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Unformatted text preview: R. 1. R. 2. R. 3. R. 4. R. 5. R. 6. CHAPTER 6 REVIEW QUESTIONS A financial option is a contract offering the holder the right to either accept a
transaction or to decline a transaction. The terms of the option usually
include a specific transaction price and a date of expiration. Call options
on common stock are options that give the holder the opportunity to purchase
shares of a particular stock at a specific transactions price before a certain
period of time. Put options on common stock are options that give the holder
the right to sell shares of a particular stock at a specific transactions
price before a certain period of time. Shareholders own the firmâ€™s assets subject to the claims of the debtholders. We
can therefore say that shareholders have a call option on the assets of the
firm. If the firmâ€™s assets turn out to be worth more than the value of the
debtholderâ€™s claim, the shareholders will exercise their option by paying the
debtholders their promised payments and taking ownership of the assets. If
the firmâ€™s assets turn out to be worth less than the debtholderâ€™s claim, the
shareholders will walk away from the firm by declaring bankruptcy, allowing
their option to the ownership of the firmâ€™s assets to expire worthless. As discussed in question 1 above, call options on common stock are options
that give the holder the opportunity to purchase shares of a particular stock
at a specific transactions price before a certain period of time. The value
of call options rise as the value of the common stock rises. However, call
options offer limited losses (their value can never go below zero) as the
holder of the option can allow the option to expire worthless. The seller of an option is the person who obligates himself to make promised
payments to the option buyer. Every dollar of gain to the option buyer
represents a dollar of loss to the option seller. Thus, the potential of
unlimited gains to buyers of options means the potential for unlimited losses
to option sellers. Option sellers enter these agreements because they are
paid a fee, known as the option premium, for taking the risks. Option sellers
hope that the options will expire worthless such that their profit will be
equal to the premium collected. The Black Scholes model estimates the value of a call option based on five
variables; (1) the price of the underlying asset, (2) the optionâ€™s exercise
price, (3) the standard deviation of the underlying assets, (4) the optionâ€™s
time to expiration, and (5) the risk free rate of interest. The optionâ€™s exercise price is one of the five variables in the Black Scholes
option pricing model. For call options, the value of the option is inversely
related to the optionâ€™s exercise price. The call option becomes profitable
as the stock price exceeds the optionâ€™s exercise price and the option 29 R. 7. R. 8. is inâ€”the~money. In situations where the exercise price is high relative to
the stock price, the stock price must rise by a large amount for the option to
be inâ€”theâ€”money. When the exercise price is low relative to the stock price,
the option is worth more. The standard deviation of the underlying asset is one of the five variables in
the Black Scholes option pricing model. The Black Scholes model illustrates
that call option prices rise as the standard deviation of the underlying asset
rises. An option is a claim on an asset which permits the holder to have
large or unlimited profit potential with limited loss potential. If the
standard deviation is high, the optionâ€™s upside potential will be increased
while the downside potential is limited. Therefore, an increase in the
standard deviation of the underlying asset benefits the call option holder. The optionâ€™s time to maturity is one of the five variables in the Black
Scholes option pricing model. Since most options allow the holder to exercise
the option at any time during the optionâ€™s lifetime, a long term option is
worth more than a short term option. Longer time to expiration allows more
time for the underlying asset to make large price movements. The benefit to more
time is similar to the benefit of more volatility as discussed in Review Question 7. 30 CHAPTER 6 PROBLEMS 1. a. The coupon .is the call option.
b. The current value of a oneâ€”week stay, the variance of the value of a oneâ€”week stay, the price at which the coupon allows the holder to purchase a stay, the
riskâ€”free rate and the time to the couponâ€™s expiration. 2. a. In the money options have exercise prices less than the underlying assets current
market value. These include both the September and October expiration dates for
both the $40 and $45 exercise (strike) prices. The out of the money options are the September and October call options with an
exercise or strike price of $50 (since it exceeds the $46 market value). b. The value of this call option, like all out of the money call options, results from
the possibility that the underlying assetâ€™s market value will rise above the exercise
or strike price before the option expires. When the option is far out of the money
and there is little time remaining, the option price will be very small to reflect
the small probability that the option will expire with value. c. The October call options have an additional month before expiration and therefore
have more value to the option holder (buyer) since there is more time for the asset
underlying the option to rise and increase the value of the option. 3. The option is out of the money since the price of the underlying asset is below the
strike price or exercise price. 4. $20. Only the $20 price is possible since the value of the underlying asset ($53)
already exceeds the exercise price by $18. Generally speaking, a call option is worth at least as much as the amount that the price of the underlying asset exceeds the strike
price. 5. a. For a $0 sales price in Reno stock, an investment in 1 share would lose $50.
For a $50 sales price in Reno stock, an investment in 1 share would lose $0.
For a $60 sales price in Reno stock, an investment in 1 share makes a $10 profit.
