$36 by the 38 slots on the wheel, the result is$36/38 = $.9474, or about 95¢. A player who pays 95¢ for the privilege of picking anumber at the roulette table can expect to break about even in the long run.
Of course, no casino will let a player buy such a bet for 95¢. Under those conditionsthe casino would make no profit. In the real world, a player who wants to purchasesuch a bet will have to pay more than the expected return, typically $1. The 5¢difference between the $1 price of the bet and the 95¢ expected return represents theprofit potential, or edge, to the casino. In the long run, for every dollar bet at the roulettetable, the casino can expect to keep about 5¢.
Given the above conditions, any player interested in making a profit would ratherswitch places with the casino so that he could be the house. Then he would have a 5¢edge on his side by selling bets worth 95¢ for $1. Alternatively, the player would liketo find a casino where he could purchase the bet for less than its expected return of 95¢,perhaps 88¢. Then the player would have a 7¢ edge over the casino.
THEORETICAL VALUE
The theoretical value of a proposition is the price one would expect to pay in order tojust break even in the long run. Thus far the only factor we have considered incetermining the value of a proposition is the expected return. We used this concept tocalculate the 95¢ fair price for the roulette bet. There may, however, be otherConsiderations,
Suppose that in our roulette example the casino decides to change the conditionsof the bet slightly. The player may now purchase the roulette bet for its expected return—_—__
1. Asis customary in the U.S., we assume a roulette wheel with 38 slots. In some parts of the world theToulette wheel may have no slot 00. This of course changes the odds.38 Chapter 3
of 95¢ and, as before, if he loses the casino will immediately collect his 95¢. Under thenew conditions, however, if the player wins the casino will send him his $36 winningsin two months. Will both the player and the casino still break even on the proposition?
Where did the player get the 95¢ he used to place his bet at the roulette wheel? Inthe immediate sense he may have taken it out of his pocket. But a closer examinationmay reveal that he withdrew the money from his savings account prior to visiting thecasino. Since he won’t receive his winnings for two months, he will have to take intoconsideration the two months interest he would have earned had he left the 95¢ in hissavings account. If interest rates are 12% annually (1% per month), the interest loss is2% Xx 95¢, or about 2¢. If the player purchases the bet for its expected return of 95¢,he will still be a 2¢ loser because of the cost of carrying a 95¢ debit for two months.The casino, on the other hand, will take the 95¢, put it in an interest-bearing account,and at the end of two months collect 2¢ in interest.
Under these new conditions the theoretical value of the bet is the expected returnof 95¢ less the 2¢ carrying cost on the bet, or about 93¢. If a player pays 93¢ for theroulette bet today and collects his winnings in two months, neither he nor the casinocan expect to make any profit in the long run.
The two most common considerations in a financial investment are the expectedreturn and carrying costs. There may, however, be other considerations. For example,suppose the casino decided to send the player a 1¢ bonus over the next two months.He could then add this additional payment to the previous theoretical value of 93¢ toget a new theoretical value of 94¢. This is similar to the dividend paid to owners of stockin a company. And, in fact, dividends are an additional consideration in evaluatingoptions on stock.
Exchanges will perhaps object to the casino analogy. They prefer that option tradingnot be thought of as gambling. There is certainly no desire here to assess the moralimplications of either gambling or option trading. The fact remains that the same lawsof probability which enable a casino to set the odds for different games of chance arethe same laws of probability which enable a trader to evaluate an option.
The concept of theoretical value based on probability is common in many aspectsof business. For those uncomfortable with the gambling analogy, one can go back to theoriginal justification for options and think of them as insurance policies which requirethe payment of a premium. Through the use of statistical data and probability theory,an actuary at an insurance company will attempt to calculate the likelihood that theinsurance company will have to make good on an insurance policy. He can then factorinto the equation what the insurance company expects to earn on premium payments,and thereby arrive at a theoretical value for the insurance policy. The policy can thenbe offered to prospective customers at an additional cost, which represents the theoreti-cal edge to the insurance company.
In the same way, the goal of option evaluation is to determine, through the use oftheoretical pricing models, the theoretical value of an option. The trader can then makean intelligent decision whether the option is overpriced or underpriced in the market-place, and whether the theoretical edge is sufficient to justify going into the marketplaceand making a trade.Introduction to Theoretical Pricing Models 39
A WORD ON MODELS
Before continuing, a few observations on models in general will be worthwhile.
