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Let the price at which the 30$ put options is priced be $M\$$.

Now suppose that an arbitrage opportunity does exist. It is easily proved that the arbitrage opportunity(AO) must consist of buying 40$ (A) put options and selling 30$ (B) put options. Let x (A) puts be bought and y (B) puts be sold for the AO.

So we spend $10x\$$ for the (A) puts and gain $My\$$ for the (B) puts. Total gain =$My - 10x$

If the stock price is above 40; both puts remain unused. Therefore net earnings = $My - 10x.$ Since earning is greater than 0 in an AO, $My - 10x \geq 0 \implies My > 10x \implies (30-M)My > 10(30-M)x$. Also, since $M < 10$, we have $y > x$

If the stock price is $= P = (0,30)$; both puts are used. Earning on (A) puts = $x(40 - P)$. Loss on B puts$= y(30 - P).$ Total earning = $x(40 - P) - y(30-P) + My - 10x = 30x - (30 - M)y + (y-x)P$

Since $y - x > 0$. Total earning is minimum when $P = 0$.

Like you said, $x = 7 , y = 9$ works. So we buy 7 puts (A) for $70 and sell 9 puts (B) for $72. Total profit now, = $72 - 70$ = 2$

If stock price is above 40; both put options remain unused. Total earning 2$

If stock price is $X = (30,40)$; we use the 7 puts (A) for profit = 7(40-x) > 0; 9 (B) puts remain unused.. Total earning = 2$ + 7(40-x)> 2$

If stock price is $X = (0,30)$; we use the 7 puts (A) for profit = 7(40 - X); 9 (B) puts are used giving a loss = 9(30 - X).
Total earning = 7(40-X) -9(30-X) + 2$ = 2X + 12$ > 0

You do not have a proof that there is no arbitrage opportunity. You have only checked one particular condition. There are more conditions to check. For example, an obvious one should be that P(30) < P(40). If that were not true, then we could arbitrage.

Well, there are really only 2 inequalities to check. First of all, assuming $K_{1}<K_{2}$, $P(K_{1})<P(K_{2})$ and $P(K_{2})-P(K_{1})\leq K_{2}-K_{1}$, since the premium for a put option increases more slowly compared to the strike price increase. If there are 3 options then we will have to compare its convexity.

@Gary Lai
–
Precisely. Now, since I've demonstrated that the assumption of "only 2 option values" is not true, you should apply your comment of "If there are 3 options, then we will have to compare its convexity", and see what happens from there.

@Calvin Lin
–
It forms another inequality! Although I still think that there is no arbitrage opportunity here, since Put option with 0 strike price wont exist.

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## Comments

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TopNewestWhen you pay the $10, your net gain is 40-10 = 30 which is better than 30-8=22

So, probably yes

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How is the net gain 40-10=30? Are you assuming that the stock is worth 0 at the end?

What is the value of "strike - put price" equal to? How does that graph (against strikes) look like? Should it be lower for lower strikes?

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Interesting question. Finally solved it.

Yes. There is an arbitrage opportunity. Buy $x$ 40$ Put options and sell $y$ 30$ Put options where $12y > 15x > 11 y$

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Assume that the $40 put is still priced at $10.

What would be the price of the $30 put option, where there will be no arbitrage opportunity?

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Let the price at which the 30$ put options is priced be $M\$$.

Now suppose that an arbitrage opportunity does exist. It is easily proved that the arbitrage opportunity(AO) must consist of buying 40$ (A) put options and selling 30$ (B) put options. Let x (A) puts be bought and y (B) puts be sold for the AO.

So we spend $10x\$$ for the (A) puts and gain $My\$$ for the (B) puts. Total gain =$My - 10x$

If the stock price is above 40; both puts remain unused. Therefore net earnings = $My - 10x.$ Since earning is greater than 0 in an AO, $My - 10x \geq 0 \implies My > 10x \implies (30-M)My > 10(30-M)x$. Also, since $M < 10$, we have $y > x$

If the stock price is $= P = (0,30)$; both puts are used. Earning on (A) puts = $x(40 - P)$. Loss on B puts$= y(30 - P).$ Total earning = $x(40 - P) - y(30-P) + My - 10x = 30x - (30 - M)y + (y-x)P$

Since $y - x > 0$. Total earning is minimum when $P = 0$.

So minimum total earning $= 30x - (30 - M)y \geq 0 \implies 30x \geq (30-M)y \implies 30Mx \geq M(30 -M)y$.

Combining both inequalities, $30Mx \geq (30 -M)y \geq 10(30-M)x$.

$\implies 30Mx \geq 10(30 -M)x \implies 40M \geq 300 \implies M \geq 7.5$

So in an AO does exist, $M \geq 7.5\$$. Therefore, if $M < 7.5\$$, than an AO does not exist.

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To check if it works.

Like you said, $x = 7 , y = 9$ works. So we buy 7 puts (A) for $70 and sell 9 puts (B) for $72. Total profit now, = $72 - 70$ = 2$

If stock price is above 40; both put options remain unused. Total earning 2$

If stock price is $X = (30,40)$; we use the 7 puts (A) for profit = 7(40-x) > 0; 9 (B) puts remain unused.. Total earning = 2$ + 7(40-x)> 2$

If stock price is $X = (0,30)$; we use the 7 puts (A) for profit = 7(40 - X); 9 (B) puts are used giving a loss = 9(30 - X). Total earning = 7(40-X) -9(30-X) + 2$ = 2X + 12$ > 0

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No, since P(40)-P(30)=2<40-30=10.

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You do not have a proof that there is no arbitrage opportunity. You have only checked one particular condition. There are more conditions to check. For example, an obvious one should be that P(30) < P(40). If that were not true, then we could arbitrage.

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Well, there are really only 2 inequalities to check. First of all, assuming $K_{1}<K_{2}$, $P(K_{1})<P(K_{2})$ and $P(K_{2})-P(K_{1})\leq K_{2}-K_{1}$, since the premium for a put option increases more slowly compared to the strike price increase. If there are 3 options then we will have to compare its convexity.

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In particular, what is the put option on the 0 strike worth?

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