over the phone. There is awide range of assets underlying the options. Over-the-counter options on foreign exchange andinterest rates are particularly popular. The chief potential disadvantage of the over-the-countermarket is that option writer may default. This means that the purchaser is subject to some creditrisk. In an attempt to overcome this disadvantage, market participants are adopting a number ofmeasures such as requiring counterparties to post collateral.
The instruments traded in the over-the-counter market are often structured by financial institu-tions to meet the precise needs of their clients. Sometimes this involves choosing exercise dates,strike prices, and contract sizes that are different from those traded by the exchange. In other casesthe structure of the option is different from standard calls and puts. The option is then referred toas an exotic option. Chapter 19 covers a number of different types of exotic options.
SUMMARY
 
 
 
 
There are two types of options: calls and puts. A call option gives the holder the right to buythe underlying asset for a certain price by a certain date. A put option gives the holder the rightto sell the underlying asset by a certain date for a certain price. There are four possible positionsin options markets: a long position in a call, a short position in a call, a long position in a put,and a short position in a put. Taking a short position in an option is known as writing it.164 CHAPTER 7
Options are currently traded on stocks, stock indices, foreign currencies, futures contracts, andother assets.
An exchange must specify the terms of the option contracts it trades. In particular, it mustspecify the size of the contract, the precise expiration time, and the strike price. In the UnitedStates one stock option contract gives the holder the right to buy or sell 100 shares. Theexpiration of a stock option contract is 10:59 p.m. Central Time on the Saturday immediatelyfollowing the third Friday of the expiration month. Options with several different expirationmonths trade at any given time. Strike prices are at $2 bs $5, or $10 intervals, depending on thestock price. The strike price is generally fairly close to the current stock price when trading in anoption begins.
The terms of a stock option are not normally adjusted for cash dividends. However, they areadjusted for stock dividends, stock splits, and rights issues. The aim of the adjustment is to keepthe positions of both the writer and the buyer of a contract unchanged.
Most options exchanges use market makers. A market maker is an individual who is prepared toquote both a bid price (the price at which he or she is prepared to buy) and an offer price (the priceat which he or she is prepared to sell). Market makers improve the liquidity of the market andensure that there is never any delay in executing market orders. They themselves make a profit fromthe difference between their bid and offer prices (known as the bid—offer spread). The exchange hasrules specifying upper limits for the bid—offer spread.
Writers of options have potential liabilities and are required to maintain margins with theirbrokers. If it is not a member of the Options Clearing Corporation, the broker will maintain amargin account with a firm that is a member. This firm will in turn maintain a margin accountwith the Options Clearing Corporation. The Options Clearing Corporation is responsible forkeeping a record of all outstanding contracts, handling exercise orders, and so on.
Not all options are traded on exchanges. Many options are traded by phone in the over-the-counter market. An advantage of over-the-counter options is that they can be tailored by afinancial institution to meet the particular needs of a corporate treasurer or fund manager.
SUGGESTIONS FOR FURTHER READING
 
 
 