A model is a scaled down or more easily managed representation of the real world.The model may be a physical one, such as a model airplane or building, or it may be amathematical one, such as a formula. In each case, the model is constructed to help uspetter understand the world in which we live. However, it is unwise, and sometimesdangerous, to assume that the model and the real world which it represents areidentical in every way. They may be very similar, but the model is unlikely to exactlyduplicate every feature of the real world.
All models, if they are to be effective, require us to make certain prior assumptionsabout the real world. Mathematical models require the input of numbers which quantifythese assumptions. If we feed incorrect data into the model, we can expect an incorrectrepresentation of the real world. Every model user must be aware: garbage in, garbageout.
These general observations about models are no less true for option pricing models.An option model is only someone’s idea of how options might be evaluated undercertain conditions. Since either the model itself, or the data which we feed into themodel, might be incorrect, there is no guarantee that model-generated values will beaccurate, nor can we be sure that these values will bear any logical resemblance to actualprices in the marketplace.
There is in fact a great deal of disagreement among traders as to the usefulness ofoption pricing models. Some traders feel that models are so much hocus-pocus, and haveno relationship to what goes on in the real world. Other traders feel that once they havea sheet of theoretical values in hand all their problems are solved. The reality liessomewhere in between.
Anew option trader is like someone entering a dark room for the first time, Withoutany guidance he will grope in the dark and may eventually find what he is looking for.The trader who is armed with a basic understanding of theoretical pricing models entersthe same room with a small candle. He can make out the general layout of the room,but the dimness of the candle prevents him from distinguishing every detail. Moreover,some of what he does see may be distorted by the flickering candle. In spite of theselimitations, a trader is more likely to find what he is looking for with a small candle thanwith no illumination at all.
The real problems with theoretical pricing models arise after the trader has acquiredsome sophistication. As he gains confidence he may begin to increase the size of histrades. When this happens, his inability to make out every detail in the room, as well asthe distortions caused by the flickering candle flame, take on increased importance.Now a misinterpretation of what he thinks he sees can lead to financial disaster, sinceany error in judgement will be greatly magnified.
The sensible approach is to make use of a model, but with a full awareness of whatit can and cannot do. Option traders will find that theoretical pricing models areinvaluable tools to understanding the pricing of options. Because of the insights gainedfrom a model, the great majority of successful option traders rely on some type of40 Chapter 3
theoretical pricing model. However, an option trader, if he is to make the best use of atheoretical pricing model, must be aware of its limitations as well as its strengths.Otherwise he may be no better off than the trader groping in the dark.2
A SIMPLE APPROACH
How might we adapt the concepts of expected return and theoretical value to thepricing of options? We might begin by calculating the expected return for an option.Let’s take a simple example.
Suppose an underlying contract is trading at $100 and that on a certain date in thefuture, which we will call expiration, the contract can take on one of five differentprices: $80, $90, $100, $110, or $120. Assume, moreover, that each of the five pricesis equally likely with 20% probability. The prices and probabilities might be representedby the line in Figure 3-1.
If we take a long position in the underlying contract at today’s price of $100, whatwill be the expected return from this position at expiration? 20% of the time we willlose $20 when the contract ends up at $80. 20% of the time we will lose $10 when thecontract ends up at $90. 20% of the time we will break even when the contract endsup at $100. 20% of the time we will make $10 when the contract ends up at $110. And20% of the time we will make $20 when the contract ends up at $120. We can writethe arithmetic:
(20% x $20) -(20% x $10) +(20% x 0} +(20% x $10) +(20% x $20) = 0
Since the profits and losses exactly offset each other, the expected return to thelong position is zero. The same reasoning will show that the expected return to a shortposition taken at the current price of $100 is also zero. Given the prices and prob-abilities, if we take either a long or short position we can expect to just break even inthe long run.
