108156.pdf

154 Mathematics for Finance

If the put is not exercised at all, then we can buy a share forXby exercising
the call at timeTand close the short position in stock. On closing the money
market position, we shall also end up with a positive amount

(−CA+PA+S(0))erT−X>XerT−X> 0.

The theorem, therefore, holds by the No-Arbitrage Principle.

Exercise 7.8

Modify the proof of Theorem 7.2 to show that

S(0)−Xe−rT≥CA−PA≥S(0)−div 0 −X

for a stock paying a dividend between time 0 and the expiry timeT,

where div 0 is the value of the dividend discounted to time 0.

Exercise 7.9

Modify the proof of Theorem 7.2 to show that

S(0)−Xe−rT≥CA−PA≥S(0)e−rdivT−X

for a stock paying dividends continuously at a raterdiv.

7.3 Bounds on Option Prices

First of all, we note the obvious inequalities

CE≤CA,PE≤PA, (7.7)

for European and American options with the same strike priceXand expiry
timeT. They hold because an American option gives at least the same rights
as the corresponding European option.
Figure 7.3 shows a scenario of stock prices in which the payoff of a European
call is zero at the exercise timeT, whereas that of an American call will be
positive if the option is exercised at an earlier timet<Twhen the stock price
S(t) is higher thanX. Nevertheless, it does not necessarily follow that the
inequalities in (7.7) can be replaced by strict ones; see Section 7.3.2, where it
is shown thatCE=CAfor call options on an asset that pays no dividends.

Exercise 7.10

Prove (7.7) by an arbitrage argument.