in the Spread Positions screen. Toselect the type of table you want, right-click on the table, and selectfrom the list of choices. The upper-left cell in the table contains thestrategy characteristic being calculated. All limitations of the Theoretical Graph feature apply to the Table feature. Figure 2-5 containsvalues for the bull call spread created in the Spread Positions screenin Figure 2-3 and graphed in Figure 2-4.
Operating the Op-Eval Pro Software • 45
TheoPrice
60 days 54 days 48 days 42 days 36 days 30 days 24 days 18 days 12 days 6 days
125
9.12
9.23
9.35
9.46
9.58
9.70
9.81
9.91
9.97
9.99
120
8.51
8.64
8.78
8.93
9.09
9.27
9.47
9.66
9.85
9.98
115
7.61
7.73
7.86
8.02
8.19
8.40
8.65
8.95
9.31
9.74
110
6.40
6.47
6.56
6.67
6.79
6.94
7.13
7.39
7.76
8.35
105
4.95
4.96
4.96
4.97
4.98
4.98
4.99
5.00
5.00
5.02
100
3.44
3.37
3.29
3.20
3.08
2.94
2.75
2.50
2.14
1.56
95
2.09
1.97
1.85
1.70
1.53
1.34
1.10
0.83
0.51
0.16
90
1.07
0.96
0.84
0.71
0.58
0.44
0.30
0.16
0.05
0.00
85
0.44
0.37
0.30
0.22
0.16
0.10
0.05
0.02
0.00
0.00
80
0.14
0.11
0.08
0.05
0.03
0.01
0.00
0.00
0.00
0.00
Figure 2-5
Theoretical Price Table
The stock price range in the left-most column and the range of daysand interval between days in the top row can be changed by rightclicking on the table and left clicking on “Change Axes.” The useralso can change the volatility assumption and set the desired numberof decimal places in the output calculation. These features make iteasy to adjust the analysis to the price and time ranges and volatilitylevel desired by the user.
The Portfolio ScreenGraphing complex positions involving up to 15 options with differentexpiration dates and one underlying is possible with the Portfolio screen.After inputting the “Underlying Parameters” and the “Option Parameters” in the upper-left portion of the screen, simply left-click on “ADD,”(does not appear in Figure, but is on the Portfolio screen) and a newcomponent will be added to the position. Note that the multiplier can
46 • Trading Options As a Professional
be changed with the “MULT” input box in the row of underlyingparameters. The “Move volatility” and “Move days to expiry” inputboxes make it possible to graph the impact of changes in these factors.Figure 2-6 shows a Portfolio screen of a six-part position.
The Distribution ScreenThe discussion of volatility in Chapter 7 explains how the impliedvolatility component in an option’s market price can be used to calculate the market’s expectation for one standard deviation of pricerange for the underlying stock between the current date and optionexpiration. The Distribution screen performs this calculation forfour time periods chosen by the user. Simply input the underlyingprice, a volatility assumption, and a time period; Op-Eval Pro willcalculate a one-standard-deviation price range. Figure 2-7 is a
Op-Eval Pro: Portfolio AnalysisUNDERLYING PARAMETERS (must be the same for all options)
X IsIndex X IsEuropeanPRICE RATE % DIVIDEN DIV DAYS MULT1004.0000100
Portfolio VALUE 600,799Greeks
Move volatility
20
Move days to expiry
100000800006000040000200000-20000-40000-60000-80000-100000-120000-140000-160000-180000-200000-220000-24000070
80
90
Qty6000-30-3050-75-20
Type Strike Vol % Days PriceStock100.0030110Call280.58301150.19Call28305695Put2.3030561.0290Put300.375685Put
Profit And Loss
OPTION PARAMETERSB/S QTY TYPE STRIKE VOL % DAYSSell 20 Put853056#123456
VEGA -414.17 THETA 1,254.95-71.17DELTA 5,260.1 GAMMA -150.58 RHO
100
110
Underlying PricePrimary Profit And Loss
Figure 2-6
The Portfolio Screen
15
120
130
Operating the Op-Eval Pro Software • 47
Op-Eval Pro: Distribution AnalysisCALCULATING ONE STANDARD DEVIATION MOVEUNDERLYING PARAMETERSPRICE
83
VOL%
33
Now
7
DECIMAL PLACES
2
83.00
6 days
79.21
77.64
14 days
76.43
21 days
28 days
DAYS
75.41
86.79
88.36
89.57
90.59
*Prices estimate a one standard deviation range for the time period indicated (365 days-a-year)
Figure 2-7
The Distribution Screen
Distribution screen showing one-standard-deviation price ranges fora stock trading at $83.00 with volatility of 33 percent for 7, 14, 21,and 28 days.
