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Unformatted text preview: T ER Prope:rties of
Stock Options In this chapter we look at the factors affecting stock option prices. We use a number of
different arbitrage arguments to explore the relationships between European option
prices, American option prices, and the underlying stock price. The most important of
these relationships is puteall 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 is never optimal to exercise an American call option on a nondividendpaying
stock prior to the option's expiration, but that under some circumstances the early
exercise of an American put option on such a stock is optimal. 9.1 FACTORS AFFECTING OPTION PRtCES
There are six factors affecting the price of a stock option:
1. The
2. The
3. The
4. The
5. The
6. The current stock price, So
strike price, K
time to expiration, T
volatility of the stock price, (j
riskfree interest rate, r
dividends expected during the life of the option In this section we consider what happens to option prices when one of these factors
changes, with all the others remaining fixed. The results are summarized in Table 9.1.
Figures 9.1 and 9.2 show how European call and put prices depend on the first five
factors in the situation where So = 50, K = 50, r = 5% per annum, (j = 30% per
annum, T = 1 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
If a call 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
?O" 206 CHAPTER 9
Table 9.1 Summary of the effect on the price of a stock option of
increasing one variable while keeping all others fixed.*
Variable Current stock price
Strike price
Time to expiration
Volatility,
Riskfree rate
Amount of future dividends European European American American
call
put
call
put + +
+ ? ? +
+ + + +
+
+ +
+
+
+ * + indicates that an increase in the variable causes the option price to increase;
 indicates that an increase in the variable causes the option price to decrease; 1 indicates that the relationship is uncertain. the stock price increases and less valuable as the strike price increases. For a put option,
the payoff on exercise is the amount by which the strike price exceeds the stock price.
Put options therefore behave in the opposite way from call options: they become less
valuable as the stock price increases and more valuable as the strik~ price increases.
Figures 9. 1(ad) illustrate the way in which put and call prices depend on the stock price
and ,strike price. Time to Expiration
Now consider the effect of the expiration date. Both put and call American options
become more valuable as the time to expiration increases. Suppose that we have two
American options that differ only as far as the expiration date is concerned. The owner
of the longlife option has all the exercise opportunities open to the owner of the shortlife optionand more. The longlife option must therefore always be worth at least as
much as the shortlife option.
.
Although European put and call options usually become more valuable as the time
to expiration increases (see, e.g., Figures 9.1(e,f)), this is not always the case. Consider
two European call options on a stock: one with an expiration date in 1 month, the other
with an expiration date in 2 months. Suppose that a very large dividend is expected in
6 weeks. The dividend will cause the stock price to decline, so that the shortlife option
could be worth more than the longlife option. Volatility
The precise way in which volatility is defined is discussed in Chapter 13. Roughly
speaking, the volatility of a stock price is a measure of how uncertain we are about
future stock price movements. As volatility increases, the chance that the stock will do
very well or very poorly increases. For the owner of a stock, these two outcomes tend to
offset each other. However, this is not so for the owner of a call or put. The owner of a
call benefits, from price increases but has limited downside risk in the event of price
decreases because the most the owner can lose is the price of the option. Similarly, the
owner of a put benefits from price decreases, but has limited downside risk in the event 207 Properties of Stock Options of price increases. The values of bQth calls and puts therefore increase as volatility
increases (see Figures 9.2(a, b)). RiskFree Interest Rate
The riskfree interest rate affects the price of an option in a less clearcut way. As interest
rates in the economy increase, the expected return required by investors from the stock
Effect of changes in stock price, strike price, and expiration date on option
prices when So = 50, K = 50, r = 5%, a = 30%, and T = 1. Figure 9.1 Call option
price,c Put option
price, p 50 50 40
30
20 0 Stock
price, So Stock
price, So 10
0 20 60 80 20 100 80 (a) 100 (b) Call option
price, c Put option
price, p 50
40
30
20
10
0 Strike
price, K 0 20 40 60 80 100 10
0 Strike
price, K 0 20 40 (c) 60 80 100 (d) Call option
price, c Put option
price, p 10 10 8 8 6 6 4 4 2
0
0.0 Time to
expiration, T 0.4 0.8 1.2
(e) 1.6 2
0
0.0 Time to
expiration, T 0.4 0.8 1.2
(f) 1.6 208 CHAPTER 9
Figure 9.2 Effect of changes in volatility and riskfree interest rate on option prices
when So = 50, K = 50, r = 5%, U = 30%, and T = l.
