prices are greater than put pricesis a relationship known as put-call parity, which is discussed in detailin Chapter 5. Put-call parity states that stock prices, call prices, andput prices must have a certain relationship with each other or therewill be arbitrage opportunities for professional traders to make nearlyriskless proﬁts. Always on the lookout for proﬁtable trades, professionalmarket makers constantly watch for market inefﬁciencies—whereprices are out of line with each other—and jump on the arbitrageopportunity. Because of the ﬁerce competition among market makers, such advantages exist only for very short periods of time, and the“inefﬁciency” generally amounts to only a few cents per share. Thelarge number of options that market makers trade every day, however,makes seizing these arbitrage opportunities proﬁtable.
Option Values and Strike PriceAs noted earlier, the strike price of an option is similar to thedeductible of an insurance policy. Increasing the deductible of aninsurance policy lowers the premium, or cost, of the policy, anddecreasing the deductible raises the premium. For options, if theunderlying stock price is at 100, the 100 Call is at the money, like a
The Basics of Option Price Behavior • 61
policy with no deductible. Raising the strike to 105 while keepingother factors such as stock price constant decreases the call value. Thishappens because with a stock price of 100, the 105 Call, being out ofthe money by ﬁve points, is like an insurance policy with a deductible.Table 3-5 shows that the value of the 100 Call falls from 6.53 to 4.37to 2.80 as the strike price rises from 100 to 105 to 110, respectively. Inthe language of insurance, as the deductible rises, that is, as the strikeprice gets further from the stock price, the premium decreases.Stock splits, such as two-for-one splits, are the most common reason that strike prices change, but these splits do not affect how far anoption is in the money or out of the money. If a stock splits two forone from 100 to 50, for example, then the original 110 Call is 10 percent out of the money, and the new 55 Call is also 10 percent out ofthe money. The change in option values after a two-for-one stock splitis more the result of the change in stock price than the change in strikeprice. There is little need to discuss changing strike price because itis extremely rare that a strike price changes while other factors remainconstant. A technical note is that in 2007, the method of handlingstock splits changed. Under the new method, with the exception ofTable 3-5
Effect of Increase in Strike PriceOriginalInputs
Stock priceStrike priceDividend yieldVolatilityInterest rateDays to exp.
1001000%30%5%60OriginalValues
100 Call:100 Put:
6.535.31
NewStrike
NewStrike
105
110
NewValues
NewValues
105 Call:105 Put:
4.378.08
110 Call:110 Put:
2.8011.45
62 • Trading Options As a Professional
two-for-one and four-for-one splits, the deliverable will be adjusted butnot the strike price or the premium-to-strike multiplier. For speciﬁcdetails refer to Securities and Exchange Commission (SEC) ReleaseNo. 34–55258.
Option Values and Time to ExpirationOption values decrease with the passage of time if all other factorsremain constant. There are two ways of illustrating this graphically.First, look back at Figures 3-1 and 3-2. These ﬁgures show optionprices at 90 days to expiration, 45 days to expiration, and at expiration.As time passes, the graphs approach the shape of the expiration proﬁtand-loss diagram, as represented by the straight lines.Figure 3-3 presents a different perspective on time erosion. Figure3-3A presents perhaps the best known illustration of time decay, that ofa nearly straight line until 30 days before expiration and then a precipitous drop to zero at expiration. This ﬁgure is based on the prices inrow 6 of Table 3-3, in which the stock price is 100. The at-the-money100 Call declines in value from 6.53 at 90 days to 5.92 at 75 days to5.25 at 60 days, etc. In this case, time decay affects the option valuerelatively little initially and relatively more as expiration approaches.Row 6 in Table 3-3 demonstrates that when one-half the time to expiration elapses, from 90 days to 45 days, only 31 percent of the 100 Callvalue erodes (6.53 to 4.50). A similar price/time relationship exists from60 days to 30 days when the 100 Call declines 31 percent from 5.25 to3.63 and from 30 days to 15 days when the decline is 30 percent from3.63 to 2.53. Row 6 in Table 3-4 reveals a similar time-decay patternfor the 100 Put when the stock price is $100.
ThetaThe name for time decay of options is theta. Theta is the theoreticalchange in option value given a one-unit change in time to expiration.The nonspeciﬁc term one unit could refer to one day, one week, orsome other time period. In the Op-Eval Pro computer program that
The Basics of Option Price Behavior • 63
7
(A)
6Theo Price
543210
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Time to Expiry
3.5
(B)
3Theo Price
2.521.510.505 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Time to Expiry
013.5
(C)
Theo Price
1312.51211.51110.510
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Time to Expiry
Figure 3-3 (A) At-the-Money Call and Time, (B) Out-of-the-Money Call andTime, (C) In-the-Money Call and Time
64 • Trading Options As a Professional
accompanies this book, the user can set the unit of time for theta.Chapter 4 discusses theta in depth with all the Greeks.