Previewing, Printing, and SavingWhat good is research and analysis if you can’t print it to show othersor save it for future use? In Op-Eval Pro, all screens can be previewedbefore printing, and all scenarios can be saved. Practice with the “SaveAs,” “Print Preview,” and “Print” control buttons on the menu bar atthe top of the screen. To retrieve a scenario saved previously, simplyclick on the “Open” button in the control bar.
48 • Trading Options As a Professional
SummaryOp-Eval Pro is designed to perform calculations, to draw graphsquickly, and to improve the analytic process. It is not designed to makedecisions for you.After installing the program, you must carefully read and thoroughly understand the Disclosures and Disclaimers before youattempt to use the program to analyze individual options or multiplepart option strategies.The Single Option Calculator, the Spread Positions screen, and thePortfolio screen all present option theoretical values and their respective deltas, gammas, thetas, and vegas. If the “Stock Price” is changedon any screen, Op-Eval Pro recalculates the volatility percentage; thisis known as the implied volatility.The graphing and table features require consistent inputs in theSpread Positions screen. The line with straight segments is a graph ofthe strategy at expiration. The middle line shows the strategy at halfthe days to expiration and may be recalculated. The third line estimates a strategy’s performance at the number of days prior to expiration indicated on the Spread Positions screen.Complex positions, including time spreads and positions withoptions with different expiration dates, can be graphed on the Portfolio screen.Practice using the many features of Op-Eval Pro. You can easilymake mistakes in inputting information if you are not thoroughlyfamiliar with the layout of the various screens. Once you becomefamiliar with its many features, however, you will ﬁnd that Op-EvalPro is a valuable tool for analyzing option prices and strategies.
Chapter 3THE BASICS OF OPTIONPRICE BEHAVIORhe subject of option price behavior requires two chapters to explainfully because traders must master two different aspects of optionprice behavior. This chapter discusses how option prices change asmarket conditions change. Option prices do not move in the short termthe same way that stock prices and prices of futures contracts move.Traders therefore must learn to think in a different way when tradingoptions. The next chapter discusses the Greeks—delta, gamma, theta,vega, and rho—how they change, how they are used to evaluate position risk, and how position risk shifts as the market ﬂuctuates.This chapter starts with a brief review of the analogy betweenoptions and insurance. Second, it describes how option values changeas the various input factors change. Third, the subject of volatility andits impact on option prices is introduced. The chapter concludes witha discussion of option price changes in a dynamic environment andthe unique way that option traders plan trades.
T
The Insurance AnalogyAn option is like an insurance policy that pays its owner if certainevents occur on or before the expiration date. If the events do notoccur, the policy expires worthless.• 49 •
50 • Trading Options As a Professional
The analogy between put options and insurance is perhaps easiestto understand. If the underlying stock declines in value, a put rises invalue. The stock decline is similar to an insured asset being damaged,and the rise in put value is similar to an insurance policy payinga claim.Why calls are like insurance may be less obvious because a call contains a right to buy rather than a right to sell. Nevertheless, calls arelike insurance policies that insure participation in a price rise; theyprotect cash or liquid investments from missing a market rally. Putsinsure against the risk of being in the market; calls insure against therisk of being out of the market. Puts limit risk from a price decline,and calls insure against missing a rally. Although one loss is a “realloss” and the other loss an “opportunity loss,” the put and call optionsthat protect against these events are similar to insurance policies inevery respect.
