Why Call Prices Are Convex in Strike, Not Concave

Summary

The document presents a proposed proof that European call prices are concave in strike and identifies an invalid inequality as its flaw. The argument considers three ordered strikes and uses the payoff bound for a call spread to bound differences in call premiums. It then applies that bound to claim the middle-strike price lies above a weighted average of the outer prices. The answer points out that one of the inequalities reverses incorrectly: for the higher strike, the stated premium difference cannot be bounded by the negative strike difference in the direction used.

The correction exposes a sign error in the proof rather than supplying a full derivation of the correct shape property. Under standard no-arbitrage assumptions, call prices as functions of strike are convex, consistent with the nonnegative value of appropriately weighted butterfly spreads. The brief answer itself does not lay out those assumptions or prove convexity, so readers should treat it as locating the argument's mistake rather than a complete treatment of option-price bounds.

Key ideas

• A bound on the payoff of a call spread must be applied with the correct strike ordering and inequality direction.
• The proposed concavity argument fails when it uses an invalid bound for the difference between prices at the two higher strikes.
• Under standard no-arbitrage assumptions, call prices are convex in strike.
• A correction that identifies a sign error does not by itself provide a full proof of the correct convexity result.

Tags

# Wrong proof that call price is concave function of strike price


# Wrong proof that call price is concave function of strike price












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?

## Answer by dm63 (score 5, accepted)

https://quant.stackexchange.com/a/38965

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.