- Options: General Properties 167
Dependence on the Underlying Asset Price.Once again, we shall con-
sider options on a portfolio ofxshares. The prices of American calls and puts
on such a portfolio will be denoted byCA(S)andPA(S), whereS=xS(0) is
the value of the portfolio, all remaining variables being fixed. The payoffs at
timetare (xS(t)−X)+for calls and (X−xS(t))+for puts.
Proposition 7.15
IfS′<S′′,then
CA(S′)<CA(S′′),
PA(S′)>PA(S′′).Proof
Suppose thatCA(S′)≥CA(S′′)forsomeS′ <S′′,whereS′=x′S(0) and
S′′=x′′S(0). We can write and sell a call on a portfolio withx′shares and buy
a call on a portfolio with pricex′′shares, both options having the same strike
priceXand expiry timeT. The balanceCA(S′)−CA(S′′) of these transactions
can be invested without risk. If the written option is exercised at timet≤T,
then we can meet the liability by exercising the other option immediately.
Becausex′ <x′′, the payoffs satisfy (x′S(t)−X)+ ≤(x′′S(t)−X)+ with
strict inequality wheneverX<x′′S(t). As a result, this strategy will provide
an arbitrage opportunity.
The proof is similar for put options.
Proposition 7.16
Suppose thatS′<S′′.Then
CA(S′′)−CA(S′)<S′′−S′,
PA(S′)−PA(S′′)<S′′−S′.Proof
By the inequalities in Theorem 7.2
CA(S′)−PA(S′)≥S′−X,
CA(S′′)−PA(S′′)≤S′′−Xe−rT.