Call option
price, c Put option
price, p 15 15 12 12 9 9 6 6 3
00 Volatility,
cr (%) 20 10 30 40 50 3
0
0 Volatility,
cr(%) 10 20 30 (a) 40 50 (b) Call option
price, c Put option
price, p 10 10 8 8 6 6 4 4 2
00 Riskfree
rate, r (%) 2 4 6 8 (c) 2
00 Riskfree
rate, r (%)
2 4 6 8 (d) tends to increase. In addition, the present value of any future cash flow received by the
holder of the option decreases. The combined impact of these two effects is to increase the
value of call options and decrease the value of put options (see Figures 9.2(c, d)).
It is important to emphasize that we are assuming that interest rates change while all
other variables stay the same, In particular we are assuming that interest rates change
while the stock price remains the same. In practice, when interest rates rise (fall), stock
prices tend to fall (rise). The net effect of an interest rate increase and the accompanying
stock price decrease can be to decrease the value of a call option and increase the value
of a put option. Similarly, .the net effect of an interest rate decrease and the accompanying stock price increase can be to increase the value of a call option and decrease the
value of a put option. Amount of Future Dividends
Dividends have the effect of reducing the stock price on the exdividend date. This is
bad news for the value of call options and good news for the value of put options. The
value of a 'call option is therefore negatively related to the size of an anticipated future
dividend, and the value of a put option is positively related to the size of an anticipated
future dividend. 209 Properties of Stock Options 9.2 ASSUMPTIONS AND NOTATION
;; In this chapter we will make assumptions similar to those made for deriving forward
and futures prices in Chapter 5. We assume that there are some market participants,
such as large investment banks, for which the following statements are true:
1. There are no transactions costs. 2. All trading profits "(net of trading losses) are subject to the same tax rate.
3. Borrowing and lending are possible at the riskfree interest rate. We assume that these market participants are prepared to take advantage of arbitrage
opportunities as they arise. As discussed in Chapters I and 5, this means that any
available arbitrage opportunities disappear very quickly. For the purposes of our
analysis, it is therefore reasonable to assume that there are no arbitrage opportunities.
We will use the following notation:
So: Current stock price K: Strike price of option
T: Time to expiration of option
ST: Stock price at maturity
r: Continuously compounded riskfree rate of interest for an investment maturing
in time T
c: Value of American call option to buy one share
P: Value of American put option to sell one share
c: Value of European call option to buy one share
p: Value of European put option to sell one share It should be noted that r is the nominal rate of interest, not the real rate of interest. We
can assume that r> O. Otherwise, a riskfree investment would provide no advantages
over cash. (Indeed, if r < 0, cash would be preferable to a riskfree investment.) 9.3 UPPER AND LOWER BOUNDS FOR OPTION PRICES
In this section we derive upper and lower bounds for option prices. These bounds do
not depend on any particular assumptions about the factors mentioned in Section 9.1
(except r > 0). If an option price is above the upper bound or below the lower bound,
then there are profitable opportunities for arbitrageurs. Upper Bounds
An American or European call option gives the holder the right to buy one share of a
stock for a certain 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:
c ~ So and C ~ So If these relationships were not true, an arbitrageur could easily make a riskless profit by
buying the stock and selling the call option. 210 CHAPTER 9
An American or European put option gives the holder the right to sell one share of a
stock for K. 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. It follows that it cannot be worth more than the present value of K today: , If this were not true, an arbitrageur could make a riskless profit by writing the option
and investing the proceeds of the sale at the riskfree interest rate. Lower Bound for Calls on NonDividendPaying Stocks
'A lower bound for the price of a European call option on a nondividendpaying stock is We" first look at a numerical example and then consider a more formal argument.
