The second General Widget example given assumed that you sold short an in-the-money option and that the price of the UI did not decline to belowthe strike price—in other words, the price
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However, you should use the annualized yields to compare two similarstrategies, not to compare one strategy with other types of investments.For example, you make 9 percent for a one-month investment, but you donot know what your return will be for the remaining 11 months of the year.You might be able to reinvest at only 5 percent and would have been betteroff investing in a certificate of deposit at 8 percent for a year
All discussions of return should also be tempered with the risk Onestrategy might make 10 percent while another strategy makes 9 percent Itmight be that the second strategy is still the best strategy because the risk
is significantly lower Think in terms of the amount of risk you are takingfor each unit of profit
Return-if-Exercised
The return-if-exercised is the return that the strategy will earn if one or
all of the short or written options are exercised The return-if-exercised isnot used if you have not sold short or written any options The return iscalculated by making the assumptions that the option is exercised and noother factor changes
The return is also affected by the type of transaction and account,which affect the carrying costs and the final position that the investor ownsafter the option is exercised
For example, in a covered call position, the return-if-exercised is thereturn on the investment if the underlying stock was called away Supposeyou are long 100 General Widget stock at $50 and short one General Widget
$45 call options at $7 The option expires in three months The return ifexercised would be the $2 profit on the option divided by the $50 price ofthe stock The annualized return would be ($2÷ $50) × (12 ÷ 3), or1/25×
4, or 16 percent
Note that the initial investment was assumed to be $50 for the stock.The return-if-exercised would be significantly different if the stock hadbeen bought on margin The cost of borrowing the money would then have
to be taken into account Also note that dividends or interest payments, ifany, should be taken into account, as well as the interest earned, if any, onthe proceeds of the short option All of these carrying-charge-type factorswill affect the return-if-exercised
Look at the same General Widget example but with these changes:the transaction is on margin, the broker loan is 12 percent, the holdingperiod is three months, the return on the short option premium is 10 per-cent, and there is a dividend of 4 percent Now, you would receive the $2profit plus an assumed $0.50 dividend (you must look closely at the chancesthat you will hold the position through the next dividend before makingthis assumption) plus an interest premium on the short option premium
of $0.175 ($7 option premium times 10 percent divided by 4), for a total
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income of $2.675 Expenses will be the cost of carrying the margin position
of $0.75 ($25 borrowed times 12 percent broker loan rate divided by 4).Thus, the net income will be $2.675− $0.75, or $1.925, on an investment of
$25, for an annualized return of 30.8 percent
The second General Widget example given assumed that you sold short
an in-the-money option and that the price of the UI did not decline to belowthe strike price—in other words, the price of the option did not change andthe stock was called away by the exercise But what if the price droppedbelow the strike price? The option would not have been exercised, and thepreceding calculation would not occur
This shows the main problem with calculating the return-if-exercised
It assumes that the option is exercised, which requires that you make anassumption on the price of the UI
Also note that there is a greater chance that the return-if-exercisedwill be an accurate description of the eventual return to you the deeperin-the-money the option is For example, writing a $40 call against an in-strument trading at $50 will give you a much greater reliability for expect-ing the return-if-exercised to be accurate than if you write a $60 call that isout-of-the-money
Return-if-Unchanged
The return-if-unchanged is the return on your investment if there is no
change in the price of the UI This calculation can be done on any optionstrategy It also assumes that the option price does not change and so de-scribes the most neutral future event For this reason, it is a popular return
to calculate It is often the starting point for the option strategist for tifying a possible investment Of course, the chances of the UI price being
iden-exactlyunchanged are very low As a result, this is just the starting pointfor analysis of the strategy, not the final analysis
The calculation is done in much the same manner as the if-exercised, except that the strategy can include multiple legs, or options.There can be different strikes and types in the calculation
return-However, the return-if-unchanged does not usually use different rities Further, it is not used in complex options strategies that use differentUIs For example, you will not see the return-if-unchanged calculated on aposition that includes options on both Treasury-bond and Treasury-notefutures
matu-Expected Return
The expected return is the possible return weighted by the probability of
the outcome Theoretically, you will receive the expected return from this
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strategy or trade You might not receive on this particular trade but shouldexpect to get in over a very large number of trades In effect, you are look-ing at the trade from the perspective of the casino owner: You know youmight lose on this particular bet, but you anticipate winning after hundreds
