Text book Problems wk 4

14 Option Markets

CHAPTER OBJECTIVES

The specific objectives of this chapter are to:

· ▪ provide a background on options,

· ▪ explain why stock option premiums vary,

· ▪ explain how stock options are used to speculate,

· ▪ explain how stock options are used to hedge,

· ▪ explain the use of stock index options, and

· ▪ explain the use of options on futures.

14-1 BACKGROUND ON OPTIONS

Options are classified as calls or puts. A  call option  grants the owner the right to purchase a specified financial instrument (such as a stock) for a specified price (called the  exercise price  or  strike price ) within a specified period of time.

   A call option is said to be  in the money  when the market price of the underlying security exceeds the exercise price,  at the money  when the market price is equal to the exercise price, and  out of the money  when it is below the exercise price.

   The second type of option is known as a  put option . It grants the owner the right to sell a specified financial instrument for a specified price within a specified period of time. As with call options, owners pay a premium to obtain put options. They can exercise the options at any time up to the expiration date but are not obligated to do so.

   A put option is said to be “in the money” when the market price of the underlying security is below the exercise price, “at the money” when the market price is equal to the exercise price, and “out of the money” when it exceeds the exercise price.

   Call and put options specify 100 shares for the stocks to which they are assigned. Premiums paid for call and put options are determined by the participants engaged in trading. The premium for a particular option changes over time as it becomes more or less desirable to traders.

   Participants can close out their option positions by making an offsetting transaction. For example, purchasers of an option can offset their positions at any time by selling an identical option. The gain or loss is determined by the premium paid when purchasing the option versus the premium received when selling an identical option. Sellers of options can close out their positions at any time by purchasing an identical option.

WEB

www.cboe.com

The volume of calls versus the volume of puts are used to assess their respective popularity.

   The stock options just described are known as American-style stock options. They can be exercised at any time until the expiration date. In contrast, European-style stock options can be exercised only just before expiration.

   In addition to options on stocks there are options on stock indexes, which allow investors the right to buy (with a call option) or sell (with a put option) a specified stock index for a specified price up to a specified expiration date. There are also options on interest rate futures contracts, which allow investors the right to buy or sell a specified interest rate futures contract for a specified price up to a specified expiration date. Options on stock indexes and on interest rate futures are covered later in this chapter.

14-1a Comparison of Options and Futures

There are two major differences between purchasing an option and purchasing a futures contract. First, to obtain an option, a premium must be paid in addition to the price of the financial instrument. Second, the owner of an option can choose to let the option expire on the expiration date without exercising it. Call options grant a right, but not an obligation, to purchase a specified financial instrument. In contrast, buyers of futures contracts are obligated to purchase the financial instrument at a specified date. If the owner does exercise the call option, the seller (sometimes called the  writer ) of the option is obligated to provide the specified financial instrument at the price specified by the option contract if the owner exercises the option. Sellers of call options receive an upfront fee (the premium) from the purchaser as compensation.

WEB

biz.yahoo.com/opt

Daily coverage of options market activity.

14-1b Markets Used to Trade Options

The Chicago Board Options Exchange (CBOE), which was created in 1973, is the most important exchange for trading options. It serves as a market for options on more than 2,000 different stocks. The options listed on the CBOE have a standardized format, as will be explained shortly. The standardization of the contracts on the CBOE proved to be a major advantage because it allowed for easy trading of existing contracts (a secondary market). With standardization, the popularity of options increased and the options became more liquid. Since there were numerous buyers and sellers of the standardized contracts, buyers and sellers of a particular option contract could be matched.

   Options are also traded at the CME Group, which was formed in July 2007 by the merger of the Chicago Board of Trade (CBOT) and the Chicago Mercantile Exchange (CME). As discussed in  Chapter 13 , the CME Group serves international markets for derivative products. To increase efficiency and reduce operating and maintenance expenses after the merger, the CME Group consolidated the CME and CBOT trading floors into a single trading floor at the CBOT and consolidated the products of the CME and CBOT on a single electronic platform. Transactions of new derivative products typically are executed by the CME Group's electronic platform.

   As the popularity of stock options increased, various stock exchanges began to list options. In particular, the American Stock Exchange (acquired by NYSE Euronext in 2008), the Nasdaq, and the Philadelphia Stock Exchange (acquired by Nasdaq in 2008) list options on many different stocks. So does the International Securities Exchange, which was the first fully electronic U.S. options exchange. Today, any particular options contract may be traded on various exchanges, and competition among the exchanges may result in more favorable prices for customers.

   Some specialized option contracts are sold “over the counter,” rather than on an exchange, whereby a financial intermediary (such as a commercial bank or an investment bank) finds a counterparty or serves as the counterparty. These over-the-counter arrangements are more personalized and can be tailored to the specific preferences of the parties involved. Such tailoring is not possible for the more standardized option contracts sold on the exchanges.

Listing Requirements  Each exchange has its own requirements concerning the stocks for which it creates options. One key requirement is a minimum trading volume of the underlying stock, since the volume of options traded on a particular stock will normally be higher if the stock trading volume is high. The decision to list an option is made by each exchange, not by the firms represented by the options contracts.

Role of the Options Clearing Corporation  Like a stock transaction, the trading of an option involves a buyer and a seller. The sale of an option imposes specific obligations on the seller under specific conditions. The exchange itself does not take positions in option contracts, but provides a market where the options can be bought or sold. The Options Clearing Corporation (OCC) serves as a guarantor on option contracts traded in the United States, which means that the buyer of an option contract does not have to be concerned that the seller will back out of the obligation.

Regulation of Options Trading  Options trading is regulated by the Securities and Exchange Commission and by the various option exchanges. The regulation is intended to ensure fair and orderly trading. For example, it attempts to prevent insider trading (trading based on information that insiders have about their firms and that is not yet disclosed to the public). It also attempts to prevent price fixing among floor brokers that could cause wider bid–ask spreads that would impose higher costs on customers.

14-1c How Option Trades Are Executed

When options exchanges were created, floor brokers of exchanges were available to execute orders for brokerage firms. They went to a specific location on the trading floor where the option was traded to execute the order. Today, computer technology allows investors to have trades executed electronically. Most small, standardized transactions are executed electronically, whereas complex transactions are executed by competitive open outcry among exchange members. Many electronic communication networks (ECNs) are programmed to consider all possible trades and execute the order at the best possible price.

   Market-makers can also execute stock option transactions for customers. They earn the difference between the bid price and the ask price for this trade, although the spread has declined significantly in recent years. Market-makers also generate profits or losses when they invest their own funds in options.

14-1d Types of Orders

As with stocks, an investor can use either a market order or a limit order for an option transaction. A market order will result in the immediate purchase or sale of an option at its prevailing market price. With a limit order, the transaction will occur only if the market price is no higher or lower than a specified price limit. For example, an investor may request the purchase of a specific option only if it can be purchased at or below some specified price. Conversely, an investor may request to sell an option only if it can be sold for some specified limit or more.

Online Trading  Option contracts can also be purchased or sold online. Many online brokerage firms, including E*Trade and TD Ameritrade, facilitate options orders. Online option contract orders are commonly routed to computerized networks on options exchanges, where they are executed. For these orders, computers handle the order from the time it is placed until it is executed.

14-1e Stock Option Quotations

Financial newspapers and other financial media publish quotations for stock options.  Exhibit 14.1  provides an example of stock options for Viperon Company stock as of May 1, when the stock was priced at about $45.62 per share. There are normally more options on each stock than what is disclosed in financial newspapers, with additional exercise prices and expiration dates. Each row represents a specific option on Viperon stock. The first data column lists the exercise (strike) price, and the second column lists the expiration date. (The expiration date for stock options traded on the CBOE is the Saturday following the third Friday of the specified month.) The third and fourth columns show the volume and the most recently quoted premium of the call option with that exercise price and expiration date. The fifth and sixth columns show the volume and the most recently quoted premium of the put option with that exercise price and expiration date.

Exhibit 14.1 Viperon Company Stock Option Quotations

	STRIKE	EXP.	VOLUME	CALL	VOLUME	PUT
Option 1	45	Jun	180	4½	60	2¾
Option 2	45	Oct	70	5¾	120	3¾
Option 3	50	Jun	360	1⅛	40	5⅛
Option 4	50	Oct	90	3½	40	6½

WEB

biz.yahoo.com/opt

Summary of the most actively traded stock options.

   A comparison of the premiums among the four options illustrates how specific factors affect option premiums. First, comparing Options 1 and 3 (to control for the same expiration date) reveals that an option with a higher exercise price has a lower call option premium and a higher put option premium. A comparison of Options 2 and 4 confirms this relationship. Second, comparing Options 1 and 2 (to control for the same exercise price) reveals that an option with a longer term to maturity has a higher call option premium and a higher put option premium. A comparison of Options 3 and 4 confirms this relationship.

14-1f Institutional Use of Options

Exhibit 14.2  summarizes the use of options by various types of financial institutions. Although options positions are sometimes taken by financial institutions for speculative purposes, they are more commonly used for hedging. Savings institutions and bond mutual funds use options to hedge interest rate risk. Stock mutual funds, insurance companies, and pension funds use stock index options and options on stock index futures to hedge their stock portfolios. Some of the large commercial banks often serve as an intermediary between two parties that take derivative positions in an over-the-counter market.

14-2 DETERMINANTS OF STOCK OPTION PREMIUMS

Stock option premiums are determined by market forces. Any characteristic of an option that results in many willing buyers but few willing sellers will place upward pressure on the option premium. Thus the option premium must be sufficiently high to equalize the demand by buyers and the supply that sellers are willing to sell. This generalization applies to both call options and put options. The specific characteristics that affect the demand and supply conditions, and therefore affect the stock option premiums, are described in what follows.

Exhibit 14.2 Institutional Use of Options Markets

TYPE OF FINANCIAL INSTITUTION	PARTICIPATION IN OPTIONS MARKETS
Commercial banks	· • Sometimes offer options to businesses.
Savings institutions	· • Sometimes take positions in options on futures contracts to hedge interest rate risk.
Mutual funds	· • Stock mutual funds take positions in stock index options to hedge against a possible decline in prices of stocks in their portfolios. · • Stock mutual funds sometimes take speculative positions in stock index options in an attempt to increase their returns. · • Bond mutual funds sometimes take positions in options on futures to hedge interest rate risk.
Securities firms	· • Serve as brokers by executing stock option transactions for individuals and businesses.
Pension funds	· • Take positions in stock index options to hedge against a possible decline in prices of stocks in their portfolio. · • Take positions in options on futures contracts to hedge their bond portfolios against interest rate movements.
Insurance companies	· • Take positions in stock index options to hedge against a possible decline in prices of stocks in their portfolio. · • Take positions in options on futures contracts to hedge their bond portfolios against interest rate movements.

