7/28/2019 corp09

1/27

1

CHAPTER 9: Derivative Securities and Risk Management

A. Risk M anagement and Derivative Securities: An Introduction

This is the first of three chapters concerned with treasury management. The focus of this

chapter is risk management for the corporation. A derivative security is simply a financialinstrument whose value is derived from that of another security, financial index or rate. A largenumber of different types of derivative securities are becoming increasingly important formanagement of a variety of different types of risks. Understanding of derivative securities is alsoimportant because the pricing of relevant derivative securities can provide useful informationregarding characteristics of the securities from which they derive their values. Thus, derivativesecurity pricing models, particularly the option pricing methodology, can very useful for thepricing of stocks, bonds and many other securities as well.

Some observers of financial markets have argued that derivative securities play noimportant productive role in our economy. It has been argued that derivative securities merelyprovide gambling opportunities to speculators, enabling them transfer wealth among themselves

without contributing to the economy. However, it can also be argued that derivative securitiesplay a very important role in improving the efficiency of capital markets. Clearly, active andefficient capital markets are essential to the output and growth of our economy. Business firmsrequire capital resources to produce goods and services and to maintain jobs for our labor force.Firms obtain this capital from investors, who, either directly or through financial institutions,purchase the securities that firms sell. Poorly functioning capital markets inhibit productioncapabilities of business, and result in economic decline. However, risk and uncertainty areunavoidable characteristics of a capitalist economy. Businesses and the capital markets in whichinvestors entrust their savings are profoundly affected by risk. Because investors and businessestend to shy away from risk, the proper functioning of capital markets is impaired by uncertaintyand volatility. Investors will be reluctant to invest and businesses will reduce output and

employment when they are unable to control their exposure to risk. Whereas consumers controlpersonal risks associated with illness or casualty losses by purchasing insurance contracts,financial risks are often controlled with positions in derivative securities.

There exist a huge variety of derivative securities. Some of the more frequently tradedderivatives are the following:

Futures contracts which provide for the transfer of a given asset at an agreed toprice at a future date,Options which confer the right but not obligation to buy or sell an asset at a pre-specified price on or before a given date,Swaps which provide for the exchange of one set of cash flows for another set of

cash flows, andHybrids combining features of two or more securities (e.g: a convertible bond).

Derivative securities in general are not an exotic invention of modern Wall Street securitydealers; derivatives have played important roles in world economies for centuries. Earlyexamples include agricultural futures contracts dating back to c.2000 B.C. in India and optioncontracts on ship cargos created by ancient Phoenicians and Greeks. Options and futurescontracts were used in 17th century markets in Amsterdam as well as in Osaka. Futures markets

7/28/2019 corp09

2/27

2

started to become more organized in the 19 th century as exchanges developed to trade contracts.Such securities have helped manufacturers, farmers, mining companies, traders and transportersmanage a multiplicity of risks ranging from stock market volatility to borrower default.

The markets for derivatives are crucial to business for the management of many types of risks. Risk factors frequently hedged with derivatives include, but are not limited to uncertainties

associated with interest and exchange rates, client default, economy-wide and industry specificoutput. Markets for explicit insurance policies on such a wide array of risks do not exist largelydue to contracting costs. Most insurance policies are fairly standardized (e.g: health, life andmany casualty policies), while customized insurance contracts are expensive and time consumingto write. Businesses must be able to act quickly to manage their risks in this environment of rapidchange. Flexibility and liquidity along with low contracting and transactions costs are key to thesuccess of the risk management operations of a firm. An active and efficient market forderivative securities can meet these important requirements.

Business firms and individual investors desiring to hedge risks are not the onlyparticipants in markets for derivatives. A second type of market participant is the speculator whotakes a position in a security based on his expectation regarding future price movement.

Although the speculator is essentially concerned with his own trading profits, he plays animportant role in maintaining liquidity in derivative markets, affording business and individualinvestors the opportunity to hedge risks quickly and efficiently. The speculator is often thecounterparty to a hedger's trade, selling or purchasing derivatives as required by hedgers.

The arbitrager, who exploits situations where derivatives are mis-priced relative to oneanother not only provides additional liquidity to derivative markets, but plays an important rolein their pricing. By constantly seeking price misalignments for a variety of types of securities,and by understanding the payoffs of securities relative to one another, arbitragers help ensurethat derivative securities are fairly priced. This activity reduces price volatility and uncertaintyfaced by hedgers.

Whereas hedgers, speculators and arbitragers all play an important role in the pricing of and liquidity maintenance for derivatives, derivatives actually play a role in the evaluation of risks and prices for other securities. For example, it is well known that futures prices providevaluable information for predicting commodity prices and interest and exchange rates. Thisinformation is most useful for business planning. In fact, one very interesting study by RichardRoll of UCLA showed that market prices of orange juice futures anticipated severe winterweather conditions more accurately than did the National Weather Bureau forecasts. In addition,sophisticated stock analysts realize that the price of a stock option is most sensitive to the risk of the underlying stock. Thus, analysts frequently rely on the option price to provide information onthe risk of the underlying stock. Market prices of derivative securities in general are quite usefulfor assessing the magnitude and pricing of a multitude of different risks.

Derivative securities are traded in the United States either on exchanges or in the so-called Over the Counter (OTC) markets. Substantial market interest is required for exchangelisting, whereas securities with smaller followings or even customized contracts can be tradedover the counter. The role of the derivatives dealer is essentially the same as that for othersecurity dealers: to facilitate transactions for clients at competitive prices. Derivative dealersmatch counterparties for derivative contracts, act as a counterparty for many of their own customcontracts and provide an array of support services including expert advice and carefullyengineered customized risk management products. I t is necessary that the dealer providing fullsupport services have a proper understanding of his client, his business and the client's needs.

7/28/2019 corp09

3/27

3

Since most clients do not have an understanding of the technical terms used in the industry, thedealer must be an effective communicator. It is equally important for the dealer to understand thenature of the securities with which he deals and how serving as a market maker for derivativesaffects the risk structure of his employer. This understanding usually requires strong analyticalskills.

While derivatives do play an important role in our economy by enabling investors tohedge risk, speculate and price other assets, a number of well-publicized difficulties have arisenfrom their use (and mis-use). First, many derivative contracts have been created to be highlyleveraged, so that very large profits or losses may result from relatively small price shifts in theunderlying asset. This feature makes it crucial for financial managers to fully understand thefeatures and implications of the contracts with which they deal. However, many of thesecontracts are somewhat complicated such that many of their users are not able to fullyunderstand their implications. This has led to some very highly publicized losses and failures,including those at Procter & Gamble (over $100 million in interest rate contracts), GibsonGreetings (over $20 million), Orange County ($1.7 billion in interest rate derivatives) andMetallgesellschaft ($1.4 billion in oil futures). Second, tracking their values and reporting them

on accounting statements are very difficult because many contracts do not conform tocharacteristics of assets typically reported by accountants. Even recent Financial AccountingStandards Board (FASB) opinions regarding "marking to the market" are not very helpfulbecause most derivative contracts either have at best a very thin market and are subject toextreme pricing volatility. Accounting difficulties make it practically impossible for investors,regulators, managers and even auditors to understand the impact of derivatives investment onfirms. These difficulties have led to a number of frauds that ultimately caused failures, includingBarings Bank ($1.4 billion in Nikkei-index derivatives). It seems that financial innovators havebeen able to develop new derivative securities faster than accountants and regulators have beenable to develop technique for tracking their values and implications.