Components of Insurance PremiumsInsurance companies consider ﬁve factors when calculating premiums.They ﬁrst look at the value of the asset being insured. If other factorsare equal, the more valuable the asset, the more expensive it is toinsure. The second factor is the deductible. The higher the deductible,the lower is the insurance premium because more of the risk is beingborn by the owner of the asset. A policy’s term, or time to expiration,is the third factor. The longer the term, the higher is the insurance premium. The fourth factor is interest rates, which inﬂuence insurancepremiums because insurance companies invest the premiums theyreceive until claims are paid. Rising interest rates cause insurance premiums to decline.The ﬁfth and ﬁnal factor is risk. Many components contribute torisk, such as the nature of the asset and the history of loss in that classof assets. For example, a company insuring a car might analyze theage and driving record of the driver, where the car is parked, andhow many miles per year it is driven. If other factors are equal, the
The Basics of Option Price Behavior • 51
higher the likelihood of damage to the car, the higher is the insurance premium.Insurance companies do not simply apply a mathematical formulato these ﬁve factors to determine the premium for a particular policy.They also consider what their competition is doing. Any differencebetween the market price of a policy and the company’s calculatedtheoretical value requires the insurance company to decide whetherto “meet the competition” and do business or to hold back and waitfor a better opportunity. Option traders also make such judgmentsall the time.
Options Compared to InsuranceCorresponding to the ﬁve components for insurance are six components for pricing options: the value of the underlying instrument,the strike price, the time to expiration, interest rates, dividends, andvolatility. With options, the price of the underlying stock or indexcorresponds to asset value in insurance. If other factors are constant,a higher price for the underlying usually translates into a higher valuefor an option.The distance between the current stock price and an option’s strikeprice corresponds to the deductible component in insurance. Thedeductible, remember, is the amount of risk borne by the insuredparty. An insurance policy with a $500 deductible, for example, meansthat a loss up to this level requires no payment from the insurancecompany. The same concept applies to options. An out-of-the-moneyoption is like a policy with a deductible: The option pays nothing ifthe underlying stock does not move beyond the strike price, like aninsurance policy expiring worthless. An at-the-money option, however,is similar to an insurance policy with no deductible; it has value—similar to a policy paying a claim—even if the stock price declines onlya little.The impact of time to expiration on option values is well known;option values decrease as expiration approaches. The same can be said
52 • Trading Options As a Professional
for insurance premiums because a six-month policy costs less than aone-year policy. What is not so obvious is exactly how time affectsoption prices. The subject of time erosion will be discussed in detaillater in this chapter.Changes in interest rates, the fourth factor, affect call and putvalues in opposite ways. An increase in interest rates causes call values to increase and put values to decrease. Fortunately, changes ininterest rates affect short-term option values minimally and thereforeusually have little impact on short-term speculative trading decisions.Nevertheless, professional market makers must be aware of the impactof interest rates on the pricing of arbitrage strategies discussed inChapter 6 because they can affect proﬁts signiﬁcantly.Dividends make up the extra component of option prices not present in the insurance analogy. The effect of dividends on option valuesis opposite that of interest rates. An increase in dividends tends todecrease call values and increase put values. Again, since dividendchanges affect option values in a minor way, they are of little consequence to speculative option traders. However, changing dividendsdo affect the pricing of arbitrage strategies for professional marketmakers, as discussed in Chapter 6.The ﬁnal component of option value, volatility, is conceptuallyidentical to the risk factor in insurance. Just as an increase in riskassessment increases insurance premiums, so too does an increase involatility increase option values. The subject of volatility is discussedin depth in Chapter 7 and in numerous other places in this book.Table 3-1 summarizes the analogy between insurance premiumsand option values, and Table 3-2 summarizes the impact of changesin the six factors on call and put values. In Table 3-2, direct effectmeans that, as the component increases and other factors remainconstant, the option value will also increase. Inverse effect meansthat as the component increases and other factors remain constant,the option value will decrease. “Price of Underlying,” for example,has a direct effect on call values and an inverse effect on putvalues.