Chance, D. M., An Introduction to Derivatives, 4th edn., Dryden Press, Orlando, FL, 1998.
Cox, J. C., and M. Rubinstein, Options Markets, Prentice Hall, Upper Saddle River, NJ, 1985.Kolb, R., Futures, Options, and Swaps, 3rd edn., Blackwell, Oxford, 1999.
McMillan, L. G., Options as a Strategic Investment, New York Institute of Finance, New York, 1992.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
7.1. Explain why brokers require margins when clients write options but not when they buy options.
7.2. A stock option is on a February, May, August, and November cycle. What options trade on(a) April 1 and (b) May 30?
7.3. A company declares a 3-for-1 stock split. Explain how the terms change for a call option with astrike price of $60.Mechanics of Options Markets 165
74.7S.
7.6.
77.78.
79,
7.10.
71
7.12,
7.13.7.14,
Explain carefully the difference between writing a put option and buying a call option.
A corporate treasurer is designing a hedging program involving foreign currency options. Whatare the pros and cons of using (a) the Philadelphia Stock Exchange and (b) the over-the-countermarket for trading?
Explain why an American option is always worth at least as much as a European option on thesame asset with the same strike price and exercise date.
Explain why an American option is always worth at least as much as its intrinsic value.
The treasurer of a corporation is trying to choose between options and forward contracts to hedgethe corporation’s foreign exchange risk. Discuss the relative advantages and disadvantages ofeach.
Consider an exchange-traded call option contract to buy 500 shares with a strike price of $40 andmaturity in four months. Explain how the terms of the option contract change when there is
a. A 10% stock dividend
b. A 10% cash dividend
c. A 4-for-1 stock split
“If most of the call options on a stock are in the money, it is likely that the stock price has risenrapidly in the last few months.” Discuss this statement.
. What is the effect of an unexpected cash dividend on (a) a call option price and (b) a put option
price?
Options on General Motors stock are on a March, June, September, and December cycle. Whatoptions trade on (a) March 1, (b) June 30, and (c) August 5?
Explain why the market maker’s bid—offer spread represents a real cost to options investors.
A United States investor writes five naked call option contracts. The option price is $3.50, thestrike price is $60.00, and the stock price is $57.00. What is the initial margin requirement?
ASSIGNMENT QUESTIONS
 
7.15.
7.16.
7A?
7.18.
A United States investor buys 500 shares of a stock and sells five call option contracts on thestock. The strike price is $30. The price of the option is $3. What is the investor’s minimum cashinvestment (a) if the stock price is $28 and (b) if it is $32?
The price of a stock is $40. The price of a one-year European put option on the stock with a strikeprice of $30 is quoted as $7 and the price of a one-year European call option on the stock witha -strike price of $50 is quoted as $5. Suppose that an investor buys 100 shares, shorts 100 calloptions, and buys 100 put options. Draw a diagram illustrating how the investor's profit or lossvaries with the stock price over the next year. How does your answer change if the investor buys100 shares, shorts 200 call options, and buys 200 put options?
“Ifa company does not do better than its competitors but the stock market goes up, executivesdo very well from their stock options. This makes no sense.” Discuss this viewpoint. Can youthink of alternatives to the usual executive stock option plan that take the viewpoint intoaccount.
Use DerivaGem to calculate the value of an American put option on a nondividend paying stockwhen the stock price is $30, the strike price is $32, the risk-free rate is 5%, the volatility is 30%,166
CHAPTER 7
 
and the time to maturity is 1.5 years. (Choose binomial American for the “option type” and 50
time
a.b.c,
d.
stens,}steps.)
What is the option’s intrinsic value?
What is the option’s time value?
What would a time value of zero indicate? What is the value of an option with zero timevalue?
Using a trial-and-error approach, calculate how low thé stock price would have to be for thetime value of the option to be zero.CHAPTER 8PROPERTIES OF STOCK OPTIONS
 
In this chapter we look at the factors affecting stock option prices. We use a number of differentarbitrage arguments to explore the relationships between European option prices, American optionprices, and the underlying stock price. The most important of these relationships is put—call parity,which is a relationship between European call option prices and European put option prices.The chapter examines whether American options should be exercised early. It shows that it isnever optimal to exercise an American call option on a non-dividend-paying stock prior to theoption’s expiration, but the early exercise of an American put option on such a stock can be optimal.
8.1 FACTORS AFFECTING OPTION PRICES
 
 
There are six factors affecting the price of a stock option:
. The current stock price, So). The strike price, K
1
2
3. The time to expiration, T
4, The volatility of the stock price, o5
. The risk-free interest rate, r2. 6. The dividends expected during the life of the option
i this section we consider what happens to option prices when one of these factors changes withall the others remaining fixed. The results are summarized in Table 8.1.
- Figures 8.1 and 8.2 show how the price of a European call and put depends on the first fivefactors in the situation where Sy = 50, K = 50, r=5% per annum, a= 30% per annum,T=) year, and there are no dividends. In this case the call price is 7.116 and the put price is 4.677.
Stock Price and Strike Price
{facall option is exercised at some future time, the payoff will be the amount by which the stock‘price exceeds the strike price. Call options therefore become more valuable as the stock pricegacteases and less valuable as the strike price increases. For a put option, the payoff on exercise ishe amount by which the strike price exceeds the stock price. Put options therefore behave in thex ite way from call options. They become less valuable as the stock price increases and moreMaluable as the strike price increases. Figures 8.1a,b,c,d illustrate the way in which put and callprices depend on the stock price and strike price.
167168 CHAPTER 8
 
tion of increasing one
"2S
variable while keepin
Table 8.1. Summary of the effect on the price of a stock og otg all ot
 