14-2a Determinants of Call Option Premiums

Call option premiums are affected primarily by the following factors:

· ▪ Market price of the underlying instrument (relative to the option's exercise price)

· ▪ Volatility of the underlying instrument

· ▪ Time to maturity of the call option

Influence of the Market Price  The higher the existing market price of the underlying financial instrument relative to the exercise price, the higher the call option premium, other things being equal. A stock's value has a higher probability of increasing well above the exercise price if it is already close to or above the exercise price. Thus a purchaser would be willing to pay a higher premium for a call option on such a stock.

   The influence of the market price of a stock (relative to the exercise price) on the call option premium can also be understood by comparing stock options with different exercise prices on the same instrument at a given time.

EXAMPLE

Consider the data shown in  Exhibit 14.3  for KSR call options quoted on February 25, with a similar expiration date. The stock price of KSR was about $140 at that time. The premium for the call option with the $130 exercise price was almost $10 higher than the premium for the option with the $150 exercise price. This example confirms that a higher premium is required to lock in a lower exercise price on call options

Influence of the Stock's Volatility  The greater the volatility of the underlying stock, the higher the call option premium, other things being equal. If a stock is volatile, there is a higher probability that its price will increase well above the exercise price. Thus a purchaser would be willing to pay a higher premium for a call option on that stock. For instance, call options on small stocks normally have higher premiums than call options on large stocks because small stocks are typically more volatile.

Exhibit 14.3 Relationship between Exercise Price and Call Option Premium on KSR Stock

EXERCISE PRICE	PREMIUM FOR APRIL EXPIRATION DATE
$130	11⅝
135	7½
140	5¼
145	3¼
150	1⅞

Influence of the Call Option's Time to Maturity  The longer the call option's time to maturity, the higher the call option premium, other things being equal. A longer time period until expiration allows the owner of the option more time to exercise the option. Thus there is a higher probability that the stock's price will move well above the exercise price before the option expires.

   The relationship between the time to maturity and the call option premium is illustrated in  Exhibit 14.4  for KSR call options quoted on February 25, with a similar exercise price of $135. The premium was $4.50 per share for the call option with a March expiration month versus $7.50 per share for the call option with an April expiration month. The difference reflects the additional time in which the April call option can be exercised.

14-2b Determinants of Put Option Premiums

The premium paid on a put option depends on the same factors that affect the premium paid on a call option. However, the direction of influence varies for one of the factors, as explained next.

Influence of the Market Price  The higher the existing market price of the underlying stock relative to the exercise price, the lower the put option premium, all other things being equal. A stock's value has a higher probability of decreasing well below the exercise price if it is already close to or below the exercise price. Thus a purchaser would be willing to pay a higher premium for a put option on that stock. This influence on the put option premium differs from the influence on the call option premium because, from the perspective of put option purchasers, a lower market price is preferable.

   The influence of the market price of a stock (relative to the exercise price) on the put option premium can also be understood by comparing options with different exercise prices on the same instrument at a given moment in time. For example, consider the data shown in  Exhibit 14.5  for KSR put options with a similar expiration date quoted on February 25, 2010. The premium for the put option with the $150 exercise price was more than $9 per share higher than the premium for the option with the $135 exercise price. The difference reflects the more favorable price at which the stock can be sold when holding the put option with the higher exercise price.

Exhibit 14.4 Relationship between Time to Maturity and Call Option Premium on KSR Stock

EXPIRATION DATE	PREMIUM FOR OPTION WITH A $135 EXERCISE PRICE
March	4½
April	7½
July	13¼

Exhibit 14.5 Relationship between Exercise Price and Put Option Premium on KSR Stock

EXERCISE PRICE	PREMIUM FOR JUNE EXPIRATION DATE
$130	1⅞
135	3⅛
140	5⅜
145	8½
150	12¼

Influence of the Stock's Volatility  The greater the volatility of the underlying stock, the higher the put option premium, all other things being equal. This relationship also held for call option premiums. If a stock is volatile, there is a higher probability of its price deviating far from the exercise price. Thus a purchaser would be willing to pay a higher premium for a put option on that stock because its market price is more likely to decline well below the option's exercise price.

Influence of the Put Option's Time to Maturity  The longer the time to maturity, the higher the put option premium, all other things being equal. This relationship also held for call option premiums. A longer time period until expiration allows the owner of the option more time to exercise the option. Thus there is a higher probability that the stock's price will move well below the exercise price before the option expires.

   The relationship between the time to maturity and the put option premium is shown in  Exhibit 14.6  for KSR put options with a similar exercise price of $135 quoted on February 25, 2010. The premium was $7.25 per share for the put option with a July expiration month versus $0.50 per share for the put option with a March expiration month. This difference reflects the additional time during which the put option with the July expiration date can be exercised.

14-2c How Option Pricing Can Be Used to Derive a Stock's Volatility

The general relationships between the determinants described previously and the option premium are explained in the text of this chapter, but see the Appendix for discussion of a well-known formula used to price stock options. Since the anticipated volatility of a stock is not observable, investors can input their own estimate of it when calculating the premium that should be paid for a particular option.

Exhibit 14.6 Relationship between Time to Maturity and Put Option Premium on KSR Stock

EXPIRATION DATE	PREMIUM FOR OPTION WITH A $135 EXERCISE PRICE
March	½
April	3⅛
July	7¼

   Some investors who are assessing a specific stock's risk adapt the option-pricing formula to derive an estimate of that stock's anticipated volatility. By plugging in values for the other factors that affect the particular stock option's premium and for the prevailing premium quoted in the market, it is possible to derive the stock's anticipated volatility, which is referred to as the implied standard deviation (or implied volatility). The implied standard deviation is derived by determining what its value must be, given the quoted option premium and the values of other factors that affect the stock option's premium. Various software packages and calculators are available that can estimate a stock's implied volatility. Such an estimate is of interest to investors because it indicates the market's view of the stock's potential volatility. Some investors may use this estimate as a measure of a stock's risk when they consider what stocks to purchase.

14-2d Explaining Changes in Option Premiums

Exhibit 14.7  identifies the underlying forces that cause option prices to change over time. Economic conditions and market conditions can cause abrupt changes in the stock price or in the anticipated volatility of the stock price over the time remaining until option expiration. Such changes would have a major impact on the stock option's premium.

Indicators Monitored by Participants in the Options Market  Since the premiums paid on stock options are highly influenced by the price movements of the underlying stocks, participants in the stock option market closely monitor the same indicators that are monitored when trading the underlying stocks. Traders of options tend to monitor economic indicators because economic conditions affect cash flows of firms and thus can affect expected stock valuations and stock option premiums. Economic conditions can also affect the premiums by influencing expected stock volatility.

EXAMPLE

During the fall of 2008, the credit crisis intensified and stock volatility increased substantially. Consequently, premiums for options increased as well. Under these conditions, more portfolio managers wanted to hedge their stock positions, but they had to pay a higher premium for put options. The sellers of put options recognized that their risk had increased because of the higher volatility and priced the put options accordingly.

   In the years after the financial crisis, stock volatility declined. Consequently, the sellers of put options were exposed to a lower level of risk, and were more willing to accept a lower premium.

14-3 SPECULATING WITH STOCK OPTIONS

Stock options are frequently traded by investors who are attempting to capitalize on their expectations. When investors purchase an option that does not cover (hedge) their existing investments, the option can be referred to as “naked” (uncovered). Since speculators trade options to gamble on price movements rather than to hedge existing investments, their positions in options are naked. Whether speculators purchase call options or put options depends on their expectations.

   In some cases, speculators borrow a portion of the funds that they use to invest in stock options. The use of borrowed funds can magnify their gains, but it can also magnify their losses. The gains and losses described in this chapter would be more pronounced if the speculators cited in the examples used borrowed funds for a portion of their investment.

Exhibit 14.7 Framework for Explaining Why a Stock Option's Premium Changes over Time

14-3a Speculating with Call Options

Call options can be used to speculate on the expectation of an increase in the price of the underlying stock.

EXAMPLE

Pat Jackson expects Steelco stock to increase from its current price of $113 per share but does not want to tie up her available funds by investing in stocks. She purchases a call option on Steelco with an exercise price of $115 for a premium of $4 per share. Before the option's expiration date, Steelco's price rises to $121. At that time, Jackson exercises her option, purchasing shares at $115 per share. She then immediately sells those shares at the market price of $121 per share. Her net gain on this transaction is measured as follows:

Pat's net gain of $2 per share reflects a return of 50 percent (not annualized).

   If the price of Steelco stock had not risen above $115 before the option's expiration date, Pat would have let the option expire. Her net loss would have been the $4 per share she initially paid for the option, or $400 for one option contract. This example reflects a 100 percent loss, since the entire amount of the investment is lost.

   The potential gains or losses from this call option are shown in the left portion of  Exhibit 14.8 , based on the assumptions that (1) the call option is exercised on the expiration date, if at all, and (2) if the call option is exercised, the shares received are immediately sold.  Exhibit 14.8  shows that the maximum loss when purchasing this option is the premium of $4 per share. For stock prices between $115 and $119, the option is exercised and the purchaser of a call option incurs a net loss of less than $4 per share. The stock price of $119 is the break-even point, because the gain from exercising the option exactly offsets the premium paid for it. At stock prices above $119, a net gain is realized.

   The right portion of  Exhibit 14.8  shows the net gain or loss to a writer of the same call option, assuming that the writer obtains the stock only when the option is exercised. Under this condition, the call option writer's net gain (loss) is the call option purchaser's net loss (gain), assuming zero transaction costs. The maximum gain to the writer of a call option is the premium received.

Exhibit 14.8 Potential Gains or Losses on a Call Option: Exercise Price = $115, Premium = $4

Exhibit 14.9 Potential Gains or Losses for Three Call Options (Buyer's Perspective)

   Several call options are available for a given stock, and the risk–return potential will vary among them. Assume that three types of call options were available on Steelco stock with a similar expiration date, as described in  Exhibit 14.9 . The potential gains or losses per unit for each option are also shown in the exhibit, assuming that the option is exercised (if at all) on the expiration date. It is also assumed that if the speculators exercise the call option, they immediately sell the stock. This comparison of different options for a given stock illustrates the various risk–return trade-offs from which speculators can choose.

   Purchasers of call options are normally most interested in returns (profit as a percentage of the initial investment) under various scenarios. For this purpose, the contingency graph can be revised to reflect returns for each possible price per share of the underlying stock. The first step is to convert the profit per unit into a return for each possible price, as shown in  Exhibit 14.10 . For example, for the stock price of $116, Call Option 1 generates a return of 10 percent ($1 per share profit as a percentage of the $10 premium paid), Call Option 2 generates a loss of about 14 percent ($1 per share loss as a percentage of the $7 premium paid), and Call Option 3 generates a loss of 75 percent ($3 per share loss as a percentage of the $4 premium paid).

   The data can be transformed into a contingency graph as shown in  Exhibit 14.11 . This graph illustrates that, for Call Option 1, both the potential losses and the potential returns in the event of a high stock price are relatively low. Conversely, the potential losses for Call Option 3 are relatively high but so are the potential returns in the event of a high stock price.

