option will bebought by another investor who does want to hold the stock. Such investors must exist; otherwisethe current stock price would not be $50. The price obtained for the option will be greater than itsintrinsic value of $10, for the reasons mentioned earlier.
For a more formal argument, we can use equation (8.1):
C2 Sy—-Ke™
Because the owner of an American call has all the exercise opportunities open to the owner of thecorresponding European call, we must have C > c. Hence,
C2 %-Ke™
Given r > 0, it follows that C > Sy — K. If it were optimal to exercise early, C would equal Sp ~ K.We deduce that it can never be optimal to exercise early.
Figure 8.3 shows the general way in which the call price varies with Sy and K. Tt indicates thatthe call price is always above its intrinsic value of max(S) — K, 0). As r or T or the volatility
' As an alternative strategy, the investor can keep the option and short the stock to lock in a better profit than $10.Properties of Stock Options 177
increases, the line relating the call price to the stock price moves in the direction indicated by thearrows (i.e., farther away from the intrinsic value).
To summarize, there are two reasons an American call on a non-dividend-paying stock shouldnot be exercised early. One relates to the insurance that it provides. A call option, when heldinstead of the stock itself, in effect insures the holder against the stock price falling below theexercise price. Once the option has been exercised and the exercise price has been exchanged for thestock price, this insurance vanishes. The other reason concerns the time value of money. From theperspective of the option holder, the later the strike price is paid out the better.
8.6 EARLY EXERCISE: PUTS ON A NON-DIVIDEND-PAYINGSTOCK
 
 
It can be optimal to exercise an American put option on a non-dividend-paying stock early. Indeed,at any given time during its life, a put option should always be exercised early if it is sufficiently deepin the money. To illustrate this, consider an extreme situation. Suppose that the strike price is $10and the stock price is virtually zero. By exercising immediately, an investor makes an immediate gainof $10. If the investor waits, the gain from exercise might be less than $10, but it cannot be more than$10 because negative stock prices are impossible. Furthermore, receiving $10 now is preferable toreceiving $10 in the future. It follows that the option should be exercised immediately.
Like a call option, a put option can be viewed as providing insurance. A put option, when held inconjunction with the stock, insures the holder against the stock price falling below a certain level.However, a put option is different from acall option in that it may be optimal for an investor to forgothis insurance and exercise early in order to realize the strike price immediately. In general, the earlyexercise of a put option becomes more attractive as Sp decreases, as r increases, and as the volatilitydecreases.
It will be recalled from equation (8.2) that
p> Kev? —S§
f do
Americanput price
       
 
11111II1111
>A K Stock price, Sg
Figure 8.4 Variation of price of an American put option with stock price, So178 CHAPTER 8
Europeanput price
    
B K
Figure 8.5 Variation of price of a European put option with the stock price, Sp
Stock price, Sp
For an American put with price P, the stronger conditionP2>K—Sy
must always hold because immediate exercise is always possible.
Figure 8.4 shows the general way in which the price of an American put varies with Sp. Providedthat r > 0, it is always optimal to exercise an American put immediately when the stock price issufficiently low. When early exercise is optimal, the value of the option is K — Sy. The curverepresenting the value of the put therefore merges into the put’s intrinsic value, K — So, for asufficiently small value of Sp. In Figure 8.4, this value of Sp is shown as point A. The line relatingthe put price to the stock price moves in the direction indicated by the arrows when r decreases,when the volatility increases, and when T increases.
Because there are some circumstances when it is desirable to exercise an American put optionearly, it follows that an American put option is always worth more than the correspondingEuropean put option. Furthermore, because an’ American put is sometimes worth its intrinsicvalue (see Figure 8.4), it follows that a European put option must sometimes be worth less than itsintrinsic value. Figure 8.5 shows the variation of the European put price with the stock price. Notethat point B in Figure 8.5, at which the price of the option is equal to its intrinsic value, mustrepresent a higher value of the stock price than point A in Figure 8.4. Point E in Figure 8.5 is whereSy = 0 and the European put price is Ke’. ,
8.7. EFFECT OF DIVIDENDS
 