 
Variable European cali European put American call American putCurrent stock price + = + —Strike price - + - +Time to expiration ? ? + +Volatility + + + 4Risk-free rate + - + _Dividends - + - +
* + indicates that an increase in the variable causes the option price to increase; — indicatesthat an increase in the variable causes the option price to decrease; ? indicates that therelationship is uncertain.
Time to Expiration
Consider next the effect of the expiration date. Both put and call American options become morevaluable as the time to expiration increases. Consider two options that differ only as far as theexpiration date is concerned. The owner of the long-life option has all the exercise opportunitiesopen to the owner of the short-life option—and more. The long-life option must therefore alwaysbe worth at least as much as the short-life option. Figures 8.le,f illustrate the way in which callsand puts depend on the time to expiration.
Although European put and call options usually become more valuable as the time to expirationincreases, this is not always the case. Consider two European call options on a stock: one with anexpiration date in one month, and the other with an expiration date in two months. Suppose that avery large dividend is expected in six weeks. The dividend will cause the stock price to decline, sothat the short-life option could be worth more than the long-life option.
Volatility
The precise way in which volatility is defined is explained in Chapters 11 and 12. Roughlyspeaking, the volatility of a stock price is a measure of how uncertain we are about future stockprice movements. As volatility increases, the chance that the stock will do very well or very poorlyincreases. For the owner of a stock, these two outcomes tend to offset each other. However, this isnot so for the owner of a call or put. The owner of a call benefits from price increases but haslimited downside risk in the event of price decreases because the most the owner can lose is theprice of the option. Similarly, the owner of a put benefits from price decreases, but has limiteddownside risk in the event of price increases. The values of both calls and puts therefore increase asvolatility increases (see Figures 8.2a, b).
Risk-Free Interest Rate
The risk-free interest rate affects the price of an option in a less clear-cut way. As interest rates inthe economy increase, the expected return required by investors from the stock tends to increase.Also, the present value of any future cash flow received by the holder of the option decreases. Thecombined impact of these two effects is to decrease the value of put options and increase the valueof call options (see Figures 8.2c,d).Properties of Stock Options 169
It is important to emphasize that we are assuming that interest rates change while all othervariables stay the same. In particular, we are assuming that interest rates change while the stockprice remains the same. In practice, when interest rates rise (fall), stock prices tend to fall (rise). Thenet effect of an interest rate increase and the accompanying stock price decrease can be to decreasethe value of a call option and increase the value of a put option. Similarly, the net effect of an
Call optionprice, ¢
 
0 20 40 60 80 100
Call optionprice, c
       
50
40
30
20
Put optionprice, p
       
504030
20
Stockprice, So
0 20 40 60 80 100
Put optionprice, p
5040 /
 
10 Strike Strikeprice, K price, K00 20 40 60 80 100 0 20 40 60 80 100(c) (d)Call option Put optionprice, c price, p
   
2 Time toexpiration, T
00 #04 O8 12 16(e)
   
2 Time toexpiration, T
00.0 0.4 0.8 1.2 1.6(f)
Figure 8.1 Effect of changes in stock price, strike price, and expiration date on optionprices when So = 50, K = 50, r=5%, o = 30%, and T= 1.170
CHAPTER 8
 
Call optionprice, c
      
1S
12
Volatility,6 (%)
  