Exhibit 14.10 Potential Returns on Three Different Call Options

	OPTION 1: EXERCISE PRICE = $105 PREMIUM = $10	OPTION 2: EXERCISE PRICE = $110 PREMIUM = $7	OPTION 3: EXERCISE PRICE = $115 PREMIUM = $4
PRICE OF STEELCO	PROFIT PER UNIT	PERCENTAGE RETURN	PROFIT PER UNIT	PERCENTAGE RETURN	PROFIT PER UNIT	PERCENTAGE RETURN
$104	−$10	−100%	−$7	−100%	−$4	−100%
106	−9	−90	−7	−100	−4	−100
108	−7	−70	−7	−100	−4	−100
110	−5	−50	−7	−100	−4	−100
112	−3	−30	−5	−71	−4	−100
114	−1	−10	−3	−43	−4	−100
116	1	10	1−	−14	−3	−75
118	3	30	1	14	−1	−25
120	5	50	3	43	1	25
122	7	70	5	71	3	75
124	9	90	7	100	5	125
126	11	110	9	129	7	175

14-3b Speculating with Put Options

Put options can be used to speculate on the expectation of a decrease in the price of the underlying stock.

EXAMPLE

A put option on Steelco is available with an exercise price of $110 and a premium of $2. If the price of Steelco stock falls below $110, speculators could purchase the stock and then exercise their put options to benefit from the transaction. However, they would need to make at least $2 per share on this transaction to fully recover the premium paid for the option. If the speculators exercise the option when the market price is $104, their net gain is measured as follows:

The net gain here is 200 percent, or twice as much as the amount paid for the put options.

   The potential gains or losses from the put option described here are shown in the left portion of  Exhibit 14.12 , based on the assumptions that (1) the put option is exercised on the expiration date, if at all, and (2) the shares would be purchased just before the put option is exercised. The exhibit shows that the maximum loss when purchasing this option is $2 per share. For stock prices between $108 and $110, the purchaser of a put option incurs a net loss of less than $2 per share. The stock price of $108 is the break-even point, because the gain from exercising the put option would exactly offset the $2 per share premium.

Exhibit 14.11 Potential Returns for Three Call Options (Buyer's Perspective)

   The right portion of  Exhibit 14.12  shows the net gain or loss to a writer of the same put option, assuming that the writer sells the stock received as the put option is exercised. Under this condition, the put option writer's net gain (loss) is the put option purchaser's net loss (gain), assuming zero transaction costs. The maximum gain to the writer of a put option is the premium received. As with call options, several put options are typically available for a given stock, and the potential gains or losses will vary among them.

14-3c Excessive Risk from Speculation

Speculating in options can be very risky. Financial institutions or other corporations that speculate in options normally have methods to closely monitor their risk and to measure their exposure to possible option market conditions. In several cases, however, a financial institution or a corporation incurred a major loss on options positions because of a lack of oversight over its options trading.

EXAMPLE

One of the most famous cases in which a financial institution took excessive risk with stock options is Barings PLC, an investment bank in the United Kingdom. In 1992, a clerk in Barings's London office named Nicholas Leeson was sent to manage the accounting at a Singapore subsidiary called Barings Futures. Shortly after he began his new position in Singapore, Leeson took and passed the examinations required to trade on the floor of the Singapore International Monetary Exchange (SIMEX). Barings Futures served as a broker on this exchange for some of its customers. Within a year of arriving in Singapore, Leeson was trading derivative contracts on the SIMEX as a broker for Barings Futures. He then began to trade for the firm's own account rather than just as a broker, trading options on the Nikkei (Japanese) stock index. At the same time, he also continued to serve as the accounting manager for Barings Futures. In this role Leeson was able to conceal losses on any derivative positions, so the financial reports to Barings PLC showed massive profits.

Exhibit 14.12 Potential Gains or Losses on a Put Option: Exercise Price = $110, Premium = $2

   By January 1995, Leeson's losses had accumulated to exceed the equivalent of $300 million. Leeson had periodically required funds to cover margin calls as his positions declined in value. Barings PLC met these funding requests, covering the equivalent of millions of dollars to satisfy the margin calls, yet did not recognize that the margin calls signaled a major problem.

   In late February 1995, an accounting clerk at Barings who noticed some discrepancies met with Leeson to reconcile the records. When Leeson was asked to explain specific accounting entries, he excused himself from the meeting and never returned. He left Singapore that night and faxed his resignation to Barings PLC from Kuala Lumpur, Malaysia. Barings PLC investigated and found that Leeson had accumulated losses totaling more than the equivalent of $1 billion, which caused Barings PLC to become insolvent. Leeson pleaded guilty to charges of fraud, and he was sentenced to prison for six and one-half years.

   Any firms that use futures or other derivative instruments can draw a few obvious lessons from the Barings collapse. First, firms should closely monitor the trading of derivative contracts by their employees to ensure that derivatives are being used within the firm's guidelines. Second, firms should separate the reporting function from the trading function so that traders cannot conceal trading losses. Third, when firms receive margin calls on derivative positions, they should recognize that there may be potential losses on their derivative instruments and should closely evaluate those positions. The Barings case was a wake-up call to many firms, which recognized the need to establish guidelines for their employees who take derivative positions and to monitor more closely the actions of these employees.

14-4 HEDGING WITH STOCK OPTIONS

Call and put options on selected stocks and stock indexes are commonly used for hedging against possible stock price movements. Financial institutions such as mutual funds, insurance companies, and pension funds manage large stock portfolios and are the most common users of options for hedging.

14-4a Hedging with Covered Call Options

Call options on a stock can be used to hedge a position in that stock.

EXAMPLE

Portland Pension Fund owns a substantial amount of Steelco stock. It expects that the stock will perform well in the long run, but it is concerned that the stock may perform poorly over the next few months because of temporary problems Steelco is experiencing. The sale of a call option on Steelco stock can hedge against such a potential loss. This is known as a covered call because the option is covered, or backed, by stocks already owned.

   If the market price of Steelco stock rises, the call option will likely be exercised and Portland will fulfill its obligation by selling its Steelco stock to the purchaser of the call option at the exercise price. But if the market price of Steelco stock declines, the option will not be exercised. Hence Portland would not have to sell its Steelco stock, and the premium received from selling the call option would represent a gain that could partially offset the decline in the price of the stock. In this case, although the market value of the institution's stock portfolio is adversely affected, the decline is at least partially offset by the premium received from selling the call option.

   Assume that Portland Pension Fund purchased Steelco stock at the market price of $112 per share. To hedge against a temporary decline in Steelco's stock price, Portland sells call options on Steelco stock with an exercise price of $110 per share for a premium of $5 per share. The net profit to Portland when using covered call writing is shown in  Exhibit 14.13  for various possible scenarios. For comparison purposes, the profit that Portland would earn if it did not use covered call writing but instead sold the stock on the option's expiration date is also shown (see the diagonal line) for various possible scenarios. The results show how covered call writing can partially offset losses when the stock performs poorly but can also partially offset gains when the stock performs well.

   The table in  Exhibit 14.13  explains the profit or loss per share from covered call writing. At any price above $110 per share as of the expiration date, the call option would be exercised and so Portland would have to sell its holdings of Steelco stock at the exercise price of $110 per share to the purchaser of the call option. The net gain to Portland would be $3 per share, determined as the premium of $5 per share (received when writing the option) minus the $2 per share difference between the price paid for the Steelco stock and the price at which the stock is sold. Comparing the profit or loss per scenarios with and without covered call writing, it is clear that covered call writing limits the upside potential return on stocks but also reduces the risk.

14-4b Hedging with Put Options

Put options on stock are also used to hedge stock positions.

EXAMPLE

Reconsider the example in which Portland Pension Fund was concerned about a possible temporary decline in the price of Steelco stock. Portland could hedge against a temporary decline in Steelco's stock price by purchasing put options on that stock. In the event that Steelco's stock price declines, Portland would likely generate a gain on its option position, which would help offset the reduction in the stock's price. If Steelco's stock price does not decline, Portland would not exercise its put option.

   Put options are typically used to hedge when portfolio managers are mainly concerned about a temporary decline in a stock's value. When portfolio managers are mainly concerned about the long-term performance of a stock, they are likely to sell the stock itself rather than hedge the position.

Exhibit 14.13 Risk-Return Trade-off from Covered Call Writing

Hedging with LEAPs   Long-term equity anticipations (LEAPs)  are options that have longer terms to expiration, usually between two and three years from the initial listing date. These options are available for some large capitalization stocks, and they may be a more effective hedge over a longer term period than using options with shorter terms to expiration. The transaction costs for hedging over a long period are lower than the costs of continually repurchasing short-term put options each time the options expire or are exercised. Furthermore, the costs of continually repurchasing put options are uncertain, whereas the costs of purchasing a put option on a long-term index option are known immediately.

14-5 OPTIONS ON ETFS AND STOCK INDEXES

Options are also traded on exchange-traded funds (ETFs) and stock indexes. Exchange-traded funds are funds that are designed to mimic particular indexes and are traded on an exchange. Thus, an ETF option provides the right to trade a specified ETF at a specified price by a specified expiration date. Since ETFs are traded like stocks, options on ETFs are traded like options on stocks. Investors who exercise a call option on an ETF will receive delivery of the ETF in their account. Investors who exercise a put option on an ETF will have the ETF transferred from their account to the counterparty on the put option.

   A  stock index option  provides the right to trade a specified stock index at a specified price by a specified expiration date. Call options on stock indexes allow the right to purchase the index, and put options on stock indexes allow the right to sell the index. If and when the index option is exercised, the cash payment is equal to a specified dollar amount multiplied by the difference between the index level and the exercise price.

   Options on stock indexes are similar to options on ETFs. However, the values of stock indexes change only at the end of each trading day, whereas ETF values can change throughout the day. Therefore, an investor who wants to capitalize on the expected movement of an index within a particular day will trade options on ETFs. An investor who wants to capitalize on the expected movement of an index over a longer period of time (such as a week or several months) can trade options on either ETFs or indexes.

   Options on indexes have become popular for speculating on general movements in the stock market. Speculators who anticipate a sharp increase in stock market prices overall may consider purchasing call options on one of the market indexes. Likewise, speculators who anticipate a stock market decline may consider purchasing put options on these indexes. A sampling of options that are traded on ETFs and on stock indexes is provided in  Exhibit 14.14 , where SPDR stands for Standard & Poor's Depositary Receipts.

   Options on sector indexes also exist, allowing investors the option to buy or sell an index that reflects a particular sector. These contracts are distinguished from stock index options because they represent a component of a stock index. Investors who are optimistic about the stock market in general might be more interested in stock index options, while investors who are especially optimistic about one particular sector may be more interested in options on a sector index. There are options available for many different sectors, including banking, energy, housing, oil exploration, semiconductors, and utilities.

14-5a Hedging with Stock Index Options

Financial institutions and other firms commonly take positions in options on ETFs or indexes to hedge against market or sector conditions that would adversely affect their asset portfolio or cash flows. The following discussion is based on the use of options on stock indexes, but options on ETFs could be used in the same manner.