Suppose that So = $20, K = $18, r = 10% per annum, and T = 1 year. In this case,
So  Ke rT = 20  18eO. 1 = 3.71 or $3.71. Consider the situation where the European call price is $3.00, which is less
than the theoretical minimum of $3.71. An arbitrageur can short the stock and buy the
call to provide a cash inflow of $20.00  $3.00 = $17.00. If invested for 1 year at 10%
per annum, the $17.00 grows to 17eo. 1 = $18.79. At the end of the year, the option
expires. If the stock price is greater than $18.00, the arbitrageur exercises the option for
$18.00, closes out the short position, and makes a profit 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 is closed out. The arbitrageur then makes an even greater profit. For example,
if the stock price is $17.00, the arbitrageur's profit is
$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 rT
Portfolio B: one share In portfolio A, the cash, if it is invested at the riskfree interest rate, will grow to K in
time T. If ST > K, the call option is exercised at maturity and portfblio A is worth ST' If
ST < K, the call option expires worthless and the portfolio is worth K. Hence, at time
T, portfolio A is worth
max(ST' K)
Portfolio B is worth ST at time T. Hence, portfolio A is always worth as much as, and 211 Properties of Stock Options can be worth more than, portfolio ~ at the option's maturity. It follows that in the
absence of arbitrage opportunities this must also be true today. !fence,
c+ Ke rT ~ So or
Because the worst that can happen to a call option is that it expires worthless, its value
cannot be negative. This means that c ~ 0 and therefore
c ~ max(So  Ke rT , 0) (9.1) Example 9.1 Consider a European call option on a nondividendpaying stock when the stock
price is $51, the strike price is $50, the time to maturity is 6 months, and the riskfree rate of interest is 12% per annum. In this case, So = 51, K = 50, T = 0.5, and
rT
r = 0.12. From equation (9.1), a lower bound for the option price is So  Ke , or
51  50eO.12x0.5 = $3.91 lower Bound for European Puts on NonDividendPaying Stocks
For a European put option on a nondividendpaying stock, a lower bound for the
price is
Again, we first consider a numerical example and then look at a more formal argument.
Suppose that So = $37, K = $40, r = 5% per annum, and T = 0.5 years. In this case,
Ke rT  So = 40eO.05xO.5  37 = $2.01 Consider the situation. where the European put price is $1.00, which is less than the
theoretical minimum of $2.01. An arbitrageur can borrow $38.00 for 6 months to buy
both the put and the stock. At the end of the 6 months, the arbitrageur will be required
to repay 38eo.05xO.5 = $38.96. If the 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, and repays the loan for an even greater profit. For example, if the stock price is
$42.00, the arbitrageur's profit 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 share
Portfolio D: an amount of cash equal to Ke rT If ST < K, then the option in portfolio C is exercised at option maturity, and the
portfolio becomes worth K. If ST > K, then the put option expires worthless, and the 212 CHAPTER 9
portfolio is worth ST at this time. Hence, portfolio C is worth
max(ST' K) in time T. Assuming the cash is invested at the riskfree interest rate, portfolio D is
worth K in time T. Hence, portfolio C is always worth as much as, and can sometimes
be worth more than, portfolio D in time T. It follows that in the absence of arbitrage
opportunities portfolio C must be worth at least as much as portfolio D today. Hence,
p + So?' Ke rT or
Because the worst that can happen to a put option is that it expires worthless, its value
cannot be negative. This means that
p ?' max(KerT  So, 0) (9.2) Example 9.2 .consider a European put option on a nondividendpaying stock when the stock
price is $38, the strike price is $40, the time to maturity is 3 months, and the
riskfree rate of interest is 10% per annum. In this case So = 38, K = 40,
T = 0.25, and r = 0.10. From equation (9.2), a lower bound for the option price
,is Ke rT  So, or
40eO.lxO.25  38 = $1.01 9.4 PUTCAll PARITY
We now derive an important relationship between p and c. Consider the following two
portfolios that were used in the previous section:
Portfolio A: one European call option plus an amount of cash equal to KerT
Portfolio C: one European put option plus one share Both are worth
max(ST, K) at expiration of the options. Because the options are European, they cannot be exercised
prior to the expiration date. The portfolios must therefore have identical values today.
