- Options: General Properties 161
Subtracting, we get
(
CE(X′)−CE(X′′)
)
+
(
PE(X′′)−PE(X′)
)
=(X′′−X′)e−rT.Since, by Proposition 7.6, both terms on the left-hand side are positive, each
is strictly smaller than the right-hand side.
Remark 7.2
In the language of mathematics the inequalities mean that the call and put
prices as functions of the strike price satisfy the Lipschitz condition with con-
stant e−rT<1,
|CE(X′′)−CE(X′)|≤e−rT|X′′−X′|,
|PE(X′′)−PE(X′)|≤e−rT|X′′−X′|.In particular, the slope of the graph of the option price as a function of the
strikepriceislessthan45◦. This is illustrated in Figure 7.5 for a call option.
Figure 7.5 Lipschitz property of call pricesCE(X)Proposition 7.8
LetX′<X′′and letα∈(0,1). Then
CE(αX′+(1−α)X′′)≤αCE(X′)+(1−α)CE(X′′),
PE(αX′+(1−α)X′′)≤αPE(X′)+(1−α)PE(X′′).In other words,CE(X)andPE(X) are convex functions ofX.
Proof
For brevity, we putX=αX′+(1−α)X′′. Suppose that
CE(X)>αCE(X′)+(1−α)CE(X′′).