Exhibit 14.14 Sampling of ETFs and Indexes on Which Options Are Traded

SAMPLING OF ETFs ON WHICH OPTIONS ARE TRADED
iShares Nasdaq Biotechnology	iShares Russell 1000 Growth Index Fund
iShares Goldman Sachs Technology Index	Energy Select Sector SPDR
iShares Goldman Sachs Software Index	Financial Select Sector SPDR
iShares Russell 1000 Index Fund	Utilities Select Sector SPDR
iShares Russell 1000 Value Index Fund	Health Care Select Sector SPDR
SAMPLING OF INDEXES ON WHICH OPTIONS ARE TRADED
Asia 25 Index	S&P SmallCap 600 Index
Euro 25 Index	Nasdaq 100 Index
Mexico Index	Russell 1000 Index
Dow Jones Industrial Average	Russell 1000 Value Index
Dow Jones Transportation Average	Russell 1000 Growth Index
Dow Jones Utilities Average	Russell Midcap Index
S&P 100 Index	Goldman Sachs Internet Index
S&P 500 Index	Goldman Sachs Software Index
Morgan Stanley Biotechnology Index

   Financial institutions such as insurance companies and pension funds maintain large stock portfolios whose values are driven by general market movements. If the stock portfolio is broad enough, any changes in its value will likely be highly correlated with market movements. For this reason, portfolio managers consider purchasing put options on a stock index to protect against stock market declines. The put options should be purchased on the stock index that most closely mirrors the portfolio to be hedged. If the stock market experiences a severe downturn, the market value of the portfolio declines. However, the put options on the stock index will generate a gain because the value of the index will be less than the exercise price. The greater the market downturn, the greater the decline in the market value of the portfolio but also the greater the gain from holding put options on a stock index. Thus, this offsetting effect minimizes the overall impact on the firm.

   If the stock market rises, the put options on the stock index will not be exercised. In this case, the firm will not recover the cost of purchasing the options. This situation is similar to purchasing other forms of insurance and then not using them. Some portfolio managers may still believe the options were worthwhile for temporary protection against downside risk.

Hedging with Long-Term Stock Index Options  Long-term equity anticipations are used by option market participants who want options with longer terms until expiration. For example, LEAPs on the S&P 100 and S&P 500 indexes are available, with expiration dates extending at least two years ahead from the initial listing date. Each of these indexes is revised to one-tenth its normal size when applying LEAPs. The result is smaller premiums, which makes the LEAPs more affordable to smaller investors.

Dynamic Asset Allocation with Stock Index Options  Dynamic asset allocation involves switching between risky and low-risk investment positions over time in response to changing expectations. Some portfolio managers use stock index options as a tool for dynamic asset allocation. For example, when portfolio managers anticipate favorable market conditions, they purchase call options on a stock index, which intensify the effects of the market conditions. Essentially, the managers are using stock index options to increase their exposure to stock market conditions. Conversely, when they anticipate unfavorable market movements, they can purchase put options on a stock index in order to reduce the effects that market conditions will have on their stock portfolios.

   Because stock options are available with various exercise prices, portfolio managers can select an exercise price that provides the degree of protection desired. For example, assume an existing stock index is quite similar to a manager's stock portfolio and that he wants to protect against any loss exceeding 5 percent. If the prevailing level of the index is 400, the manager should purchase put options with an exercise price of 380 because that level is 5 percent lower than 400. If the index declines to a level below 380, the manager will exercise the options and the gain from doing so will partially offset the reduction in the stock portfolio's market value.

   This strategy is essentially a form of insurance, where the premium paid for the put option is similar to an insurance premium. Because the index must decline by 5 percent before there is any possibility that the option will be exercised, this is similar to the “deductible” that is typical of insurance policies. If portfolio managers desire to protect against even smaller losses, they can purchase a put option that specifies a higher exercise price on the index, such as 390. To obtain the extra protection, however, they would have to pay a higher premium for the option. In other words, the cost of the portfolio insurance would be higher because of the smaller “deductible” desired.

   In another form of dynamic asset allocation, portfolio managers sell (write) call options on stock indexes in periods when they expect the stock market to be stable. This strategy does not create a perfect hedge, but it can enhance the portfolio's performance in periods when stock prices are stagnant or declining.

   Portfolio managers can adjust the risk–return profile of their investment position by using stock index options rather than restructuring their existing stock portfolios. This form of dynamic asset allocation avoids the substantial transaction costs associated with restructuring the stock portfolios.

14-5b Using Index Options to Measure the Market's Risk

Just as a stock's implied volatility can be derived from information about options on that stock, a stock index's implied volatility can be derived from information about options on that stock index. The same factors that affect the option premium on a stock affect the option premium on an index. Thus, the premium on an index option is positively related to the expected volatility of the underlying stock index. If investors want to estimate the expected volatility of the stock index, they can use software packages to insert values for the prevailing option premium and all the other factors (except volatility) that affect an option premium.

   The CBOE volatility index (VIX) represents the implied volatility derived from options on the S&P 500 index (an index of 500 large stocks). This index is closely monitored by many investors because it indicates the market's anticipated volatility for the market (with the S&P 500 serving as proxy for the market). The VIX is sometimes referred to as the “fear index” because high values are perceived to reflect a high degree of fear that stock prices could decline.

14-6 OPTIONS ON FUTURES CONTRACTS

In recent years, the concept of options has been applied to futures contracts to create options on futures contracts (sometimes referred to as “futures options”). An option on a particular futures contract gives its owner the right (but not an obligation) to purchase or sell that futures contract for a specified price within a specified period of time. Thus, options on futures grant the power to take the futures position if favorable conditions occur but the flexibility to avoid the futures position (by letting the option expire) if unfavorable conditions occur. As with other options, the purchaser of options on futures pays a premium.

   Similarly, options are available on stock index futures. They are used for speculating on expected stock market movements or hedging against adverse market conditions. Individuals and financial institutions use them in a manner similar to the way stock index options are used.

   Options are also available on interest rate futures, such as Treasury note futures or Treasury bond futures. The settlement dates of the underlying futures contracts are usually a few weeks after the expiration date of the corresponding options contracts.

   A call option on interest rate futures grants the right to purchase a futures contract at a specified price within a specified period of time. A put option on financial futures grants the right (again, not an obligation) to sell a particular financial futures contract at a specified price within a specified period of time. Because interest rate futures contracts can hedge interest rate risk, options on interest rate futures might be considered by any financial institution that is exposed to that risk, including savings institutions, commercial banks, life insurance companies, and pension funds.

14-6a Speculating with Options on Futures

Speculators who anticipate a change in interest rates should also expect a change in bond prices. They could take a position in options on Treasury bond futures to capitalize on their expectations.

Speculation Based on an Expected Decline in Interest Rates  If speculators expect a decline in interest rates, they may consider purchasing a call option on Treasury bond futures. If their expectations are correct, the market value of Treasury bonds will rise and the price of a Treasury bond futures contract will rise as well. The speculators can exercise their option to purchase futures at the exercise price, which will be lower than the value of the futures contract.

EXAMPLE

Kelly Warden expects interest rates to decline and purchases a call option on Treasury bond futures. The exercise price on Treasury bond futures is 94-32 (9432/64 percent of $100,000, or $94,500). The call option is purchased at a premium of 2-00 (i.e., 2 percent of $100,000), which equals $2,000. Assume that interest rates do decline and, as a result, the price of the Treasury bond futures contract rises over time to a value of 99-00 ($99,000) shortly before the option's expiration date. At this time, Kelly decides to exercise the option and closes out the position by selling an identical futures contract (to create an offsetting position) at a higher price than the price at which she purchased the futures. Kelly's net gain from this speculative strategy is

This net gain of $2,500 represents a return on investment of 125 percent.

   The seller of the call option will have the opposite position to the buyer. Thus the gain (or loss) to the buyer will equal the loss (or gain) to the seller of the call option.

EXAMPLE

Ellen Rose sold the call option purchased by Kelly Warden in the previous example. Ellen is obligated to purchase and provide the futures contract at the time the option is exercised. Her net gain from this speculative strategy is

In the absence of transaction costs, Ellen's loss is equal to Kelly's gain. If the Treasury bond futures price had remained below the exercise price of 94-32 ($94,500) until the expiration date, the option would not have been exercised. In that case, the net gain from purchasing the call option on Treasury bond futures would have been −$2,000 (the premium paid for the option) and the net gain from selling the call option would have been $2,000.

   When interest rates decline, the buyers of call options on Treasury bonds may simply sell their previously purchased options just before expiration. If interest rates rise, the options will not be desirable. Then buyers of call options on Treasury bond futures will let their options expire, and their loss will be the premium paid for the call options on futures. Thus the loss from purchasing options on futures is more limited than the loss from simply purchasing futures contracts.

   Some speculators who expect interest rates to remain stable or decline may be willing to sell a put option on Treasury bond futures. If their expectations are correct, the price of a futures contract will likely rise and so the put option will not be exercised. Therefore, sellers of the put option would earn the premium that was paid to them when they sold the option.

Speculation Based on an Expected Increase in Interest Rates  If speculators expect interest rates to increase, they can benefit from purchasing a put option on Treasury bond futures. If their expectations are correct, the market value of Treasury bonds will decline and so the price of a Treasury bond futures contract will also decline. The speculators can exercise their option to sell futures at the exercise price, which will be higher than the value of the futures contract. They can then purchase futures (to create an offsetting position) at a lower price than the price at which they sold futures. If interest rates decline, the speculators will likely let the options expire and their loss will be the premium paid for the put options on futures.

EXAMPLE

John Drummer expects interest rates to increase and purchases a put option on Treasury bond futures. Assume the exercise price on Treasury bond futures is 97-00 ($97,000) and the premium paid for the put option is 3-00 ($3,000). Assume that interest rates do increase and, as a result, the price of the Treasury bond futures contract declines over time to a value of 89-00 ($89,000) shortly before the option's expiration date. At this time, John decides to exercise the option and closes out the position by purchasing an identical futures contract. John's net gain from this speculative strategy is

The net gain of $5,000 represents a return on investment of about 167 percent.

   The person who sold the put option on Treasury bond futures to John in this example incurred a loss of $5,000, assuming that the position was closed out (by selling an identical futures contract) on the same date that John's position was closed out. If the Treasury bond futures price had remained above the exercise price of 97–00 until the expiration date, then the option would not have been exercised and John would have lost $3,000 (the premium paid for the put option).

   Some speculators who anticipate an increase in interest rates may be willing to sell a call option on Treasury bond futures. If their expectations are correct, the price of the futures contract will likely decline and hence the call option will not be exercised.

14-6b Hedging with Options on Interest Rate Futures

Options on futures contracts are also used to hedge against risk. Financial institutions commonly hedge their bond or mortgage portfolios with options on interest rate futures contracts. The position they take on the options contract is designed to create a gain that can offset a loss on their bond or mortgage portfolio while still allowing some upside potential.