or thousands of bets have been made
The most common way to calculate the expected return is to takethe implied volatility and compute the probability of various prices based
on the implied volatility (see Chapter 5 for more details) It is assumedthat prices will describe a normal bell-shaped curve (though scientificstudies suggest this is not accurate, it is usually close enough for vir-tually all option strategies) The precise math is beyond the scope ofthis book, but the following is a simple illustration of the principle: As-sume that the expected distribution of prices, as suggested by the im-plied volatility, suggests that the chances are 66 percent that prices ofWidgeteria will stay within a range of $50 to $60 Your position has beenconstructed to show a profit of $1,000 if prices stay within that range.There is a 16.5 percent chance of prices trading above $60 and a sim-ilar chance of prices trading below $50 You will lose $1,000 if pricesmove above 60 or below 50 Your expected return is, therefore, the sum
of the potential profits and losses multiplied by their respective chances
of happening: (0.66 × 1,000) + (0.165 × −1,000) + (0.165 × −1,000),
or $330
Another example looks at the expected return from the perspective ofjust the price of the UI and what it implies for the price of the option Makethe absurd assumption that the price of Widgets R Us can only trade at aprice of $50 or $60 at expiration and that the current price is $55 Furtherassume that your study of implied volatility suggests that there is a 60 per-cent chance of prices ending at $60 and a 40 percent chance of ending at
$50 The expected return from this position is (0.60× $5) + (0.40 × −$5),
or $3− $2, or $1 This would then be a good value for an option, given allother things being irrelevant
The delta of an option is a very good approximation of the chance that
an option will end in-the-money This is not technically true but is closeenough for even the most picky of arbitrageurs
This type of analysis has the advantage of acknowledging that ferent strategies will have different variability of returns The return-if-unchanged can look identical for two completely different strategies thatdiverge wildly as soon as the price of the UI moves away from unchanged
dif-At the same time, it has the same advantage of being neutral to the ture direction of the market It assumes that there are equal chances of themarket climbing as falling As a result, it is recommended that option strate-gists try to concentrate on using this form of analysis if they have the capa-bility to calculate the expected return
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Return-per-Day
The return-per-day is the expected return each day until either expiration
or the day you expect to liquidate the trade For example, you might becomparing two covered call writing programs and want to know whichone is best Take the expected return and divide by the number of daysuntil expiration That way, you can compare two investments of differinglengths
Once again, the variability of possible returns can vary widely from thesimple case presented here The return-per-day should only be considered
a starting point, much the same way that the return-if-unchanged is a ing point
start-The best strategies to use the return-per-day are the strategies thatare more arbitrage or financing related, such as boxes or reversals Thevariability of the possible outcomes is fairly limited, so the return-per-daymakes more sense
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Trang 6C H A P T E R 4
Advanced Option Price Movements
ADVANCED OPTION PRICE MOVEMENTS
The concepts outlined in this chapter form the basis for the option gies in Part Two These concepts expand on the basics in Chapter 3 Theyare not necessary for most traders who are mainly looking at option strate-gies to hold to expiration
strate-The first topic in this chapter will be a quick introduction to optionpricing models, particularly the Black-Scholes Model Also discussed will
be the greeks and how they affect the price of an option; probability
dis-tributions and how they affect options; option pricing models and their vantages, disadvantages, and foibles and using them The final major topic
ad-will be the concept of delta neutral, which is a key concept for many of the
advanced strategies in this book
Which option should you buy? What if you are looking for the price ofWidget futures to move from 50 to 60 over the next four months? Do youbuy the option that expires in three months and roll it over near expiration?
Or do you buy the six-month option and liquidate it in four months? Theanswer to these questions is whichever option maximizes profit for a givenlevel of risk
To decide on an option, you need to find the fair value and teristics of the various options available for your preferred strategy Youneed to find out which option provides the best value, which requires
charac-an ability to determine the fair value of charac-an option charac-and to monitor thechanges in that fair value You must be able to determine the likely future
39
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price of that option, given changes in such critical components of optionsprices as time, volatility, and the change in the price of the underlyinginstrument (UI)
OPTION PRICING MODELS
Option pricing models help you answer key questions:
r What is a particular option worth?
r Is the option over- or undervalued?
r What will the option price be under different scenarios?
