TITLE: Price of a European Call option is a convex function of strike price K
QUESTION [16 upvotes]: I'm trying to show that the price of a European call option (payoff function is $(S_1-K)^+$) in a no-arbitrage market is a decreasing and convex function of K.
That it shall be decreasing makes sense; as $K$ increases, $S_1-K$ decreases and we make less profit. But why shall it be convex?

REPLY [5 votes]: There is a much simpler solution to the question regarding the convexity of the call option function. Let $K$ be the strike price and $x$ the stock price. Since the function from $K$ to $\max(0, x-K)$ is convex and conditional expectations are linear, the function $K$ to the conditional expectation of $\max(0, x-K)$ (under the risk-neutral density function) is also convex. Therefore we have shown that the call option price function is convex.