0 10 20 30 40 30(a)
Call optionprice, ¢
 
Put optionprice, p
        
15
12
Volatility,o(%)
(b)
Put optionprice, p
 
(c)
(d)
Figure 8.2 Effect of changes in volatility and risk-free interest rate on optionprices when Sy = 50, K = 50, r= 5%, ¢ = 30%, and T=1.
pri when $5 = 30, K =
interest rate decrease and the accompanying stock price increase can be to increase the value of acall option and decrease the value of a put option.
Dividends
Dividends have the effect of reducing the stock price on the ex-dividend date. This is bad news forthe value of call options and good news for the value of put options. The value of a call option istherefore negatively related to the size of any anticipated dividends, and the value of a put option ispositively related to the size of any anticipated dividends.
8.2 ASSUMPTIONS AND NOTATION
 
 
In this chapter we will make assumptions similar to those made for deriving forward and futuresprices in Chapter 3. We assume that there are some market participants, such as large investmentbanks, for which
1. There are no transactions costs.Properties of Stock Options 171
2. All trading profits (net of trading losses) are subject to the same tax rate.3. Borrowing and lending are possible at the risk-free interest rate.We assume that these market participants arc prepared to take advantage of arbitrage opportun-ities as they arise. As discussed in Chapters 1 and 3, this means that any available arbitrageopportunities disappear very quickly. For the purposes of our analyses, it is therefore reasonable toassume that there are no arbitrage opportunities.We will use the following notation:So: Current stock priceK: Strike price of optionT: Time to expiration of optionSr: Stock price at maturity: Continuously compounded risk-free rate of interest for an investment maturing in time T
: Value of American call option to buy one share
wr Ss
: Value of American put option to sell one sharec: Value of European call option to buy one sharep: Value of European put option to sell one shareIt should be noted that r is the nominal rate of interest, not the real rate of interest. We can assume
that r > 0; otherwise, a risk-free investment would provide no advantages over cash. (Indeed, ifr <0, cash would be preferable to a risk-free investment.)
8.3 UPPER AND LOWER BOUNDS FOR OPTION PRICES
 