Option pricing models provide guidance, not certainty The output of
an option pricing model is based on the accuracy of the model itself as well
as the accuracy and timeliness of the inputs
Option pricing models provide a compass to aid in evaluating an option
or an option strategy However, no option model has yet been designed thattruly takes into account the totality of reality Corners are cut, so only anapproximation of reality is represented in the models The model is notreality but only a guide to reality Thus, the compass is slightly faulty, buthaving it is better than wandering blindly in the forest
Option pricing models allow the trader to deal with the complexity
of options rather than be overwhelmed Option pricing models provide aframework for analysis of specific options and option strategies They givethe strategist an opportunity to try out “what if” scenarios Although op-tion pricing models are not 100 percent accurate, they provide more thanenough accuracy for nearly all option trading styles The inability to ac-count for the last tick in the price of an option is essentially irrelevant fornearly all traders On the other hand, arbitrageurs, who are looking to makevery small profits from a large number of trades, need to be keenly aware
of the drawbacks and inaccuracies of option pricing models They mustlook at every factor through a microscope
One early book that was related to options pricing was Beat the
Mar-ket by Sheen Kassouf and Ed Thorp This book sold very well and lined a method of evaluating warrants on stocks, which are essentiallylong-term options on stocks However, these models that came beforethe Black-Scholes Model are rarely mentioned today mainly because oftwo factors: (1) they were not arbitrage models; and (2) options were notpopular, so few traders or academics were paying attention to optionspricing problems
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Arbitrage Models
An arbitrage model is a pricing model in which all the components of the
model are related to each other in such a way that if you know all of thecomponents of the model but one, you can solve for the unknown compo-nent This applies to all of the components It ties up all the factors relating
to the pricing of an option in one tidy package
Furthermore, an arbitrage model is a model that prices the option,given certain inputs, at a price where the buyer or seller would be am-bivalent between the UI and the option For example, a thoroughly rationalbettor would be ambivalent between being given $1 or putting up $1 withanother bettor and flipping a coin to see who wins the $2 The expectedreturn from both of these deals is $1
An arbitrage model attempts to do the same thing The expected returnfrom, say, owning 100 shares of Widgetmania at $50 should be exactly thesame as owning an option to buy the same shares
There are many different option pricing models The most popular isthe Black-Scholes Model Other models for pricing options are:
r Cox-Ross-Rubenstein (or Binomial) Model
r Garman-Kohlhagen Model
r Jump Diffusion Model
r Whalley Model
r Value Line Model
Each model takes a look at evaluating options from a different tive Usually the goal of the model is to better estimate the fair value of anoption Sometimes the goal is to speed up computation of the fair value
perspec-Black-Scholes Model
The first arbitrage model is the most famous and most popular optionpricing model—the Black-Scholes Model Professors Stanley Black andMyron Scholes were fortunate that they published their revolutionarymodel just as the Chicago Board Options Exchange (CBOE) was founded.The opening of the CBOE shifted the trading of options from a small over-the-counter backwater of the financial community to a huge and growingmarket and created a demand for greater information about options pric-ing The Black-Scholes was deservedly at the right place at the right time.The initial version of the Black-Scholes Model was for European op-tions that did not pay dividends They added the dividend component soonafter Mr Black made modifications to the model so that it could be usedfor options on futures This model is often called the Black Model Mark
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Garman and Steven Kohlhagen then created the Garman-Kohlhagen Model
by modifying the Black-Scholes Model so that it gave more accurate pricing
of options on foreign exchange All of these versions of the Black-ScholesModel are similar enough that they are often simply described generically
as the Black-Scholes Model
Another popular model is the Cox-Ross-Rubenstein, or Binomial,Model This model takes a different approach to the pricing of options.However, many option traders feel that it is generally more accurate thanthe Black-Scholes Models The main drawback, however, is that it is com-putationally more time consuming
The Black-Scholes Model is used only for pricing European options.Yet most options traded in the world are American options, which allowfor early exercise It has been found, however, that the increase in accuracyfrom using a true American-pricing model is usually not worth the greatercost in computational time and energy This is particularly true with op-tions on futures
Arbitrageurs will sometimes shift to an American pricing model when
a stock option gets near expiration or becomes deep in-the-money Theseare the circumstances when the chances of early exercise become morelikely and the greater accuracy of a model that prices American-style op-tions becomes more important
Another apparent oddity is that the Black-Scholes Model does not priceput options, only calls However, the price of a put can be found by usingthe model to price a call and using the put-call parity principle
The Black-Scholes Model assumes that two positions can be structed that have essentially the same risk and return The assumption
con-is that, for a very small move in either of the two positions, the price of theother position will move in essentially the same direction and magnitude
This was called the riskless hedge and the relationship between the two positions was known as the hedge ratio.
