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?

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestNo, since P(40)-P(30)=2<40-30=10.

Log in to reply

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.

Log in to reply

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.

Log in to reply

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

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Interesting question. Finally solved it.

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

Log in to reply

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

Log in to reply

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?

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.

Log in to reply

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

So, probably yes

Log in to reply

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?

Log in to reply