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I've somehow proved that European call price $C(K)$ is a concave function of strike price $K$, but I can't spot where the mistake is.

Suppose $K_1 < K_2 < K_3$ and thus $K_2 = \lambda K_1 + (1 - \lambda) K_3$ for some 0 < λ < 1.

If we buy a call struck at $K_1$ and sell a call struck at $K_2$ then the net premium paid is $C(K_1) - C(K_2)$ while the payoff at expiry is at most $K_2 - K_1$

Therefore we have a lemma: $C(K_1) - C(K_2) \leq K_2 - K_1$

This lemma apparently implies that $C(K)$ is concave:

$$ \begin{align*} \lambda C(K_1) + (1 - \lambda) C(K_3) - C(K_2) &= \lambda [C(K_1) - C(K_3)] + C(K_3) - C(K_2) \\ &\leq \lambda (K_3 - K_1) + K_2 - K_3 \\ &= K_2 - \lambda K_1 - (1 - \lambda) K_3 = 0 \end{align*} $$

Where is the error?

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It took me a while, but I think the statement $C(K_3) - C(K_2) \leq K_2 - K_3$ is not true if $K_3 > K_2$

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  • $\begingroup$ Indeed, we can have $C(K_2)-C(K_3) \le K_3-K_2$, but not $C(K_3)-C(K_2) \le K_2-K_3$. $\endgroup$ – Gordon Mar 24 '18 at 20:54

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