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Answers to Questions and Problems
1. What is the maximum theoretical value for a call? Under what conditions does a call reach this maximum
value? Explain.
The highest price theoretically possible for a call option is to equal the value of the underlying stock. This
happens only for a call option that has a zero exercise price and an infinite time until expiration. With such a
call, the option can be instantaneously and costlessly exchanged for the stock at any time. Therefore,
the call must have at least the value of the stock itself. Yet it cannot be worth more than the stock, because the
option merely gives access to the stock itself. As a consequence, the call must have the same price as the stock.
2. What is the maximum theoretical value for an American put? When does it reach this maximum? Explain.
The maximum value of a put equals the potential inflow of the exercise price minus the associated outflow of
the stock price. The maximum value for this quantity occurs when the stock price is zero. At that time, the
value of the put will equal the exercise price. In this situation, the put gives immediate potential access to the
exercise price because it is an American option.
3. Answer question 2 for a European put.
As with the American put, the European put attains its maximum value when the stock price is zero.
However, before expiration, the put cannot be exercised. Therefore, the maximum price for a European put
is the present value of the exercise price, when the exercise price is discounted at the riskfree rate from
expiration to the present. This discounting reflects the fact that the owner of a European put cannot exercise
now and collect the exercise price. Instead, he or she must wait until the option expires.
4. Explain the difference in the theoretical maximum values for an American and a European put.
The exercise value of a put option equals the exercise price (an inflow) minus the value of the stock at
the time of exercise (an outflow). In our notation, this exercise value is
X–S
. For any put, the maxi
mum value occurs when the stock is worthless,
S
5
0. The American and European puts have different
maximum theoretical values because of the different rules governing early exercise. Because an American
put can be exercised at any time, its maximum theoretical value equals the exercise price,
X
. If the stock
price is zero at any time, an American put gives its owner immediate access to amount
X
through exer
cise. This is not true of a European put, which can be exercised only at expiration. If the option has
time remaining until expiration and the stock is worthless, the European put holder must wait until
expiration to exercise. With the stock worthless, the exercise will yield
X
to the European put holder.
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 Spring '08
 Danısoglu
 Strike price

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