EXAMPLE

Emory Savings and Loan Association has a large number of long-term, fixed-rate mortgages that are mainly supported by short-term funds and would therefore be adversely affected by rising interest rates. As shown in the previous chapter, sales of Treasury bond futures can partially offset the adverse effect of rising interest rates in such a situation. Recall that if interest rates decline instead, the potential increase in Emory's interest rate spread (difference between interest revenues and expenses) would be partially offset by the loss on the futures contract.

   One potential limitation of selling interest rate futures to hedge mortgages is that households may prepay their mortgages. If interest rates decline and most fixed-rate mortgages are prepaid, Emory will incur a loss on the futures position without an offsetting gain on its spread. To protect against this risk, Emory can purchase put options on Treasury bond futures. Suppose that Emory purchases put options on Treasury bond futures with an exercise price of 98-00 ($98,000) for a premium of 2-00 ($2,000) per contract. The initial Treasury bond futures price is 99-00 at the time. First, assume that interest rates rise, causing a decline in the Treasury bond futures price to 91-00. In this scenario, Emory will exercise its right to sell Treasury bond futures and offset its position by purchasing identical futures contracts and thereby generate a net gain of $5,000 per contract, as shown in  Exhibit 14.15 . The gain on the futures position helps to offset the reduction in Emory's spread that occurs because of the higher interest rates.

   Now consider a second scenario in which interest rates decline, causing the Treasury bond futures price to rise to 104-00. In this scenario, Emory does not exercise the put options on Treasury bond futures because the futures position would result in a loss.

   This example shows how a put option on futures offers more flexibility than simply selling futures. However, a premium must be paid for the put option. Financial institutions that wish to hedge against rising interest rate risk should compare the possible outcomes from selling interest rate futures contracts versus purchasing put options on interest rate futures in order to hedge interest rate risk.

Exhibit 14.15 Results from Hedging with Put Options on Treasury Bond Futures

	SCENARIO 1: · • INTEREST RATES RISE · • T-BOND FUTURES PRICE DECLINES TO 91–00	SCENARIO 2: · • INTEREST RATES DECLINE · • T-BOND FUTURES PRICE INCREASES TO 104–00
Effect on Emory's spread	Spread is reduced.	Spread is increased, but mortgage prepayments may occur.
Effect on T-bond futures price	Futures price decreases.	Futures price increases.
Decision on exercising the put option	Exercise put option.	Do not exercise put option.
Selling price of T-bond futures	$98,000	Not sold
−Purchase price of T-bond futures	−$91,000	Not purchased
−Price paid for put option	−$2,000	−$2,000
=Net gain per option	$5,000	−$2,000

14-6c Hedging with Options on Stock Index Futures

Financial institutions and other investors commonly hedge their stock portfolios with options on stock index futures contracts. The position they take on the options contract is designed to create a gain that can offset a loss on their stock portfolio while still allowing some upside potential.

EXAMPLE

You currently manage a stock portfolio that is valued at $400,000, and you plan to hold these stocks over a long-term period. However, you are concerned that the stock market may experience a temporary decline over the next three months and that your stock portfolio will probably decline by about the same degree as the market. You want to create a hedge so that your portfolio will decline no more than 3 percent from its present value, but you would like to maintain any upside potential. You can purchase a put option on index futures to hedge your stock portfolio. Put options on S&P 500 index futures are available with an expiration date about three months from now.

   Assume that the S&P 500 index level is currently 1600 and that one particular put option on index futures has a strike price of 1552 (3 percent less than the prevailing index level) and a premium of 10. Since the options on S&P 500 index futures are priced at $250 times the quoted premium, the dollar amount to be paid for this option is 10 × $250 = $2,500. If the index level declines below 1552 (reflecting a decline of more than 3 percent), then you may exercise the put option on index futures, which gives you the right to sell the index for a price of 1552. At the settlement date of the futures contract, you will receive $250 times the difference between the futures price of 1552 and the prevailing index level. If the market declines by 5 percent, for example, the index will decline from 1600 to 1520. There will be a gain on the index futures contract of (1552 − 1520) × $250 = $8,000. Meanwhile, a 5 percent decline in the value of the portfolio reflects a loss of $20,000 (0.05 − $400,000 = $20,000). The $8,000 gain (excluding the premium paid) from the options contract reduces the overall loss to $12,000, or 3 percent of the portfolio.

Determining the Degree of the Hedge with Options on Stock Index Futures  In the previous example, losses of less than 3 percent are not hedged. When using put options to hedge, various strike prices exist for an option on a specific stock index and for a specific expiration date. For example, put options on the S&P 500 index may be available with strike prices of 1760, 1800, 1840, and so on. The higher the strike price relative to the prevailing index value, the higher the price at which the investor can lock in the sale of the index. However, a higher premium must be paid to purchase put options with a higher strike price. From a hedging perspective, this simply illustrates that a higher price must be paid to “insure” (hedge) against losses resulting from stock market downturns. This concept is analogous to automobile insurance, where a person must pay a higher premium for a policy that features a lower deductible.

Selling Call Options to Cover the Cost of Put Options  In the previous example, the cost of hedging with a put option on index futures is $2,500. Given your expectations of a weak stock market over the next three months, you could generate some fees by selling call options on S&P 500 index futures to help cover the cost of purchasing put options.

EXAMPLE

Assume that there is a call option on S&P 500 index futures with a strike price of 1648 (3 percent above the existing index level) and a premium of 10. You can sell a call option on index futures for $2,500 (10 × $250) and use the proceeds to pay the premium on the put option. The obvious disadvantage of selling a call option to finance the purchase of the put option is that it limits your upside potential. For example, if the market rises by 5 percent over the three-month period, the S&P 500 index level will rise to 1680. The difference between this level and the strike price of 1648 on the call option forces you to make a payment of (1680 − 1648) × $250 = $8,000 to the owner of the call option. This partially offsets the gain to your portfolio that resulted from the favorable market conditions.

   When attempting to hedge larger portfolios than the one in the previous example, additional put options would be purchased to hedge the entire portfolio against a possible decline in the market. For example, if your stock portfolio were $1.2 million, you would need to purchase three put options on S&P 500 index futures contracts. Since each index futures contract would have a value of $400,000, you would need a short position in three index futures contracts to hedge the entire stock portfolio (assuming that the index and the stock portfolio move in tandem).

14-7 OPTIONS AS EXECUTIVE COMPENSATION

Many firms distribute stock options to executives and other managers as a reward for good performance. For example, a manager may receive a salary along with call options on 10,000 shares of stock that have an exercise price above the prevailing price and an expiration date of five years from today. The purpose of awarding options as compensation is to increase the executive's incentive to make decisions that increase the value of the firm's stock, as their compensation from options is tied to the stock price.

14-7a Limitations of Option Compensation

Many option compensation programs do not account for general market conditions. Thus executives who owned stock options when general stock market conditions were more favorable may have earned high compensation because their firms' stock prices increased substantially during that period, even for firms that performed poorly relative to others in the same industry. Because compensation from holding options is driven more by general stock market conditions than by the firm's relative performance, stock options are not always effective at aligning executives' incentives with actual firm performance.

   Another concern with using options as compensation is that executives with substantial options may be tempted to manipulate the stock's price upward in the short term, even though doing so adversely affects the stock price in the long term. For example, they might use accounting methods that defer the reporting of some expenses until next year and accelerate the reporting of some revenue. In this way, short-term earnings will appear favorable but earnings in the following period will be reduced. When these executives believe that the stock price has peaked in the short term, they can exercise their options by selling their shares in the secondary market. Firms can prevent the wrongful use of options by requiring that executives hold them for several years before exercising them.

   Until 2006, firms did not have to report their options as an expense but could claim them as a tax deduction. If Enron Corporation had reported its option compensation as expenses on its income statements over the five-year period preceding its 2001 bankruptcy, its net income would have been reduced by $600 million. Because options did not have to be reported as an expense on the income statements, some firms were overly generous in awarding options. Global Crossing's CEO earned $730 million from options before the firm filed for bankruptcy. Largely as a result of options, U.S. CEO compensation is more than 500 times the average compensation of a firm's employees.

Backdating Options  In the late 1990s and early 2000s, some firms allowed their CEOs to backdate options they had already been granted to an earlier period when the stock price was lower. This enabled the CEOs to use a lower exercise price on their call options, generating larger gains for them when they exercised their options. Thus option compensation to some CEOs was mostly influenced by the backdating process and not by how well the stock performed. When the practice of backdating options was publicized in 2006, firms that allowed backdating terminated the practice.

14-8 GLOBALIZATION OF OPTIONS MARKETS

The globalization of stock markets has resulted in the need for a globalized market in stock options. Options on stock indexes representing various countries are now available. Options exchanges have been established in numerous countries, including Australia, Austria, Belgium, France, Germany, and Singapore. U.S. portfolio managers who maintain large holdings of stocks from specific countries are heavily exposed to the conditions of those markets. Rather than liquidate the portfolio of foreign stocks to protect against a possible temporary decline, the managers can purchase put options on the foreign stock index of concern. Portfolio managers residing in these countries can also use this strategy to hedge their stock portfolios.

   Portfolio managers desiring to capitalize on the expectation of temporary favorable movements in foreign markets can purchase call options on the corresponding stock indexes. Thus the existence of options on foreign stock indexes allows portfolio managers to hedge or speculate based on forecasts of foreign market conditions. The trading of options on foreign stock indexes avoids the transaction costs associated with buying and selling large portfolios of foreign stocks.

14-8a Currency Options Contracts

A  currency call option  provides the right to purchase a specified currency for a specified price within a specified period of time. Corporations involved in international business transactions use currency call options to hedge future payables. If the exchange rate at the time payables are due exceeds the exercise price, corporations can exercise their options and purchase the currency at the exercise price. If the prevailing exchange rate is lower than the exercise price, corporations can purchase the currency at the prevailing exchange rate and let the options expire.

   Speculators purchase call options on currencies that they expect to strengthen against the dollar. If the foreign currency strengthens as expected, they can exercise their call options to purchase the currency at the exercise price and then sell the currency at the prevailing exchange rate.

   A  currency put option  provides the right to sell a specified currency for a specified price within a specified period of time. Corporations involved in international business transactions may purchase put options to hedge future receivables. If the exchange rate at the time they receive payment in a foreign currency is less than the strike price, they can exercise their option by selling the currency at that price. But if the prevailing exchange rate is higher than the exercise price, they can sell the currency at the prevailing exchange rate and let the options expire.

   Speculators purchase put options on currencies they expect to weaken against the dollar. If the foreign currency weakens as expected, the speculators can purchase the currency at the prevailing spot rate and exercise their put options to sell the currency at the exercise price.

   For every buyer of a currency call or put option, there must be a seller (or writer). A writer of a call option is obligated to sell the specified currency at the specified strike price if the option is exercised. A writer of a put option is obligated to purchase the specified currency at the specified strike price if the option is exercised. Speculators are inclined to write call options on foreign currencies that they expect to weaken against the dollar or to write put options on those they expect to strengthen against the dollar. If a currency option expires without being exercised, the writer earns the up-front premium received.

