Problem $3\mathrm{.}3$

Suppose that $c_1,c_2$ and $c_3$ are the prices of European call options with strike prices $K_1,K_2$ and $K_3,$ respectively, where $K_3>K_2>K_1$ and $K_3-K_2=K_2-K_1.$ All option have the same maturity. Show that

$c_{2}0\mathrm{.}5(c_1{c}_3)$

This condition must be satisfied by all European option prices with the same maturity, and is known as the butterfly arbitrage condition.

Hint: Consider a portfolio that is long one option with strike $K_1,$ long one option with strike $K_3,$ and short two options with strike price $K_2.$ Plot the payoff of this portfolio as a function of the stock price at maturity $S(T),$ and convince yourself that it is alwnys positive.