 
In this section we derive upper and lower bounds for option prices. These bounds do not dependon any particular assumptions about the factors mentioned in the previous section (except r > 0).If an option price is above the upper bound or below the lower bound, there are profitableopportunities for arbitrageurs.
Upper Bounds
An American or European call option gives the holder the right to buy one share of a stock for acertain price. No matter what happens, the option can never be worth more than the stock. Hence,the stock price is an upper bound to the option price:
e<Sy and C<S
If these relationships were not true, an arbitrageur could easily make a riskless profit by buying thestock and selling the call option.
An American or European put option gives the holder the right to sell one share of a stock forK. No matter how low the stock price becomes, the option can never be worth more than K.
Hence,p<K and P<K
For European options, we know that at maturity the option cannot be worth more than K. It172 CHAPTER 8
follows that it cannot be worth more than the present value of K today:p< Ke"
If this were not true, an arbitrageur could make a riskless profit by writing the option and investingthe proceeds of the sale at the risk-free interest rate.
Lower Bound for Calls on Non-Dividend-Paying Stocks
A lower bound for the price of a European call option on a non-dividend-paying stock isSy — Keo?
We first look at a numerical example and then consider a more formal argument.Suppose that Sy = $20, K = $18, r= 10% per annum, and T = 1 year. In this case,
So — Ke"? = 20 - 18¢°'! =3.71
or $3.71. Consider the situation where the European call price is $3.00, which is less than thetheoretical minimum of $3.71. An arbitrageur can buy the call and short the stock to provide acash inflow of $20.00 — $3.00 = $17.00. If invested for one year at 10% per annum, the $17.00grows to 17¢°' = $18.79. At the end of the year, the option expires. If the stock price is greaterthan $18.00, the arbitrageur exercises the option to buy the stock for $18.00, closes out the short
sosition and makes a oroft ofposition, ana Maxes a promt of
$18.79 — $18.00 = $0.79
If the stock price is less than $18.00, the stock is bought in the market and the short position isclosed out. The arbitrageur then makes an even greater profit. For example, if the stock price is
$17.00, the arbitrageur’s profit is
arbDitrageur's p
$18.79 — $17.00 = $1.79
For a more formal argument, we consider the following two portfolios:
Portfolio A: one European call option plus an amount of cash equal to Ke"Portfolio B: one shareIn portfolio A, the cash, if it is invested at the risk-free interest rate, will grow to K in time T. If
Sr; > K, the call option is exercised at maturity and portfolio A is worth S,. If Sy < K, the calloption expires worthless and the portfolio is worth K. Hence, at time 7, portfolio A is worth
max(Sr, K)
Portfolio B is worth S; at time 7. Hence, portfolio A is always worth as much as, and can be worthmore than, portfolio B at the option’s maturity. It follows that in the absence of arbitrageopportunities this must also be true today. Hence,
c+ Ke" > Soorc > Sy— Ke"
Because the worst that can happen to a call option is that it expires worthless, its value cannot beProperties of Stock Options 173
negative. This means that c > 0, and therefore that¢ 2 max(Sy ~ Ke~’, 0) (8.1)
Example 8.1 Consider a European call option on a non-dividend-paying stock when the stockprice is $51, the exercise price is $50, the time to maturity is six months, and the risk-free rateof interest is 12% per annum. in this case, Sy) = Si, K = 50, T = 0.5, andr = 0.12. Fromequation (8.1), a lower bound for the option price is Sy — Ke7’, or
SI — 50e7° 12% — $3.91
Lower Bound for European Puts on Non-Dividend-Paying Stocks
For a European put option on a non-dividend-paying stock, a lower bound for the price isKe — S,
Again, we first consider a numerical example and then look at a more formal argument.Suppose that Sy = $37, K = $40, r= 5% per annum, and T = 0,5 years. In this case,
Ke™ — S$, = 40e°°*°5 — 37 = $2.01
Consider the situation where the European put price is $1.00, which is less than the theoreticalminimum of $2.01. An arbitrageur can borrow $38.00 for six months to buy both the put and thestock. At the end of the six months, the arbitrageur will be required to repay 38¢°°*°> = $38.96. Ifthe stock price is below $40.00, the arbitrageur exercises the option to sell the stock for $40.00,repays the loan, and makes a profit of
$40.00 — $38.96 = $1.04
If the stock price is greater than $40.00, the arbitrageur discards the option, sells the stock, andtepays the loan for an even greater profit. For example, if the stock price is $42.00, the arbitrageur’sprotit is
$42.00 — $38.96 = $3.04
For a more formal argument, we consider the following two portfolios:
Portfolio C: one European put option plus one sharePortfolio D: an amount of cash equal to Ke?
if Sp < K, the option in portfolio C is exercised at option maturity, and the portfolio becomesworth K. If S; > K, the put option expires worthless, and the portfolio is worth S; at this time.
Hence, portfolio C is worthmax(Sy, K)
at time 7. Assuming the cash is invested at the risk-free interest rate, portfolio D is worth K attime 7. Hence, portfolio C is always worth as much as, and can sometimes be worth more than,portfolio D at time T. It follows that in the absence of arbitrage opportunities portfolio C must beworth at least as much as portfolio D today. Hence,
ptS2Ke™174 CHAPTER 8 |
orpe Ke — So
Because the worst that can happen to a put option is that it expires worthless, its value cannot benegative. This means thatp 2 max(Ke™! — Sp, 0) (8.2)
Example 8.2. Consider a European put option on a non-dividend-paying stock when the stockprice is $38, the exercise price is $40, the time to maturity is three months, and the risk-free rateof interest is 10% per annum. In this case S, =38, K=40, T=0.25, and r=0.10. Fromequation (8.2), a lower bound for the option price is Ke? — Sp, or
407.5 _ 38 = $1.01
8.4 PUT-CALL PARITY
 
