this is the amount paid atexpiration when the in-the-money options are exercised or assigned.The gross proﬁt (3) of 0.04 per share is the difference between revenue and cost. The interest income (4) is based on the funds invested(net credit) and the borrowing rate and come to 0.07. The fundsinvested of 10.00 are the revenue of 10.04 minus the cost of establishing the position of 0.04. The proﬁt before transaction costs (5) of0.11 is the sum of the gross proﬁt and the interest income. Finally,subtracting transaction costs of 6 cents per share (6) results in the netproﬁt (7) of 5 cents per share.Lines 8 through 10 in Table 6-28 analyze the relationship of thevalues of the two spreads. In this case, the call spread has a value ofTable 6-28
1234567
Pricing a Short 100–110 Box Spread—Part 3: Analysis of CashFlows, Net Proﬁt, and Spread Values
Revenue  net credit per share for establishing the position  Cost  net debit paid at expiration  110  100  Gross proﬁt Interest income  10.00  (0.04  60/365) Proﬁt before transaction costs Transaction costs Net proﬁt
Analysis of Vertical Spread Values8 Value of 100-110 call spread  9.10  3.46 9 Value of 100-110 put spread  6.70  2.3010 Sum of spread values
Conclusion: For a short box spread, the sum of the credit call spreadplus the credit put spread (10) equals the net credit per share (1).
10.0410.00 0.04 0.07 0.110.06 0.055.64 4.40 10.04
202 • Trading Options As a Professional
5.64 (8), and the put spread has a value of 4.40 (9). The sum of thespread values (10) therefore is 10.04, which equals the revenue (1) andis the net credit received for establishing the short box spread.The conclusion of this three-part exercise is stated at the bottom ofTable 6-28. For a short box spread, the sum of the credit call spreadplus the credit put spread equals the net credit required per share.
Motivations for Establishing a Short Box SpreadWhether paid on borrowed funds or earned on funds invested, interest is an essential component of arbitrage strategies. Option marketmakers who trade in several stocks frequently will ﬁnd that theyhave conversion positions in one stock and reverse-conversion positions in another. As a result, they might borrow from or lend to themselves. After all, a borrower who pays 5 percent to borrow fundsfrom a bank and receives 4 percent on funds invested with the bankwould save 1 percent by using the invested funds directly instead ofborrowing.Consider a market maker who prices conversions and long boxspreads assuming a borrowing rate of 5 percent and who prices reverseconversions and short box spreads assuming a lending rate of 4 percent.Further assume that this market maker has accumulated a conversionposition in stock 1 that requires $1 million in borrowings. The question is, “Does this position affect how reverse conversions and short boxspreads should now be priced?” The answer is yes because fundsreceived from these positions will reduce the funds that need to be borrowed. In theory, one might think that the lending rate on net creditpositions could be dropped to zero because the borrowed funds on netdebit positions cost 5 percent, but there is a complicating factor—themarket maker’s capital requirement.Every position involves risk, and therefore, every position requires asupporting equity requirement to cover that risk. Stock purchased onmargin or sold short, for example, requires an equity deposit of 50 percent. While option arbitrage positions involve signiﬁcantly less risk than
Arbitrage Strategies • 203
long or short stock positions, they still require some equity. Therefore,the decision of how low to lower the lending rate when pricing a netcredit arbitrage to fund a net debit arbitrage depends partly on the availability of equity. When the size of trading positions expands to the limitof available equity, it may not be possible to create additional positions,let alone positions at lower-than-normal interest rates. At other times,when equity is plentiful because positions are small, pricing short boxspreads and reverse conversion positions at near-zero lending rates maybe practical. This decision will depend on a market maker’s individualcircumstances.
SummaryArbitrage, conceptually, involves trading in two different markets withthe goal of proﬁting from small price differences. Options arbitrageinvolves buying real stock and selling synthetic stock or, vice versa,buying synthetic stock and selling real stock.A strategy known as the conversion consists of buying stock, buyingputs, and selling calls on a share-for-share basis. The call and put havethe same strike price and the same expiration date. All options arbitrage strategies are based on the conversion concept. A conversionmakes a proﬁt if the time value of the call exceeds the time value ofthe put by an amount sufﬁcient to cover transaction costs, includingborrowing costs.The reverse conversion, as its name implies, is the opposite of aconversion. It consists of short stock, short puts, and long calls on ashare-for-share basis and is established for a net credit. A reverse conversion will make proﬁt if the interest earned exceeds transaction costsplus the difference of call time value minus put time value.Box spreads are four-part options-only arbitrage strategies. They consist of a long synthetic stock position at one strike price and a shortsynthetic stock position at another strike price. A long box spread isestablished for a net cost, or net debit, and a short box spread is established for a net credit.
