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
3 years, 2 months ago

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

- 2 years, 10 months ago

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.

Staff - 2 years, 10 months ago

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.

- 2 years, 10 months ago

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?

Staff - 2 years, 10 months ago

It will be zero.

- 2 years, 10 months ago

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.

Staff - 2 years, 10 months ago

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

- 2 years, 10 months ago

Interesting question. Finally solved it.

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

- 3 years, 2 months ago

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

- 3 years, 2 months ago

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

What would be the price of the $30 put option, where there will be no arbitrage opportunity? Staff - 3 years, 2 months ago Log in to reply Let the price at which the 30$ 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.

- 3 years, 2 months ago

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

So, probably yes

Staff - 3 years, 2 months ago

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?

Staff - 3 years, 2 months ago