B. Options Contracts

A stock option is a legal contract that grants its owner the right (though, not obligation) toeither buy or sell a given stock. There are two types of stock options: puts and calls. A call grantsits owner to purchase stock (called underlying shares) for a specified exercise price (also knownas a striking price or exercise price) on or before the expiration date of the contract. In a sense, acall is similar to a coupon that one might find in a newspaper enabling its owner to, for example,purchase a roll of paper towels for one dollar. If the coupon represents a bargain, it will beexercised and the consumer will purchase the paper towels. If the coupon is not worth exercising,it will simply be allowed to expire. The value of the coupon when exercised would be theamount by which value of the paper towels exceeds one dollar (or zero if the paper towels areworth less than one dollar). Similarly, the value of a call option at exercise equals the differencebetween the underlying market price of the stock and the exercise price of the call.

Suppose, for example, that a call option with an exercise price of $90 currently exists onone share of stock. The option expires in one year. This share of stock is expected to be wortheither $80 or $120 in one year, but we do not know which at the present time. If the stock wereto be worth $80 when the call expires, its owner should decline to exercise the call. It wouldsimply not be practical to use the call to purchase stock for $90 (the exercise price) when it canbe purchased in the market for $80. The call would expire worthless in this case. If, instead, the

7/28/2019 corp09

4/27

4

stock were to be worth $120 when the call expires, its owner should exercise the call. Its ownerwould then be able to pay $90 for a share that has a market value of $120, representing a $30profit. In this case, the call would be worth $30 when it expires. Let T designate the options termto expiry, S T the stock value at option expiry and c T be the value of the call option at expiry. Thevalue of this call at expiry is determined as follows:

(1) ],0[ XSMAXc T T = When S T =80, C T =MAX[0, 80 90] =0When S T=120, C T =MAX[0, 120 90] =30

A put grants its owner the right to sell the underlying stock at a specified exercise priceon or before its expiration date. A put contract is similar to an insurance contract. For example,an owner of stock may purchase a put contract ensuring that he can sell his stock for the exerciseprice given by the put contract. The value of the put when exercised is equal to the amount bywhich the put exercise price exceeds the underlying stock price (or zero if the put is neverexercised).

To continue the above example, suppose that a put option with an exercise price of $90currently exists on one share of stock. The put option expires in one year. Again, this share of stock is expected to be worth either $80 or $120 in one year, but we do not know which at thepresent time. If the stock were to be worth $80 when the put expires, its owner should exercisethe put. In this case, its owner could use the put to sell stock for $90 (the exercise price) when itcan be purchased in the market for $80. The put would be worth $10 in this case. If, instead, thestock were to be worth $120 when the put expires, its owner should not exercise the put. Itsowner should sell for $90 for a share that has a market value of $120. In this case, the call wouldbe worth nothing when it expires. Let p T be the value of the put option at expiry. The value of this put at expiry is determined as follows:

(2) p T =MAX[0, X S T]When S T=80, p T =MAX[0, 90 80] =10When S T=120, p T =MAX[0, 90 120] =0

The owner of the option contract may exercise his right to buy or sell; however, he is notobligated to do so. Stock options are simply contracts between two investors issued with the aidof a clearing corporation, exchange and broker which ensure that investors honor theirobligations to each other. The corporation whose stock options are traded will probably not issueand does not necessarily trade these options. Investors, typically through a clearing corporation,exchange and brokerage firm, create and trade option contracts amongst themselves.

For each owner of an option contract, there is a seller or "writer" who creates the

contract, sells it to a buyer and must satisfy an obligation to the owner of the option contract. Theoption writer sells (in the case of a call exercise) or buys (in the case of a put exercise) the stockwhen the option owner exercises. The owner of a call is likely to profit if the stock underlyingthe option increases in value over the exercise price of the option (he can buy the stock for lessthan its market value); the owner of a put is likely to profit if the underlying stock declines invalue below the exercise price (he can sell stock for more than its market value). Since the optionowner's right to exercise represents an obligation to the option writer, the option owner's profitsare equal to the option writer's losses. Therefore, an option must be purchased from the option

7/28/2019 corp09

5/27

5

writer; the option writer receives a "premium" from the option purchaser for assuming the risk of loss associated with enabling the option owner to exercise.

Most stock options in the United States are traded on an exchange. The largest optionsexchange is the Chicago Board Options Exchange. The American, New York, Philadelphia andPacific Exchanges also maintain stock options trading.

The Options Clearing Corporation (OCC), jointly owned by the options exchanges, istechnically the counterparty for all options transactions in the United States. This means that alloption buyers own options that were written by the OCC and all option writers are obligated tothe OCC. This arrangement, given the financial stability of the OCC and all of the exchangeswhich own and back its obligations and all of the brokerage firms that back the obligations of theexchanges effectively eliminated default risk in listed options trading.

Options are also traded on a variety of different commodities, currencies and otherfinancial instruments. For example, option positions can be taken on commodities like oil, ormore easily taken on futures positions on such commodities or other instruments. Options tradeon stock market and other indices as well as on currencies and fixed income instruments.

Options may be classified into either the European variety or the American variety.

European options may be exercised only at the time of their expiration; American options maybe exercised any time before and including the date of expiration. Most option contracts traded inthe United States (and Europe as well) are of the American variety. We will demonstrate in thenext section that American options can never be worth less than their otherwise identicalEuropean counterparts.

An Introduction to Option PricingBlack and Scholes provided a derivation for an option-pricing model based on the

assumption that the natural log of stock price relatives will be normally distributed. 1 Theassumptions on which the Black-Scholes Options Pricing Model and its derivation are based areas follows:

1. There exist no restrictions on short sales of stock or writing of call options.2. There are no taxes or transactions costs.3. There exists continuous trading of stocks and options.4. There exists a constant riskless interest rate that applies for both borrowing and

lending.5. The range of potential stock prices is continuous.6. The underlying stock will pay no dividends during the life of the option.7. The option can be exercised only on its expiration date; that is, it is a European

Option.8. Shares of stock and option contracts are infinitely divisible.9. Stock prices follow an to process; that is, they follow a continuous time random

walk in two dimensional continuous space. This simply means that stock pricesare randomly distributed (in a manner somewhat similar to a normal distribution)and can take on any positive value at any time.

1 The stock price relative for a given period t is defined as (P t-Pt-1)P t. Thus, the log of the stock price relative isdefined as ln[(P t-Pt-1)P t].

7/28/2019 corp09

6/27

6

From an applications perspective, one of the most useful aspects of the Black-ScholesModel is that it only requires five inputs. All of these inputs with the exception of the variance of underlying stock returns are normally quite easily obtained: 2

1. The current stock price (S 0): Use the most recent quote.

2. The variance of returns on the stock (F 2

): Several methods will be discussed later.3. The exercise price of the option (X): Given by the contract4. The time to maturity of the option (T): Given by the contract5. The risk-free return rate (r f ): Use a treasury issue rate with an appropriate term to

maturity.

It is important to note that the following less easily obtained factors are not required as modelinputs:

1. The expected or required return on the stock or option and2. Investor attitudes toward risk

If the assumptions given above hold, the Black-Scholes model specifies that the value of a call option is given as follows:(8) )()( 2100 dN

eX

dNSc Tr f =

(9) T

TrXSd f

)5.()/ln( 201

++=

(10) Tdd = 12

where N(d*) is the cumulative normal distribution function for (d*). This function is frequentlyreferred to in a statistics setting as the "z" value for (d*). From a computational perspective, onewould first work through Equation (9), then Equation (10) before valuing the call with Equation(8).