Tllis means that
(9.3)
c+Ke rT = p+So
This relationship is known as putcall parity. It shows that the value of a European call
with a certain strike price and exercise date can be deduced from the value of a
European put with the same strike price and exercise date, and vice versa.
If equation (9.3) does not hold, there are arbitrage opportunities~ Suppose that the
stock price is $31, the strike price is $30, the riskfree interest rate is 10% per annum,
the price of a 3month European call option is $3, and the price of a threemonth
European plit option is $2.25. In this case,
c + Ke rT = 3 + 30eO.lx3jI2 = $32.26 213 Properties of Stock Options and
p + So = 2.25 + 31 = $33.25 Portfolio C is overpriced relative to portfolio A. The correct arbitrage strategy is to buy
the securities in portfolio A and short the securities in portfolio C. The strategy involves
buying the call 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 riskfree interest rate, this amount grows to
30.25eo.lxO.25 = $31.02 in 3 months.
If the stock price at expiration of the option is greater than $30, the call will be
exercised; and if it is less 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 = $1.02 For an alternative situation, suppose that the call price is $3 and the put price is $1. In
this case,
c + Ke rT = 3 + 30eO.lx3/12 = $32.26
and
p + So = 1 + 31 = $32.00
Portfolio A is overpriced relative to portfolio C. An arbitrageur can short the securities in
portfolio A and buy the securities in portfolio C to lock in a profit. The strategy involves Table 9.2 Arbitrage opportunities when puteall parity does not hold. Stock
price = $31; interesf rate = 10%;call price = $3. Both put and call have a
strike price of $30 and 3 months to maturity.
Threemollth put price = $2.25 Tlll'eemollth put price = $1 Action noll':
Buy call for $3
Short put to realize $2.25
Short the stock to realize $31
Invest $30.25 for 3 months Action noll':
Borrow $29 for 3 months
Short call to realize $3
Buy put for $1
Buy the stock for $31 Action in 3 months if ST > 30:
Receive $31.02 from investment
Exercise call to buy stock for $30
Net profit = $1.02 Action in 3 months if ST > 30:
Call exercised: sell stock for $30
Use $29.73 to repay loan
Net profit = $0.27 Action in 3 months if ST < 30:
Receive $31.02 from investment
Put exercised: buy stock for $30
Net profit = $1.02 Action in 3 months if ST < 30:
Exercise put to sell stock for $3
Use $29.73 to repay loan
Net profit = $0.27 214 CHAPTER 9 Business Snapshot 9.1 PutCall Parity and Capital Structure The pioneers of option pricing were Fischer Black, Myron Scholes, and Robert
Merton. In the early 1970s, they showed that options can be used to characterize
the capital structure of a company. Today this model is widely used by financial
institutions to assess a company's credit risk.
To illustrate the model, consider a company that has assets that are financed with
zerocoupon bonds and equity. Suppose that the bonds mature in 5 years at which
time a principal payment of K is required. The company pays no dividends. If the
assets are worth more than K in 5 years, the equity holders choose to repay the
bondholders. If the assets are worth less than. K, the equity holders choose to declare
bankruptcy and the bondholders end up owning the company.
The value of the equity in 5 years is therefore max(A T  K, 0), where AT is the
_value of the company's assets at that time. This shows that the equity holders have a
5year European call option on the assets of the company with a strike price of K.
