- Options: General Properties 159
all, the final balance will also be positive,PAerT>0, at expiry. These results
canbesummarisedasfollows.
Proposition 7.5
The prices of American call and put options on a stock paying no dividends
satisfy the inequalities
max{ 0 ,S(0)−Xe−rT}≤CA<S(0),
max{ 0 ,−S(0) +X}≤PA<X.Next we consider options on dividend-paying stock. The lower bounds for
European options imply thatS(0)−div 0 −Xe−rT≤CE≤CAand−S(0) +
div 0 +Xe−rT ≤PE ≤PA. But because the price of an American option
cannot be less than its payoff at any time, we also haveS(0)−X≤CAand
X−S(0)≤PA. Moreover, the upper boundsCA<S(0) andPA<Xcan
be established in a similar manner as for non-dividend paying stock. All these
inequalities can be summarised as follows: For dividend-paying stock
max{ 0 ,S(0)−div 0 −Xe−rT,S(0)−X}≤CA<S(0),
max{ 0 ,−S(0) + div 0 +Xe−rT,−S(0) +X}≤PA<X.Exercise 7.13
Prove by an arbitrage argument thatCA<S(0) for an American call
on dividend-paying stock.7.4 Variables Determining Option Prices
The option price depends on a number of variables. These can be variables
characterising the option, such as the strike priceXor expiry timeT, variables
describing the underlying asset, for example, the current priceS(0) or dividend
raterdiv, variables connected with the market as a whole such as the risk-free
rater, and of course the running timet.
We shall analyse option prices as functions of one of the variables, keeping
the remaining variables constant. This is a significant simplification because
usually a change in one variable is accompanied by changes in some or all
of the other variables. Nevertheless, even the simplified situation will provide
interesting insights.