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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.

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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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  • $\begingroup$ Thanks. Can a similar line of reasoning always be used when there are two/three puts, to see if there is arbitrage? $\endgroup$
    – Jojo
    Commented Sep 18, 2015 at 1:00
  • $\begingroup$ @Jojo: Correct. $\endgroup$
    – Gordon
    Commented Sep 18, 2015 at 1:47
  • $\begingroup$ @Gordon: Did you mean $K_1=10?$, how do you get $1/9K_1?$ If so, why do you pick that value of $K_1?$ Also, how do you go from $< RHS $ to $=8/9 Put(K_3)?$ $\endgroup$
    – user12348
    Commented Oct 29, 2016 at 18:26
  • $\begingroup$ @user12348: No, $K_1=0$ so that the put with strike $K_1$ has a value $0$. The inequality is based on convexity. $\endgroup$
    – Gordon
    Commented Oct 29, 2016 at 18:57
  • $\begingroup$ @Gordon How do we prove that put option is convex with respect to their strike price? $\endgroup$
    – Idonknow
    Commented Dec 5, 2019 at 4:04

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