What about the bondholders? They get min(A T , K) in 5 years. This is the same as
K  max(K  AT, 0). The bondholders have given the equity holders the right to sell
the·company's assets to them for K in 5 years. The bonds are therefore worth the
present value of K minus the value of a 5year European put option on the assets
with a strike price of K.
To summarize, if c and p are the value of the call and put options, respectively,
then
Value of equity = c
Value of debt = PV(K)  P
Denote the value of the assets of the company today by A o. The value of the assets
must equal the total value of the instruments used to finance the assets. This means
that it must equal the sum of the value of the equity and the value of the debt, so that
Ao = Rearranging this equation, we c + [PV(K)  p] ~ave c + PV(K) = p + A o
This is the puteall parity result in equation (9.3) for call and put options on the
assets of the company.
shorting the call and buying both the put and the stock with an initial investment of
$31 + $1  $3 = $29 When the investment is financed at the riskfree interest rate, a repayment of2geo.lxo.25 =
$29.73 is required at the end of the 3 months. As in the previous case, either the call or the
put will be exercised. 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 These examples are illustrated in Table 9.2. Business Snapshot 9.1 shows how options
and· puteall parity can help us understand the positions of the debt and equity holders
In a company. 215 Properties of Stock Options American Options
Putcall parity holds only for European options. However, it is p6ssible to derive some
results for American option prices. It can be shown (see Problem 9.18) that, when there
are no dividends,
So  K ~ C  p ~ So  Ke rT (9.4) Example 9.3 An American call option on a nondividendpaying stock with strike price $20.00
and maturity in 5 months is worth $1.50. Suppose that the current stock price is
$19.00 and the riskfree interest rate is 10% per annum. From equation (9.4), we
have
19  20 ~ C  p ~ 19  20eO.lx5jI2
or
1~PC~0.18 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 strike price and expiration date as the American
call are $2.50 and $1.68. 9.5 EARLY EXERCISE: CAllS ON A NONDIVIDENDPAYING STOCK
This section demonstrates that it is never optimal to exercise an American call option on
a nondividendpaying stock before the expiration date.
To illustrate the general nature of the argument, consider an American call option on
a nondividendpaying stock with 1 month to expiration when the stock price is $50 and
the strike price is $40. The option is deep in the money, and the investor who owns the
option might well be tempted to exercise it immediately. However, if the investor plans
to hold the stock obtaiped by exercising the option for more than 1 month, this is not
the best strategy. A better course of action is to keep the option and exercise it at the
end of the month. The $40 strike price is then paid out 1 month later than it would be if
the option were exercised immediately, so that interest is earned on the $40 for 1 month.
Because the stock pays no dividends, no income from the stock is sacrificed. A further
advantage of waiting rather than exercising immediately is that there is some chance
(however remote) that the stock price will fall below $40 in 1 month. In this case, the
investor will not exercise in 1 month 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 keep the stock for the remaining life of the option (1 month, in this case). What
if the investor thinks the 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. l The option will be bought by another investor who does
want to hold the stock. Such investors must exist: otherwise the current stock price
would not be $50. The price obtained for the option will be greater than its intrinsic
value of $10, for the reasons mentioned earlier.
I As an alternative strategy, the investor can keep the option and short the stock to lock in a better profit
than SIO. 216 CHAPTER 9
Variation of price of an American or European call option on a nondividendpaying stock with the stock price, So. Figure 9.3
Call option
price K Stock price, So For a more formal argument, we can use equation (9.1):
c ~ So  Ke rT Beca.use the owner of an American call has all the exercise opportunities open to the
owner of the corresponding European call, we must have
Hence, C~c C ~ So  Ke rT Given r > 0, it follows that C > So  K. If it were optimal to exercise early, C would
equal So  K. We deduce that it can never be optimal to exercise early.
Figure 9.3 shows the general way in which the call price varies with So. It indicates
that the call price is always above its intrinsic value ofmax(So  K, 0). As r or T or the
volatility increases, the line relating the call price to the stock price moves in the
direction indicated by the arrows (i.e., farther away from the intrinsic value).
