Using Put-Price Convexity to Identify a Strike-Price Arbitrage

Summary

The document applies convexity of European put prices with respect to strike to compare two puts on the same non-dividend-paying stock. It introduces a zero-strike put as an anchor, then expresses the lower strike as a weighted average of that anchor and the higher strike. Convexity implies that the put at the intermediate strike should be worth no more than the corresponding weighted average of the endpoint values; the zero-strike put has no payoff, leaving a bound based on the higher-strike put.

For the stated prices, the put at strike 80 is priced above that convexity bound implied by the put at strike 90, indicating an inconsistency with the assumed no-arbitrage relationship. The argument explains why the weight is determined by the relative strikes. It presents a pricing condition rather than a full trade construction, and does not discuss transaction costs, bid-ask spreads, or other market frictions.

Key ideas

• European put prices are convex as a function of strike under the stated setup.
• The strike 80 put can be expressed as an intermediate strike between zero and 90.
• The strike weights follow from expressing the intermediate strike as a weighted average of the endpoints.
• A price above the convexity bound signals a potential arbitrage inconsistency.

Tags

# Finding Arbitrage in two Puts


# Finding Arbitrage in two Puts












A European Put Option on a non-dividend paying stock with strike price 80 is currently priced at 8 and a put option on the same stock with strike price 90 is priced at 9. Is there an arbitrage opportunity existing in these two Options?

I know we have to used the fact that Put Options values are convex with respect to their Strike Prices and could use the equation $P(\lambda K) < \lambda P(K)$? But, in the solution book that I have, they take $\lambda$ to be 8/9 and I don't know why this is.

## Answer by Gordon (score 11, accepted)

https://quant.stackexchange.com/a/20773

Let $K_1=0$, $K_2=80$, and $K_3=90$. Then \begin{align*} K_2 = 1/9 \, K_1 + 8/9 \, K_3. \end{align*} Moreover, \begin{align*} Put(K_2) &= Put(1/9 \, K_1 + 8/9 \, K_3)\\ &< 1/9 \, Put (K_1) + 8/9\, Put(K_3)\\ &= 8/9 \, Put(K_3). \end{align*} Taking $K=K_3$ and $\lambda = 8/9$, we have that $$ Put(\lambda K) < \lambda Put(K).$$

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