168 Mathematics for Finance
On subtracting, we obtain
(
CA(S′′)−CA(S′)
)
+
(
PA(S′)−PA(S′′)
)
≤S′′−S′+X(1−e−rT)
≤S′′−S′.Each of the two terms on the left-hand side is positive, so it must be strictly
less thanS′′−S′, which completes the proof.
Proposition 7.17
LetS′<S′′and letα∈(0,1). Then
CA(αS′+(1−α)S′′)≤αCA(S′)+(1−α)CA(S′′),
PA(αS′+(1−α)S′′)≤αPA(S′)+(1−α)PA(S′′).Proof
LetS=αS′+(1−α)S′′and letS′=x′S(0),S′′=x′′S(0) andS=xS(0).
Suppose that
CA(S)>αCA(S′)+(1−α)CA(S′′).
We can write and sell a call on a portfolio withxshares, and purchaseαcalls
on a portfolio withx′shares and 1−αcalls on a portfolio withx′′shares, all
three options sharing the same strike priceXand expiry timeT.Thepositive
balanceCA(S)−αCA(S′)−(1−α)CA(S′′) of these transactions can be invested
without risk. If the written option is exercised at timet≤T, then we shall
have to pay (xS(t)−X)+,wherex=αx′+(1−α)x′′. We can exercise the
other two options to cover the liability. This is an arbitrage strategy because
(xS(t)−X)+≤α(x′S(t)−X)++(1−α)(x′′S(t)−X)+.The proof for put options is similar.
Dependence on the Expiry Time.For American options we can also for-
mulate a general result on the dependence of their prices on the expiry timeT.
To emphasise this dependence, we shall now writeCA(T)andPA(T) for the
prices of American calls and puts, assuming that all other variables are fixed.
Proposition 7.18
IfT′<T′′,then
CA(T′)≤CA(T′′),
PA(T′)≤PA(T′′).