To summarize, there are two reasons an American call on a nondividendpaying
stock should not be exercised early. One relates to the insurance that it provides. A call
option, when held instead of the stock itself, in effect insures the holder against the
stock price falling below the strike price. Once the option has been exercised and the
strike price has been exchanged for the stock price, this insurance vanishes. The other
reason concerns the time value of money. From the perspective of the option holder,
the later the strike price is paid out, the better. 9.6 EARLY EXERCISE: PUTS ON A NONDIVIDENDPAYING STOCK
It can be optimal to exercise an American put option on a nondividendpaying stock eady. Indeed, at any given time during its life, a put option should always be exercised
early if it is sufficiently deep in the money. 217 Properties of Stock Options To illustrate this, consider an extreme situation. Suppose that the strike price is $10
and the stock price is virtually zero. By exercising immediately, an investor makes an
immediate gain of $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 to receiving $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 in conjunction with the stock, insures the holder against the stock price
falling below a certain level. However, a put option is different from a call option in that
it may be optimal for an investor to forgo this insurance and exercise early in order to
realize the strike price immediately. In general, thy early exercise of a put option
becomes more attractive as So decreases, as r increases, and as the volatility decreases.
It will be recalled from equation (9.2) that For an American put with price P, the stronger condition
P ~ K  So must always hold because immediate exercise is always possible.
Figure 9.4 shows the general way in which the price of an American put varies with
So. Provided that r> 0, it is always optimal to exercise an American put immediately
when the stock price is sufficiently low. When early exercise is optimal, the value of the
option is K  So. The curve representing the value of the put therefore merges into the
put's intrinsic value, K  So, for a sufficiently small value of So. In Figure 9.4, this value
of So is shown as point A. The line relating the 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 option early, it follows that an American put option is always worth more than the
Figure 9.4 Variation of price of an American put option with stock price, So. American
put price ",, , A ,, ,, ,, ,, ,, ,
K Stock price, So 218 CHAPTER 9
Figure 9.5 Variation of price of a European put option with the stock price, So. European
put price E ""... ... ... ... ... ... ... ... ... ... ... ... "... " B K Stock price, So corresponding European put option. Furthermore, because an American put is sometimes worth its intrinsic value (see Figure 9.4), it follows that a European put option
must sometimes be worth less than its intrin.sic value. Figure 9.5 shows the variation of
the European put price with the stock price. Note that point B in Figure 9.5, at which
the price of the option is equal to its intrinsic value, must represent a higher value of the
stock price than point A in Figure 9.4. Point E in Figure 9.5 is where So = 0 and the
European put price is Ker~ 9.7 EFFECT OF DIVIDENDS
The results produced so far in this chapter have assumed that we are dealing with
options on a nondividendpaying stock. In this section we examine the impact of
dividends. In the United States most exchangetraded stock options have a life of less
than 1 year and dividends 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 the calculation of D, a dividend is assumed to occur at
the time of its exdividend date. lower Bound for Calls and Puts
We can redefine portfolios A and B as follows:
Portfolio A: one European call option plus an amount of cash equal to D + Ke rT
Portfolio B: one share A similar argument to the one used to derive equation (9.1) shows that
c ~ So  D  Ke rT (9.5) 219 Properties of Stock Options 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 rT A similar argument to the one used to derive equation (9.2) shows that
p~D + KerT  So (9.6) Early Exercise
When dividends are expected, we can no longer assert than an American call option will
not be exercised early. Sometimes it is optimal to exercise an American call immediately
prior to an exdividend date. It is never optimal to exercise a call at other times. This
point is discussed further in the appendix to Chapter 13. PutCall Parity
Comparing the value at option maturity of the redefined portfolios A and C shows that,
with dividends, the puteall parity result in equation (9.3) becomes
c+D+KerT =p+So (9.7) Dividends cause equation (9.4) to be modified (see Problem 9.19) to
So  D  K::::; C  p::::; So  Ke rT (9.8) 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 riskfree interest rate, and
the dividends expected during the life of the option. The value of a call generally
increases as the current stock price, the time to expiration, the volatility, and the riskfree interest rate increase. The value of a call decreases as the strike price and expected
dividends increase. The value of a put generally increases as the strike price, the time to
expiration, the volatility, and the expected dividends increase. The value of a put
decreases as the current stock price and the riskfree interest rate increase.