Time Decay Is ComplicatedUnfortunately for newcomers to options, the effect of time decay onoption values is not as simple as it may ﬁrst appear. Figure 3-3B andC shows that time decay for in-the-money and out-of-the-moneyoptions differ from time decay for at-the-money options (Figure 3-3A).Figure 3-3B demonstrates how time decay affects out-of-the-moneycalls. The assumptions are a call strike of 110, a stock price of 100,and a time period of the last 90 days to expiration. The nearly straightline contrasts sharply with the line in Figure 3-3A. Note in particularthat the least amount of time decay occurs during the last week,whereas in Figure 3-3A the greatest amount of time decay occurs during that week.Figure 3-3C shows that time erosion of in-the-money calls nearlymatches the time erosion of out-of-the-money calls. The ﬁgure graphsa 90 Call with a stock price of 100 during the last 90 days to expiration. Remember, only the time-value portion of an option decays.Consequently, with a stock price of 100, 90 days to expiration, and a90 Call value of 12.82, time decay will affect only 2.82 of this in-themoney option’s value. At expiration, the value of this call will equalits intrinsic value of 10.00.
Time Decay and VolatilityAn examination of Table 3-6 reveals the impact of volatility on timedecay. There are nine columns and three sections in the table. Eachcolumn represents a different number of days to expiration. The threemain sections of rows contain values of a 100 Call, a 105 Call, and a110 Call. Each section contains three different volatility assumptions.The percentages under the values will help you to see the differingrates of time decay.
The Basics of Option Price Behavior • 65
Table 3-6
Values of At-the-Money and Out-of-the-Money Calls at VariousLevels of Volatility*
Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Col 7
Col 8
Col 9
56Days
49Days
42Days
35Days
28Days
21Days
14Days
7Days
Exp.
2.0359%
1.6448%
1.1433%
00%
2.4248%
1.7033%
00%
3.2048%
2.2533%
00%
0.4329%
0.2215%
0.050%
00%
1.1539%
0.7225%
0.269%
00%
1.3530%
0.6114%
00%
0.012%
0.000%
00%
0.149%
0.021%
00%
0.4616%
0.103%
00%
100 Call3.43100%
3.1993%
2.9486%
20% Volatility2.6678%
2.3669%
30% Volatility4.98100%
4.6493%
4.2886%
3.9078%
3.4669%
2.9859%
40% Volatility6.54100%
6.1093%
5.6386%
105 Call1.48100%
1.2886%
1.0873%
5.1278%
4.5769%
3.9459%
20% Volatility0.8759%
0.6544%
30% Volatility2.92100%
2.6189%
2.2878%
1.9366%
1.5553%
40% Volatility4.44100%
4.0190%
3.5680%
110 Call0.52100%
0.4179%
0.3058%
3.0869%
2.5658%
1.9945%
20% Volatility0.2038%
0.1121%
0.0510%
30% Volatility1.59100%
1.3484%
1.0969%
0.8352%
0.5635%
0.3522%
40% Volatility2.90100%
2.5387%
2.1474%
1.7360%
1.3246%
0.8931%
Assumptions: Stock price, 100; no dividends; interest rate, 5%.* Values shown in percentages indicate rate of time decay.