The Basics of Option Price Behavior • 53
Table 3-1
Components of Value: Option Values Compared with InsurancePremiums
Insurance Policy
Option
Asset ValueDeductibleTimeInterest RatesRisk Premium
Price of UnderlyingStrike PriceTimeInterest Rates and DividendsVolatility Value
Table 3-2
Changing Components and Changing Option Values
Component
Effect on Call Value
Effect on Put Value
DirectInverseDirectDirectInverseDirect
InverseDirectDirectInverseDirectDirect
Price of UnderlyingStrike PriceTimeInterest RateDividendsVolatility
Note: Direct effect means that as the component increases and other factors remain constant, the optionvalue also will increase. Inverse effect means that as the component increases and other factors remainconstant, the option value will decrease.
Option-Pricing FormulasTheoretical option values can be calculated using one of a few mathematical formulas. Two professors at the University of Chicago, FisherBlack and Myron Scholes, developed the ﬁrst formula, commonlyknown as the Black-Scholes option-pricing model in 1973. This formula involves advanced calculus. Subsequently, other mathematicians created additional formulas that are generically known asbinomial option-pricing models. The mathematics behind these formulas are beyond the scope of this book but can be found in booksby J. C. Cox and S. A. Ross, M. Rubenstein, and J. Hull. The Op-EvalPro software that accompanies this text allows the user to choosebetween four formulas, the Black-Scholes model and three binomial
54 • Trading Options As a Professional
models. For educational purposes, you should compare option valuesgenerated by the different formulas with the same inputs. In mostcases, however, you will ﬁnd that the results vary little. Unless otherwise indicated, option values in exhibits in this book are calculatedwith a standard Black-Scholes model and presented in decimalsrounded to the second place.The following examples review how changes in the inputs to theoption-pricing formula affect call and put values. Each exampleis static; only one factor changes at a time, while all other factorsremain constant. Some dynamic examples will be presented later inthis chapter.
Call Values and Stock PricesTable 3-3 illustrates how call values change when the price of theunderlying and time to expiration change. Table 3-3 contains 11 rowsand 7 columns. The rows indicate different stock prices; the columnsindicate different days prior to expiration. By looking up and down thecolumns and across the rows, you can see how changes in underlyingprice or time to expiration or both cause changes in an option’s theoretical value.An example of how call prices change starts in row 6, column 1 ofTable 3-3. The stock price is 100, the call has 90 days to expiration, andthe theoretical value of the 100 Call is 6.53. If the stock price rises by1 to 101 (row 7, column1) and the other factors are unchanged, thetheoretical value of the 100 Call increases by 0.58 to 7.11. If the stockprice decreases by 1 to 99, the call value decreases by 0.54 to 5.99. Inboth cases, the call price changes less than the stock price. Lookinganywhere on the table, this relationship between stock price and optionvalue always holds. Assuming that factors other than price remain constant, with any time left until expiration, an option’s theoretical valuealways changes less than one-for-one with a change in the stock price.Furthermore, the ratio of option-price change to stock-price changevaries with the stock price and with time.
