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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≥0⟹My>10x⟹(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≥0⟹30x≥(30−M)y⟹30Mx≥M(30−M)y.
Combining both inequalities, 30Mx≥(30−M)y≥10(30−M)x.
⟹30Mx≥10(30−M)x⟹40M≥300⟹M≥7.5
So in an AO does exist, M≥7.5$. Therefore, if M<7.5$, than an AO does not exist.
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 K1<K2, P(K1)<P(K2) and P(K2)−P(K1)≤K2−K1, 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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When you pay the $10, your net gain is 40-10 = 30 which is better than 30-8=22
So, probably yes
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How is the net gain 40-10=30? Are you assuming that the stock is worth 0 at the end?
What is the value of "strike - put price" equal to? How does that graph (against strikes) look like? Should it be lower for lower strikes?
Interesting question. Finally solved it.
Yes. There is an arbitrage opportunity. Buy x 40$ Put options and sell y 30$ Put options where 12y>15x>11y
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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≥0⟹My>10x⟹(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≥0⟹30x≥(30−M)y⟹30Mx≥M(30−M)y.
Combining both inequalities, 30Mx≥(30−M)y≥10(30−M)x.
⟹30Mx≥10(30−M)x⟹40M≥300⟹M≥7.5
So in an AO does exist, M≥7.5$. Therefore, if M<7.5$, than an AO does not exist.
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
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 K1<K2, P(K1)<P(K2) and P(K2)−P(K1)≤K2−K1, 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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