162 Mathematics for Finance
We can write and sell an option with strike priceX, and purchaseαoptions with
strike priceX′and 1−αoptions with strike priceX′′, investing the balance
CE(X)−
(
αCE(X′)+(1−α)CE(X′′))
0 without risk. If the option with
strike priceXis exercised at expiry, then we shall have to pay (S(T)−X)+.
We can raise the amountα(S(T)−X′)++(1−α)(S(T)−X′′)+by exercising
αcalls with strikeX′and 1−αcalls with strikeX′′. In this way we will realise
an arbitrage profit because of the following inequality, which is easy to verify
(the details are left to the reader, see Exercise 7.15):
(S(T)−X)+≤α(S(T)−X′)++(1−α)(S(T)−X′′)+. (7.9)
Convexity for put options follows from that for calls by put-call parity (7.1).
Alternatively, an arbitrage argument can be given along similar lines as for call
options.
Exercise 7.15
Verify inequality (7.9).Remark 7.3
According to Proposition 7.8,CE(X)andPE(X) are convex functions ofX.
Geometrically, this means that if two points on the graph of the function are
joined with a straight line, then the graph of the function between the two
points will lie below the line. This is illustrated in Figure 7.6 for call prices.
Figure 7.6 Convexity of call pricesCE(X)Dependence on the Underlying Asset Price.The current priceS(0) of the
underlying asset is given by the market and cannot be altered. However, we can
consider an option on a portfolio consisting ofxshares, worthS=xS(0). The
payoff of a European call with strike priceXon such a portfolio to be exercised
at timeTwill be (xS(T)−X)+. For a put the payoff will be (X−xS(T))+.We
shall study the dependence of option prices onS. Assuming that all remaining
variables are fixed, we shall denote the call and put prices byCE(S)andPE(S).