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

$p[K_1+(1-)K_2]p(K_1)+(1-)p(K_2),$ for all

Hints:

$($a$)$ suppose by contradiction that $[K_1+(1-)K_2]>p(K_1)+(1-)p(K_2),$ can you construct an arbitrage strategy?

$($b$)$ following hint $($i$),$ the right$-$hand side has a lower value, so you want to buy it $($ unitsof put option options with $K_1$ and $1-$ units of put option options with $K_2)$ and sell the left$-$hand side $($a put option with strike $K_1+(1-)K_2).$