It is possible to reach some co.ndusions about the value of stock options without
making any assumptions about the volatility of stock prices. For example, the price of a
call option on a stock must always be worth less than the price of the stock itself.
Similarly, the price of a put option on a stock must always be worth less than the
option's strike price.
A European call option on a nondividendpaying stock must be worth more than
max(So  Ke rT , 0)
where So is the stock price, K is the strike price, r is the riskfree interest rate, and Tis
the time to expiration. A European put option on a nondividendpaying stock must be 220 CHAPTER 9
worth more than max(KerT  So, 0) When dividends with present value D will be paid, the lower bound for a European call
option becomes
max(So  D  Ke rT , 0)
and the lower bound for a European put option becomes max(KerT +D So, 0) Puteall parity is a relationship between the price, c, of a European call option on a
stock and the price, p, of a European put option on a stock. For a nondividendpaying
stock, it is
c+KerT =p+So
For a dividendpaying stock, the puteall parity relationship is c+D+Ke rT =p+So
Putcall parity does not hold for American options. However, it is possible to use
arbitrage arguments to obtain upper and lower bounds for the difference between the
price of an American call and the price of an American put.
In Chapter 13, we will carry the analyses in this chapter further by making specific
assumptions about the probabilistic behavior of stock prices. The analysis will enable us
to derive exact pricing formulas for European stock options. In Chapters 11 and 17, we
will see how numerical procedures can be used to price American options. FURTHER READING
Black, F., and M. Scholes. "The Pricing of Options and Corporate Liabilities," Journal oj
Political Economy, 81 (MayjJune1973): 63759.
Broadie, M., and J. Detemple. "American Option Valuation: New Bounds, Approximations, and
a Comparison of Existing Methods," Review oj Financial Studies, 9, 4 (1996): 121150.
Merton, R. c.. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,"
Journal oj Finance, 29, 2 (1974): 44970.
Merton, R. C. "Theory of Rational Option Pricing," Bell Journal oj Economics and Management
Science, 4 (Spring 1973): 14183.
Merton, R. C. "The Relationship between Put and Call Prices: Comment," Journal oj Finance,
28 (March 1973): 18384.
Stoll, H. R. "The Relationship between Put and Call Option Prices," Journal oj Finance, 31 (May
1969): 31932. Questions and Problems (Answers in Solutions Manual)
9.1. List the six factors that affect stock option prices.
9.2. What is a lower bound for the price of a 4month call option on a nondividendpaying
stock when the stock price is $28, the strike price is $25, and the riskfree interest rate is
8% per annum? Properties of Stock Options 221 9.3. What is a lower bound for the pric~ of a Imonth European put option on a nondividendpaying stock when the stock price is $12, the strike price i~ $15, and the riskfree
interest rate is 6% per annum?
9.4. Give two reasons why the early exercise of an American call option on a nondividendpaying stock is not optimal. The first reason should involve the time value of money. The
second should apply even if interest rates are zero.
9.5. "The early exercise of an American put is a tradeoff between the time value of money
and the insurance value of a put." Explain this statement.
9.6. Explain why an American call option on a dividendpaying stock is always worth at least
as much as its intrinsic value. Is the same true of a European call option? Explain your
answer.
9.7. The price of a nondividendpaying stock is $19 and the price of a 3month European call
option on the stock with a strike price of $20 is $1. The riskfree rate is 4% per annum.
What is the price of a 3month European put option with a strike price of $20?
9.8. Explain why the arguments leading to putcall parity for Eur()pean options cannot be
used to give a similar result for American options.