 
The results produced so far in this chapter have assumed that we are dealing with options on anon-dividend-paying stock. In this section we examine the impact of dividends. In the UnitedStates, exchange-traded stock options generally have less than cight months to maturity, Thedividends payable during the life of the option can usually be predicted with reasonable accuracy.We will use D to denote the present value of the dividends during the life of the option. In thecalculation of D, a dividend is assumed to occur at the time of its ex-dividend date.Properties of Stock Options 179
Lower Bound for Calls and PutsWe can redefine portfolios A and B as follows:
Portfolio A: one European call option plus an amount of cash equal to D+ Ke’?Portfolio B: one share
A similar argument to the one used to derive equation (8.1) shows that¢2S—-D-—Ke™ (8.5)
We can also redefine portfolios C and D as follows:
Portfolio C: one European put option plus one share
Portfolio D: an amount of cash equal to D+ Ke?
A similar argument to the one used to derive equation (8.2) shows thatp>D+Ke™-Sy (8.6)
Early Exercise
When dividends are expected, we can no longer assert that an American call option will not beexercised early. Sometimes it is optimal to exercise an American call immediately prior to an ex-dividend date. It is never optimal to exercise a call at other times. This point is discussed further inChapter 12.
Put-Call Parity
Comparing the value at option maturity of the redefined portfolios A and C shows that, withdividends, the put—call parity result in equation (8.3) becomes
c+D4+Ke™ = p+Sy (8.7)Dividends cause equation (8.4) to be modified (see Problem 8.18) to
Sy -D-K<C—-P<S—Ke"™ (8.8)
8.8 EMPIRICAL RESEARCH
 
 
Empirical research to test the results in this chapter might seem to be relatively simple to carry outonce the appropriate data have been assembled. In fact, there are a number of complications:
1. Itis important to be sure that option prices and stock prices are being observed at exactly thesame time. For example, testing for arbitrage opportunities by looking at the price at whichthe last trade is done each day is inappropriate. This point was made in Chapter 7 inconnection with the data in Table 7.1.
2. It is important to consider carefully whether a trader can take advantage of any observedarbitrage opportunity. If the opportunity exists only momentarily, there may in practice beno way of exploiting it.180 CHAPTER 8
3. Transactions costs must be taken into account when determining whether arbitrageopportunities are possible.
4. Put-call parity holds only for European options. Exchange-traded stock options areAmerican,
5. Dividends to be paid during the life of the option must be estimated.
Some of the empirical research carried out in the early days of exchange-traded options markets isdescribed in the papers by Bhattacharya (1983), Galai (1978), Gould and Galai (1974), Klemkoskyand Resnick (1979, 1980), and Stoll (1969), all referenced at the end of this chapter. Galai andBhattacharya test whether option prices are ever less than their theoretical lower bounds; Stoll,Gould and Galai, and the two papers by Klemkosky and Resnick test whether put—call parityholds. We will consider the results of Bhattacharya and of Klemkosky and Resnick.
Bhattacharya examined whether the theoretical lower bounds for call options applied in practice.He used data consisting of the transactions prices for options on 58 stocks over a 196-day periodbetween August 1976 and June 1977. The first test examined whether the options satisfied thecondition that price be at least as great as the intrinsic value—that is, whetherC > max(Sp — K, 0). More than 86,000 option prices were examined and about 1.3% were foundto violate this condition. In 29% of the cases, the violation disappeared by the next trade,indicating that in practice traders would not have been able to take advantage of it. Whentransactions costs were taken into account, the profitable opportunities created by the violationdisappeared. Bhattacharya’s second test examined whether options sold for less than the lowerbound Sy — D ~ Ke~” (see equation (8.5)). He found that 7.6% of his observations did in fact sellfor less than this lower bound. However, when transactions costs were taken into account, these didnot give rise to profitable opportunities.
Klemkosky and Resnick’s tests of put-call parity used data on option prices taken from tradesbetween July 1977 and June 1978. They subjected their data to several tests to determine thelikelihood of options being exercised early, and they discarded data for which early exercise wasconsidered probable. In doing so, they felt they were justified in treating American options asEuropean. They identified 540 situations where the call price was too low relative to the put priceand 540 situations where the reverse was true. After transactions costs were allowed for, 38 of thefirst set of situations and 147 of the second set of situations were still profitable. The opportun-ities persisted when either a 5- or a 15-minute delay between the opportunity being noted andtrades being executed was assumed. Klemkosky and Resnick’s conclusion was that arbitrageopportunities were available to some traders, particularly market makers, during the period theystudied.
SUMMARY
 
