Arbitrage from a Convexity Violation in Call Prices

Summary

The document poses a one-period option-pricing problem in which three calls share an underlying and maturity, and their strikes are evenly spaced. It assumes the middle-strike call costs more than the average price of the two outer-strike calls, then asks for an explicit arbitrage strategy. This price pattern violates the convexity condition expected of call prices across strikes under standard no-arbitrage assumptions.

The setup points toward a butterfly spread: combine the outer-strike calls against two middle-strike calls, choosing the direction that receives cash upfront. The spread’s expiration payoff is nonnegative across possible stock prices, while the initial price inequality supplies a positive credit. The document itself does not provide the construction or proof, so those details are an implication of the stated setup rather than a worked answer. It gives no empirical test and does not address transaction costs or trading constraints.

Key ideas

• Call prices should be convex as a function of strike under standard no-arbitrage assumptions.
• The setup uses three evenly spaced strikes and an overpriced middle-strike call relative to the outer calls.
• A butterfly position can express the convexity violation as a candidate arbitrage.
• The prompt provides assumptions but no explicit portfolio or payoff proof.
• The argument is theoretical and does not address trading frictions.

Tags

# Arbitrage strategy one-period model


# Arbitrage strategy one-period model












Consider a one-period model with a stock $S_0=1$ and $S_1>0$. Introduce call options with strikes $K_1<K_2<K_3$ maturing at $T=1$. Assume further that $$ C(K_2)>\frac12(C(K_1)+C(K_3)) $$ and $K_2=\frac12(K_1+K_3)$. Here, $C(K_i)$ is the initial price of the call option with strike $K_i$. I want to find an explicit arbitrage strategy.

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