- Options: General Properties 155
Figure 7.3 Scenario in which an American call can bring a positive payoff,
but a European call cannot
It is also obvious that the price of a call or put option has to be non-negative
because an option of this kind offers the possibility of a future gain with no
liability. Therefore,
CE≥ 0 ,PE≥ 0.
Similar inequalities are of course valid for the more valuable American options.
In fact the prices of options are nearly always strictly positive, except for some
very special circumstances, for example,CE= 0 for a call option with strike
priceX = 120 dollars one day prior to exercise when the underlying stock
is trading at $100 and daily price movements are limited by stock exchange
regulations to±10%.
In what follows we are going to discuss some further simple bounds for the
prices of European and American options. The advantage of such bounds is
that they are universal. They are independent of any particular model of stock
prices and follow from the No-Arbitrage Principle alone.
7.3.1 European Options
We shall establish some upper and lower bounds on the prices of European call
and put options.
On the one hand, observe that
CE<S(0).If the reverse inequality were satisfied, that is, if CE ≥ S(0), then we
could write and sell the option and buy the stock, investing the balance
on the money market. On the exercise dateTwe could then sell the stock
for min(S(T),X), settling the call option. Our arbitrage profit would be
(CE−S(0))erT+ min(S(T),X)>0. This proves thatCE <S(0). On the