9.9. What is a lower bound for the price of a 6month call option on a nondividendpaying
stock when the stock price is $80, the strike price is $75, and the riskfree interest rate is
10% per annum?
9.10. What is a lower bound for the price of a 2month European put option on a nondividendpaying stock when the stock price is $58, the strike price is $65, and the riskfree
interest rate is 5% per annum?
9.11. A 4month European call option on a dividendpaying stock is currently selling for $5.
The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in
1 month. The riskfree interest rate is 12% per annum for all maturities. What opportunities are there for an arbitrageur?
9.12. A Imonth European put option on a nondividendpaying stock is currently selling
for $2.50. The stock price is $47, the strike price is $50, and the riskfree interest rate is
6% per annum. What opportunities are there for an arbitrageur?
9.13. Give an intuitive explanation of why the early exercise of an American put becomes more
attractive as the riskfree rate increases and volatility decreases.
9.14. The price of a European call that expires in 6 months and has a strike price of $30 is $2.
The underlying stock price is $29, and a dividend of $0.50 is expected in 2 months and
again in 5 months. The term structure is flat, with all riskfree interest rates being 10%.
What is the price of a European put option that expires in 6 months and has a strike price
of $30?
9.15. Explain carefully the arbitrage opportunities in Problem 9.14 if the European put price
is $3.
9.16. The price of an American call on a nondividendpaying stock is $4. The stock price is
$31, the strike price is $30, and the expiration date is in 3 months. The riskfree interest
rate is 8%. Derive upper and lower bounds for the price of an American put on the same
stock with the same strike price and expiration date.
9.17. Explain carefully the arbitrage opportunities in Problem 9.16 if the American put price is
greater than the calculated upper bound. 222 CHAPTER 9 9.18. Prove the result in equation (9.4). (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 an American put option plus one share.)
9.19. Prove the result in equation (9.8). (Hint: For the first part of the relationship, consider
(a) a portfolio consisting of a European call plus an amount of cash equal to D + K,
and (b) a portfolio consisting of an American put option plus one share.)
9.20. Regular call options on nondividendpaying stocks should not be exercised early. However, there is a tendency for executive stock options to be exercised early even when the
company pays no dividends (see Business Snapshot 8.3 for a discussion of executive stock
options). Give a possible reason for this.
9.21. Use the software DerivaGem to verify that Figures 9.1 and 9.2 are correct. Assignment Questions
9.22. A European call option and put option on a stock both have a strike price of $20 and an
expiration date in 3 months. Both sell for $3. The riskfree interest rate is 10% per annum,
the current stock price is $19, and a $1 dividend is expected in I month. Identify the
arbitrage opportunity open to a trader.
9.23. Suppose that Cl, Cl, and C3 are the prices of European call options with strike prices K 1,
K2, and K3, respectively, where K 3 > K2> Kt and K 3  K2 = K2  K 1• All options have
the same maturity. Show that
C2 ::s;; 0.5(cl + C3)
(Hint: Consider a portfolio that is long one option with strike price KI, long one option
with strike price K3, and short two options with strike price K2')
9.24. What is the result corresponding to that in Problem 9.23 for European put options?
9.25. Suppose that you are the manager and sole owner of a highly leveraged company. All the
debt will mature in 1 year. If at that time the value of the company is greater than the face
value of the debt, you will payoff 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?
9.26. Consider an option on a stock when the stock price is $41, the strike price is $40, the riskfree rate is 6%, the volatility is 35%, and the time to maturity is 1 year. Assume that a
dividend of $0.50 is expected after 6 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 puteall parity holds.
(d) Explore using DerivaGem what happens to the price of the options as the time to
maturity becomes very large. For this purpose, assume there are no dividends.
Explain the results you get. ...
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This note was uploaded on 01/30/2012 for the course MATH 174 taught by Professor Donblasius during the Spring '11 term at UCLA.
 Spring '11
 DonBlasius
 Math, Factors

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