N(d1) and N(d 2) are areas under the standard normal distribution curves (Z values).Simply locate the Z value on an appropriate table corresponding to the N(d 1) and N(d 2) values.Consider the following simple example of a Black-Scholes Model application: An investor hasthe opportunity to purchase a six month call option for $7.00 on a stock which is currently sellingfor $75. The exercise price of the call is $80 and the current riskless rate of return is 10% perannum. The variance of annual returns on the underlying stock is 16%. At its current price of

$7.00, does this option represent a good investment? First, we note the model inputs in symbolicform:

t =.5 r f =.10 e . 2.71828X =80 2 =.16

2 These five inputs are the only that are necessary if the assumptions underlying the model hold. The sample sourcesfor deriving input values may or may not be the most appropriate for a given contract.

7/28/2019 corp09

7/27

7

=.4 S 0 =75

Our first steps are to find d 1 from Equation (9) and d 2 from Equation (10):

d1 ={ln(75/80) +(.1 +.5*.16)*.5} {.4 * %.5}={ln(.9375) +.09} .2828 =.09

d2 =d1 - .4*%.5 =.09 - .2828 =-.1928

Next, by either using a Z-table or by using an appropriate polynomial estimating function from astatistics manual, we find normal density functions for d 1 and d 2:

N(d1) =N(.09) =.536 ; N(d 2) =N(-.1928) =.420

Finally, we use N(d 1) and N(d 1) in Equation (8) to value the call:

c0 =75(.536) - [80 @.9512] @(.42) =8.20

Since the 8.20 value of the call exceeds its 7.00 market price, the call seems to be a goodpurchase.

Put-Call ParityBefore proceeding with pricing models applicable to the valuation of call options, we will

first discuss a simple model concerning the relationship between put and call values. When thisrelationship holds, one is able to value a put based on knowledge of a call with exactly the sameterms. First, assume that there exists a European put (with a current value of p 0) and a Europeancall (with a value of c 0) written on the same underlying stock that currently has a value equal toX. Both options expire at time T and the riskless return rate is r f . The basic Put-Call EquivalenceFormula is as follows:

(1)000 pSXec

Tr f +=+

That is, a portfolio consisting of one call with an exercise price equal to X and a purediscount riskless note with a face value equal to X must have the same value as a secondportfolio consisting of a put with exercise price equal to X and one share of the stock underlyingboth options.

.......

Proof:

Assume that portfolio A consists of one call with an exercise price equal to X and a purediscount riskless note with a face value equal to X. Portfolio B consists of a put with exerciseprice equal to X and one share of the stock underlying both options. Regardless of the final stockprice, the two portfolios will have the same terminal values:

7/28/2019 corp09

8/27

8

OUTCOME If XS 1 If XS >1 Ending stock price (S1): 1S 1S Ending Put Price (p1): 1SX 0

Ending Call Price (c1): 0XS 1 Ending Treasury Bill Value: X X

Ending Value for Portfolio A: X1S

Ending Value for Portfolio B: X1S

The values of both portfolios are X if the stock's ending value is low; if the stock's endingvalue is high, the value of both portfolios is S 1. Since the value of portfolio A must equal thevalue of portfolio B in each case at time 1, their current (time 0) values must be identical.

.......

A very useful implication of the put call parity relation, we can easily derive the price of a put given a stock price, call price, exercise price and riskless return:

(2)

C. Forward and Futures Contracts

A forward contract represents an agreement specifying delivery of given quantity of anasset at a later date at a given price. Essentially, a forward contract is an agreement providing fora seller (an entity said to be taking the short position on the asset) to deliver the specified asset at

a future date to a purchaser (an entity said to be taking the long position on the asset). The actualexchange of the asset for cash occurs at a date subsequent to the date the forward contract isinitiated. A large number of forward transactions involve direct negotiations between banks,brokerage firms and other financial institutions.

A futures contract also represents an agreement specifying delivery of given quantity of an asset at a later date at a given price. However, the futures contract differs from a forwardcontract in several important ways. First, a forward contract is created by its long and shortparticipants according to whatever terms they specify. However, a futures contract is created by aclearing house which acts as a middleman between the contract participants. The clearing servesas counterparty on all transactions which effectively eliminates default risk. A futures contract istraded on an exchange. To facilitate trading and to ensure that a reasonably large number of

investors will be interested in the futures contract, the futures contract is standardized withrespect to the exact quantity and nature of the asset to be delivered along with the date that theactual sale of the asset will take place (settlement date). To ensure that both parties to the futurescontract will honor their commitments, participants are expected to post margin , which is, ineffect, collateral required by the brokerage firm. Furthermore, futures contracts provide formarking to the market , which involves daily re-computations of the margin requirement based onupdated asset value. Several exchanges exist for trading of futures contracts, including theChicago Board of Trade, the Chicago Mercantile Exchange and the New York Mercantile

000 SXecp Tr f +=

7/28/2019 corp09

9/27

9

Exchange. Futures markets are tightly regulated and overseen by the Commodities Futures Trading Commission (CFTC) and to a lesser extent, the Securities Exchange Commission (SEC).

Futures contracts are traded on a variety of different types of assets. Futures contractshave been traded for many centuries on a multiplicity of commodities, including agriculturalproducts such as corn, wheat and pork bellies, metals such as gold and copper, minerals such as

diamonds, petroleum and phosphates, as well as paper, timber. Futures contracts are traded on anassortment of financial securities such as treasury bonds, market indices and interest rates.Futures contracts also trade on a large number of different currencies.

How might a futures contract help a business control risk? Consider the following simpleexample involving a wheat farmer and a cereal manufacturer. Suppose that the manufacturerneeds to purchase wheat in three months for cereal production and the farmer, who is just nowplanting seed wishes to sell wheat at harvest in three months. Since the market price of wheat islikely to be affected by general economic and trade conditions, weather, and many other factorsbeyond the control of the two trading partners, neither knows what the price of wheat will be inthree months. Hence, the manufacturer faces uncertainty with respect to cereal production costsand the farmer cannot know what revenues he will derive from the sale of his produce. Such

uncertainty clearly affects the abilities of the trading partners to make appropriate businessdecisions. In fact, in the face of such uncertainties, businesses often scale back their levels of operations to avoid losses. I t would seem simple for the farmer and the cereal manufacturer toagree on a transaction price of wheat in advance. In fact, this is exactly what the futures marketallows trading partners to do; the party wishing to lock in a purchase price for wheat takes whatis called a long position in the futures contract and the party wishing to lock in a selling price forwheat takes a short position in the futures contract. The futures contract, in effect, obliges eachof its participants to transact for wheat at the agreed upon price (settlement price) at the pre-specified date. However, since the manufacturer never knows exactly whose wheat it willpurchase, and the farmer never knows exactly who will be the end user of his wheat, they simplytake positions in futures contracts with anonymous counterparties. Although the farmer actuallysells wheat at the prevailing market price at harvest, his short position in the futures contractenables to him offset decreases (increases) in the market price of wheat with gains (losses) in hisshort position in the futures contract. Similarly, the cereal manufacturer purchases wheat at theprevailing market price at harvest, though its long position in the futures contract enables to himoffset increases (decreases) in the market price of wheat with gains (losses) in its long position inthe futures contract.

