Finance Quant Interview - I

The Put option on the $40 strike is priced at $10.
The Put option on the $30 strike is priced at $8.

Is there an arbitrage opportunity?

Note by Calvin Lin
4 years, 3 months ago

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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

Agnishom Chattopadhyay Staff - 4 years, 3 months ago

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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?

Calvin Lin Staff - 4 years, 3 months ago

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Interesting question. Finally solved it.

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

Siddhartha Srivastava - 4 years, 3 months ago

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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?

Calvin Lin Staff - 4 years, 3 months ago

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Let the price at which the 30$ put options is priced be M$ 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$ 10x\$ for the (A) puts and gain My$ My\$ for the (B) puts. Total gain =My10x My - 10x

If the stock price is above 40; both puts remain unused. Therefore net earnings = My10x. My - 10x. Since earning is greater than 0 in an AO, My10x0    My>10x    (30M)My>10(30M)x My - 10x \geq 0 \implies My > 10x \implies (30-M)My > 10(30-M)x . Also, since M<10 M < 10 , we have y>x y > x

If the stock price is =P=(0,30) = P = (0,30) ; both puts are used. Earning on (A) puts = x(40P) x(40 - P) . Loss on B puts=y(30P). = y(30 - P). Total earning = x(40P)y(30P)+My10x=30x(30M)y+(yx)P x(40 - P) - y(30-P) + My - 10x = 30x - (30 - M)y + (y-x)P

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

So minimum total earning =30x(30M)y0    30x(30M)y    30MxM(30M)y = 30x - (30 - M)y \geq 0 \implies 30x \geq (30-M)y \implies 30Mx \geq M(30 -M)y .

Combining both inequalities, 30Mx(30M)y10(30M)x 30Mx \geq (30 -M)y \geq 10(30-M)x .

    30Mx10(30M)x    40M300    M7.5 \implies 30Mx \geq 10(30 -M)x \implies 40M \geq 300 \implies M \geq 7.5

So in an AO does exist, M7.5$ M \geq 7.5\$ . Therefore, if M<7.5$ M < 7.5\$ , than an AO does not exist.

Siddhartha Srivastava - 4 years, 3 months ago

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To check if it works.

Like you said, x=7,y=9 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) 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) 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

Siddhartha Srivastava - 4 years, 3 months ago

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

Gary Lai - 3 years, 11 months ago

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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.

Calvin Lin Staff - 3 years, 11 months ago

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Well, there are really only 2 inequalities to check. First of all, assuming K1<K2K_{1}<K_{2}, P(K1)<P(K2)P(K_{1})<P(K_{2}) and P(K2)P(K1)K2K1P(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 - 3 years, 11 months ago

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@Gary Lai Great analysis! I question your assumption of "there are only 2 options given".

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

Calvin Lin Staff - 3 years, 11 months ago

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@Calvin Lin It will be zero.

Gary Lai - 3 years, 11 months ago

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@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 Staff - 3 years, 11 months ago

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@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.

Gary Lai - 3 years, 11 months ago

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