trader B will agree on theimplied volatility of an option, but if their expectations for volatilityare different, they will disagree on whether that option is overvaluedor undervalued.
234 • Trading Options As a Professional
An Alternative FocusRather than focusing on whether an option is overvalued or undervalued, a trader’s time is spent most productively on reﬁning the threepart forecast—for the stock price, the time period, and the level of