D. Exchange Options and Futures

This section is concerned with some of the instruments that the corporation can use tohedge its exchange risks. Exchange transactions can occur in either spot or forward markets. Inthe spot market, the exchange of one currency for another occurs when the agreement is made.For example, dollars may be exchanged for euro now in an agreement made now. This would bea spot market transaction. In a forward market transaction, the actual exchange of one currencyfor another actually occurs at a date later than that of the agreement. Thus, traders could agreenow on an exchange rate for two currencies at a later date.

A forward exchange contract is a contract specifying delivery of one currency for anotherat a later date. Thus, a forward exchange contract is simply a contract providing for the exchangeof currency at some future date. Forward exchange contracts involve one position in each of two

7/28/2019 corp09

10/27

10

currencies:

1. Long: An investor has a "long" position in that currency that he will accept at thelater date.

2. Short: An investor has a "short" position in that currency that he must deliver in

the exchange.Forward exchange contracts will usually involve a third party to act as an intermediary betweenthe exchange parties. This intermediary, generally a bank, is referred to as a market maker. Theforward exchange rate, or forward rate, is the number of currency units that must be given up forone unit of a second currency at a later date. This rate may be quite different from the spot rate,which designates the number of currency units that must be exchanged for one unit of anothercurrency now. Forward rates are likely to differ from spot rates simply because supply anddemand conditions are likely to change over time. Thus, anticipated changes in the five factorslisted in section B will cause forward rates to differ from spot rates.

There are a number of reasons for participating in forward exchange markets. Many

investors participate for the purpose of speculation; they may feel that they know whichdirections exchange rates will change, and therefore, take positions in contracts enabling them toprofit from anticipated changes. Such speculation is often regarded as quite risky. Corporationsoften take short positions in forward contracts to lock in exchange rates for the dates that theymust deliver cash to companies in other countries. Corporations may take long or short positionsin contracts to diminish their exchange rate risk. Corporations may take short positions inforward exchange contracts to lock in the exchange rates for foreign-denominated cash flowsthey anticipate receiving. A smaller group of investors, known as arbitrageurs purchase forwardcontracts, and then invest in one of a variety of offsetting securities (including offsetting forwardcontracts) for the purpose of profiting from market price discrepancies.

Participants in forward exchange markets face a number of risks. Among these are:

1. Rate risk: Exchange rates may change in directions opposite to those anticipatedby participants.

2. Credit risk: the other party to the contract may default by not delivering thecurrency specified in the contract. In many instances, an intermediary such as alarge reputable commercial bank may act to ensure that one or both contractingparties will honor their agreements.

3. Liquidity risk: The market participant may have difficulty obtaining the currencyhe must deliver; he may be "stuck" with a currency that will be difficult to sell.Again, a number of intermediaries may improve liquidity by trading and makingmarkets for various currencies.

Futures contracts are merely standardized forward contracts. They are typically traded onexchanges such as the Chicago Board of Trade and the New York Mercantile Exchange in theU.S. These exchanges and their associated clearing corporations eliminate much of the credit riskand some of the liquidity risk associated with futures contracts. Of course, the rate risk and otherforms of risk remain. In addition, government, exchange and brokerage houses require thatfutures traders maintain deposits (margin) to ensure that they will honor their obligations. Often,an investor must vary his margin, depending on the trading environment and the prices of the

7/28/2019 corp09

11/27

11

underlying currencies. This variance of margin, which increases as the value of the investor'sobligation increases and decreases as the obligation decreases, is sometimes referred to as"marking to the market."

Because a forward rate specifies the prices investors will pay for currencies at a laterdate, one might expect that the prevailing forward rate of exchange equals the expected spot rate

at the time of currency delivery:(2) F0 =E[S1]

Equation (2) states that the forward rate of exchange prevailing at time zero equals the rate of exchange expected for time one. Speculators will tend to force Equation (2) to hold, at leastapproximately, through their trading activity.

Exchange Futures ContractsFutures contracts might be regarded as being standardized forward contracts. As

discussed earlier, they are typically traded on exchanges such as the International Monetary

Market (IMM) on the Chicago Mercantile Exchange (CME) as opposed to being traded in theover-the-counter markets. One takes a position in a futures contract by placing an order through abroker, who then executes the order on an exchange. An important difference between forwardand futures markets involves margin requirements and marking to the market. This processessentially requires a futures market participant to post a deposit known as margin. Marking tothe market occurs when this margin requirement changes as the participants position changes invalue. Marking to the market often involves contract settlement on a daily basis throughout thelife of the contract.

Forward markets typically involve dealers executing individually negotiated transactionsdirectly with one another through computer trading systems and by telephone. Futures contractstypically involve standardized transactions with standardized settlement dates on organizedexchanges. While forward contracts frequently settle with delivery of the underlying asset,futures contracts typically settlement by cash equivalence.

In the U.S., a clearinghouse serves as the intermediary for all futures transactions. Technically, contract participants are all obligated to the clearinghouse to settle their contracts.Futures trading in the United States is regulated by the Commodities Futures TradingCommission (CFTC) which functions in a manner similar to the S.EC. The forward markets,which generally include only well-known institutions, are self-regulated.

Futures (and forward) contracts are used for hedging and for speculation. Arbitrageurshelp maintain an efficient pricing system by seeking situations where contracts and currenciesare mispriced with respect to one another. Businesses with foreign customers and suppliersfrequently use futures contracts to hedge their exchange rate risk. For example, consider a

Japanese firm wishing to order a communications system from an American manufacturer for$5,000,000 payable upon delivery in six months. The Japanese firm is obliged to pay in dollars,but has no control over the price of dollars in yen; thus, it faces the risk that the value of thedollar will increase from its current price of, say, 100 per dollar. Since most U.S. firms expectto be paid for their products with dollars, many foreign firms and individuals with this type of exchange rate risk would be discouraged from purchasing American products. However, therepresently exist a number of types of derivative securities to control this risk. For example, the

Japanese firm could purchase in American markets put options on 500,000,000 with an exercise

7/28/2019 corp09

12/27

12

price of $.01 per yen. This purchase of puts would guarantee the Japanese firm a minimumselling price for 500,000,000 needed to purchase $5,000,000. If the price of dollars were todecline, the Japanese would simply purchase $5,000,000 in cash markets for fewer yen.

Currency and interest rate futures are traded on numerous exchanges throughout theworld. Among the U.S. exchanges listing currency futures are the International Monetary

Market (IMM), a subsidiary of the Chicago Mercantile Exchange (CME), the Philadelphia Boardof Trade (PBOT - a subsidiary of the Philadelphia Exchange) and the MidAmerica CommoditiesExchange. Non-U.S. exchanges include the London International Financial Futures Exchange,the March Terme International de France (MATIF), The Singapore International MoneyExchange (SIMEX) and the Tokyo International Financial Futures Exchange (TIFFE). TheGLOBEX2 trading system serves as an extended hours facility from 2:30PM to 7:05AM Chicagotime.

There are also active markets for Eurodollar contracts as well as for Euroyen, EuroSwissand EURIBOR contracts. These contracts are often used to manage interest rate risk.