204 • Trading Options As a Professional
Although market makers can price arbitrage strategies with a targetproﬁt in mind, competition in the marketplace often inﬂuenceswhether the target proﬁt can be achieved. Frequently, market makersmust choose between accepting a lower proﬁt and not making a trade.Deciding whether or not to accept a lower proﬁt is part of the art oftrading as a market maker.Knowledge of arbitrage strategies and synthetic pricing relationshipshelps all traders to evaluate trading strategies and make trading decisions. Knowing that puts are in line with calls or that the options arein line with the stock gives a trader conﬁdence that prices are fair inthe current market environment.
Chapter 7VOLATILITYt has been said that volatility is the most used and least understoodword in the options business. On the most basic level, the termvolatility means movement in general, not movement in a particulardirection. Because traders tend to think in terms of up or down, however, this concept can be confusing. Nevertheless, option traders needto gain an accurate understanding of volatility because it signiﬁcantlyaffects option prices, trading decisions, and risk analysis. While adetailed knowledge of option-pricing formulas is not required to tradeoptions, there are some simple mathematical relationships that arehandy to know because they help traders to recognize good tradingopportunities.This chapter has six parts, three of which discuss volatility as itrelates to stock prices and three of which discuss volatility as it relatesto option prices. First, the volatility of stock prices is examined fromthe mathematician’s point of view. The discussion, however, is conceptual, not technical. Volatility is deﬁned, and the notion of standarddeviation is introduced by comparing the price action of two stocks.The next section demonstrates how traders might use volatility to estimate stock price ranges and the probabilities of those ranges occurring. Realized volatility and expected volatility then are deﬁned andillustrated with some examples. The fourth part discusses volatilityas it relates to option prices, speciﬁcally, implied volatility. The ﬁfth
I
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206 • Trading Options As a Professional
section discusses how to evaluate option prices and the meaning andusefulness of the terms overvalued option and undervalued option.Finally, the chapter introduces the phenomenon of volatility skew.
Volatility DeﬁnedVolatility is a measure of price changes without regard to direction.Thus, for example, in volatility terms, a 1 percent price rise will equala 1 percent price decline. With volatility, it is the percentage changethat matters, not the absolute amount of change, the stock price, orthe direction.This nondirectional nature of volatility can be difﬁcult to grasp fortraders who tend to think in terms of direction and in terms of goodand bad. A trader with a bullish opinion, for example, views a pricerise as good and a price decline as bad. A trader with a bearish opinion thinks the opposite. Regardless of the size of the movement, amovement in the “right direction” is good. Years of trading with thismind-set can impede one’s full understanding of the nondirectionalnature of volatility.A second complicating aspect of volatility is that one price change,in and of itself, is not important. Rather, only a series of price changesover several trading days, evaluated together, determines a stock’s volatility. Just as a “shallow river” that averages 6 inches in depth can have oneor two places that are 9 feet deep, so too can a “low-volatility stock” havean occasional big-price-change day. Similarly, a “high-volatility stock”can have some days when there is very little or no price change. Oneday’s price change is just one number; volatility describes a seriesof numbers.
Historic VolatilityMathematicians look at a series of price changes over several days,weeks, or months and derive what is called the standard deviation ofmovement. Mathematically, for option traders, historic volatility is the
Volatility • 207
annualized standard deviation of daily returns over a speciﬁc timeperiod. A standard deviation is the average difference between each ofthe daily returns and the mean return over the period observed. Donot let this deﬁnition intimidate you because the following discussionis conceptual, not mathematical.Price observations typically are made over 30 days, 90 days, or oversome other deﬁned period. To make meaningful comparisons of volatility, the exact observation period must be speciﬁed. Daily closing pricestypically are used, but daily opening prices or weekly closing prices orsome other consistent method of observation also could be used.Comparing one speciﬁc price change with another seems like asimple process, but comparing two series of prices changes is moredifﬁcult. Figure 7-1, for example, contains graphs of daily closing
115.00
Stock 1
110.00105.00100.0095.0090.00
Stock 2115.00110.00105.00100.0095.0090.00
Figure 7-1
Which Stock is More Volatile?