 
There are six factors affecting the value of a stock option: the current stock price, the strike price,the expiration date, the stock price volatility, the risk-free interest rate, and the dividends expectedduring the life of the option. The value of a call generally increases as the current stock price, thetime to expiration, the volatility, and the risk-free interest rate increase. The value of a calldecreases as the strike price and expected dividends increase. The value of a put generally increasesas the strike price, the time to expiration, the volatility, and the expected dividends increase. Thevalue of a put decreases as the current stock price and the risk-free interest rate increase,
It is possible to reach some conclusions about the value of stock options without making anyProperties of Stock Options 181
assumptions about the volatility of stock prices. For example, the price of a call option on a stockmust always be worth less than the price of the stock itself. Similarly, the price of a put option on astock must always be worth less than the option’s strike price.
A call option on a non-dividend-paying stock must be worth more than
max(S, — Ke’, 0)
where Sp is the stock price, K is the exercise price, r is the risk-free interest rate, and 7 is the time toexpiration. A put option on a non-dividend-paying stock must be worth more than
max(Ke~’? — So, 0)When dividends with present value D will be paid, the lower bound for a call option becomesmax(Sp — D — Ke~’, 0)and the lower bound for a put option becomesmax(Ke~? + D— Sp, 0)
Put-call parity is a relationship between the price, c, of a European call option on a stock and theprice, p, of a European put option on a stock. For a non-dividend-paying stock, it is
c+ Ke = ptyFor a dividend-paying stock, the put-call parity relationship isc+D+Ke"™ =p+S
Put-call parity does not hold for American options. However, it is possible to use arbitragearguments to obtain upper and lower bounds for the difference between the price of an Americancall and the price of an American put.
In Chapter 12, we will carry the analyses in this chapter further by making specific assumptionsabout the probabilistic behavior of stock prices. The analysis will enable us to derive exact pricingformulas for European stock options. In Chapter 18, we will see how numerical procedures can beused to price American options.
 
 
Bhattacharya, M., “Transaction Data Tests of Efficiency of the Chicago Board Options Exchange,”Journal of Financial Economics, 12 (1983), 161-85.
Galai, D., “Empirical Tests of Boundary Conditions for CBOE Options,” Journal of FinancialEconomics, 6 (1978), 187-211.
Gould, J. P., and D. Galai, ‘Transactions Costs and the Relationship between Put and Call Prices,”Journal of Financial Economics, 1 (1974), 105-29.
Klemkosky, R. C., and B. G. Resnick. ““Put-Call Parity and Market Efficiency,”’ Journal of Finance, 34(December 1979), 1141-1155.182
CHAPTER 8
Klemkosky, R. C., and B. G. Resnick, “An Ex-Ante Analysis of Put—Call Parity,”” Journal of FinancialEconomics, 8 (1980), 363-78.
Merton, R. C., “The Relationship between Put and Call Prices: Comment,” Journal of Finance, 28(March 1973), 183-84.
Merton, R. C., “Theory of Rational Option Pricing,” Bell Journal of Economics and ManagementScience, 4 (Spring 1973), 141-83
Stoll, H. R., “The Relationship between Put and Call Option Prices,” Journal of Finance, 31 (May 1969),
 
 
 