E. Swap Contracts

As defined earlier, a derivative security may be simply defined as an instrument whosepayoff or value is a function of that of another security, index or value. There exist many types of derivative securities other than futures and options that can be used to manage a diversity of risks. One such security, a swap, which provides for the exchange of one set of cash flows foranother set of cash flows. Commercial banks are very active in the creation and marketing of swap contracts and are frequently referred to as swap banks. Swap banks serve their clients aseither brokers or dealers, depending on whether they act merely as matchmakers or actuallytake positions in the contracts that they deal. Most major participants in swaps markets belong tothe International Swap and Derivatives Association. Because swap contracts are often highlycustomized for performance of specific functions, secondary markets are not as well developedas currency futures and options markets. Nonetheless, survey data indicated that the totalnotional principal in currency swap markets had grown from $682 billion in 1987 to $32,942billion by 1998. 3

Consider, for example, a currency swap, which is a contract to exchange fixed streams of cash flows denominated in two currencies. The cash flow streams associated with each side of the swap are tied to securities (typically coupon bonds) denominated in one of the currencies. Forexample, a Japanese investor can agree to swap the payments associated with a yen denominated10% ten year par 10,000,000 note for the payments associated with a dollar denominated 12%ten year par $100,000 note.

Consider the case where Japanese regulations have restricted investment in many types of securities; in particular, Japanese institutions have been restricted with respect to non-yen bondpurchases. Suppose that a firm wished to borrow dollars to purchase American products.

Japanese tax code often makes borrowing less expensive in Japan. The borrower could sell to a Japanese institution a yen denominated bond (resulting in an attractive interest rate due topreferential tax treatment of Japanese zero coupon notes) then execute a dollar/yen currencyswap such that its initial loan receipts and loan repayments are denominated in dollars. Thus, allof the borrower's net cash flows are denominated in dollars (it has synthesized a dollar loan) andthe Japanese institution fulfills regulatory requirements by issuing a yen denominated note. Thus,

3International Swaps and Derivatives Association, Inc.

7/28/2019 corp09

13/27

13

many derivative securities such as the currency swap enable financial market participants tosynthesize other securities which are either unavailable or inappropriately priced.

A number of contracts are designed to pay off when the underlying asset remains within aspecified range. For example, the FX range floater pays off for each day that the exchange rateremains within a specified range. Boost structures are floating rate notes, swaps or options whose

payoffs are greater (boosted) for each day that the price of the underlying security remains in (ordeparts from) a predetermined range. Corridor accrual notes (also known as fairway or rangenotes) accrue interest only if a pre-specified LIBOR rate remains within a given range. Thepayoff function of the do-nothing option is inversely related to the price range of its underlyingsecurity. This security allows its owner to directly short the volatility of the underlying securitywithout assuming unlimited liability associated with shorting a straddle.

An interest rate swap is a contract to exchange streams of cash flows that are based ondifferent interest rates denominated in the same currency. Interest rate swaps frequently involvethe exchange of cash flows associated with a variable rate note for those associated with a fixedrate note. Such swaps are known as plain vanilla interest rate swaps. The London Interbank OfferRate (LIBOR) is frequently used as the variable index for such a swap. For example, to deal with

balance sheet risk, J apanese institutions tend to favor variable interest rate debt while theirAmerican counterparts deal more with fixed rate securities. A Japanese bank wishing to lendmoney at a variable rate to an American business preferring a fixed rate can execute an interestrate swap. The bank lends money to the American firm at a fixed interest rate. Separately, the

Japanese bank engages in a swap contract with a swap dealer where it agrees to exchange thecash flows associated with its fixed-rate note for the cash flows associated with a variable rateindex such as the LIBOR. The swap dealer arranging this interest rate swap has helped enabledfirms in both countries to obtain their desired cash flow and risk structures. The swap dealermust be aware of risks associated with potential default. Normally this risk is greater for fixedrate notes, leading to a quality spread differential (QSD) or interest rate difference whichincreases as the term of the notes increases.

A currency-interest rate swap is simply the exchange of floating and fixed rate streamsdenominated in two currencies. Thus, a currency-interest rate swap is the combination of thecurrency and interest rate swaps. In part because swaps are frequently customized for specificpurposes, a variety of other more complex and exotic swaps are marketed as well.

An equity swap is a contract providing for the delivery of cash flows associated withshares of equity in exchange for the cash flows associated with another asset (such as a debt orindex instrument). Equity swaps permit investors to reduce their risk in an equity investmentwithout actually selling shares. Most frequent participants in this market have been corporatemanagers. In a well-publicized case involving Autotote Company, the CEO arranged to deliverdividends and any capital gains (which would be negative in the event of a capital loss)associated with Autotote stock in exchange for certain cash flows associated with treasurysecurities. Thus, technically, the CEO did not sell his shares, though he divested himself of anyof the return risk associated with share ownership. By engaging this equity swap, the CEOreduces his risk in the employing company without having to report a sale of shares. This meansthat the CEO is not subject to capital gains taxes at the time of the transaction. 4 The CEO is notrequired to report to the SEC a transaction as an insider, or bear the selling price consequencesassociated with an insider sell transaction. Furthermore, the CEO maintains his voting control in

4 This tax benefit may be eliminated by the IRS in the future.

7/28/2019 corp09

14/27

14

the company's shares. Thus, in a sense, the equity swap permits the CEO the opportunity to, ineffect, execute a sale of shares without bearing the undesirable consequences associated with thesale.

F. Exotic Options

As described earlier, options which confer the right but not obligation to buy or sell anasset at a pre-specified price on or before a given date. We discussed earlier the "plain vanilla"options with the most simple terms. A variety of other options exist to perform various functions.

AnAsian Option (average rate) has a payoff function that is based on the average price(or average exchange rate) of the underlying asset (or currency). For example, an Asian call oncurrency permits its owner to receive the difference between the average currency exchange rateover the life of the option (A T) and the exercise price (E) associated with the option: C A,T=MAX(0, A T-E) and p A,T=MAX(0, E-A T) where A T - E St/n. A potential user of the Asian optionmight be an importer who purchases from a particular country the same number of units of itsresource each day. For example, an electric utility company purchasing oil from Mexico each

day using pesos may wish to use Asian options to reduce its currency risk. Since the exchangerate will vary daily, the oil expenses incurred by the oil importer will vary. The Asian optionhelps enable the exporter to stabilize its cash flows without entering the derivatives market on adaily basis. The cash flow structures of these options vary from contract to contract. Forexample, some contracts call for the payoff to be related to the difference between the time Tspot rate and the average exchange rate realized during the life of the option. The payoff functions for the Average Strike Price options are: C A,T=MAX(0, S T -A T) and p A,T=MAX(0,A T-S T) where S T is the spot rate prevailing at the time the option expires.

A Lookback Option enables its owner to purchase (or sell in the case of a put) theunderlying currency at the lowest rate (or highest rate in the case of a put) realized over the lifeof the option. The payoff function for lookback options might be C A,T=MAX(0, S T -SMIN) and

pA,T=MAX(0, SMAX-S T) where S MINand S MAXare the minimum and maximum exchange ratesrealized over the lives of the contracts.A Zero Cost Collar is a package of options designed to require zero net investment.

Typically, the collar consists of a package with a long position in a put enabling its owner to sellthe underlying security if its price drops to a specified price along with a short position in a callwhose exercise price is set so that it exactly offsets what is paid for the put. Hence, such a collarrequires no net investment. Similarly, the Range Forward Contract enables (and obliges) itsowner to purchase the underlying security with a time T value for the following price: (X 1 if S T$X1; S T if X1>S T$X2; or X2 if S T

7/28/2019 corp09

15/27

15

arrangement at a later date.