66 • Trading Options As a Professional
The ﬁrst row in the upper section of Table 3-6, shows that withvolatility of 20 percent, the 100 Call declines in value from 3.43 at 56days to 2.36 at 28 days and to zero at expiration. Owing to timeerosion, this call loses 31 percent of its initial value of 3.43 in the ﬁrsthalf of its life and 69 percent in the second half. With volatility of30 and 40 percent, the rate of decay is nearly identical. The conclusion is that for at-the-money options, regardless of the level of volatility and assuming that other factors remain constant, loss of valueowing to time decay in the ﬁrst half of an option’s life equals onlyabout one-third of the initial value.For the 105 Call, which is 5 percent out of the money, the rate oftime decay is noticeably different from that of the at-the-money call andchanges when volatility changes. The ﬁrst row in the middle section ofTable 3-6 shows that with volatility of 20 percent, the 105 Call declinesin value from 1.48 at 56 days to 0.65 at 28 days and to zero at expiration. Thus 56 percent of the initial value is lost owing to time erosionin the ﬁrst half of the 105 Call’s life, and 44 percent is lost in the second half. With volatility of 30 percent, the rate of decay is slightly lessin the ﬁrst half and slightly more in the second half. The decline from56 days to 28 days is 47 percent versus 56 percent at 20 percent volatility. From 28 days to expiration, the 105 Call loses 53 percent of its valueat 56 days versus 44 percent when volatility was 20 percent. With volatility of 40 percent, the 105 Call declines even less in the ﬁrst half(42 percent) and more in the second half (58 percent).The 110 Call is 10 percent out of the money in this example andreﬂects a third impact of volatility on time decay. The ﬁrst row in thelower section of Table 3-6 shows that with volatility of 20 percent, the110 Call declines 79 percent from 56 days to 28 days and 21 percentfrom 28 days to expiration. With volatility of 30 percent, the rate ofdecay is less in the ﬁrst half (65 versus 79 percent) and more in thesecond half (35 versus 21 percent). With volatility of 40 percent,the 110 Call declines 54 percent in the ﬁrst half of its 56-day life and46 percent in the second half.
The Basics of Option Price Behavior • 67
Table 3-6 leads to three conclusions. First, out-of-the-money optionsdecay differently than at-the-money options. Out-of-the-money optionsdecay more in the ﬁrst half of their lives and less in the second half,whereas at-the-money options decay less initially and more as expiration approaches. Second, the further out of the money an option is, thegreater is the amount of time decay in the ﬁrst half of its life. Third,increasing volatility causes less value to erode in the ﬁrst half of an outof-the-money option’s life and more in the second half.
Alternatives for Premium SellersTable 3-6 should give pause to traders and investors who employ strategies that sell options consistently, especially if the sold options are 5 or10 percent out of the money. These option users, who are commonlyknown as premium sellers, should study the evidence in Table 3-6 thatsuggests an alternative strategy to selling one-month options that are 5or 10 percent out of the money. Table 3-6 suggests that under certaincircumstances, selling a two-month option, covering it one monthbefore expiration, and then selling the next two-month option can bringin more time premium than selling one-month options every month.
Option Values and Interest RatesFigure 3-4A shows that call values rise when interest rates rise, andFigure 3-4B shows that put values decline. These rises and declinesare a direct consequence of put-call parity explained in Chapter 5.Fortunately, the impact of interest rates is small. When interest ratesrise from 3 to 5 percent, the value of a 90-day at-the-money 100 Callrises from 6.29 to 6.53. These values assume a stock price of 100,volatility of 30 percent, and no dividends. Although rare, interest ratescan change 2 percent in fairly short periods of time. Such dramaticchanges in interest rates, however, always have been accompanied byother macroeconomic or global political events having equally dramatic impacts on stock prices and volatility. When compared with the
68 • Trading Options As a Professional
9
(A)
87
Theo Price
6543210
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Interest Rate
6(B)5
Theo Price
4
3
2
1
00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Interest RateFigure 3-4 (A) Call Values and Interest Rates, (B) Put Values andInterest Rates
The Basics of Option Price Behavior • 69
dramatic changes in stock prices and volatility, the impact of changing interest rates on option prices has been minor. Finally, althoughchanges in interest rates affect option values the least of all inputs, theycan have a signiﬁcant impact on professional traders who engage inarbitrage strategies, as explained in Chapter 6.
Option Values and DividendsThe impact of dividends on option prices is opposite that of interestrates. With no dividends, the call value exceeds the put value by thecost of money (interest rate). Dividends, however, effectively reducethe cost of money because the dividend proceeds can be used to paythe interest. Consequently, as dividends rise, the call value decreases,and the put value increases. Call and put values are equal when thedividend yield equals the interest rate.
Option Values and VolatilityVolatility, as will be discussed in detail in Chapter 7, is a measure ofstock price ﬂuctuation without regard to direction. The greaterthe volatility, the higher is an option’s price. Volatility is stated inpercentage terms. For example, the past price action of a particularunderlying security is said to “trade at 25 percent volatility,” or anoption’s theoretical value is said to be calculated “using 30 percentvolatility.”Figure 3-5B gives an example of how theoretical values of an atthe-money 100 Call change as the volatility changes from 0 to 50 percent, assuming a stock price of $100, a 4 percent interest rate, and nodividends. The upper line demonstrates values at 90 days to expiration,and the lower line shows values at 45 days to expiration. The ﬁgureshows that changing volatility is nearly linear for at-the-money optionsregardless of time to expiration.