Now suppose that we take a long position in a $100 call. Forgetting for a momentabout what we might pay for the call, what will be the expected return given the pricesand probabilities in Figure 3-1? If the underlying contract finishes at $80, $90, or $100the call will expire worthless. If the underlying contract finishes at $110 or $120 thecall will be worth $10 and $20, respectively. The arithmetic is:
(20% x 0) +(20% x 0} +(20% x 0) +(20% x $10) +(20% x $20) = +86
Figure 3-1
$80 $90 $100 $110 $12020% 20% 20% 20% 20%
 
 
2. Two interesting articles discuss these limitations:Figlewski, Stephen; “What Does an Option Pricing Model Tell Us about Option Prices?,” FinaneialAnalysts Journal, Septeraber/October 1989, pages 12-15.Black, Fischer; “Living Up to the Model,” Risk, Vol. 3, No. 3., March 1990, pages 11-13.Introduction to Theoretical Pricing Models 41
The call can never be worth less than zero, so the expected return from the callposition is always a non-negative number, in this case $6.
If we want to develop a theoretical pricing model using this approach, we mightpropose a series of possible prices and probabilities for the underlying contract atexpiration. Then, given an exercise price, we can calculate the value of the option ateach price outcome, multiply the value by its associated probability, add up all thesenumbers, and thereby obtain an expected return for the option.
In the foregoing example we took a very simple situation with only five possibleprice outcomes, each with identical probability. What changes might we make in orderto develop a more realistic model? For one thing, we would have to know the settlementprocedure for the option. In the United States, all options are subject to stock-typesettlement, which requires full payment for the option. If the $100 call will have anexpected return of $6 at expiration, we will have to deduct the carrying costs to get itsvalue today. If interest rates are 12% annually (1% per month) and the option will expirein two months, we will have to discount the $6 expected return by the 2% carryingcost, or about 12¢. The theoretical value of the option will then be $5.88.
What other factors might we have to consider? We assumed that all five priceoutcomes were equally likely. Is this a realistic assumption? Suppose you were told thatonly two possible prices were possible at expiration, $110 and $250. With theunderlying contract at $100 today, which do you think is more likely? Based onexperience, most traders would probably agree that extreme price changes which arefar away from today’s price are less likely than small changes which remain close totoday’s price. For this reason, $110 is more likely than $250. To take this intoconsideration, perhaps our price outcomes, in terms of probability, ought to be concen-trated around the present price of the underlying contract. Such a distribution is shownin Figure 3-2. Now the expected return from a $100 call is:
(10% x 0) +(20% x 0) +(40% x 0) +(20% x $10) +(10% x $20} = $4.00
If, as before, the option is subject to stock-type settlement and carrying costs are2%, the theoretical value of the option will now be $3.92.
Note that in Figure 3-2 all outcomes and probabilities are arranged symmetrically.Even though the new probabilities altered the expected return for the $100 call, theexpected return from any position taken in the underlying contractis still zero. For eachupward price move, there is a downward move of equal magnitude and probability. WeMight, however, believe that the expected return to an underlying contract is not zero,that there is a greater chance that the contract will move one direction rather thananother. Look at the price outcomes and probabilities in Figure 3-3. Using these newprobabilities, the expected return from a long position in the underlying contract is:
—(10% x $20) -(20% x $10} +(30% x 0) +(25% x $10) +(15% x $20} = +$1.50and the expected return for the $100 call is:(10% x 0} +(20% x 0) +(30% x 0) +(25% x $10) +(15% x $20) = +$5.5042 Chapter 3
 
 
 
Figure 3-2$80 $90 $100 $110 $12010% 20% 40% 20% 10%Figure 3-3$80 $90 $100 $110 $12010% 20% 30% 25% 15%
Note that the underlying contract now has a positive expected return, so it may seemthat there is money to be made simply by purchasing the underlying contract. Thiswould be true if there were no other considerations. But suppose the underlyingcontract is a stock, and therefore subject to stock-type settlement. If we purchase thestock at today’s price of $100 and hold it for some period, there is a carrying costassociated with the investment. If the carrying cost is exactly equal to the expectedreturn of $1.50, we will just break even. For a long stock position to be profitable, thestock must appreciate by at least the amount of carrying costs over the holding period.Therefore, the expected return from the stock must be some positive number. If weassume that any stock trade will just break even, the expected return must be equal tothe carrying costs.
Some stocks also pay dividends. If the dividend is paid during the holding period,it will affect the expected return. A trader who buys stock will have to pay out carryingcosts, but he will receive the dividends. If we again assume thata stock trade will breakeven, the expected return at the end of the holding period must be identical to thecartying costs less the dividend. If the carrying cost for the stock over some period is$3.50, and a $1.00 dividend is expected during this period, the expected return at theend of the period must be $2.50. A trader who purchases the stock today will incur aninterest debit of $3.50 at the end of the holding period, but this will be exactly offset bythe $1.00 dividend which he receives during the holding period,® as well as the $2.50expected return at the end of the period.