SUMMARY

· ▪ Stock options are traded on exchanges, just as many stocks are. Orders submitted by a brokerage firm are transmitted to a trading floor, where floor brokers execute the trades. Many trades are executed electronically.

· ▪ The premium of a call option is influenced by the characteristics of the option and of the underlying stock that can affect the potential gains. In particular, the premium is higher when the market price of the stock is high relative to the exercise price, when the stock's volatility is greater, and when the term until expiration is longer. For put options, the higher the market price of the stock relative to the exercise price, the lower the premium. The volatility of the underlying stock and the term to expiration are related to the put option premium in the same manner as they are to the call option premium.

· ▪ Speculators purchase call options on stocks whose prices are expected to rise and purchase put options on those expected to decrease.

· ▪ Financial institutions can hedge against adverse movements in a stock by selling call options on that stock. Alternatively, they can purchase put options on that stock.

· ▪ Financial institutions commonly hedge their stock portfolios by purchasing put options on stock indexes. They may also use stock index options as a tool for dynamic asset allocation, increasing their exposure when they have optimistic views about the stock market and reducing their exposure (buying put options on stock indexes) when they have pessimistic views.

· ▪ Speculators purchase call options on interest rate futures contracts when they expect interest rates to decrease. Financial institutions with large holdings of long-term debt securities hedge against interest rate risk by purchasing put options on interest rate futures. Index options can be used to speculate on movements in stock indexes and require only a small investment. Put options on stock indexes can be purchased to hedge a stock portfolio whose movements are similar to that of the stock index. Options on stock index futures can be used to speculate on movements in the value of the stock index futures contract. Put options on stock index futures can be purchased to hedge portfolios of stocks that move in tandem with the stock index.

POINT COUNTER-POINT

If You Were a Major Shareholder of a Publicly Traded Firm, Would You Prefer That Stock Options Be Traded on That Stock?

Point  No. Options can be used by investors to speculate, and excessive trading of the options may push the stock price away from its fundamental price.

Counter-Point  Yes. Options can be used by investors to temporarily hedge against adverse movements in the stock, so they may reduce the selling pressure on the stock in some periods.

Who Is Correct?  Use the Internet to learn more about this issue and then formulate your own opinion.

QUESTIONS AND APPLICATIONS

· 1. Options versus Futures Describe the general differences between a call option and a futures contract.

· 2. Speculating with Call Options How are call options used by speculators? Describe the conditions under which their strategy would backfire. What is the maximum loss that could occur for a purchaser of a call option?

· 3. Speculating with Put Options How are put options used by speculators? Describe the conditions under which their strategy would backfire. What is the maximum loss that could occur for a purchaser of a put option?

· 4. Selling Options Under what conditions would speculators sell a call option? What is the risk to speculators who sell put options?

· 5. Factors Affecting Call Option Premiums Identify the factors affecting the premium paid on a call option. Describe how each factor affects the size of the premium.

· 6. Factors Affecting Put Option Premiums Identify the factors affecting the premium paid on a put option. Describe how each factor affects the size of the premium.

· 7. Leverage of Options How can financial institutions with stock portfolios use stock options when they expect stock prices to rise substantially but do not yet have sufficient funds to purchase more stock?

· 8. Hedging with Put Options Why would a financial institution holding the stock of Hinton Co. consider buying a put option on that stock rather than simply selling it?

· 9. Call Options on Futures Describe a call option on interest rate futures. How does it differ from purchasing a futures contract?

· 10. Put Options on Futures Describe a put option on interest rate futures. How does it differ from selling a futures contract?

Advanced Questions

· 11. Hedging Interest Rate Risk Assume a savings institution has a large amount of fixed-rate mortgages and obtains most of its funds from short-term deposits. How could it use options on financial futures to hedge its exposure to interest rate movements? Would futures or options on futures be more appropriate if the institution is concerned that interest rates will decline, causing a large number of mortgage prepayments?

· 12. Hedging Effectiveness Three savings and loan institutions (S&Ls) have identical balance sheet compositions: a high concentration of short-term deposits that are used to provide long-term, fixed-rate mortgages. The S&Ls took the following positions one year ago.

NAME OF S&L	POSITION
LaCrosse	Sold financial futures
Stevens Point	Purchased put options on interest rate futures
Whitewater	Did not take any position in futures

· Assume that interest rates declined consistently over the last year. Which of the three S&Ls would have achieved the best performance based on this information? Explain.

· 13. Change in Stock Option Premiums Explain how and why the option premiums may change in response to a surprise announcement that the Fed will increase interest rates, even if stock prices are not affected.

· 14. Speculating with Stock Options The price of Garner stock is $40. There is a call option on Garner stock that is at the money with a premium of $2.00. There is a put option on Garner stock that is at the money with a premium of $1.80. Why would investors consider writing this call option and this put option? Why would some investors consider buying this call option and this put option?

· 15. How Stock Index Option Prices May Respond to Prevailing Conditions Consider the prevailing conditions that could affect the demand for stocks, including inflation, the economy, the budget deficit, the Fed's monetary policy, political conditions, and the general mood of investors. Based on these conditions, would you consider purchasing stock index options at this time? Offer some logic to support your answer. Which factor do you think will have the biggest impact on stock index option prices?

· 16. Backdating Stock Options Explain what backdating stock options entails. Is backdating consistent with rewarding executives who help to maximize shareholder wealth?

· 17. CBOE Volatility Index How would you interpret a large increase in the CBOE volatility index (VIX)? Explain why the VIX increased substantially during the credit crisis.

Interpreting Financial News

Interpret the following comments made by Wall Street analysts and portfolio managers.

· a. “Our firm took a hit because we wrote put options on stocks just before the stock market crash.”

· b. “Before hedging our stock portfolio with options on index futures, we search for the index that is most appropriate.”

· c. “We prefer to use covered call writing to hedge our stock portfolios.”

Managing in Financial Markets

Hedging with Stock Options As a stock portfolio manager, you have investments in many U.S. stocks and plan to hold these stocks over a long-term period. However, you are concerned that the stock market may experience a temporary decline over the next three months and that your stock portfolio will probably decline by about the same degree as the market. You are aware that options on S&P 500 index futures are available. The following options on S&P 500 index futures are available and have an expiration date about three months from now:

STRIKE PRICE	CALL PREMIUM	PUT PREMIUM
1372	40	24
1428	24	40

The options on S&P 500 index futures are priced at $250 times the quoted premium. Currently, the S&P 500 index level is 1400. The strike price of 1372 represents a 2 percent decline from the prevailing index level, and the strike price of 1428 represents an increase of 2 percent above the prevailing index level.

· a. Assume that you want to take an options position to hedge your entire portfolio, which is currently valued at about $700,000. How many index option contracts should you take a position in to hedge your entire portfolio?

· b. Assume that you want to create a hedge so that your portfolio will lose no more than 2 percent from its present value. How can you take a position in options on index futures to achieve this goal? What is the cost to you as a result of creating this hedge?

· c. Given your expectations of a weak stock market over the next three months, how can you generate some fees from the sale of options on S&P 500 index futures to help cover the cost of purchasing options?

PROBLEMS

· 1. Writing Call Options A call option on Illinois stock specifies an exercise price of $38. Today, the stock's price is $40. The premium on the call option is $5. Assume the option will not be exercised until maturity, if at all. Complete the following table:

ASSUMED STOCK PRICE AT THE TIME THE CALL OPTION IS ABOUT TO EXPIRE	NET PROFIT OR LOSS PER SHARE TO BE EARNED BY THE WRITER (SELLER) OF THE CALL OPTION
$37
39
41
43
45
48

· 2. Purchasing Call Options A call option on Michigan stock specifies an exercise price of $55. Today, the stock's price is $54 per share. The premium on the call option is $3. Assume the option will not be exercised until maturity, if at all. Complete the following table for a speculator who purchases the call option:

ASSUMED STOCK PRICE AT THE TIME THE CALL OPTION IS ABOUT TO EXPIRE	NET PROFIT OR LOSS PER SHARE TO BE EARNED BY THE SPECULATOR
$50
52
54
56
58
60
62

· 3. Purchasing Put Options A put option on Iowa stock specifies an exercise price of $71. Today, the stock's price is $68. The premium on the put option is $8. Assume the option will not be exercised until maturity, if at all. Complete the following table for a speculator who purchases the put option (and currently does not own the stock):

ASSUMED STOCK PRICE AT THE TIME THE PUT OPTION IS ABOUT TO EXPIRE	NET PROFIT OR LOSS PER SHARE TO BE EARNED BY THE SPECULATOR
$60
64
68
70
72
74
76

· 4. Writing Put Options A put option on Indiana stock specifies an exercise price of $23. Today, the stock's price is $24. The premium on the put option is $3. Assume the option will not be exercised until maturity, if at all. Complete the following table:

ASSUMED STOCK PRICE AT THE TIME THE PUT OPTION IS ABOUT TO EXPIRE	NET PROFIT OR LOSS PER SHARE TO BE EARNED BY THE WRITER (OR SELLER) OF THE PUT OPTION
$20
21
22
23
24
25
26

· 5. Covered Call Strategy

· a. Evanston Insurance, Inc., has purchased shares of Stock E at $50 per share. It will sell the stock in six months. It considers using a strategy of covered call writing to partially hedge its position in this stock. The exercise price is $53, the expiration date is six months, and the premium on the call option is $2. Complete the following table:

POSSIBLE PRICE OF STOCK E IN SIX MONTHS	PROFIT OR LOSS PER SHARE IF A COVERED CALL STRATEGY IS USED	PROFIT OR LOSS PER SHARE IF A COVERED CALL STRATEGY IS NOT USED
$47
50
52
55
57
60

· b. Assume that each of the six stock prices in the table's first column has an equal probability of occurring. Compare the probability distribution of the profits (or losses) per share when using covered call writing versus not using it. Would you recommend covered call writing in this situation? Explain.

· 6. Put Options on Futures Purdue Savings and Loan Association purchased a put option on Treasury bond futures with a September delivery date and an exercise price of 91–16. The put option has a premium of 1–32. Assume that the price of the Treasury bond futures decreases to 88–16. Should Purdue exercise the option or let it expire? What is Purdue's net gain or loss after accounting for the premium paid on the option?

· 7. Call Options on Futures Wisconsin, Inc., purchased a call option on Treasury bond futures at a premium of 2–00. The exercise price is 92–08. If the price of the Treasury bond futures rises to 93–08, should Wisconsin exercise the call option or let it expire? What is Wisconsin's net gain or loss after accounting for the premium paid on the option?

· 8. Call Options on Futures DePaul Insurance Company purchased a call option on an S&P 500 futures contract. The option premium is quoted as $6. The exercise price is 1430. Assume the index on the futures contract becomes 1440. Should DePaul exercise the call option or let it expire? What is the net gain or loss to DePaul after accounting for the premium paid for the option?

