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I have seen questions involving the convexity of the price of put options as functions of strike price. https://quant.stackexchange.com/questions/50308/how-to-take-advantage-of-arbitrage-opportunity-of-two-options and https://quant.stackexchange.com/questions/22328/arbitrage-opportunity-interview-question?noredirect=1&lq=1

Here is an example question: "A European put option on a non-dividend paying stock with strike price 80 dollars is currently priced at 8 dollars and a put option on the same stock with strike price 90 dollars is priced at 9 dollars. Is there an arbitrage opportunity existing in these two options?"

In the problem they seem to know that there is an arbitrage opportunity because of convexity being violated: apparently we should expect $P(λK)<λP(K)$ by convexity, but in the problem we have $P(λK)=λP(K)$. However, I thought that convexity only told us that $P(λK)\leq λP(K)$, and therefore it's not violated in the problem? Am I mistaken on the definition of convexity here, or is there another way of determining there is an arbitrage opportunity?

J. W. Tanner
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Zonova
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    An epsilon away from the maturity the put price is strictly convex in the strike price. – Kurt G. Sep 13 '23 at 06:18
  • Ahh, thank you! Would you have a reference I could look at? – Zonova Sep 13 '23 at 11:55
  • What reference? It is well known (and easy to verify) that the derivative of the price w.r.t. the strike is the put's exercise probability $N(-d_2),.$ This is clearly increasing when the strike $K$ is increasing. – Kurt G. Sep 13 '23 at 12:43

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