Inan arbitrage-jree market, where no profit can be made by either buying or sellingacontract, all credits and debits, including the expected return, must exactly cancel out.If we assume an arbitrage-free market, we must necessarily assume that the forwardprice, the average price of the contract at the end of the holding period, is the currentprice, plus an expected return which will exactly offset all other credits and debits. Ifthe holding costs on a $100 stock over some period are $4, the forward price must be$104, If the stock also pays a $1 dividend, the forward price must be $103. In bothcases the credits and debits will exactly cancel out.
The calculation of a forward price depends on the characteristics of the contract aswell as market conditions. In the case of a stock, the considerations are the price of the
3. The trader can also earn interest on the dividend from the time he receives it until the end of the holdingperiod. Since this will usually be a very small amount in relation to the other factors, we will ignore it.Introduction to Theoretical Pricing Models 43
stock, the length of the holding period, interest rates, and dividends. In the case of afutures contract, the situation is much simpler. A futures contract requires no initial cashoutlay, since it is subject to futures-type settlement. Moreover, a futures contract doesnot pay dividends. This means that the forward price of a futures contract in anarbitrage-free market is simply the current price of the futures contract. If a trader buysa futures contract at $100, the break even price for the contract at the end of the holdingperiod is $100.
Going back to our very simple pricing model, we might make the assumption thatthe underlying market is arbitrage-free,4 that there is no money to be made from tradingthe underlying contract. The expected return must then be equal to the differencebetween the current price of the underlying market and its forward price. In the case ofstock, the expected return will be carrying costs less dividends. In the case of futures,the expected return will be zero.
Even if we assume an arbitrage-free market in the underlying, with appropriateprobabilities associated with each price outcome, we still have one major problem. Inour simplified model there were only five possible price outcomes, while in the realworld there are an infinite number of possibilities. To enable our model to more closelyapproximate real world conditions we will have to construct a probability line withevery possible price outcome and its associated probability. This may seem an impossibletask, but it is the basis for all theoretical pricing models.
We can now summarize the necessary steps in developing a model:
1. Propose a series of possible prices at expiration for the underlying contract2. Assign an appropriate probability to each possible price
3. Maintain an arbitrage-free underlying market4
From the prices and probabilities in steps 1, 2, and 3, calculate the expectedreturn for the option
5. From the option’s expected return, deduct the carrying cost
If we can accomplish all this, we will finally have a theoretical value from which we canbegin to trade.
Prior to 1973, evaluation of options required the solution of complex mathematicalequations. Since such methods were slow and tedious, a trader who tried to use themquickly found that profit opportunities disappeared faster than the evaluation methodscould identify them. In 1973, concurrent with the opening of the Chicago BoardOptions Exchange, Fischer Black and Myron Scholes introduced the first practicaltheoretical pricing model for options. The Black-Scholes Model, with its relativelysimple arithmetic and limited number of inputs, most of which were easily observable,proved an ideal tool for traders in the newly opened U.S. option market. Although otherModels have since been introduced to overcome some of its original deficiencies, theBlack-Scholes Model remains the most widely used of all option pricing models.
 
4. We need not necessarily asstime an arbitrage free underlying market. But we shall see that this is animportant assumption in most theoretical pricing models.44 Chapter 3
In its original form, the Black-Scholes Model was intended to evaluate Europeanoptions (no early exercise permitted) on non-dividend paying stocks. Shortly after itsintroduction, realizing that most stocks do pay dividends, Black and Scholes added adividend component. In 1976, Fischer Black made slight modifications to the model toallow for the evaluation of options on futures contracts. And in 1983, Mark Garmanand Steven Kohlhagen made several other modifications to allow for the evaluation ofoptions on foreign currencies.° The futures version and the foreign currency version areknown officially as the Black Model and the Garman-Kohthagen Model, respectively.But the evaluation method in each version, whether the original Black-Scholes Modelfor stock options, the Black Model for futures options, or the Garman-Kohthagen Modelfor foreign currency options, is so similar that they have all come to be known as simplythe Black-Scholes Model. The various forms of the model differ primarily in how theycalculate the forward price of the underlying contract, and an option trader will simplychoose the form appropriate to the underlying instrument.
