166 Mathematics for Finance
Proposition 7.13
Suppose thatX′<X′′.Then
CA(X′)−CA(X′′)<X′′−X′,
PA(X′′)−PA(X′)<X′′−X′.Proof
Suppose thatX′<X′′, butCA(X′)−CA(X′′)≥X′′−X′. We write and
sell a call with strike priceX′, buy a call with strike priceX′′and invest the
balanceCA(X′)−CA(X′′) without risk. If the written option is exercised at
timet≤T, then we shall have to pay (S(t)−X′)+. Exercising the other option
immediately, we shall receive (S(t)−X′′)+. Observe that
(S(t)−X′′)+−(S(t)−X′)+≥−(X′′−X′)with strict inequality wheneverS(t)<X′′. Together with the risk-free invest-
ment, amounting to at leastX′′−X′, we shall therefore end up with a non-
negative sum of money, and in fact realise an arbitrage profit ifS(t)<X′′.
The proof is similar for put options.
Theorem 7.14
Suppose thatX′<X′′and letα∈(0,1). Then
CA(αX′+(1−α)X′′)≤αCA(X′)+(1−α)CA(X′′),
PA(αX′+(1−α)X′′)≤αPE(X′)+(1−α)PA(X′′).Proof
For brevity, we putX=αX′+(1−α)X′′. Suppose that
CA(X)>αCA(X′)+(1−α)CA(X′′).We write an option with strike priceXand buyαoptions with strike priceX′
and (1−α) options with strike priceX′′, investing without risk the positive
balance of these transactions. If the written option is exercised at time t≤T,
then we exercise both options held. In this way we shall achieve arbitrage
because
(S(t)−X)+≤α(S(t)−X′)++(1−α)(S(t)−X′′)+.The proof for put options is similar.