April 2012 Question 7 iii)

[s16455441]

Hi - I was hoping for some help.

First to clarify, my understanding is that the question is asking us.. " what are a and b, such that the net profit of this portfolio is the same as just having the 5,000,000 call options with strike price of 100 - whilst taking tax into consideration ".

The first part I think I understand.

When 100 < St < 120, are -b call options are equal to 0 as we have max(St-120, 0) = 0. Therefore

0.6 * a * (St-100) = 0.6 * 5,000,000 * (St - 100)

Equating the coefficients, I understand a = 5,000,000.

The next part is where I am struggling to understand.

I suspect the 600,000 is coming from... " suppose the share price is 120 (the minimum of this band). From the original call options we now have 5,000,000 * 0.6 * (120 - 100) = 60,000,000 p = £ 600,000 .

I can't wrap my head around what the 0.2 * 5,000,000 * (S - 120) is doing.

It is confusing me that we have a call option with a strike of 120 and that side of the equation, because my understanding was it should only be relating to the 5,000,000 calls with strike of 100.

I see that this must be something to do with allowing for " the profit is now greater than 1,000,000 so you are being taxed at 80% ".

Through an example, I think I can see how this works

Suppose St = 140.

The original call would be the following...

(140 - 100) * 5,000,000 = 200,000,000 p = £2,000,000.

Now thinking about net profit after tax..

1,000,000 * 0.6 + 1,000,000 * 0.2 = £800,000 net profit.

Thinking about what the solution gives us..

converting to £ because 600,000 is in £

600,000 + 0.2 * 5,000,000 * (1.4 - 1.2) = 800,000.

So I think my is question is am 1) I along the right lines, and 2) how could I think about this in the exam / what is the interpreting of this side of the equation: 600,000 + 0.2 × 5,000,000(S − 120)

 

[John Potter]

We're told that she realises that if she purchases a portfolio of call options with a strike of 100p and call options with a strike of 120p, then the proceeds will be liable for tax at only 40%.

So, we can match the payoffs at S = 120 and any number S > 120. Let's use your idea of S = 140 - you are thinking along very good lines!

If S = 120, we need the payoff from a call options to
0.01*a * max(120-100,0) = 50,000 *max(120-100,0)
(since £50,000 per penny) and this leads to a = 5,000,000.
You've got this bit - great!

If S= 140, the bonus is 50,000*max(140-100,0) = 2,000,000
After tax, this is
1,000,000 * (1 - 0.4) + 1,000,000 * (1 - 0.8) = 800,000

Now, the payoff from a call options bought and b call options sold, will be taxed at 40%

(a*max(140-100,0) - b * max(140-120,0)) * (1 - 0.4)

= £1,200,000 - £0.12 b = £800,000

b = 400,000 / 0.12 = £ 3,333,333

This question is hard BTW!

John

 

[s16455441]

Thanks very much John - this makes sense