164 Mathematics for Finance
Proof
We employ put-call parity (7.1):
CE(S′′)−PE(S′′)=S′′−Xe−rT,
CE(S′)−PE(S′)=S′−Xe−rT.Subtracting, we get
(CE(S′′)−CE(S′)) + (PE(S′)−PE(S′′)) =S′′−S′.Both terms on the left-hand side are non-negative by the previous theorem, so
each is strictly less than the right-hand side.
Remark 7.5
A consequence of Proposition 7.10 is that the slope of the straight line joining
two points on the graph of the call or put price as a function ofSis less than 45◦.
This is illustrated in Figure 7.7 for call options. In other words, the call and
Figure 7.7 Lipschitz property of call pricesCE(S)put pricesCE(S)andPE(S) satisfy the Lipschitz condition with constant 1,
∣∣
CE(S′′)−CE(S′)
∣∣
≤|S′′−S′|,
∣∣
PE(S′′)−PE(S′)
∣∣
≤|S′′−S′|.
Proposition 7.11
LetS′<S′′and letα∈(0,1). Then
CE(αS′+(1−α)S′′)≤αCE(S′)+(1−α)CE(S′′),
PE(αS′+(1−α)S′′)≤αPE(S′)+(1−α)PE(S′′).This means that the call and put prices are convex functions ofS.