208 • Trading Options As a Professional
prices of two stocks over 31 days. Both stocks start at $100 and end at$100, but they have very different price actions in between. Stock 1trades in a narrow range around $100, whereas stock 2 falls fairlyquickly below $94, then rises to $110, and then falls back to $100.The question is: “Is stock 1 or stock 2 more volatile?” Take a momentto reﬂect on this question, and then compare your answer, which isbased on your own subjective, visual evaluation with the technicalanswer that is presented below.The historic volatility of stock 1 is calculated from the informationin Table 7-1. The left column, “Day,” simply assigns a number to eachclosing price; in the real world, this number would be a date. Themiddle column, “Closing Price,” contains the 31 closing prices thatare plotted in Figure 7-1. The right column, “Daily Return,” containspercentage changes in price from the previous day’s price.The daily return takes two steps to calculate. The closing price of theprevious day is subtracted from the closing price of the current day, andthen the difference is divided by the closing price of the previous day.The daily return for day 1 of 1.80 percent, for example, is calculated asfollows: The closing price on day 0 of 100 is subtracted from the closingprice on day 1 of 101.80 to yield a difference of 1.80. This differencethen is divided by the closing price on day 0 of 100. The result is 1.80,or 1.80 percent. There is no daily return for day 0 because this daymarks the ﬁrst price observation; the previous price is unknown.Using the data in the right column in Table 7-1, the standard deviation of these daily returns can be calculated. A standard deviationis a measure of the spread of values in a set of data. In practice, thestandard deviation of these daily numbers is converted to an annualstandard deviation by multiplying it by the square root of the numberof days in a year. This calculation produces 37.55 percent, which isshown at the bottom of the table. The calculation of a standard deviation is a standard spreadsheet function, so the mathematicallyinclined may easily do their own research. However, if you are notmathematically inclined, do not worry; Op-Eval Pro performs themany important volatility calculations.
Volatility • 209
Table 7-1
Stock 1: Calculation of Historic Volatility
Day
Closing Price
Daily Return (Day 2 – Day 1)/Day 1
0123456789101112131415161718192021222324252627282930
100.00101.8099.8098.10100.90102.1099.5097.9099.70102.6099.1098.4097.6099.10101.30102.70100.2099.3096.7099.70102.8098.3098.90101.5099.10100.6099.1096.9099.90102.80100.00
1.80%1.96%1.70%2.85%1.19%2.55%1.61%1.84%2.91%3.41%0.71%0.81%1.54%2.22%1.38%2.43%0.90%2.62%3.10%3.11%4.38%0.61%2.63%2.36%1.51%1.49%2.22%3.10%2.90%2.72%
Note: Annualized standard deviation of daily returns  37.55-–%.
While option traders do not need to know advanced calculus, they doneed to understand what 37.55 percent volatility means and what a standard deviation is. These concepts can best be explained by the followingdiscussion that compares the price action of the two stocks in Figure 7-1.
210 • Trading Options As a Professional
Historic Volatility—Comparison 1Table 7-2 shows one method of comparison; it lists daily closing pricesand daily returns for both stocks. As you look down the “Daily Return”column for each stock, you will ﬁrst observe that the absolute value ofevery percentage change of stock 1 exceeds the corresponding percentage change of stock 2. This difference indicates that the volatility of stock1 is higher than the volatility of stock 2. The second indication appearsat the bottom of Table 7-2. The annualized standard deviation for stock1 is 37.55 percent, and for stock 2 it is 22.11 percent.The conclusion is clear: Stock 1 is more volatile than stock 2. Somemay ﬁnd this result surprising because stock 2 fell $7.00, then rose$18.00, and then fell $8.00, whereas stock 1 traded within a ﬁve-pointrange. Remember, though, that volatility is a statistical measure ofdaily price action, not the accumulated size of price change or direction. Stock 2 experiences several smaller percentage changes in thesame direction relative to stock 1. When discussing volatility, severalsmaller percentage changes is the operative term.
Historic Volatility—Comparison 2In the real world, of course, every percentage change in one stock willhardly ever be larger or smaller than every corresponding percentagechange in another stock. Table 7-3 presents another method of comparing the volatilities of stocks 1 and 2—ranking the absolute valuesof the percentage changes (daily returns) from smallest to largest.Table 7-3 compares two samples of the 30 price-change observationsfor each stock. The following discussion illustrates some basic probability concepts that are helpful in understanding why volatility is important when trading options. The ﬁrst sample consists of the smallest 20observations, or two-thirds of the total. For stock 1, the smallest 20changes are less than 2.63 percent and average 1.67 percent. In contrast, the smallest 20 changes for stock 2 are less than 1.53 percent andaverage 0.86 percent. The comparison of this subgroup indicates thatstock 1 has a higher volatility than stock 2.