70 • Trading Options As a Professional
1110
(A)
9
Theo Price
8765432100
2
4 6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50Volatility
7
(B)
6
Theo Price
543210–10 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50Volatility
Figure 3-5 (A) Volatility and At-the-Money Calls, (B) Volatility and 10% Outof-the-Money Calls
Figure 3-5B illustrates that changing volatility has a different impacton the value of out-of-the-money options. The lines in the ﬁgure showvalues of a 110 Call (10 percent out of the money) at 90 days (theupper line) and at 45 days (the lower line). The impact of volatilitydepends on both the distance to the strike price and the level ofvolatility.
The Basics of Option Price Behavior • 71
Extreme VolatilityExtreme volatility means that option values rise to their limit. The limitof value for a call is the stock price because no rational investor wouldpay more for a call than for the stock. For a put, the limit of value isthe strike price because prices cannot fall below zero. Figure 3-6 showshow prices of at-the-money calls rise when volatility rises to 1,000 percent. Although not shown, put values rise in a similar manner. Figures 3-5A and 3-6 both illustrate an at-the-money call but look differentbecause of the range on the x axis. The range of volatility for Figure3-5 is 0 to 50 percent, and for Figure 3-6 it is 0 to 1,000 percent.
Dynamic MarketsThe discussion to this point has assumed that only one component ofvalue changes while the rest stay constant. In the real world, of course,more than one component changes at a time. Many market forecastsdo not generally call for a stock to move up or down on the same day11010090
Theo Price
807060504030201000
100
200
300
400
500
600
700
800
900
Volatility
Figure 3-6 Call Values and Extreme Volatility 100 Call at 90 days and45 days; stock price 100
1000
72 • Trading Options As a Professional
while volatility remains unchanged. Rather, stock prices and volatility both can change over a period of several days or weeks, and changesin each of the three factors—stock price, time, and volatility—willaffect an option’s price differently. While interest rates do change, thechanges are typically small, and the impact on option values is negligible. Dividend changes typically occur only once each year and aresomewhat predictable, although special dividends, dividend suspensions, and deviations from past practices can cause option pricesto “adjust” when the news hits the market. The topic of dividends isdiscussed in Chapter 6.
Three-Part ForecastsWhile stock traders need to focus only on the direction of the stockprice, option traders must add two components—a forecast for timeand a forecast for the level of implied volatility. Chapter 7 discussesthe topic of implied volatility in depth.Back to Table 3-1. If a forecast called for the stock price to rise from$97 at 90 days to $101 at 75 days, the 100 Call price might be expectedto rise from 4.97 to 6.49. If, however, the stock price rise was expectedto take 15 days longer, until 60 days to expiration, then the call wouldbe expected to rise to 5.82, an outcome that is 0.67, or approximately40 percent, less proﬁtable.And what if volatility changed? The trading scenarios discussed nextexamine a forecast with changing volatility.
Trading ScenariosAssume that Joe is anticipating a bullish earnings report from Jumpco,a children’s playground equipment distributor. The report is due inthree days, and Joe believes that good news could send the stock price,currently $67, up 10 percent to $74 shortly thereafter. To analyze how
The Basics of Option Price Behavior • 73
much he might make if he buys the Jumpco April 75 Call and if hisforecast proves accurate, Joe creates Table 3-7. He starts with what heknows, a stock price of $67, a strike price of 75, interest rates of 4 percent, dividends of 2 percent, and 16 days to April expiration. Seeingthat the market price of the call is 50 cents, Joe calculates the impliedvolatility at 38 percent using the method that will be described inChapter 7.Scenario 1 in Table 3-7 forecasts a stock price rise from $67 to $74,or approximately 10 percent, in seven days, with volatility staying atTable 3-7
Need for a Three-Part Forecast
Scenario 1: Stock price up 10% in 7 days with volatility unchanged at 38%Original Inputs
New Inputs
Stock priceStrike priceDividend yieldVolatilityInterest ratesDays to expiration
67.0075.002.0%38.0%4.0%25
Outputs
Outputs
75 call value
0.50
74.0075.002.0%38.0%4.0%18
← Increase in stock price
2.10
← Call value up 320%
← Volatility constant← Decrease in days
Scenario 2: Stock price up 10% in 7 days with volatility declining to 25%Original Inputs
New Inputs
Stock priceStrike priceDividend yieldVolatilityInterest ratesDays to expiration
67.0075.002.0%38.0%4.0%25
Outputs
Outputs
75 call value
0.50
← Increase in stock price
74.0075.002.0%25.0%4.0%18
← Decrease in days
1.25
← Call value up 150%
← Decrease in volatility
74 • Trading Options As a Professional
Scenario 3: Stock price up 5% in 7 days with volatility declining to 25%Original Inputs