G. M anaging Exchange ExposureMultinational companies face a number of risks not experienced by companies operating

in only one country. Among the most significant of these risks is exchange rate uncertainty and

fluctuations. Corporate financial managers face three primary types of risk or exposure due toexchange rate fluctuations:

1. Accounting or translation exposure: the impact of exchange rate changes on thefirm's financial statements

2. Transaction exposure: potential gains or losses on the satisfaction of currentobligations due to exchange rate shifts. We will focus on Transaction exposure inthis section.

3. Operating or economic exposure: changes in operating revenues or costs inducedby exchange rate shifts

While the exchange risks faced by multinational firms are substantial, there still existsmuch controversy over whether these risks should be hedged. It is not always clear that hedgingexposure risk adds to firm value. First, investors may be drawn to the multinational firm toimprove the diversification of their portfolios. Hedging exchange exposure might actuallyhamper investors portfolio diversification efforts. Furthermore, shareholders should be able toemploy a variety of portfolio management techniques themselves to manage exchange exposure,rendering the management of exchange exposure at the corporate level redundant. Finally, manyanalysts that only the management of systematic risk enhances firm value while exchange raterisk for American investors tends to be largely unsystematic.

On the other hand, there are several arguments favoring exposure management. First,because of economies of scale and stronger creditworthiness, corporations are often able toemploy many risk management techniques less expensively than individual investors. Inaddition, if a firm were to file for bankruptcy, or risk such a filing, it is likely to face substantialdirect and indirect bankruptcy costs. Hedging exchange exposure may reduce the likelihood of incurring such costs. In addition, the progressive corporate income taxation system used in theU.S. imposes tax penalties on firms whose taxable income levels fluctuate from year to year.

Transaction exposure is a firm's risk arising from settlement of obligations denominatedin foreign currencies. Transaction exposure is usually a more serious immediate risk thantranslation exposure. The following are sources of transaction exposure:

1. Purchasing or Selling Goods or Services on Foreign Denominated Credit2. Borrowing or Lending with Foreign Denominated Notes3. Taking Positions in Forward Exchange Contracts4. Buying or Selling Assets Denominated in Foreign Currencies

The following are strategies for contending with transactions exposure:

1. Do Nothing - Accept the Risk2. Hedge in Forward Markets3. Hedge in Money Market: Take Offsetting Position in Local Currency Debt Instruments

7/28/2019 corp09

16/27

16

4. Hedge in Option Markets: Incomplete Hedge Except with "Conversions"

We will use examples to examine each of these strategies for dealing with transactions exposure.

Example I:Consider this first example regarding the management of transactions exposure. TheDayton Company of America invested in a British construction project and expects to receive apayoff of 1,000,000 in three months. The company wishes to realize its revenues in dollars. Theproblem is that one does not know what the payoff will be in dollars due to exchange rateuncertainty. Assume relevant data as follows:

Spot exchange rate: $1.7640/ Three month forward exchange rate: $1.7540/U.K. Borrowing interest rate: 10.0%U.S. Borrowing interest rate: 8.0%

U.K. Lending interest rate: 8.0%U.S. Lending interest rate: 6.0 %

Also assume that there exist call and put options and forward contracts with the following terms:

Size of contract: 31,250 Term to expiration: 3 monthsExercise price: $1.75 per Put Premium: $0.025 per poundCall Premium: $.065 per poundBrokerage cost per options contract on 1,000,000: $50

Transactions cost on 1,000,000 forward contract: $500

Our problem is to evaluate methods of managing the transaction risk associated with thisextension of credit and the implications of each. We should determine which hedging strategy islikely to be optimal and why. First, we will consider the alternative of doing nothing:

1. Unhedged alternative:

Strategy: Wait 3 months then sell 1,000,000 for dollars at the then prevailing spot rate.

Result: Risk is unlimited. The expected value of the transaction is $1,754,000 based on theforward rate of exchange used as a predictor for the future spot rate.

2. Forward market hedge: Strategy: Sell 1,000,000 forward for dollars at once.

Result: $1,754,000 will certainly be received in three months. Transactions costs at timezero will total $500. Foregone interest on this $500 over three months totals $7.50=.06 @.25 @$500. The total amount (net of forgone interest) to be received in

7/28/2019 corp09

17/27

17

three months is $1,753,492.50. This amount is certain.

3. Money market hedge:

Strategy: Borrow 975,609.76 in U.K. for three months @ 10% p.a.

Exchange 975,609.76 for $1,720,975.60 nowInvest $1,720,975.60 for three months @ 6% p.a.

Result: Pound loan is repaid by receipts from sale in three months. $1,746,790.20 isobtained from U.S. investment. This amount is certain. Notice that this strategy inthis example is inferior to the forward market hedge.

4. Options market hedges: There are two strategies here. The first is the put hedge strategy which involves the

purchase of a put on pounds enabling the firm to protect itself against devaluation of pounds. If the value of pounds increases, the firm realizes a greater profit. However, the firm must pay the

full cost of the put. With the call and put strategy, the proceeds from the sale of a call are used tooffset the purchase price of the put. This strategy acts as a collar, locking in the value of poundsat the originations of the options contracts. Hence, the firm does not benefit from anyappreciation in the value of the pound.

a. Put Hedge Strategy: Purchase three month put option on 1,000,000 with anexercise price of $1.75/ with a premium of $25,000. Time zero brokerage coststotal $1,600 (32 contracts at $50 per contract). Thus the total time zero cashoutlay is $26,600. Foregone interest on the sum of the premium and brokeragecosts totals $399. Expressed in terms of future value, the total cash outlay is$26,999.

Result: Receive one of the following in three months:1. An unlimited maximum less the $26,999 premium and brokerage fees. Thedollar value of this strategy increases as the value of the dollar drops against thepound.2. A minimum of $1,750,000 less $26,999 for a net of $1,723,001. This minimumvalue to be received may be unacceptably low; however, there is upside cash flowpotential.

b. Call and Put Hedge Strategy: Through the combination of calls and puts, totalrisk can be eliminated. Consider the writing of a call with an exercise price of $1.75 expiring in three months along with the purchase of a put with the sameterms. The time zero cash flows are summarized as follows:

Put Premium....... - $25,000 Call Premium....... +$65,000Put brokerage fee. - $ 1,600 Call brokerage fee. - $ 1,600

Net Time zero cash flows +$36,800

Result: The interest earned on the net time zero outlay is $552. If the three month

7/28/2019 corp09

18/27

18

exchange rate is less than $1.75/, the exchange rate of $1.75/ is locked in by theput. If the exchange rate exceeds $1.75/, the obligation incurred by the shortposition in the call is activated. Thus, the exchange rate of $1.75/ is locked in nomatter what the exchange rate is. The cash flows in three months are summarizedas follows:

Put cash flows (1,000,000 * MAX[1.75-S 1,0])Call cash flows (1,000,000 * MIN[1.75-S 1,0]) Total of option transactions:1,000,000 * (1.75 - S 1) =$1,750,000 - (1,000,000 * S 1)Exchange of Currency = (1,000,000 * S 1)

Time zero cash flows =$ 36,800Interest on Time zero flows =$ 552

TOTAL TIME ONE CASH FLOWS =$1,787,352

This strategy seems superior to either the forward market hedge or the money markethedge. Its Time One cash flows are locked in at a higher level.