The great majority of options currently traded are American options, carrying withthem the right of early exercise. For this reason, it may seem that the Black-Scholesmodel, with its assumption of no early exercise, is poorly suited for use in most markets.However, the Black-Scholes Model has proven so easy to use that many traders do notbelieve the more accurate values derived from an American option pricing model, whichallows for the possibility of early exercise, is worth the additional effort. In somemarkets, particularly futures options markets, the additional early exercise value is sosmall that there is virtually no difference between values obtained from the Black-Scholes model and values obtained from an American pricing model.
Due to its widespread use and its importance in the development of other pricingmodels, we will for the moment restrict ourselves to a discussion of the Black-Scholesmodel and its various forms. In later chapters we will consider the question of earlyexercise. We will also look at alternative methods for pricing options when we questionsome of the basic assumptions in the Black-Scholes Model.
The reasoning which led to the development of the Black-Scholes Model dependson the five steps we listed earlier in this chapter when we proposed a simple methodfor evaluating options. Black and Scholes worked originally with call values, but putvalues can be derived in much the same way. Alternatively, we will see in Chapter 11that in an arbitrage-free market there is a unique relationship between an underlyingcontract, and a call and put with the same exercise price and expiration date. Thisrelationship enables us to derive a put value simply by knowing the associated call value.
In order to calculate an option’s theoretical value using the Black-Scholes Model,we need to know at a minimum five characteristics of the option and its underlyingcontract. These are:
1. The option’s exercise price
2. The amount of time remaining to expiration
5. We are speaking here of options on a physical foreign currency, rather than options on a foreigncurrency futures contract. The latter may be evaluated using the Black Model for futures options.Introduction to Theoretical Pricing Models 45
3. The current price of the underlying contract4, The risk-free interest rate over the life of the option5. The volatility of the underlying contract
The last input, volatility, may be unfamiliar to the new trader. While we will putoff a detailed discussion of this input to the next chapter, from our previous discussionone can reasonably infer that volatility is related to the speed of the market.
If we know each of the required inputs, we can feed them into the theoreticalpricing model and thereby generate a theoretical value.
Black and Scholes also incorporated into their model the concept of the risklesshedge. For every option position there is a theoretically equivalent position in theunderlying contract such that, for small price changes in the underlying contract, theoption position will gain or lose value at exactly the same rate as the underlying position.To take advantage of a theoretically mispriced option, it is necessary to establish a hedgeby offsetting the option position with this theoretically equivalent underlying position.That is, whatever option position we take, we must take an opposing market positionin the underlying contract. The correct proportion of underlying contracts needed toestablish this riskless hedge is known as the hedge ratio.
Why is it necessary to establish a riskless hedge? Recall that in our simplifiedapproach an option’s theoretical value depended on the probability of various priceoutcomes for the underlying contract. As the underlying contract changes in price, theprobability of each outcome will also change. If the underlying price is currently $100and we assign a 25% probability to $120, we might drop the probability for $120 to10% if the price of the underlying contract falls to $80. By initially establishing a risklesshedge, and then by adjusting this hedge as market conditions change, we are taking intoconsideration these changing probabilities.
In this sense an option can be thought of as a substitute for a similar position in theunderlying contract. A call is a substitute for a long position; a put is a substitute for ashort position. Whether it is better to take the position in the option or in the underlying
Figure 3-4
Exercise Price
Time to Expiration
Theoretical » Theoreticai
Underlying Price Pricing Model Value
Interest Rates
Volatility
 
 46 Chapter 3
contract depends on the theoretical value of the option and its price in the marketplace,If a call can be purchased (sold) for less (more) than its theoretical value, it will, in thelong run, be more profitable to take a long (short) market position by purchasing (selling)calls than by purchasing (selling) the underlying contract. In the same way, if a put canbe purchased (sold) for less (more) than its theoretical value, it will, in the long run, bemore profitable to take a short (long) market position by purchasing (selling) puts thanby selling (buying) the underlying contract.
Since the theoretical value obtained from a theoretical pricing model is no betterthan the inputs into the model, a few comments on each of the inputs will beworthwhile.
EXERCISE PRICE
There ought never be any doubt about the exercise price of an option, since it is fixedin the terms of the contract and does not vary over the life of the contract.© ADeutschemark March 58 call traded on the Chicago Mercantile Exchange cannotsuddenly turn into March 59 call or a March 57 call. An IBM July 55 put traded on theChicago Board Options Exchange cannot turn into a July 50 put or a July 60 put.