The Basics of Option Price Behavior • 55
Table 3-3
Row 11Row 10Row 9Row 8Row 7Row 6Row 5Row 4Row 3Row 2Row 1
Theoretical Values of 100 Call at Various Stock Prices and Days toExpiration (Interest Rate, 5%; Volatility, 30%; No Dividends)Col1
Col2
Col3
Col4
Col5
Col6
Col7
Stock 90Price Days
75Days
60Days
45Days
30Days
15Days
0Days
1051041031021011009998979695
9.058.377.727.096.495.925.384.864.383.933.51
8.397.717.056.425.825.254.724.213.743.312.91
7.676.976.305.665.074.503.983.493.042.632.26
6.866.135.444.804.193.633.122.652.231.861.53
5.915.124.393.713.092.532.041.611.250.940.70
5.004.003.002.001.000.000.000.000.000.000.00
9.668.998.347.717.116.535.995.464.974.504.06
For example, when the stock price rises by 1 from 97 to 98 at 45days, the call value rises by 0.45 from 3.04 to 3.49, or approximately45 percent of the stock-price change. In another situation, whenthe stock rises from 102 to 103 at 60 days, the call value rises by0.63 from 6.42 to 7.05, or approximately 63 percent of the stockprice change.Two conclusions can be drawn from Table 3-3. First, as stated earlier, option prices generally change less than stock prices. Second,the amount by which option prices change depends on the timeto expiration and the relationship of the stock price to the option’sstrike price.Figure 3-1 shows how call values change as stock prices change.The upper line (curved) contains call values at 90 days to expiration,like column 1 in Table 3-3. The middle line (curved) contains callvalues at 45 days to expiration, like column 4 in Table 3-3; and thelower line (two straight sections) contains call values at expiration, likecolumn 7 in Table 3-3. Figure 3-1 illustrates the same two conceptsfrom Table 3-3—that call values are directly correlated with stock
56 • Trading Options As a Professional
prices and that the level of correlation varies depending on therelationship of the stock price to the strike price.Call values are near zero when the stock price is signiﬁcantly belowthe strike price. Call values rise gradually at ﬁrst as the stock pricemoves up toward the strike price. The values then rise faster and fasteras the stock price reaches and then moves above the strike price.Finally, as the stock price soars signiﬁcantly above the strike price, thechange in call value approaches a one-for-one relationship withchange in stock price. In theory, however, call values never changeexactly one for one with stock prices because, in theory, the value ofthe call always will contain at least a slight time premium.
Put Values and Stock PricesThe same price-change characteristics apply to puts that apply to callswith one major difference: Put values are inversely correlated with181614
Theo Price
12108642086 88 90 92 94 96 98 100 102 104 106 108 110 112 114UnderlyingFigure 3-1
100 Call – Values at 90 Days, 45 Days, and Expiration
The Basics of Option Price Behavior • 57
stock prices. Thus put values fall as stock prices rise and rise as stockprices fall. Table 3-4 contains theoretical values of a 100 Put at various stock prices and days to expiration.An example of how put prices change starts in row 6, column 1 ofTable 3-4. The stock price is 100, the put has 90 days to expiration,and the theoretical value of the 100 Put is 5.31. If the stock price fallsby 1 to 99 (row 5, column1) and the other factors are unchanged, thetheoretical value of the 100 Put increases by 0.45 to 5.76. If the stockprice rises by 1 to 101, the put value decreases to 4.88. In both cases,the put price changes less than the stock price, and this relationshipis always true for puts, just as it is for calls. Furthermore, the ratio ofthe change in put price to change in stock price varies with stock priceand with time.For example, when the stock price falls by 1 from 99 to 98 at 45days, the put value rises from 4.36 to 4.89, or approximately 53 percent of the stock-price change. In another situation, when the stockrises from 103 to 104 at 60 days, the put falls from 3.23 to 2.89, orapproximately 34 percent of the stock-price change.Table 3-4
Row 11Row 10Row 9Row 8Row 7Row 6Row 5Row 4Row 3Row 2Row 1
Theoretical Values of 100 Put at Various Stock Prices and Days toExpiration (Interest Rate, 5%; Volatility, 30%; No Dividends)Col1
Col2
Col3
Col4
Col5
Col6
Col7
Stock 90Price Days
75Days
60Days
45Days
30Days
15Days
0Days
1051041031021011009998979695
3.033.353.704.074.474.905.365.846.366.917.48
2.572.893.233.604.004.434.905.405.936.497.09
2.062.352.693.053.453.894.364.895.436.026.64