For a $100 sales price in Reno stock, an investment in 1 share makes a $50 profit. 31 Profit/Loss 6. a. P refit/Loss Profit/Loss of Stock in Reno (10 40 20 â€”60 l 60
Reno Stock Price 40 80 120 For a $0 price in Reno stock, a $10 investment in a call option would lose $10.
For a $50 price in Reno stock, a $10 investment in a call option would lose $10.
For a $60 price in Reno stock, a $10 investment in a call option would break even.
For a $100 price in Reno stock, a $10 investment in a call option makes $40 profit Profit/Loss of $10 Call Option on Reno 40 20 / I 60
Reno Stock Price 20 40 120 32 c. The stock investment performs better by $10 for all Reno stock price values of
greater than $50. However, the call option performs much better for the lowest
outcomes in Reno. The call option exposes the investor to a smaller dollar loss
since the largest possible loss is $10. However, this reduced loss potential is
offset by the reduced profits when Reno goes up or stays the same. 7. a. For a $0 price in Reno stock, writing a $10 call option makes a $10 profit.
For a $50 price in Reno stock, writing a $10 call option makes a $10 profit.
For a $60 price in Reno stock, writing a $10 call option breaks even.
For a $100 price in Reno stock, writing a $10 call option loses $40. Profit/Loss of Writing $10 Call Option on Reno
60 Profit/Loss â€”20 " _60 i l t I l
0 20 40 60 80 100 120
Reno Stock Price c. The graph in 7(b) is the mirror image to the graph in 6(b). This reflects the
fact that an option is a contract between two people and therefore any profit or
loss to one of the participants must be offset by an equal and opposite effect to
the other participant. Note that option buying offers limited loss potential with,
in theory, unlimited profit potential. Call option writing offers limited profit
potential with virtually unlimited loss potential. 8. a. $12 b. $2 c. $0 d. $0 The call option at expiration will be worth the stock price minus the strike price
or $0, whichever is greater. 33 9. a. $2 b. greater than $2 c. $0 01. greater than $0 e. $0 f. greater than $0 The call option at expiration will be worth the stock price minus the strike price
or $0, whichever is greater. Prior to expiration, the call prices will be greater. 10.
Value of Call Option at Expiration 330 ~ * mÂ» $25 a
/} $20 A l/ $15 â€˜ Option Price $10 A ss 1 so: I u I 1 /1 I i 4â€” t 25 30 35 40 45 50 55 60 65 70 75
Stock Price 11.a. Up b. Up c. Up d. Up 6. Down f. Nothing (beta is in Chapter 12) 12. a. $100. The most a call option can be worth is the price of the underlying asset.
b. $0. The least a call option can be worth is zero. 13. The owner of the call option hopes that the underlying stock price will rise. The
writer of the call option would hope that the price falls (or at least does not rise). 14. a â€” call option â€” on the
the ~ stockholders  wish to
firmâ€™s â€” debtholders â€” by paying them
firrnâ€™s â€” debt ~ which represents
or â€” strike price â€” of the option.
optionâ€™s â€” time to expiration â€” The
as a â€” call option price â€” and it
firmâ€™s â€” price of underlying assets  or
the â€” variance of the firmâ€™s underlying assets â€” to 34 15. a. Underlying Asset Strike Price Time Std. Dev. Riskâ€”free Rate
$16 $20 0.25 0.40 10.00%
From formula as in appendix:
d1= â€”0.8907 d2= â€” 1.0907
From Table 6.2: (and interpolating when appropriate)
N(d1)= 0.1865 N(d2)= 0.1377
From Formula A61: exp(â€”â€”rt) = 0.97531
CALL PRICE = $0.30
b. Underlying Asset Strike Price Time Std. Dev. Riskâ€”free Rate
$18 $20 0.25 0.40 10.00%
d1= â€”â€”0.3018 d2= â€”0.5018
N(d1)= 0.3814 N(d2)= 0.3079
exp(â€”rt) = 0.97531 CALL PRICE = $0.86
c. Underlying Asset Strike Price Time Std. Dev. Riskâ€”free Rate
$20 $20 0.25 0.40 10.00%
d1 = 0.2250 d2= 0.0250
N(d1)= 0.5890 N(d2)= 0.5100
exp(â€”rt) = 0.97531 CALL PRICE = $1.83
(I. Underlying Asset Strike Price Time Std. Dev. Riskâ€”free Rate
$22 $20 0.25 0.40 10.00%
d1= 0.7016 d2= 0.5016
N(d1)= 0.7585 N(d2)= 0.6920
exp(â€”rt) = 0.97531 CALL PRICE = $3.19
e. Underlying Asset Strike Price Time Std. Dev. Riskfree Rate
$24 $20 0.25 0.40 10.00%
d1= 1.1366 d2= 0.9366
N(d1)= 0.8721 N(d2)= 0.8255
exp(â€”rt) = 0.97531 CALL PRICE = $4.83 35 16â€˜ Value of Call Option $6 $5â€” / $2â€œ Option Price â€˜1â€œ /I/ so I l I I I
16 18 20 22 24 Stock Price 17. This problem must be solved by trial and error. In other words, various values of the volatility must be inserted into the option price model until the volatility
is found that will generate the option price ($2.00).