New Inputs
Stock priceStrike priceDividend yieldVolatilityInterest ratesDays to expiration
67.0075.002.0%38.0%4.0%25
Outputs
Outputs
75 call value
0.50
70.5075.002.0%25.0%4.0%18
← Increase in stock price
0.30
← Call value down 40%
← Decrease in volatility← Decrease in days
38 percent. Given these assumptions, Joe estimates that the JumpcoApril 75 Call will rise 320 percent to 2.10. While such a scenario isenticing, Joe does not immediately buy a call because he also realizesthat market action might differ from his forecast.Even if his stock-price and time forecasts are accurate, Joe wants toconsider the consequences of implied volatility declining. A littleresearch, as explained in Chapter 7, shows that 25 percent is a moretypical level of volatility for Jumpco options than the current level of38 percent. Joe therefore creates Scenario 2 to estimate the impactof volatility returning to 25 percent. As a result of these calculations,the estimated price of the April 75 Call falls to 1.25, a still impressive150 percent increase from the current price of 0.50.Joe is not through with his analysis yet. Although he is bullish onJumpco, Joe is curious about what would happen to the price of thecall if the stock price rose only 5 percent and not the 10 percent thathe is hoping for. Scenario 3 in Table 3-7 estimates that with the stockprice rising to $70.50, up approximately 5 percent from $67, in sevendays and with volatility decreasing to 25 percent, the Jumpco April 75Call declines from 0.50 to 0.30, for a loss of 20 cents, or 40 percent.The maximum risk of buying a call, of course, is the full cost ofthe call plus commissions, or 100 percent of the investment. Nevertheless, given the conﬁdence he has in his forecast and his willingness
The Basics of Option Price Behavior • 75
to risk the full investment, Joe decides to buy some Jumpco April75 Calls.The three-part forecast and multiple-scenario analysis that Joe usedto analyze the Jumpco April 75 Calls is what distinguishes speculativeoption trading from speculative stock trading. Option traders need toconsider the impact of both time and volatility when making tradingdecisions. Creating tables of theoretical values, such as presented inthis chapter, helps traders to develop realistic expectations aboutoption price behavior. The Op-Eval Pro software that accompaniesthis book and is explained in Chapter 2 was used to create these tables.The program can help traders to develop realistic expectations aboutoption prices and to select a strategy that matches their forecasts. Withpractice, any trader can master making these multiple scenarios.
SummaryOptions have value, in part, because they are similar to insurance. Putsinsure owned assets against a market decline, and calls insure cash orliquid assets from missing a rally. The factors that actuaries considerin determining insurance premiums correspond to the componentsused by formulas such as the Black-Scholes option pricing model tocalculate option theoretical values.The asset-value component of insurance premiums corresponds tothe underlying price component used to value options. The deductiblein insurance corresponds to the strike price. The time factor is the samefor both, but interest rates in insurance correspond to a combinationof both interest rates and dividends for options. Finally, the risk factorin insurance is similar to volatility in options.Option prices, prior to expiration, always move less than one forone with underlying stock-price changes. Delta represents theexpected option-price change given a one-unit change in the stock’sprice. The passage of time causes option prices to decrease—all otherpricing factors remaining constant. At-the-money options decrease
76 • Trading Options As a Professional
with the passage of time in a nonlinear manner, less initially and moreas expiration nears. The time decay of in-the-money and out-of-themoney options is different, increasing for a while and then slowing asexpiration nears. Interest rates directly affect call prices and inverselyaffect put prices. Dividends are the opposite of interest rates: Dividends up, call prices down, and put prices up, and vice versa.Volatility has a direct impact on option prices. The higher thevolatility, the higher both put and call prices will be. Volatility is ameasure of stock-price ﬂuctuation without regard to direction. Whileit is a statistical concept not grasped easily by nonmathematicians,volatility can be understood intuitively and incorporated, subjectively,into trading decisions.Markets are dynamic, not static, and the impact on option pricesof the factors discussed in this chapter can be confusing at ﬁrst. Nevertheless, realistic expectations about option-price behavior are essential, and analyzing multiple scenarios helps to better understand thepotential proﬁt and risks of contemplated strategies.
Chapter 4THE GREEKShis chapter discusses ﬁve indicators that traders use to estimatehow option prices and position risk change in dynamic marketconditions. The ﬁve indicators—delta, gamma, theta, vega,