- Options: General Properties 157
Exercise 7.12
For dividend-paying stock sketch the regions of call and put prices de-
termined by the bounds.7.3.2 European and American Calls on Non-Dividend
Paying Stock
Consider European and American call options with the same strike priceXand
expiry timeT. We know thatCA≥CE, since the American option gives more
rights than its European counterpart. If the underlying stock pays no dividends,
thenCE≥S(0)−Xe−rTby Proposition 7.3. It follows thatCA>S(0)−Xif
r>0. Because the price of the American option is greater than the payoff, the
option will sooner be sold than exercised at time 0.
The choice of 0 as the starting time is of course arbitrary. Replacing 0 by
any givent<T, we can show by the same argument that the American option
will not be exercised at timet. This means that the American option will in
fact never be exercised prior to expiry. This being so, it should be equivalent
to the European option. In particular, their prices should be equal, leading to
the following theorem.
Theorem 7.4
The prices of European and American call options on a stock that pays no
dividends are equal,CA=CE, whenever the strike priceXand expiry timeT
are the same for both options.
Proof
We already know thatCA≥CE, see (7.7) and Exercise 7.10. IfCA>CE,then
write and sell an American call and buy a European call, investing the balance
CA−CEat the interest rater. If the American call is exercised at timet≤T,
then borrow a share and sell it forXto settle your obligation as writer of the
call option, investingXat rater. Then, at timeTyou can use the European
call to buy a share forXand close your short position in stock. Your arbitrage
profit will be (CA−CE)erT+Xer(T−t)−X>0. If the American option is not
exercised at all, you will end up with the European option and an arbitrage
profit of (CA−CE)erT>0. This proves thatCA=CE.
Theorem 7.4 may seem counter-intuitive at first sight. While it is possible