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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.

Bob Jansen
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Jojo
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1 Answers1

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

Gordon
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  • Thanks. Can a similar line of reasoning always be used when there are two/three puts, to see if there is arbitrage? – Jojo Sep 18 '15 at 01:00
  • @Jojo: Correct. – Gordon Sep 18 '15 at 01:47
  • @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 $=8/9 Put(K_3)?$ – user12348 Oct 29 '16 at 18:26
  • @user12348: No, $K_1=0$ so that the put with strike $K_1$ has a value $0$. The inequality is based on convexity. – Gordon Oct 29 '16 at 18:57
  • @Gordon How do we prove that put option is convex with respect to their strike price? – Idonknow Dec 05 '19 at 04:04
  • @Idonknow: In $(K-S)^+$, hold $S$ fixed and draw a graph with respect $K$, you will see it. – Gordon Dec 05 '19 at 13:24
  • Actually we have $Put(\lambda K) = \lambda Put(K)$ in his case, or is it too late and I need to sleep?:) – Yian Pap Dec 28 '20 at 21:34
  • You can ask as another question. – Gordon Dec 29 '20 at 14:22
  • What I meant was that in this case we don't stirctly have $Put(\lambda K) < \lambda Put(K)$ as the answer seems to suggest, but rather $Put(\lambda K) = \lambda Put(K)$, i.e.we are "on the edge" of arbitrage. Since $Put(K_2) = 8$ and $Put(K_3) = 9$ as the question states. – Yian Pap Jan 23 '21 at 20:22
  • I am also confused about this. I thought that convexity tells us that $P(\lambda K) \leq \lambda P(K)$, and not necessarily $P(\lambda K) < \lambda P(K)$. In this example, as mentioned by Yian Pap's comment, we have them as equal, so convexity is still satisfied. So how should we know that an arbitrage opportunity exists? – Zonova Sep 12 '23 at 16:05