· 9. Covered Call Strategy Coral, Inc., has purchased shares of Stock M at $28 per share. Coral will sell the stock in six months. It considers using a strategy of covered call writing to partially hedge its position in this stock. The exercise price is $32, the expiration date is six months, and the premium on the call option is $2.50. Complete the following table:

POSSIBLE PRICE OF STOCK M IN SIX MONTHS	PROFIT OR LOSS PER SHARE IF COVERED CALL STRATEGY IS USED
$25
28
33
36

· 10. Hedging with Bond Futures Smart Savings Bank desired to hedge its interest rate risk. It considered two possibilities: (1) sell Treasury bond futures at a price of 94–00, or (2) purchase a put option on Treasury bond futures. At the time, the price of Treasury bond futures was 95–00. The face value of Treasury bond futures was $100,000. The put option premium was 2–00, and the exercise price was 94–00. Just before the option expired, the Treasury bond futures price was 91–00, and Smart Savings Bank would have exercised the put option at that time, if at all. This is also the time when it would have offset its futures position, if it had sold futures. Determine the net gain to Smart Savings Bank if it had sold Treasury bond futures versus if it had purchased a put option on Treasury bond futures. Which alternative would have been more favorable, based on the situation that occurred?

FLOW OF FUNDS EXERCISE

Hedging with Options Contracts

Carson Company would like to acquire Vinnet, Inc., a publicly traded firm in the same industry. Vinnet's stock price is currently much lower than the prices of other firms in the industry because it is inefficiently managed. Carson believes that it could restructure Vinnet's operations and improve its performance. It is about to contact Vinnet to determine whether Vinnet will agree to an acquisition. Carson is somewhat concerned that investors may learn of its plans and buy Vinnet stock in anticipation that Carson will need to pay a high premium (perhaps a 30 percent premium above the prevailing stock price) in order to complete the acquisition. Carson decides to call a bank about its risk, as the bank has a brokerage subsidiary that can help it hedge with stock options.

· a. How can Carson use stock options to reduce its exposure to this risk? Are there any limitations to this strategy, given that Carson will ultimately have to buy most or all of the Vinnet stock?

· b. Describe the maximum possible loss that may be directly incurred by Carson as a result of engaging in this strategy.

· c. Explain the results of the strategy you offered in the previous question if Vinnet plans to avoid the acquisition attempt by Carson.

INTERNET/EXCEL EXERCISES

· 1. Go to  www.cboe.com  and, under “Quotes & Data,” select “Delayed Quotes Classic.” Insert the ticker symbol for a stock option in which you are interested. Assess the results. Did the premium (“Net”) on the call options increase or decrease today? Did the premium on the put options increase or decrease today?

· 2. Based on the changes in the premium, do you think the underlying stock price increased or decreased? Explain.

ONLINE ARTICLES WITH REAL-WORLD EXAMPLES

Find a recent practical article available online that describes a real-world example regarding a specific financial institution or financial market that reinforces one or more concepts covered in this chapter.

   If your class has an online component, your professor may ask you to post your summary of the article there and provide a link to the article so that other students can access it. If your class is live, your professor may ask you to summarize your application of the article in class. Your professor may assign specific students to complete this assignment or may allow any students to do the assignment on a volunteer basis.

   For recent online articles and real-world examples related to this chapter, consider using the following search terms (be sure to include the prevailing year as a search term to ensure that the online articles are recent):

· 1. stock options AND profits

· 2. stock options AND losses

· 3. stock options AND speculators

· 4. stock options AND hedge

· 5. stock option premium AND volatility

· 6. stock option AND ETFs

· 7. stock index option

· 8. LEAPs AND speculate

· 9. selling stock options AND cover

· 10. stock options AND compensation

APPENDIX 14 Option Valuation

THE BINOMIAL PRICING MODEL

The binomial option-pricing model was originally developed by William F. Sharpe. An advantage of the model is that it can be used to price both European-style and American-style options with or without dividends. European options are put or call options that can be exercised only at maturity; American options can be exercised at any time prior to maturity.

Assumptions of the Binomial Pricing Model

The following are the main assumptions of the binomial pricing model:

· 1. The continuous random walk underlying the Black-Scholes model can be modeled by a discrete random walk with the following properties:

· ▪ The asset price changes only at discrete (noninfinitesimal) time steps.

· ▪ At each time step, the asset price may move either up or down; thus there are only two returns, and these two returns are the same for all time steps.

· ▪ The probabilities of moving up and down are known.

· 2. The world is risk-neutral. This allows the assumption that investors' risk preferences are irrelevant and that investors are risk-neutral. Furthermore, the return from the underlying asset is the risk-free interest rate.

Using the Binomial Pricing Model to Price Call Options

The following is an example of how the binomial pricing model can be employed to price a call option (i.e., to determine a call option premium). To use the model, we need information for three securities: the underlying stock, a risk-free security, and the stock option.

   Assume that the price of Gem Corporation stock today is $100. Furthermore, it is estimated that Gem stock will be selling for either $150 or $70 in one year. That is, the stock is expected either to rise by 50 percent or to fall by 30 percent. Also assume that the annual risk-free interest rate on a one-year Treasury bill is 10 percent, compounded continuously. Assume that a T-bill currently sells for $100. Since interest is continuously compounded, the T-bill will pay interest of $100 × (e10 − 1), or $10.52.

   Currently, a call option on Gem stock is available with an exercise price of $100 and an expiration date one year from now. Since the call option is an option to buy Gem stock, the option will have a value of $50 if the stock price is $150 in one year. If the stock price in one year is $70, the call option will have a value of $0. Our objective is to value this call option using the binomial pricing model.

   The first step in applying the model to this call option is to recognize that three investments are involved: the stock, a risk-free security, and the call option. Using the information already given, we have the following payoff matrix in one year:

SECURITY	PRICE IF STOCK IS WORTH $150 IN ONE YEAR	PRICE IF STOCK IS WORTH $70 IN ONE YEAR	CURRENT PRICE
Gem stock	$150.00	$70.00	$100.00
Treasury bill	110.52	110.52	100.00
Call option	50.00	0.00	?

   The objective of the binomial pricing model is to determine the current price of the call option. The key to understanding the valuation of the call option using the binomial pricing model is that the option's value must be based on a combination of the value of the stock and the T-bill. If this were not the case, arbitrage opportunities would result. Consequently, in either the up or the down state, the payoff of a portfolio consisting of Ns shares of Gem stock and Nb T-bills must be equal to the value of the call option in that state. Using the payoff matrix just given, we derive the following system of two equations:

150Ns + 110.52Nb = 50

70Ns + 110.52Nb = 0

Since we are dealing with two linear equations with two unknowns, we can easily solve for the two variables by substitution. Doing so gives the following values for the number of shares and the number of T-bills in the investor's so-called replicating portfolio:

Ns = 0.625

Nb = −0.3959

   In other words, the payoffs of the call option on Gem stock can be replicated by borrowing $39.59 at the risk-free rate and buying 0.625 shares of Gem stock for $62.50 (since one share currently sells for $100). Since the payoff of this replicating portfolio is the same as that for the call, the cost to the investor must be the value of the call. In this case, since $39.59 of the outlay of $62.50 is financed by borrowing, the outlay to the investor is $62.50 − $39.59 = $22.91. Thus the call option premium must be $22.91.

   In equation form, the value of the call option (Vc) can therefore be written as

Vc = NsPs + NbPb

   Computation of the Ns can be simplified somewhat. More specifically,

Ns = h = 

Pou − Pod

Psu − Psd

where

· Pou  = value of the option in the up state

· Pod  = value of the option in the down state

· Psu  = price of Gem stock in the up state

· Psd  = price of Gem stock in the down state

This is also referred to as the hedge ratio (h).

In this example, the amount borrowed (BOR) is equal to the product of the number of risk-free securities in the replicating portfolio and the price of the risk-free security:

NbPb = BOR = PV (hPsd − Pod)

where

· PV  = present value of a continuously compounded sum

· h  = hedge ratio

Thus the value of the call option can be expressed more simply as

Vc = hPs + BOR

   To illustrate why the relationships discussed so far should hold, we assume for the moment that a call option on Gem stock is selling for a premium of $25 (i.e., the call option is overpriced). In this case, investors are presented with an arbitrage opportunity to make an instantaneous, riskless profit. In particular, investors could write a call, buy the stock, and borrow at the risk-free rate. Now assume that the call option on Gem stock is selling for a premium of only $20 (i.e., the call option is underpriced). Using arbitrage, investors would buy a call, sell the stock short, and invest at the risk-free rate.

Using the Binomial Pricing Model to Price Put Options

Continuing with the example of Gem Corporation, the only item that changes in the payoff matrix when the option is a put option (i.e., an option to sell Gem stock) is the value of the option at expiration in the up and down states. If Gem stock is worth $150 in one year, the put option will be worthless; if Gem stock is worth only $70 in one year, the put option will be worth $30. Since the value of the risk-free security is contingent only on the T-bill interest rate, it will be unaffected by the fact that we are now dealing with a put option.

   The hedge ratio and the amount borrowed can be easily determined using the formulas introduced previously. The hedge ratio is

h = 

0 − 30

150 − 70

 = −0.375

The amount borrowed is

BOR = PV [−0.375(70) − 30] = 

−56.25

eRT

 =−50 90

Therefore, to replicate the put option, the investor would sell short 0.375 Gem shares and lend $50.90 at the risk-free rate. Thus the net amount the investor must put up is $50.90 − $37.50 = $13.40. Accordingly, this is the fair value of the put.

Put-Call Parity

With some mathematical manipulation, the following relationship can be derived between the prices of puts and calls (that have the same exercise price and time to expiration):

Pp = Pc + 

E

eRT

 − Ps

where

· Pp  = price of a put option

· Pc  = price of a call option

· E  = exercise price of the option

· eRT  = present value operator for a continuously compounded sum, discounted at interest rate R for T years

· Ps  = price of the stock

Using the example of Gem Corporation,

Pp = $22.91 + 

$100

e0.10

 − $100 = $13.39

THE BLACK-SCHOLES OPTION-PRICING MODEL FOR CALL OPTIONS

In 1973, Black and Scholes devised an option-pricing model that motivated further research on option valuation that continues to this day.

Assumptions of the Black-Scholes Option-Pricing Model

The following are some of the key assumptions underlying the Black-Scholes option-pricing model:

· 1. The risk-free rate is known and constant over the life of the option.

· 2. The probability distribution of stock prices is lognormal.

· 3. The variability of a stock's return is constant.

· 4. The option is to be exercised only at maturity, if at all.

· 5. There are no transaction costs involved in trading options.

· 6. Tax rates are similar for all participants who trade options.

· 7. The stock of concern does not pay cash dividends.