We now derive an important relationship between p and c. Consider the following two portfoliosthat were used in the previous section: ‘
Portfolio A; one European call option plus an amount of cash equal to Ke?Portfolio C: one European put option plus one share
Both are worthmax(S;, K)
at expiration of the options. Because the options are European, they cannot be exercised prior tothe expiration date. The portfolios must therefore have identical values today. This means that
T
e+ Ke" =pt+Sy (8.3)
This relationship is known as put—call parity. It shows that the value of a European call with acertain exercise price and exercise date can be deduced from the value of a European put with thesame exercise price and exercise date, and vice versa. .If equation (8.3) does not hold, there are arbitrage opportunities. Suppose that the stock price
is $31, the exercise price is $30, the risk-free interest rate is 10% per annum, the price of a three-month European call option is $3, and the price of a three-month European put option is $2.25. Inthis case,
| eavT
e| Ke
p+ Sy = 2.25 + 31 = $33.25
31 We ObA/l2 _ 039 96| 302 = $32.26
Portfolio C is overpriced relative to portfolio A. The correct arbitrage strategy is to buy thesecurities in portfolio A and short the securities in portfolio C. The strategy involves buying thecall and shorting both the put and the stock, generating a positive cash flow of
—3 + 2.25 + 31 = $30.25
up front. When invested at the risk-free interest rate, this amount grows to 30.25e"!*°> = $31.02in three months.Properties of Stock Options 175
If the stock price at expiration of the option is greater than $30, the call will be exercised. If it isless than $30, the put will be exercised. In either case, the investor ends up buying one share for$30. This share can be used to close out the short position. The net profit is therefore
$31.02 — $30.00 = $1.02For an alternative situation, suppose that the call price is $3 and the put price is $1. In this case,
c+ Ke =3 +4300 0/2? = $39.96pt Sp = 1431 = $32.00
_ Portfolio A is overpriced relative to portfolio C. An arbitrageur can short the securities in portfolio Aand buy the securities in portfolio C to lock in a profit. The strategy involves shorting the call andbuying both the put and the stock with an initial investment of
$31+$1 — $3 = $29
When the investment is financed at the risk-free interest rate, a repayment of 29e"!*°> — $29.73 is
required at the end of the three months. As in the previous case, either the call or the put will beexercised. The short call and long put option position therefore leads to the stock being sold for
$30.00. The net profit is therefore$30.00 — $29.73 = $0.27
American Options
Put-call parity holds only for European options. However, it is possible to derive some results forAmerican option prices. It can be shown (see Problem 8.17) that
So-K<C-P<Sy—Ke™ (8.4)
Example 8.3. An American call option on a non-dividend-paying stock with exercise price $20.00and maturity in five months is worth $1.50. Suppose that the current stock price is $19.00 and therisk-free interest rate is 10% per annum. From equation (8.4),
19-20<C-—-P<19 — 2 e701 *5/12or1>P-cC20.18
showing that P—C lies between $1.00 and $0.18. With C at $1.50, P must lie between $1.68 and$2.50. In other words, upper and lower bounds for the price of an American put with the same strikeprice and expiration date as the American call are $2.50 and $1.68.
8.5 EARLY EXERCISE: CALLS ON A NON-DIVIDEND-PAYINGSTOCK
 
 
This section demonstrates that it is never optimal to exercise an American call option on a non-dividend-paying stock before the expiration date. To illustrate the general nature of the argument,consider an American call option on a non-dividend-paying stock with one month to expirationwhen the stock price is $50 and the strike price is $40. The option is deep in the money, and the176 CHAPTER 8
Call optionprice
   
 
>K Stock price, Sy
Figure 8.3 Variation of price of an American or European call optionon a non-dividend-paying stock with the stock price, Sp.
investor who owns the option might well be tempted to exercise it immediately. However, if theinvestor plans to hold the stock for more than one month, this is not the best strategy. A bettercourse of action is to keep the option and exercise it at the end of the month. The $40 strike price isthen paid out one month later than it would be if the option were exercised immediately, so thatinterest is earned on the $40 for one month. Because the stock pays no dividends, no income fromthe stock is sacrificed. A further advantage of waiting rather than exercising immediately is thatthere is some chance (however remote) that the stock price will fall below $40 in one month. In thiscase the investor will not exercise and will be glad that the decision to exercise early was not taken!
This argument shows that there are no advantages to exercising early if the investor plans to keepthe stock for the remaining life of the option (one month, in this case), What if the investor thinksthe stock is currently overpriced and is wondering whether to exercise the option and sell the stock?In this case, the investor is better off selling the option than exercising it.! The