1.451.722.032.392.783.223.714.244.825.456.12
0.700.931.181.501.882.322.833.404.044.745.49
0.000.000.000.000.000.001.002.003.004.005.00
3.443.764.114.494.885.315.766.246.747.287.84
58 • Trading Options As a Professional
Figure 3-2 shows how put values change as stock prices change.The upper line (curved) reﬂects put values at 90 days to expiration,like column 1 in Table 3-4. The middle line (curved) contains putvalues at 45 days to expiration, like column 4 in Table 3-4; and thelower line (two straight sections) contains put values at expiration, likecolumn 7 in Table 3-4. Like Table 3-4, Figure 3-2 illustrates that putvalues are inversely correlated with stock prices and that the level ofcorrelation varies depending on the relationship of the stock price tothe strike price.Put values are near zero when the stock price is signiﬁcantly abovethe strike price. Put values rise gradually as the stock price declinestoward the strike price. The values then rise faster and faster as thestock price reaches and then falls below the strike price. Finally, as thestock price drops signiﬁcantly below the strike price, the change in putvalue approaches a one-for-one relationship with change in stockprice. In theory, however, put values never change exactly one for onewith stock prices because, in theory, the value of a put always will contain at least a slight time premium.181614
Theo Price
12108642086 88 90 92 94 96 98 100 102 104 106 108 110 112 114UnderlyingFigure 3-2
100 Put – Values at 90 Days, 45 Days and Expiration
The Basics of Option Price Behavior • 59
DeltaThe ratio of option-price change to stock-price change is an important aspect of option-price behavior, and it is referred to as the deltaof an option. Speciﬁcally, delta is the change in option theoreticalvalue given a one-unit change in price of the underlying stock. Thischapter introduces delta, but Chapter 4 will discuss it in detail.Recall the example from Table 3-3. When the stock price rosefrom 100 to 101 at 90 days to expiration, and the 100 Call rose 0.58from 6.53 to 7.11, this call would be described as having a “delta of58.” Actually, the delta is 0.58, or 58 percent. This means that the100 Call is estimated to change in price by an amount equal to58 percent of the stock-price change. Look at a different examplefrom Table 3-4, where the stock price declined from 100 to 99, andthe 100 Put rose by 53 cents. This put would be described as havinga “delta of 53,” or negative 53 percent. This means that the 100 Putis estimated to change in price by 53 cents when the stock pricechanges by $1.
Call Values Relative to Put ValuesThe relationship between call and put values confuses many traders.Intuition may tell you that calls and puts with the same strike priceand the same expiration should have the same value if the stockprice is equal to the strike price. Actually, however, this parity doesnot happen. Assuming no dividends, call prices always will be greaterthan put prices because they contain an interest component thatputs lack. Evidence of this disparity appears from a comparison ofTables 3-3 and 3-4. Row 6 in both tables shows option values withthe stock price at $100. At 90 days (column 1), the 100 Call has avalue of 6.53, but the 100 Put has a value of 5.31. The call value isalso greater than the put value in every cell of row 6. These differences stem from an interest factor that is part of the call pricebut which is not part of the put price. When arbitrage strategiesare explained in Chapter 6, the reason for the interest factor willbecome clear.
60 • Trading Options As a Professional
In rows other than row 6 in Tables 3-3 and 3-4, the time value ofthe 100 Call is greater than the time value of the 100 Put. Comparethe values in row 4, column 2 of both tables, for example, where thestock price is $98, and there are 75 days to expiration. The value ofthe 100 Call is 4.86, all of which is time value. The value of the 100Put of 5.84 consists of $2.00 of intrinsic value and $3.84 of time value.In this example, the time value of the 100 Put is 1.02 less than thetime value of the 100 Call. As another example, consider row 9, column 5 in both tables, which is a stock price of $103 and 30 days toexpiration. The value of the 100 Call of 5.44 consists of $3.00 of intrinsic value and $2.44 of time value. The value of the 100 Put of 2.03 isall time value and 0.41 less than the time value of the 100 Call.The general concept of why call