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Prove using a no - arbitrage argument that a European put option on a non - dividend - paying stock is a convex function of

Prove using a no-arbitrage argument that a European put option on a non-dividend-paying
stock is a convex function of the strike price; that is, if we denote by p(K) the put price when
the strike price is K, then
p[K1+(1-)K2]p(K1)+(1-)p(K2),
for all 01
Hints:
(a) suppose by contradiction that p[K1+(1-)K2]>p(K1)+(1-)p(K2), can you
construct an arbitrage strategy?
(b) following hint (i), the right-hand side has a lower value, so you want to buy it ( units
of put option options with K1 and 1- units of put option options with K2) and sell
the left-hand side (a put option with strike K1+(1-)K2).
Answer.
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