Volatility • 211
Table 7-2
Comparing Historic Volatility: Method 1Stock 1
Stock 2
Day
Closing Price
Daily Return
Closing Price
Daily Return
0123456789101112131415161718192021222324252627282930
100.00101.8099.8098.10100.90102.1099.5097.9099.70102.6099.1098.4097.6099.10101.30102.70100.2099.3096.7099.70102.8098.3098.90101.5099.10100.6099.1096.9099.90102.80100.00
1.80%1.96%1.70%2.85%1.19%2.55%1.61%1.84%2.91%3.41%0.71%0.81%1.54%2.22%1.38%2.43%0.90%2.62%3.10%3.11%4.38%0.61%2.63%2.36%1.51%1.49%2.22%3.10%2.90%2.72%
100.0099.1097.9096.4094.3093.5095.3096.3097.85100.20103.00103.60104.10105.30106.90107.80109.50109.10108.80110.80109.20108.10108.00107.40105.90106.50105.40103.80101.90100.85100.00
0.90%1.21%1.53%2.18%0.85%1.93%1.05%1.61%2.40%2.79%0.58%0.48%1.15%1.52%0.84%1.58%0.37%0.27%1.84%1.44%1.01%0.09%0.56%1.40%0.57%1.03%1.52%1.83%1.03%0.84%
Note: Annualized standard deviation of daily returns: stock 1  37.55%; stock 2  22.11%.
The second sample compares the smallest 29 observations, or approximately 96 percent of the total observations. Again, the measures forstock 1 are greater than those for stock 2. The absolute values of 29
212 • Trading Options As a Professional
Comparing Historic Volatility: Method 2
221817122325111530512129267132242027143168281964910
0.09%0.27%0.37%0.48%0.56%0.57%0.58%0.84%0.84%0.85%0.90%1.01%1.03%1.03%1.05%1.15%1.21%1.40%1.44%1.52%1.53%1.53%1.58%1.61%1.83%1.84%1.93%2.18%2.40%2.79%
95% less than 2.41%, avg. 1.16%
0.61%0.71%0.81%0.90%1.19%1.38%1.49%1.51%1.54%1.61%1.70%1.80%1.84%1.96%2.22%2.22%2.36%2.43%2.55%2.62%2.63%2.72%2.85%2.90%2.91%3.10%3.10%3.11%3.41%4.38%
95% less than 3.42%, avg. 2.08%
221112175152625137318227142416618233042992819201021
Stock 2
68% less than 2.63%, avg. 1.67%
Stock 1
68% less than 1.53%, avg. 0.86%
Table 7-3
changes are less than 3.42 percent for stock 1 but less than 2.41 percentfor stock 2. Also, the average of these changes is 2.08 percent for stock1 and 1.16 percent for stock 2. Statistically, stock 1 is more volatile thanstock 2 during the period observed.
Volatility • 213
Another Look at Daily ReturnsFigure 7-2 is a bar graph of the daily returns in chronological orderfor stocks 1 and 2. Although a visual comparison of these data indicates that there are more large daily returns for stock 1 than for stock2, visual observation for larger amounts of data is both impractical andinaccurate. For this reason, mathematicians prefer to organize dailyreturn data in the manner presented in Figure 7-3.Figure 7-3 contains two bar graphs that show the frequency ofreturns for stocks 1 and 2. The frequency is the percent of observationsthat fall within a range. The tallest bar for stock 2, for example, indicates that approximately 20 percent of the outcomes are very close to
Daily Returns - Stock 1
Daily Returns
6.00%4.00%2.00%0.00%–2.00%–4.00%–6.00%
Days (in chronological order)Daily Returns - Stock 2
6.00%Daily Returns
4.00%2.00%0.00%–2.00%–4.00%–6.00%Figure 7-2
Days (in chronological order)Bar Graph of Daily Returns in Chronological Order
214 • Trading Options As a Professional
Stock 1
20%15%10%5%0%–4%
–2%
0%
2%
4%
2%
4%
Daily Returns
Stock 2
20%15%10%5%0%–4%
–2%
0%Daily Returns
Figure 7-3
Frequency of Daily Returns
0 percent return. In both graphs, most returns are close to 0 percent,but the distributions are clearly different. The peak of the distributionfor stock 1 is lower than that for stock 2, and the bars are spread outin a wider and ﬂatter pattern for stock 1 than for stock 2. A wider andﬂatter pattern means that more daily deviations from the mean arelarger for stock 1, and this is expected because it is more volatile thanstock 2. If a line were drawn from the tops of the bars, that line wouldform an imperfect bell-shaped curve. For mathematicians, the shape
Volatility • 215
Frequency
of the bell-shaped curve describes a stock’s volatility: The broader thecurve, the higher is the volatility; the narrower the curve, the lower isthe volatility.Figure 7-4 is a stylized graph of a normal distribution. A normaldistribution means that the left and right halves of the distribution areidentical in shape. When returns are normally distributed, inferencescan be made about the frequency of returns occurring in the future.Therefore, if returns are normally distributed, approximately 68 percentwill occur within one standard deviation of the mean, 95 percent willoccur within two standard deviations, and 99 percent occur within threestandard deviations.A short, broad line at the bottom center of the diagram in Figure7-4 indicates the mean of the distribution, and the text under the curveexplains how returns will be distributed around the mean within one,two, or three standard deviations.The concept of Figure 7-4 is expanded in Table 7-4, which lists thestatistical percentages of the occurrence of events out to six standarddeviations. Statistically, a stock-price change equal to six standard deviations is not impossible, just unlikely.
