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.


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 $$ Put(\lambda K) < \lambda Put(K).$$

  • $\begingroup$ Thanks. Can a similar line of reasoning always be used when there are two/three puts, to see if there is arbitrage? $\endgroup$
    – Jojo
    Sep 18 '15 at 1:00
  • $\begingroup$ @Jojo: Correct. $\endgroup$
    – Gordon
    Sep 18 '15 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
    Oct 29 '16 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
    Oct 29 '16 at 18:57
  • $\begingroup$ @Gordon How do we prove that put option is convex with respect to their strike price? $\endgroup$
    – Idonknow
    Dec 5 '19 at 4:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.