319-32.QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)8.1. List the six factors affecting stock option prices.
8.2.
8.3.
8.4.
8.5.
8.6.
8.7.
8.8.
8.9.
8.10.
What is a lower bound for the price of a four-month call option on a non-dividend-paying stockwhen the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% perannum?
What is a lower bound for the price of a one-month European put option on a non-dividend-paying stock when the stock price is $12, the strike price is $15, and the risk-free interest rate is6% per annum?
Give two reasons why the carly exercise of an American call option on a non-dividend-payingstock is not optimal. The first reason should involve the time value of moncy. The sccond reasonshould apply even if interest rates are zero.
“The early exercise of an American put is a tradeoff between the time value of money and theinsurance value of a put.” Explain this statement.
Explain why an American call option is always worth at least as much as its intrinsic value. Is thesame true of a European call option? Explain your answer.
Explain why the arguments leading to put-call parity for European options cannot be used togive a similar result for American options.
What is a lower bound for the price of a six-month call option on a non-dividend-paying stockwhen the stock price is $80, the strike price is $75, and the risk-free interest rate is 10% perannum?
What is a lower bound for the price of a two-month European put option on a non-dividend-paying stock when the stock price is $58, the strike price is $65, and the risk-free interest rate is5% per annum?
A four-month European call option on a dividend-paying stock is currently selling for $5. Thestock price is $64, the strike price is $60, and a dividend of $0.80 is expected in one month. Therisk-free interest rate is 12% per annum for all maturities. What opportunities are there for anarbitrageur?
. A one-month European put option on a non-dividend-paying stock is currently selling for $2.50,
The stock price is $47, the strike price is $50, and the risk-free interest rate is 6% per annum.
What opportunities are there for an arbitrageur?
. Give an intuitive explanation of why the early exercise of an American put becomes more
attractive as the risk-free rate increases and volatility decreases.Properties of Stock Options 183
8.13.
8.14.8.15.
8.18.
8.19.
8.20.
The price of a European call that expires in six months and has a strike price of $30 is $2. Theunderlying stock price is $29, and a dividend of $0.50 is expected in two months and again in fivemonths. The term structure is flat, with all risk-free interest rates being 10%. What is the price ofa European put option that expires in six months and has a strike price of $30?
Explain carefully the arbitrage opportunities in Problem 8.13 if the European put price is $3.The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, thestrike price is $30, and the expiration date is in three months. The risk-free interest rate is 8%.Derive upper and lower bounds for the price of an American put on the same stock with the samestrike price and expiration date.
. Explain carefully the arbitrage opportunities in Problem 8.15 if the American put price is greater
than the calculated upper bound.
. Prove the result in equation (8.4). (Hint: For the first part of the relationship consider (a) a portfolio
consisting of a European call plus an amount of cash equal to K and (b) a portfolio consisting of anAmerican put option plus one share.)
Prove the result in equation (8.8). (Hint: For the first part of the relationship consider (a) a portfolioconsisting of a European call plus an amount of cash equal to D + K and (b) a portfolio consistingof an American put option plus one share.)
Even when the company pays no dividends, there is a tendency for executive stock options to beexercised early (see Section 7.12 for a discussion of executive stock options). Give a possiblereason for this.
Use the software DerivaGem to verify that Figures 8.1 and 8.2 are correct.
ASSIGNMENT QUESTIONS
 
8.21.
A European call option and put option on a stock both have a strike price of $20 and anexpiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the
" current stock price is $19, and a $1 dividend is expected in one month. Identify the arbitrage
8.22.
8.23.8.24.
opportunity open to a trader.
Suppose that c,, ¢2, and cz are the prices of European call options with strike prices K,, K2,and K;, respectively, where K; > Ky > K, and K, — K; = Ky — K,. All options have the samematurity. Show that
€, $ 0.5(c; +¢3)
(Hint: Consider a portfolio that is long one option with strike price K,, long one option withstrike price K3, and short two options with strike price K,.)
What is the result corresponding to that in Problem 8.22 for European put options?
Suppose that you are the manager and sole owner of a highly leveraged company. All the debt willmature in one year. If at that time the value of the company is greater than the face value of thedebt, you will pay off the debt. If the value of the company is less than the face value of the debt,you will declare bankruptcy and the debt holders will own the company.
a. Express your position as an option on the value of the company.
b. Express the position of the debt holders in terms of options on the value of the company.
c. What can you do to increase the value of your position?184
8.25.
CHAPTER 8
Consider an option on a stock when the stock price is $41, the strike price is $40, the risk-free rate
is 69%, the volatility is 259%. and the timc to maturity is one vear. Assume that a dividend of 90.5018 6%, the volatility is 35%, and the time to maturity is one year. Assume that a dividend of $0.50
is expected after six months.a. Use DerivaGem to value the option assuming it is a European call.b. Use DerivaGem to value the option assuming it is a European put.c. Verify that put-call parity holds.d. Explore using DerivaGem what happens to the price of the options as the time to maturitybecomes very large. For this purpose assume there are no dividends. Explain the resultsyou get. 
CHAPTER Q
OF e ™ 2
TRADING STRATEGIES INVOLVINGOPTIONS
 