Example II:Consider the following second example concerned with management of transactions

exposure. The Smedley Company has sold products to a Japanese client for 15,000,000.Payment is due six months later. Relevant data is as follows:

Spot exchange rate: 105/$Six month forward exchange rate: 112/$

Japanese Borrowing interest rate: 9.0%U.S. Borrowing interest rate: 7.0%

Japanese Lending interest rate: 7.0%U.S. Lending interest rate: 5.0 %

There exists call and put options and forward contracts with the following terms:

Size of futures and options contracts: 1,000,000 Term to expiration/settlement of all contracts: 6 monthsExercise price of put and call: $.009/Put Premium: $0.00001/Call Premium:$.0001/Brokerage cost per options contract: $50

Transactions cost on 15,000,000 forward contract: $500

Our problem is to evaluate methods of managing the transaction risk associated with thisextension of credit and the implications of each. We should determine which hedging strategy islikely to be optimal and why. First, we will consider the alternative of doing nothing:

1. Unhedged alternative: Strategy:Wait 6 months then sell 15,000,000 for dollars at the then prevailing spot rate.

7/28/2019 corp09

19/27

19

Result:Risk is unlimited. The expected value of the transaction is $133,928.57.

2. Forward market hedge:

Strategy:Sell 15,000,000 forward for dollars at once.

Result:$133,928.57 will certainly be received in six months. Transactions costs at timezero will total $500. Foregone interest over six months totals $12.50. The totalamount (net of transactions costs and foregone interest) to be received in sixmonths is $133,416.07. This amount is certain.

3. Money market hedge: Strategy:

Borrow 14,354,067 in Japan for six months @ 9% p.a.Exchange 14,354,067 for $136,705.40 nowInvest $136,705.40 for six months @ 5% p.a.

Result: Yen loan is repaid by receipts from sale in six months. $140,123.03 is obtainedfrom U.S. investment. This amount is certain. Notice that this strategy is superiorto the forward market hedge.

4. Options market hedges: a. Put Hedge Strategy: Purchase six month put option on 15,000,000 withan exercise price of $.009/ with a premium of $150. Time zero brokerage coststotal $750 (15 contracts at $50 per contract - pretty high, given the premiumsinvolved). Thus the total time zero cash outlay is $900. Forgone interest on thesum of the premium and brokerage costs totals $22.50. Expressed in terms of future value, the total cash outlay is $922.50.

Result: Receive one of the following in six months:1. An unlimited maximum less the $922.50 premium and brokerage fees. Thedollar value of this strategy increases as the value of the dollar drops.2. A minimum of $135,000 less $922.50 for a net of $134,077.50. This minimumvalue to be received may be unacceptably low; however, there is upside cash flowpotential.

b. Call and Put Hedge Strategy: Through the combination of calls and puts,total risk can be eliminated. Consider the writing of a call with an exercise priceof $.009 expiring in six months along with the purchase of a put with the sameterms. The time zero cash flows are summarized as follows:

Put Premium........ - $ 150 Call Premium....... +$ 1,500Put brokerage fee. - $ 750 Call brokerage fee. - $ 750

7/28/2019 corp09

20/27

20

Net Time zero cash flows -$150

Result:The foregone interest on the net time zero outlay is $3.75. If the six monthexchange rate is less than $.009/, the exchange rate of $.009/ is locked in by theput. If the exchange rate exceeds $.009/, the obligation incurred by the short

position in the call is activated. Thus, the exchange rate of $.009/ is locked in nomatter what the exchange rate is. The cash flows in six months are summarized asfollows:

Put cash flows (15,000,000 * MAX[.009-S 1,0])Call cash flows (15,000,000 * MIN[.009-S 1,0])

Total of option transactions:(15,000,000 * (.009 - S 1) = $135,000 - (15,000,000 * S 1)Exchange of Currency = (15,000,000 * S 1)

Time zero cash flows = $ -150Interest on Time zero flows = $ 3.75

TOTAL TIME ONE CASH FLOWS = $134,846.25 This riskless strategy seems superior to the forward market hedge and inferior to the moneymarket hedge.

Other Hedging Strategies: The hedging strategies described in the above examples are more effective when firms

can easily enter into forward, money market and options contracts. This is normally quite easilyaccomplished in countries with major currencies. However, in many instances, firms will need tohedge exchange rate volatility in countries where such opportunities are not available. Forexample, it is often very difficult to employ the above strategies in smaller Asian countries,Africa or Latin America. In many instances, firms might modify these strategies with crosshedges. Typically, these cross hedging strategies involve the use of contracts denominated incurrencies strongly correlated with the currency to be hedged. For example, rather than attemptto directly hedge exchange rate risk involving Philippine pesos, a firm might use contractsdenominated in yen, which are fairly highly correlated with pesos. An alternative cross hedgingstrategy is to hedge currency risk with commodity contracts whose values are strongly correlatedwith that of the currency. For example, the price of oil is strongly correlated with the value of theSaudi Arabia rival. The firm needing to hedge risk associated with the rival can use futurescontracts on oil as an imperfect substitute for the currency itself.

Survey data suggests that the forward market hedge is most commonly used by large U.S.corporations. 5 Currency swaps and options contracts are also frequently used. Such contractusage is most prevalent in the financial services industries and by firms engaged in the largestextent of international business.

5 Jesswein, Kurt, Chuck K.Y. Kwok and William Folks, Jr. Corporate Use of Innovative Foreign Exchange RiskManagement Products, Columbia J ournal of World Business , Fall 1995. Pp. 70-82.

7/28/2019 corp09

21/27

21

QUESTI ONS AND PROBLEMS

1. Call and put options with an exercise price of $30 are traded on one share of Company Xstock.a. What is the value of the call and the put if the stock is worth $33 when the options

expire?b. What is the value of the call and the put if the stock is worth $22 when the optionsexpire?

c. What is the value of the call writer's obligation stock is worth $33 when the optionsexpire? What is the value of the put writer's obligation stock is worth $33 when theoptions expire?

d. What is the value of the call writer's obligation stock is worth $22 when the optionsexpire? What is the value of the put writer's obligation stock is worth $22 when theoptions expire?

e. Suppose that the purchaser of a call in part a paid $1.75 for his option. What was hisprofit on his investment?

f. Suppose that the purchaser of a call in part b paid $1.75 for his option. What was hisprofit on his investment?

2. Arkin Company stock currently sells for $12 per share and is expected to be worth either $10or $16 in one year. The current riskless return rate is .125. What would be the value of a one-yearcall with an exercise price of $8?

3. Consider a one time period, two potential outcome framework where there exists Company Qstock currently selling for $50 per share and a riskless $100 face value T-Bill currently sellingfor $90. Suppose Company Q faces uncertainty, such that it will pay its owner either $30 or $70in one year. Further assume that a call with an exercise price of $55 exists on one share of Qstock.a. What are the two potential values the call might have at its expiration?b. What is the riskless rate of return for this example? Remember, the Treasury Bill pays

$100 and currently sells for $90.c. What is the hedge ratio for this call option?d. What is the current value of this option?

4. Rollins Company stock currently sells for $12 per share and is expected to be worth either$10 or $16 in one year. The current riskless return rate is .125. What would be the value of aone-year call with an exercise price of $8?