We start with a volatility of 0.20 and compute the option price: Underlying Asset Strike Price Time Volatility Riskâ€”free Rate
$20 $20 0.25 0.20 10.00%
d1 = 0.3000 d2: 0.2000
N(d1)= 0.6179 N(d2)= 0.5793
exp(â€”rt) = 0.97531 CALL PRICE = $1.06 As a shortcut, we can note that problem 15 c found that the call option price for
a volatility of 0.4 was $1.83. Thus by increasing the volatility by 0.2 (from 0.2 to
0.4), the call option price increased from $1.06 to $1.83, or by $0.77. In order
to increase the call option price another $0.17, a volatility of .44 appears reasonable: Underlying Asset Strike Price Time Volatility Riskâ€”free Rate
$20 $20 0.25 0.44 10.00%
d1: 0.2236 d2: 0.0036
N(d1)= 0.5885 N(d2)= 0.5015
exp(â€”rt) = 0.97531 CALL PRICE = $1.99 We are interpolating and rounding the values from Table 6.2 and therefore it is reasonable to assume that 0.44 produces an adequate approximation. The precise
answer is 0.443033. 36 CHAPTER 6 DISCUSSION QUESTIONS 1. Call options do offer virtually unlimited potential gains and losses are limited to
the amount invested, however, there are no "free lunches" in life and there is a
cost to the benefits of owning a call option. If the price of the asset underlying the
option remains approximately the same, the call option will fall in value. Thus,
generally speaking, call options lose money whenever the price of the underlying
asset falls or remains level. A call option will only be a winner of the price of
the underlying asset rises substantially, in general. Thus, call options tend to offer
small probabilities of large gains and large probabilities of losses. 2. The chapter discussion assumes that the firmâ€™s assets become riskier but that the
value of the assets remains constant â€” for example if a company could costlessly
trade their low risk assets for a group of high risk assets with identical total
values. In such a case, their would be no net increase or net decrease in total value.
However, there would be a wealth transfer from the firmâ€™s bondholders to the firmâ€™s
stockholders. The bondholders would lose because the probability that they will
be repaid in full has fallen due to the higher risk of the assets. The stockholders
will gain because the added risk increases their potential profits without an offsetting
increase in their potential losses since bankruptcy limits the amount that the
stockholders can lose. 3. No. Since a call option gives its owner a right to buy something rather than an
obligation, the worst thing that can happen is that the owner allows the option to
expire without using it. 4. Option trading may or not be speculation depending upon how the particular option
trader uses the options and whether the risks of the options add or hedge away the
total risks that the trader faces. The question of whether or not it should be legal
is helped by an analysis of Chapter 3. Brieï¬‚y, some people would argue that option
trading is mostly or completely speculation, that speculation can hurt people and
therefore it should be illegal. Others argue that speculators help an economy by
providing liquidity and more accurate prices. Still others argue that people have
the right to do whatever they want with their money including speculation. Speculation and government attempts to limit or prevent speculation have existed
for centuries. Time has shown that speculation that was vehemently opposed when
begun has produced enormous benefits to society when allowed to continue. The
trading of agricultural futures contracts in the U. S. is an excellent example. 5. It is generally true that options expire, the transaction is completed and therefore
options force the recognition of losses (and profits) on tax returns and other
accounting statements. Alternatively, stocks do not in general force similar
recognition. However, it is important to realize that market prices are usually the
best indication of true value and therefore when a stock price falls the investor 37 has lost money whether or not the stock is sold. The idea that money is not lost
until the stock is sold is an extremely commonly held but very dangerous viewpoint. If you wreck your sportscar into a tree is it reasonable to concloude that no loss
has taken place unless you are forced to sell it? There is no reason to believe that a stock that has fallen greatly in value recently is any more likely to rise in the
future than any other equally volatile stock. . Both statements are correct. Whether options or stocks are riskier depends upon
the basis for comparison. In terms of equal dollar investments, options are riskier
than stocks because options fluctuate more in percentage terms. Thus it is riskier
to purchase $10,000 of call options on IBM rather than $10,000 of IBM common
stock. In terms of equal units of invesments, options are less risky than stocks
because options ï¬‚uctuate fewer dollars per unit than do the underlying shares. Thus,
it is generally less risky to purchase 2 call options (on 200 shares of IBM common
stock) than it is to purchase 200 shares of IBM stock. Note that call options are
generally priced much lower than stocks. Thus, each unit requires a relatively small
investment and therefore exposes the investor to less risk. However, for equal dollar
investments, call options are almost always much riskier than the underlying stocks. 38 ...
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