Using Call-Price Convexity to Identify a Vertical-Strike Arbitrage

Summary

For European calls on the same underlying and with the same maturity, the document tests whether prices are convex as strike changes. It uses three strikes, with the middle strike halfway between the outer strikes, so convexity requires the middle call price not to exceed the average of the outer call prices. The quoted prices violate that condition, indicating an inconsistency under the stated setup.

The proposed trade buys half a call at each outer strike and sells one call at the middle strike. Its initial cost is negative at the quoted prices, and the answer recommends checking the payoff across regions separated by the strikes to establish nonnegative terminal value. This is a theoretical arbitrage argument that assumes matching contract terms and prices at which all legs can be traded. The source sketches, rather than details, the payoff verification and does not discuss transaction costs, liquidity, or execution risk.

Key ideas

• Call prices with a common maturity and underlying must satisfy convexity across strikes under standard no-arbitrage assumptions.
• When the middle strike is halfway between the outer strikes, its price is bounded by the average prices of the outer calls.
• The suggested position buys half of each outer-strike call and sells one middle-strike call.
• Arbitrage conclusions require consistent contract terms and executable prices for every leg.

Tags

# Show that convexity of call price as a function of the strike is violated


# Show that convexity of call price as a function of the strike is violated












European call options with strikes 90, 100 and 110 on the same underlying asset and with the same maturity are trading for 22.50, 18.84 and 13.97 respectively. show that the convexity of the call price as a function of the strike is violated, hence leading to an arbitrage opportunity. Describe in detail a trading strategy that makes a riskless profit.

## Answer by Alexey Kalmykov (score 2, accepted)

https://quant.stackexchange.com/a/4472

The price of payoff is convex if for every $0\le\lambda\le 1$:

$V(\lambda K_1 + (1-\lambda)K_3 )\le \lambda V(K_1) + (1-\lambda)V(K3)$

, where $V(K)$ is the price of an option with strike $K$.

We want that $\lambda K_1 + (1-\lambda)K_3 = K_2$. Solving it for $\lambda$ we get $\lambda=0.5$. Substituting $\lambda$ back to our inequality, we see that the convexity property doesn't hold.

We see that the portfolio of options with strike $K_1$ and $K_3$ is too cheap. Therefore, we can form an arbitrage portfolio by purchasing 0.5 of $K_1$-strike options and 0.5 of $K_3$-strike options. To finance this purchase we sell one option with strike $K_2$. By checking the payouts of our portfolio for stock prices $S$ in intervals $S<K_1,K_1\le S < K_2, K_2 \le S< K_3$ and $S \ge K_3$ we convince ourselves that this is indeed an arbitrage opportunity.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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