160 Mathematics for Finance
7.4.1 European Options
Dependence on the Strike Price. We shall consider options on the same
underlying asset and with the same exercise timeT, but with different val-
ues of the strike priceX. The call and put prices will be denoted byCE(X)
and, respectively,PE(X) to emphasise their dependence onX. All remaining
variables such as the exercise timeT, running timetand the underlying asset
priceS(0) will be kept fixed for the time being.
Proposition 7.6
IfX′<X′′,then
CE(X′)>CE(X′′),
PE(X′)<PE(X′′).This means thatCE(X) is a strictly decreasing andPE(X) a strictly increasing
function ofX.
These inequalities are obvious. The right to buy at a lower price is more
valuable than the right to buy at a higher price. Similarly, it is better to sell
an asset at a higher price than at a lower one.
Exercise 7.14
Give a rigorous arbitrage argument to prove the inequalities in Proposi-
tion 7.6.Proposition 7.7
IfX′<X′′,then
CE(X′)−CE(X′′)<e−rT(X′′−X′),
PE(X′′)−PE(X′)<e−rT(X′′−X′).Proof
By put-call parity (7.1)
CE(X′)−PE(X′)=S(0)−X′e−rT,
CE(X′′)−PE(X′′)=S(0)−X′′e−rT.