The Black-Scholes Partial Differential Equation

The Black-Scholes option-pricing model was one of the first to introduce the concept of a riskless hedge. Assume that an amount is invested in a risk-free asset. Thus the investor would earn a return of rπdt over a time interval dt. If an appropriately selected portfolio— with a current value of π consisting of a company's stock and an offsetting position in an option on that stock—returns more than rπdt over a time interval dt, an investor could conduct arbitrage by borrowing at the risk-free rate and investing in the portfolio. Conversely, if the portfolio returns less than rπdt, the investor would short the portfolio and invest in the risk-free asset. In either case, the arbitrageur would make a riskless, no-cost, instantaneous profit. Thus the return on the portfolio and on the riskless asset must be roughly equal.

   Using this argument, Black and Scholes developed what has become known as the Black-Scholes partial differential equation:

where

The Black-Scholes Option-Pricing Model for European Call Options

In order to price an option, the partial differential equation must be solved for V, the value of the option. Assuming that the risk-free interest rate and stock price volatility are constant, solving the Black-Scholes partial differential equation results in the Black- Scholes formula for European call options:

V = SN(d1) − Ee− rTN (d2)

where

· V  = value of the call option

· S  = stock price

· N(·) = cumulative distribution function for a standardized normal random variable

· E  = exercise price of the call option

· e  = base e antilog or 2.7183

· r  = risk-free rate of return for one year assuming continuous compounding

· T  = time remaining to maturity of the call option, expressed as a fraction of a year

The terms N(d1) and N(d2) deserve further elaboration. The term N represents a cumulative probability for a unit normal variable, where

and

Here ln(S/E) is the natural logarithm and σ denotes the standard deviation of the continuously compounded rate of return on the underlying stock.

Using the Black-Scholes Option-Pricing Model to Price a European Call Option

To illustrate how the Black-Scholes equation can be used to price a call option, assume you observe a call option on MPB Corporation stock expiring in six months with an exercise price of $70. Thus T = 0.50 and E = $70. Furthermore, the current price of MPB stock is $72 and the stock has a standard deviation of 0.10. The annual risk-free rate is 7 percent. Solving for d1 and d2, we obtain

   A table that identifies the area under the standard normal distribution function (see  Exhibit 14A.1 ) can now be used to determine the cumulative probability. Because d1 is 0.9287, the cumulative probability from zero to 0.9287 is about 0.3235 (from  Exhibit 14A.1 , using linear interpolation). Because the cumulative probability for a unit normal variable from negative infinity to zero is 0.50, the cumulative probability from negative infinity to 0.9287 is 0.50 + 0.3235 = 0.8235. For d2, the cumulative probability from zero to 0.8580 is 0.3045. Therefore, the cumulative probability from negative infinity to 0.8580 is 0.50 + 0.3042 = 0.8042.

   Now that N(d1) and N(d2) have been calculated, the call option value can be estimated as follows:

   Thus a call option on MPB stock should sell for a premium of $4.91.

Put-Call Parity

Using the Black-Scholes option-pricing model, the same relationship exists between the price of a call and that of a put as in the binomial option-pricing model. Thus we have

Pp = Pc +

E

eRT

 − Ps

Deriving the Implied Volatility

In the example of deriving the value of a call option, an estimate of the stock's standard deviation was used. In some cases, investors want to derive the market's implied volatility rather than a valuation of the call option. The volatility is referred to as “implied” under these circumstances because it is not directly observable in the market. The implied volatility can be derived with some software packages by inputting values for the other variables (prevailing stock price, exercise price, option's time to expiration, and interest rate) in the call option-pricing model and also inputting the market premium of the call option. Instead of deriving a value for the call option premium, the prevailing market premium of the call option is used along with these other variables to derive the implied volatility. The software package will also show how the stock's implied volatility is affected for different values of the call option premium. An increase in the market premium of the call option, when other variables have not changed, reflects an increase in the implied volatility. This relationship can be verified by entering a slightly higher market premium into the software program and checking how the implied volatility changes in response to that increase. The point is that investors can detect changes in the implied volatility of a stock by monitoring how the stock's call option premium changes over short intervals of time.

Exhibit 14A.1 Institutional Use of Options Markets

d	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.0000	0.0040	0.0080	0.0120	0.0160	0.0199	0.0239	0.0279	0.0319	0.0359
0.1	0.0398	0.0438	0.0478	0.0517	0.0557	0.0596	0.0636	0.0675	0.0714	0.0753
0.2	0.0793	0.0832	0.0871	0.0910	0.0948	0.0987	0.1026	0.1064	0.1103	0.1141
0.3	0.1179	0.1217	0.1255	0.1293	0.1331	0.1368	0.1406	0.1443	0.1480	0.1517
0.4	0.1554	0.1591	0.1628	0.1664	0.1700	0.1736	0.1772	0.1808	0.1844	0.1879
0.5	0.1915	0.1950	0.1985	0.2019	0.2054	0.2088	0.2123	0.2157	0.2190	0.2224
0.6	0.2257	0.2291	0.2324	0.2357	0.2389	0.2422	0.2454	0.2486	0.2517	0.2549
0.7	0.2580	0.2611	0.2642	0.2673	0.2704	0.2734	0.2764	0.2794	0.2823	0.2852
0.8	0.2881	0.2910	0.2939	0.2967	0.2995	0.3023	0.3051	0.3078	0.3106	0.3133
0.9	0.3159	0.3186	0.3213	0.3238	0.3264	0.3289	0.3315	0.3340	0.3365	0.3389
1.0	0.3413	0.3438	0.3461	0.3485	0.3508	0.3531	0.3554	0.3577	0.3599	0.3621
1.1	0.3643	0.3665	0.3686	0.3708	0.3729	0.3749	0.3770	0.3790	0.3810	0.3830
1.2	0.3849	0.3869	0.3888	0.3907	0.3925	0.3944	0.3962	0.3980	0.3997	0.4015
1.3	0.4032	0.4049	0.4066	0.4082	0.4099	0.4115	0.4131	0.4147	0.4162	0.4177
1.4	0.4192	0.4207	0.4222	0.4236	0.4251	0.4265	0.4279	0.4292	0.4306	0.4319
1.5	0.4332	0.4345	0.4357	0.4370	0.4382	0.4394	0.4406	0.4418	0.4429	0.4441
1.6	0.4452	0.4463	0.4474	0.4484	0.4495	0.4505	0.4515	0.4525	0.4535	0.4545
1.7	0.4554	0.4564	0.4573	0.4582	0.4591	0.4599	0.4608	0.4616	0.4625	0.4633
1.8	0.4641	0.4649	0.4656	0.4664	0.4671	0.4678	0.4686	0.4693	0.4699	0.4706
1.9	0.4713	0.4719	0.4726	0.4732	0.4738	0.4744	0.4750	0.4756	0.4761	0.4767
2.0	0.4773	0.4778	0.4783	0.4788	0.4793	0.4798	0.4803	0.4808	0.4812	0.4817
2.1	0.4821	0.4826	0.4830	0.4834	0.4838	0.4842	0.4846	0.4850	0.4854	0.4857
2.2	0.4861	0.4866	0.4868	0.4871	0.4875	0.4878	0.4881	0.4884	0.4887	0.4890
2.3	0.4893	0.4896	0.4898	0.4901	0.4904	0.4906	0.4909	0.4911	0.4913	0.4916
2.4	0.4918	0.4920	0.4922	0.4925	0.4927	0.4929	0.4931	0.4932	0.4934	0.4936
2.5	0.4938	0.4940	0.4941	0.4943	0.4945	0.4946	0.4948	0.4949	0.4951	0.4952
2.6	0.4953	0.4955	0.4956	0.4957	0.4959	0.4960	0.4961	0.4962	0.4963	0.4964
2.7	0.4965	0.4966	0.4967	0.4968	0.4969	0.4970	0.4971	0.4972	0.4973	0.4974
2.8	0.4974	0.4975	0.4976	0.4977	0.4977	0.4978	0.4979	0.4979	0.4980	0.4981
2.9	0.4981	0.4982	0.4982	0.4982	0.4984	0.4984	0.4985	0.4985	0.4986	0.4986
3.0	0.4987	0.4987	0.4987	0.4988	0.4988	0.4989	0.4989	0.4989	0.4990	0.4990

AMERICAN VERSUS EUROPEAN OPTIONS

American Call Options

An American call option is an option to purchase stock that can be exercised at any time prior to maturity. However, the original Black-Scholes model was developed to price European call options, which are options to purchase stock that can be exercised only at maturity. Naturally, European puts could be directly derived using the put–call parity relationship. Consequently, the question is whether the Black-Scholes equation can be used to price American call options.

   In general, early exercise of a call may be justified only if the asset makes a cash payment such as a dividend on a stock. If there are no dividends during the life of the option, early exercise would be equivalent to buying something earlier than you need it and then giving up the right to decide later whether you really wanted it. However, if it is possible to save a little money in doing so, early purchase/exercise can sometimes be justified. Early exercise is not appropriate every time there is a dividend, but if early exercise is justified then it should occur just before the stock's ex-dividend date. This minimizes the amount of time value given up without sacrificing the dividend. If there are no dividends on the underlying stock, the Black-Scholes model can be used to price American call options. If the underlying stock pays dividends, however, the Black-Scholes model may not be directly applicable. The binomial option-pricing model can be used to price American call options that pay dividends.

American Put Options

Suppose you purchase a European put option. If the value of the underlying asset goes to zero, then the option has reached its maximum value. It allows you to sell a worthless asset for the exercise price. Yet because the option is European, it cannot be exercised prior to maturity. Its value will simply be the present value of the exercise price, and this value will gradually rise as a function of the time value of money until expiration, when the option will be exercised. This is clearly a situation where you would want an American option, so you could exercise it as soon as the asset value goes to zero.

   The previous example indicates that there would be a demand for an option that allowed early exercise. It is not necessary, however, for the asset price to go to zero. The European put price must be at least the present value of the exercise price minus the asset price. Clearly, the present value of the exercise price minus the asset price is less than the exercise price minus the asset price, which is the amount that could be claimed if the put could be exercised early. Thus an American put will sell for more than the European put and so the Black-Scholes option-pricing model cannot be used to price an American put option. There is a point where the right to exercise early is at its maximum value, and at that point the American put would be exercised. Finding that point is difficult, but to do so is to unlock the mystery of pricing the American put.

   The only surefire way to price the American put correctly is to use a numerical procedure such as the binomial model. The procedure is similar to partitioning a two-dimensional space of time and the asset price into finer points and then solving either a difference or a differential equation at each time point. The process starts at expiration and successively works its way back to the present by using the solution at the preceding step. Thus a closed-form solution does not exist. American puts can be viewed as an infinite series of compound options (that is, an option on an option). At each point in time, the holder of the put has the right to decide whether to exercise it or not. A decision not to exercise the put is tantamount to exercising the compound option and obtaining a position in a new compound option, which can be exercised an instant later. The sequence reiterates until the expiration day. This logic can lead to an intuitive but complex mathematical formula that contains an infinite number of terms. Thus, the holders of American puts face an infinite series of early exercise decisions. Only at an instant before the asset goes ex-dividend do the holders know that they should wait to exercise. This is because they know the asset will fall in value an instant later, so they might as well wait an instant and benefit from the decline in value.