**5. A stock currently selling for $50 has a variance of returns equal to .36. The riskless returnrate equals .08. Under the binomial framework, what would be the value of nine-month (.75year) European calls and European puts with striking prices equal to $80 if the number of treesteps (n) were:a. 2b. 3c. 8

7/28/2019 corp09

22/27

22

6. Ibis Company stock is currently selling for $50 per share and has a multiplicative upwardmovement equal to 1.2776 and a multiplicative downward movement equal to .7828. What is thevalue of a nine-month (.75 year) European call and a European put with striking prices equal to$60 if the number of tree steps were 2? Assume a riskless return rate equal to .081.

7. Kestrel Company stock is currently selling for $40 per share. Its historical standard deviationof returns is .5. The one-year Treasury Bill rate is currently 5%. Assume that all of the standardBlack-Scholes Option Pricing Model assumptions hold.

a. What is the value of a put on this stock if it has an exercise price of $35 and expires inone year?

b. What is the implied probability that the value of the stock will be less than $30 in oneyear?

8. Evaluate calls and puts for each of the following European stock option series:

Option 1 Option 2 Option 3 Option 4 T =1 T =1 T =1 T =2S =30 S =30 S =30 S =30 =.3 =.3 =.5 =.3r =.06 r =.06 r =.06 r =.06X =25 X =35 X =35 X =35

9. Evaluate each of the European options in the series on ABC Company stock. Prices for eachof the options are listed in the table. Determine whether each of the options in the series shouldbe purchased or sold at the given market prices. The current market price of ABC stock is 120,the August options expire in nine days, September options in 44 days and October options in 71days. The stock variances prior to expirations are projected to be .20 prior to August, .25 prior toSeptember, and .20 prior to October. The treasury bill rate is projected to be .06 for each of thethree periods prior to expiration. Do not forget to convert the number of days given to fractionsof 365-day years.

CALLSX AUG SEP OCT110 9.500 10.500 11.625 =.20 FOR AUG115 4.625 7.000 8.125 =.25 FOR SEP120 1.250 3.875 5.250 =.20 FOR OCT125 .250 2.125 3.125 r =.06130 .031 .750 1.625 S =120

PUTSX AUG SEP OCT110 .031 .750 1.500115 .375 1.750 2.750120 1.625 6.750 4.500125 5.625 6.750 7.875130 10.625 10.750 11.625

Exercise prices for 15 calls and 15 puts are given in the left columns. Expiration dates aregiven in column headings and current market prices are given in the table interiors.

10. Emu Company stock currently trades for $50 per share. The current riskless return rate is

7/28/2019 corp09

23/27

23

.06. Under the Black-Scholes framework, what would be the standard deviations implied by six-month (.5 year) European calls with current market values based on each of the followingstriking prices:a. X =40; c 0 =11.50b. X =45; c 0 = 8.25c. X =50; c 0 = 4.75d. X =55; c 0 = 2.50e. X =60; c 0 = 1.25

7/28/2019 corp09

24/27

24

Solutions

1. a. c T =$33 - $30 =$3; p T =0b. c T =0; p T =$30 - $22 =$8c. c T =-$3; p T =0d. c T =0; p T =-$8e. $3 - $1.75 =$1.25f. $0 - $1.75 =-$1.75

2. The hedge ratio for the call equals 1. Since the riskless return rate is .125, the calls currentvalue must be $4.8888889.

3. a. c T =MAX[0,S T-X]; c T =$0 or $15b. $100/$90 - 1 =.1111c.

)(0 udS

cc du

=

)6.4.1(50015

=

d.)1(

)1( 000

f

df

r

dSCSrc

+

++=

625.8)1111.1(

506.375.050375.)1111.1(0 =

+

++=c

4. First, find the hedge ratio:

)(0 udScc du

=

)83333.3333.1(1228

=

Now, value the call:

)1()1( 00

0f

df

r

dSCSrc

+

++=

88889.4)125.1(

1283333.12121)125.1(0 =

+

++=c

7/28/2019 corp09

25/27

25

4. Value the calls using the Black-Scholes Model:c0 =S0N(d1) - Xe-rTN(d2)

d1 =[ln(SX) +(r +.5 F 2)T] F % T

d2 =d1 - F % T

Thus, we will first compute d 1, d2, N(d1), N(d2) for each of the calls; then we will compute eachcall's value. Finally, we will use putcall parity to value each of the puts.

5. The following are call and put values:n c0 p0 2 4.62 29.963 3.91 29.258 3.87 29.21

6. c0 =$5.10; p 0 =$11.61

7. a. d 1 =.6172; d 2 =.1178; N(d 1) =.7314; N(d 2) =.5469c0 =11.05; with put-call parity: p 0 =4.34

b. Use X=30; d 1 =.925; d 2 =.4245; N(d 2) =.66441-N(d2) =.3356

8. The options are valued with the Black-Scholes Model in a step-by-step format in the followingtable:

OPTION 1 OPTION 2 OPTION 3 OPTION 4d(1) .957739 -.163836 .061699 .131638d(2) .657739 -.463836 -.438301 -.292626N[d(1)] .830903 .434930 .524599 .552365N[d(2)] .744647 .321383 .330584 .384904

Call 7.395 2.455 4.841 4.623Put 0.939 5.416 7.803 5.665

9. Value the calls using the Black-Scholes Model:

c0 =S0N(d1) - Xe-rTN(d2)

d1 =[ln(SX) +(r \+.5 F 2)T] F % T

7/28/2019 corp09

26/27

26

d2 =d1 - F % T

Thus, we will first compute d 1, d2, N(d1), N(d2) for each of the calls; then we will compute eachcall's value. We will then use put-call parity to value each put.

First find for each of the 15 calls values for d 1:X AUG SEP OCT

110 2.833394 1.129163 1.162841115 1.417978 .617046 .658904120 .062811 .126728 .176418125 -1.237028 -.343571 -.286369130 -2.485879 -.795423 -.731003

Next, find for each of the 15 calls values for d 2:X AUG SEP OCT

110 2.801988 1.042362 1.074632115 1.386572 .530245 .570695120 .031405 .039928 .088208125 -1.268433 -.430371 -.374578130 -2.517284 -.882222 -.819212

Now, find N(d1) for each of the 15 calls:X AUG SEP OCT

110 .997697 .870585 .877553115 .921901 .731398 .745021120 .525041 .550422 .570017125 .108038 .365584 .387298130 .006462 .213184 .232388

Next, determine N(d 2) for each of the 15 calls:

X AUG SEP OCT110 .997461 .851378 .858730115 .917214 .702029 .715897120 .512527 .515925 .535145125 .102322 .333463 .353987130 .005913 .188828 .206333

Now use N(d 1) and N(d 2) to value the calls and put-call parity to value the puts.

CALLSX AUG SEP OCT

110 10.165 11.494 11.942115 5.305 7.616 8.030

120 1.593 4.586 4.930125 .193 2.488 2.741130 .008 1.211 1.375

PUTSX Aug Sep Oct

110 0.003 .701 0.666115 0.134 1.787 1.685120 1.415 3.721 3.537125 5.009 6.587 6.290

7/28/2019 corp09

27/27

130 9.816 10.274 9.866

The options whose values are underlined are overvalued by the market; they should besold. Other options are undervalued by the market; they should be purchased.

10. Implied volatilities are given as follows:a. X =40; F =.2579b. X =45; F =.3312c. X =50; F =.2851d. X =55; F =.2715e. X =60; F =.2704

These values are obtained through a process of substitution and iteration.