- Options: General Properties 163
Remark 7.4
Even though options on a portfolio of stocks are of little practical significance,
the functionsCE(S)andPE(S) are important because they also reflect the de-
pendence of option prices on very sudden changes of the price of the underlying
such that the remaining variables remain almost unaltered.
Proposition 7.9
IfS′<S′′,then
CE(S′)<CE(S′′),
PE(S′)>PE(S′′),that is,CE(S) is a strictly increasing function andPE(S) a strictly decreasing
function ofS.
Proof
Suppose thatCE(S′)≥CE(S′′)forsomeS′<S′′,whereS′=x′S(0) andS′′=
x′′S(0). We can write and sell a call on a portfolio withx′shares and buy a call
on a portfolio withx′′shares, the two options sharing the same strike priceX
and exercise timeT, and we can invest the balanceCE(S′)−CE(S′′) without
risk. Sincex′<x′′, the payoffs satisfy (x′S(T)−X)+≤(x′′S(T)−X)+with
strict inequality wheneverX<x′′S(T). If the option sold is exercised at timeT,
we can, therefore, exercise the other option to cover our liability and will be
left with an arbitrage profit.
The inequality for puts follows by a similar arbitrage argument.
Exercise 7.16
Prove the inequality in Proposition 7.9 for put options.Proposition 7.10
Suppose thatS′<S′′.Then
CE(S′′)−CE(S′)<S′′−S′,
PE(S′)−PE(S′′)<S′′−S′.