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Call options inthemoney have an intrinsic value equal to the difference between the market price and
the strike price.
Value of call = (Market price of stock  Exercise price of call) X 100
The market price of stock is at the current date. Of course, the market price will typically change
on a stock each day. The exercise (strike) price of the call is fixed for its life. For example, the exercise
(strike) price for a 3month call is the same for the entire period.
EXAMPLE 16.19 The market price per share of a stock is $45, with a strike price of $40. Remember that one call
is for 100 shares of stock. The value of the call is
$45  $40 = $5 X 100 shares = $500
Outofthemoney call options have no intrinsic value.
In effect, the total premium consists of the intrinsic value plus speculative premium (time value)
based on factors such as risk, variability, forecasted future prices, expiration date, leverage, and
dividend.
Total premium = Intrinsic value + Speculative premium
The call purchaser takes the risk of losing the entire investment price for the option if a price increase
does not take place.
EXAMPLE 16.20 A 2month call option allows you to buy 500 shares of ABC Company at $20 per share. Within
that time period, you exercise the option when the market price is $38. Your gain is $9,000($38  $20 = $18X 500
shares). If the market price had declined from $20 you would not have exercised the call option, and you would
have lost your entire investment.
By purchasing a call you can own common stock for a fraction of the cost of purchasing regular
shares. Calls cost significantly less than common stock. Leverage exists because a little change in
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CHAP. 161 WARRANTS, CONVERTIBLES, OPTIONS, AND FUTURES
common stock price can result in a major change in the call optionâ€™s price. A part of the percentage gain
in the price of the call is the speculative premium attributable to the remaining life on the call.
EXAMPLE 16.21 A stock has a current market price of $35. A call can be purchased for $300 allowing the
acquisition of 100 shares at $35 each. If the price of the stock increases, the call will also be worth more. Assume
that the stock is at $55 at the callâ€™s expiration date.
The profit is $20 ($55  $35) on each of the 100 shares of stock in the call, or a total of $2,000 on an investment
of $300. A return of 667 percent ($2,000/$3,000)is earned.
1 . THE BLACKSCHOLES OPTION PRICING MODEL (OPM)
65
The model provides the relationship between call option value and the five factors that determine
the premium of an optionâ€™s market value over its expiration value:
1. Time to maturity. The longer the option period, the greater the value of the option.
2. Stock price volatility. The greater the volatility of the underlying stockâ€™s price, the greater its
value.
3. Exercise price. The lower the exercise price, the greater the value.
4. Stock price. The higher the price of the underlying stock, the greater the value.
5. Riskfree rate. The higher the riskfree rate, the higher the value.
The formula is:
V = PIN(dl)] Xe*[N(d2)]
where V = Current value of a call option
P = current price of the underlying stock
N(d) = cumulative normal probability density function = probability that a deviation less than
d will occur in a standard normal distribution.
X = exercise or strike price of the option
t = time to exercise date (for example, 3 months means t = 3/12= 1/4= 0.25)
r = (continuously compounded) riskfree rate of interest
e = 2.71828
In(P/X) + [ r + s2/2]t
d1 =
S˜
s2 = variance per period of (continuously compounded) rate of return on the stock
The formula, while somewhat imposing, actually requires readily available input data, with
the exception of s2, or volatility. P,X,r, and t are easily obtained.
The implications of the option model are the following:
1. The value of the option increases with the level of stock price relative to the exercise price (PIX),
the time to expiration times the interest rate (rt),and the time to expiration times the stockâ€™s
variability (s2t ) .
2. Other properties:
a. The option price is always less than the stock price.
b. The optional price never falls below the payoff to immediate exercise ( P  X or zero,
whichever is larger).
c. If the stock is worthless, the option is worthless.
d. As the stock price becomes very large, the option price approaches the stock price less the
present value of the exercise price.
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WARRANTS, CONVERTIBLES, OPTIONS, AND FUTURES
EXAMPLE 16.22 You are evaluating a call option which has a $20 exercise price and sells for $1.60. It has 3
months to expiration. The underlying stock price is also $20 and its variance is 0.16. The riskfree rate is 12 percent.
The option's value is:
First, calculate dl and d2:
In(P/X) + [r + s2/2]t
dl =
Sdi
 ln($20/$20)+ [0.12 + (0.16/2)](0.25)

(0.40)m
 0 +0.05 o.25
=
0.20
 S˜ = 0.25  0.20 = 0.05
= dl
d2
Next, look up the values for N ( d l ) and N(d2):
N(d1) = N(0.25) = 1 0.4013 = 0.5987
N(d2) = N(0.05) 1 0.4801 = 0.5199
=
Finally, use those values to find the option's value:
V = PIN(dl)]  Xe"[N(dz)]
= $20[0.5987]  $20e('.' 2)(o.25) [O. 5 1991
= $11.97  $19.41(0.5199)
= $11.97  $10.09 = $1.88
At $1.60, the option is undervalued according to the BlackScholes model. The rational investor would buy one
option and sell 0.5987 shares of stock short.
16.6 FUTURES
A futures is a contract to purchase or sell a given amount of an item for a given price by a certain
date (in the future  thus the name "futures market"). The seller of a futures contract agrees to deliver
the item to the buyer of the contract, who agrees to purchase the item. ?he contract specifies the amount,
valuation, method, quality, month and means of delivery, and exchange to be traded in. The month of
delivery is the expiration date; in other words, the date on which the commodity or financial
instrument must be delivered.
Commodity contracts are guarantees by a seller to deliver a commodity (e.g., cocoa or cotton).
Financial contracts are a commitment by the seller to deliver a financial instrument (e.g., a
Treasury bill) or a specific amount of foreign currency.
Future markets can be used for both hedging and speculating.
EXAMPLE 16.23 Investors use hedging to protect their position in a commodity. For example, a citrus grower
(the seller) will hedge to get a higher price for his products while a processor (or buyer) of the item will hedge to
obtain a lower price. By hedging an investor minimizes the risk of loss but loses the prospect of sizable profit.
Review Questions
1. The is the cash paid when a warrant is given up to acquire common stock.
2. Dividends are not received on holding
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WARRANTS, CONVERTIBLES, OPTIONS, AND FUTURES
CHAP. 161
warrant is sold separately from the bond.
3.
dividend.
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