We discussed the profit pattern from an investment in a single stock option in Chapter 1. In this’ chapter we cover more fully the range of profit patterns obtainable using options. We will assumethat the underlying asset is a stock. Similar results can be obtained for other underlying assets,such as foreign currencies, stock indices, and futures contracts. We will also assume that theoptions used in the strategies we discuss are European. American options may lead to slightlydifferent outcomes because of the possibility of early exercise.In the first section we consider what happens when a position in a stock option is combined witha position in the stock itself. We then move on to examine the profit patterns obtained when aninvestment is made in two or more different options on the same stock. One of the attractions ofoptions is that they can be used to create a wide range of different payoff functions. If Europeanoptions were available with every single possible strike price, any payoff function could in theorybe created.For ease of exposition we ignore the time value of money when calculating the profit from atrading strategy in this chapter. The profil is calculated as the final payoff minus the initial cost, notas the present value of the final payoff minus the initial cost.
9.1 STRATEGIES INVOLVING A SINGLE OPTION AND A STOCK
 
 
There are a number of different trading strategies involving a single option on a stock and the stockitself. The profits from these are illustrated in Figure 9.1. In this figure and in other figuresthroughout this chapter, the dashed line shows the relationship between profit and the stock pricefor the individual securities constituting the portfolio, whereas the solid line shows the relationshipbetween profit and the stock price for the whole portfolio.
In Figure 9.1a, the portfolio consists of a long position in a stock plus a short position in a calloption. This is known as writing a covered call. The long stock position ‘‘covers” or protects theinvestor from the payoff on the short call that becomes necessary if there is a sharp rise in the stockprice. In Figure 9.1b, a short position in a stock is combined with a long position in a call option.This is the reverse of writing a covered call. In Figure 9.1c, the investment strategy involves buyinga put option on a stock and the stock itself. The approach is sometimes referred to as a protectiveput strategy. In Figure 9.1d, a short position in a put option is combined with a short position inthe stock. This is the reverse of a protective put.
185186
Profit
A Profit
CHAPTER 9
 
 
A Profit
(a)_
A Profit
 
(b)
 
 
(c)
 
 
(d)
Figure9.1__ Profit patterns (a) long position in a stock combined with short position in acall; (b) short position in a stock combined with long position in a call; (c) long position ina put combined with long position in a stock; (d) short position in a put combined with
short position in a stock
The profit patterns in Figures 9.1a, b, c, d have the same general shape as the profit patternsdiscussed in Chapter | for short put, long put, long call, and short call, respectively. Put—call parityprovides a way of understanding why this is so. Recall from Chapter 8 that the put-call parity
relationship is
ptS=c+Ke 74D
(9.1)
where p is the price of a European put, Sp is the stock price, c is the price of a European call,K is the strike price of both call and put, r is the risk-free interest rate, T is the time to maturityTrading Strategies Involving Options 187
of both call and put, and D is the present value of the dividends anticipated during the life ofthe option.
Equation (9.1) shows that a long position in a put combined with a long position in the stock isequivalent to a long call position plus a certain amount (= Ke’ + D) of cash. This explains whythe profit pattern in Figure 9.lc is similar to the profit pattern from a long call position. Theposition in Figure 9.1d is the reverse of that in Figure 9.1c and therefore leads to a profit patternsimilar to that from a short call position.
Equation (9.1) can be rearranged to become
Sy-c= Ke +D-—p
In other words, a long position in a stock combined with a short position in a call is equivalent toa short put position plus a certain amount (= Ke~’? + D) of cash. This equality explains why theprofit pattern in Figure 9.1a is similar to the profit pattern from a short put position. The positionin Figure 9.1b is the reverse of that in Figure 9.la and therefore leads to a profit pattern similar tothat from a long put position.
9.2 SPREADS
A spread trading strategy involves taking a position in two or more options of the same type(ie., two or more calls or two or more puts). ~
Bull Spreads
One of the mosi popular types of spreads is a bull spread. It can be created by buying a cali optionona stock with a certain strike price and selling a call option on the same stock with a higher strikeprice. Both options have the same expiration date. The strategy is illustrated in Figure 9.2. Theprofits from the two option positions taken separately are shown by the dashed lines. The profitfrom the whole strategy is the sum of the profits given by the dashed lines and is indicated by thesolid line. Because a call price always decreases as the strike price increases, the value of the optionsold is always less than the value of the option bought. A bull spread, when created from calls,therefore requires an initial investment.
Profit
 