TIME TO EXPIRATION
Like the exercise price, the option’s expiration date is fixed and will not vary. Our DMMarch 58 call will not suddenly turn into an April 58 call, nor will our IBM July 55 putturn into a June 55 put. Of course, each day that passes brings us closer to expiration,so in that sense the time to expiration is constantly growing shorter. However, theexpiration date, like the exercise price, is fixed by the exchange and will not change.
Time to expiration, like all inputs in the Black-Scholes Model, is entered as anannualized number. If we are entering raw data directly into the model we must makethe appropriate annualization. With 91 days remaining to expiration, we would enteran input of .25 (91/365 = .25). With 36 days remaining, we would enter .10 (36/365= 10). However, most option evaluation computer programs already have this transfor-mation incorporated into the software so that we need only enter the correct numberof days remaining to expiration.
It may seem that we have a problem in deciding what number of days to enter intothe model. We need the amount of time remaining to expiration for two purposes, tocalculate the interest considerations and to calculate the likelihood of movement in theunderlying contract. For volatility purposes in assessing the “speed” of the market weare only interested in trading days. Only on those days can the price of the underlyingcontract actually change. This might lead us to drop weekends and holidays from ourcalculations. On the other hand, for interest rate purposes we must include every day.
6. It ds true that an exchange may adjust the exercise price of a stock option if there is a stock split. Inpractical terms this is not really a change in the exercise price because the exercise price retains thesame relationship to the stock price. The characteristics of the option contract remain essentiallyunchanged.Introduction to Theoretical Pricing Models 47
If we borrow or lend money we expect the interest to accrue every day, no matter thatsome of the days are not business days.
It turns out that this is not really a problem. In calculating the “speed” of the marketwe observe only the price changes that occur on business days. But we can make slightchanges to this observed value and annualize the number before feeding it into thetheoretical pricing model, The result is that we can feed into our model the actualnumber of days remaining to expiration knowing that the model will interpret thenumber correctly.
PRICE OF THE UNDERLYING
Unlike the exercise price and time to expiration, the correct price of the underlying isnot always obvious. At any one time there is usually a bid price and an asked price, andit may not be clear whether we ought to use one or the other of these prices, or perhapssome price in between.
We have noted that the correct use of an option’s theoretical value requires us tohedge the option position with an opposing trade in the underlying contract. Thereforethe underlying price we feed into our theoretical pricing model ought to be the price atwhich we believe we can make the opposing trade. If we intend to purchase calls or sellputs, both of which are long market positions, we will have to hedge by selling theunderlying contract. In that case we ought to use the bid price since that is the price atwhich we can sell the underlying. On the other hand, if we intend to sell calls or buyputs, both of which are short market positions, we will have to hedge by purchasing theunderlying contract. Now we ought to use the asked price since that is the price atwhich we can buy the underlying,
In practice, the bid and offer are constantly changing, and many traders will simplyuse the last trade price as the basis for theoretical evaluation. But the last trade priceMay not always reflect the present market. Even the settlement price quoted in anewspaper may not accurately reflect the market at the close of business. The last tradeprice may show 751% for a contract, but the market at the close may have been 75%bid, 7514 offered. A trader who hoped to buy at 7514 would have very little chance ofbeing filled because of the difficulty of buying at the bid price. Even a purchase at somemiddle price, say 7594, may be unlikely if the market is very unbalanced with manymore contracts being bid for at 75'/4 than offered at 7514. For all of these reasons, anexperienced trader will rarely enter an option market without knowing the exact bidand offer in the underlying market.
INTEREST RATES
Since an option trade may result in either a cash credit or debit to a trader’s account,the interest considerations resulting from this cash flow must also play a role in optionevaluation. This is a function of interest rates over the life of the option.
The interest rate component plays two roles in the theoretical evaluation of options.First, it may affect the forward price of the underlying contract. If the underlying48 Chapter 3
contract is subject to stock-type settlement, as we raise interest rates we raise theforward price, increasing the value of calls and decreasing the value of puts. Secondly,the interest rate may affect the cost of carrying the option. If the option is subject tostock-type settlement, as we raise interest rates we decrease the value of the option. Inspite of the fact that the interest rate