68% ofdailyreturnswithin1 std dev95% of daily returnswithin 2 std devs99% of daily returnswithin 3 std devs
MeanDaily ReturnsFigure 7-4
Distribution of Returns
216 • Trading Options As a Professional
Table 7-4
Percentage of Events by Standard DeviationsPercentage of EventsOccurring Within theNumber of StandardDeviations Indicated
1 sd2 sd3 sd4 sd5 sd6 sd
68.26894921371%95.44997361036%99.73002039367%99.99366575163%99.99994266969%99.99999980268%
Approx. 2 out of 3Approx. 19 out of 20Approx. 369 out of 370Approx. 15,999 out of 16,000Approx 999,999 out of 1,000,000You get the idea!
sd  standard deviation.Source: Wikipedia.
Summary of Historic VolatilityHistoric volatility is a measure of observed stock-price changes in thepast. While most price changes are small, large ones can occur. Thedistribution of price changes can be graphed, mathematically, by abell-shaped, or normal, curve. A narrower curve with a higher peakindicates lower volatility, and a wider curve with a lower peak indicates higher volatility.
Realized VolatilityRealized volatility is a measure of stock-price ﬂuctuations between todayand some date in the future. If one could observe stock prices from todayuntil that future date and calculate historic volatility using those prices,that calculation would produce the realized volatility. Future volatilityis another name for realized volatility because it is unknown today.
The Meaning of “30 Percent Volatility”An option’s volatility is stated as a percentage based on the annualizedstandard deviation. Applying the percentages from Table 7-4, it followsthat if a stock’s volatility is 30 percent, then there is a 68 percent chance
Volatility • 217
that one year from today the stock price will lie between 30 percentbelow today’s price and 30 percent above today’s price. There is alsoa 95 percent chance that it will lie between 60 percent below and60 percent above today’s price (two standard deviations). And thereis a 99 percent chance that it will lie between 90 percent below and90 percent above today’s price (three standard deviations).
No Direction ImpliedRemember, the volatility percentage does not give an indication ofdirection, only the range. Therefore, the volatility cannot help a traderforecast market direction. However, given a trader’s forecast for direction, volatility can help to estimate a proﬁt target.A trader who is very bullish, for example, might forecast a twostandard-deviation price rise. For a stock currently trading at $100 and at30 percent volatility, then a two-standard-deviation price expectationimplies that the stock price will reach $160 some time in the next year(100 plus 60 percent, or two standard deviations). The bullish component of the forecast and the expectation for a two-standard-deviation pricechange are the trader’s subjective opinion based on his or her experienceand interpretation of market conditions. They are not objective calculations derived from the options market. The $160 price target, however,is an objective calculation based on the stock’s current price and volatility. Because making a one-year forecast may seem impractical to many,a simple method of converting the annual price-range expectation intoa shorter time period may be more useful.
Converting Annual Volatilityto Different Time PeriodsThe one-year standard deviation of stock price range can be convertedto a standard deviation for any time period using the formula presented in Table 7-5. The formula is: Annual volatility  the squareroot of time. This formula can be used to estimate both a price rangeand a probability that the price will lie within that range.
218 • Trading Options As a Professional
Table 7-5
Converting a Stock’s Annual Volatility to Different Time Periods
The formula:Annual volatility  square root of time  SD for time periodAnnual volatility  √days to exp./days per year  volatility for days to exp.where volatility  standard deviation of returns.Assumption: Volatility 35%Standard deviation for one year:0.35  √365/365  0.350For a stock at 78.50, one standard deviation for one year is 27.47 (78.50  0.35)Standard deviation for four weeks: 0.35  √28/365  0.097For a stock at 78.50, one