 
Figure 9.2 Bull spread created using call options188 CHAPTER 9
Table 9.1 Payoff from a bull spread
 
Stock price Payoff from Payoff from Totalrange long call option short call option payoffS7 2 K2 S7— Ky Ky — Sr Ky — KyK, < Sr < Ko S7— Ki 0 Sr— kySr < Ky 0 0 0
Suppose that K, is the strike price of the call option bought, K, is the strike price of the calloption sold, and S; is the stock price on the expiration date of the options. Table 9.1 shows thetotal payoff that will be realized from a bull spread in different circumstances. If the stock pricedoes well and is greater than the higher strike price, the payoff is the difference between the twostrike prices, or K, — K,. If the stock price on the expiration date lies between the two strike prices,the payoff is S; — K,. If the stock price on the expiration date is below the lower strike price, thepayoff is zero. The profit in Figure 9.2 is calculated by subtracting the initial investment from thepayoff.
A bull spread strategy limits the investor’s upside as well as downside risk. The strategy can bedescribed by saying that the investor has a call option with a strike price equal to K, and haschosen to give up some upside potential by selling a call option with strike price K, (Ky > K,). Inreturn for giving up the upside potential, the investor gets the price of the option with strikeprice K>. Three types of bull spreads can be distinguished:
1. Both calls are initially out of the money.2. One call is initially in the money; the other call is initially out of the money.3. Both calls are initially in the money.
The most aggressive bull spreads are those of type 1. They cost very little to set up and have a smallprobability of giving a relatively high payoff (= K, — K,). As we move from type | to type 2 andfrom type 2 to type 3, the spreads become more conservative.
Example9.1_ An investor buys for $3 a call with a strike price of $30 and sells for $1 a call withastrike price of $35. The payoff from this bull spread strategy is $5 if the stock price is above $35 andzero if it is below $30. If the stock price is between $30 and $35, the payoff is the amount by whichthe stock price exceeds $30. The cost of the strategy is $3 — $1 = $2. The profit is therefore asfollows:
Stock price range ProfitSr < 30 -230 < Sy < 35 Sr — 32Sr > 35 3
Bull spreads can also be created by buying a put with a low strike price and selling a put with ahe atetlen pean ae Hhaeteated in Elguen O27 Tlaiike the bill ceeesd crested from calle boll carecadshigh strike pricc, as illustrated in Figure 9.3. Unlike the bull spread created from calls, bull spreads
created from puts involve a positive cash flow to the investor up front (ignoring margin require-ments) and a payoff that is either negative or zero.Trading Strategies Involving Options 189
 
Figure 9.3 Bull spread created using put options
Bear Spreads
An investor who enters into a bull spread is hoping that the stock price will increase. By contrast,an investor who enters into a bear spread is hoping that the stock price will decline. As with a bullspread, a bear spread can be created by buying a call with one strike price and selling a call withanother strike price. However, in the case of a bear spread, the strike price of the option purchasedis greater than the strike price of the option sold. In Figure 9.4 the profit from the spread is shownby the solid line. A bear spread created from calls involves an initial cash inflow (when marginrequirements are ignored), because the price of the call sold is greater than the price of the callpurchased.
Assume that the strike prices are K, and Ky, with K, < K. Table 9.2 shows the payoff that willbe realized from a bear spread in different circumstances. If the stock price is greater than K>, thepayoff is negative at —(K, — K,). If the stock price is less than K,, the payoff is zero. If the stockprice is between K, and K3, the payoff is —(S; — K,). The profit is calculated by adding the initialcash inflow to the payoff.
 
Figure 9.4 Bear spread created using call options190 CHAPTER 9
Table 9.