Generally speaking, the hedge ratio describes the number of the lying instrument for each option For example, a hedge ratio of 0.50 meansthat one half of the value of one option is needed to hedge the option Inthe case of a stock option, a hedge ratio of 0.50 would mean that 50 shares
under-of the underlying stock are needed to hedge one option In the case under-of anoption on a futures contract, a hedge ratio of 0.50 would mean that onehalf of a futures contract is needed to hedge the option Clearly, one can-not hold only one half of a futures contract, but that is how many would beneeded to theoretically hedge the option on that futures contract
The Black-Scholes Model assumes that the two sides of the positionare equal and that an investor would be indifferent as to which one he
or she wished to own You would not care whether you owned a call orthe UI if the call were theoretically correctly priced In the same way, a
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put would be a substitute for a short position in the UI This was a majorintellectual breakthrough Previously, option pricing models were basedmore on observing the past rather than strictly and mathematically looking
at the relationship of the option to the UI
An arbitrage model relies heavily on the inputs into the model for itsaccuracy Designing a model using gibberish for inputs will lead to a modelthat outputs gibberish The Black-Scholes Model takes these factors intoaccount:
r Current price of the UI
r Strike price of the option
r Current interest rates
r Expected volatility of the UI until expiration
r The possible distribution of future prices
r The number of days to expiration
r Dividends (for options on stocks and stock indexes)
Given this information, the model can be used to find the fair price ofthe option But suppose the current price of the option was known, andwhat was wanted was the expected volatility that was implied in the price
of the option No problem The Black-Scholes Model could be used to solvefor the expected volatility The model can be used to solve for any of thelisted factors, given that the other factors are known This is a powerfulflexibility
A further advantage of the model is that the calculations are easy Thevarious factors in the model lend themselves to easy calculation using asophisticated calculator or a simple computer The calculations with othermodels, which might give better results, take so long that they have limiteduse Option traders are usually willing to give up a little accuracy to obtain
an answer before the option expires!
The Black-Scholes Model is the standard pricing model for options Ithas stood the test of time All of the examples in this book, and virtuallyall other books, are derived using the Black-Scholes Model However, themodel has some drawbacks As a result, the model is no longer the standardfor options on bonds, foreign exchange, and futures, though the standardmodels for these three items are modifications of the original
Assumptions of the Black-Scholes Model
Examining the assumptions of the Black-Scholes Model is not done to icize the model but to identify its strengths and weaknesses so that thestrategist does not make a wrong move based on a false assumption
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Current Price of the UI The current price of the UI is usually knownwith some certainty for most option traders They can look on the screen
or call their broker and get a price for the UI It usually does not matter ifthe price quote is a little wrong
However, arbitrageurs often have a problem determining exactly whatthe price of the UI is They ask: How wide is the bid/ask spread? Is thelast trade on the bid, in the middle, or on the ask? Has the bid/ask spreadmoved since the last trade? Are prices extremely volatile, and will I have ahard time executing a trade at the current bid or ask because the bids andoffers are moving so much?
The Strike Price of the Option Fortunately, this one factor is stableand does not change significantly Strike prices for stock options do changewhenever there is a stock split or a stock dividend
Interest Rates The Black-Scholes Model assumes that setting up theright relationship between the UI and the option will lead to a neutral pref-erence by the investor The value of the UI and the value of the option will
be balanced because the Black-Scholes Model is an arbitrage model.The model assumes that the so-called risk-free rate is the proper rate.Traditionally, the risk-free rate is considered the rate paid on U.S govern-ment securities, specifically, Treasury bills, notes, and bonds
To make the model work, it is assumed that interest is being paid orreceived on balances It is assumed that all positions are financed, an as-sumption that is reasonable because there is always an opportunity costeven if the position is not financed The Black-Scholes Model assumes thatyou would invest your money in Treasury bills if you did not invest it in anoption
The term of the interest rate used in the model should be the term
to expiration of the option For example, if you are pricing an option thatmatures in 76 days, then you should theoretically use the interest rate cor-responding to a Treasury bill that matures in 76 days In the real world, ofcourse, you would simply select a Treasury bill that matures close to thatperfect number of days
The problem is that the model assumes that you both invest yourmoney and borrow money at the risk-free rate It is quite reasonable
to assume that you will invest your money in Treasury bills in the realworld However, only the U.S government can borrow at the Treasury-billrate All other borrowers must pay more, sometimes much more As aresult, some options traders assume that they invest at the Treasury-billyield but that they borrow at the Eurodollar yield or at the prime rate Ingeneral, the rate assumed in the model will have little effect on the price ofthe option The level of interest rates mainly affects the price of multiyearoptions