Index Arbitrage

[angelo]

[Solved] Thanks everybody.
Hi everybody,

I have the following question:

My lecturer gave me the following answer, which I think it's wrong:

The problem I think he did wrong for the pay-off of the long stock position. In my opinion, after 40 days, the number of stocks has to grow to: 45,336 * exp(0.030 * 40 / 365) = 45485 (Nt) due to share repurchase (0.030 here is the dividend yield). So the pay-off should be:
Current Share Price (St) * Number of stocks (Nt) - Money Borrowed ($20 million * exp(0.035 * 40/365) = $439 * 45485 - $20.0769 million = - $108,944 which is equal to: his result * exp(-0.03 * 40 / 365)

Could everybody clarify?

Thanks a lot!

 

[angelo]

That has been taken into account while you have calculated your cost of the position... You have subtracted dividend yield from the interest you paid... So you can not take the benefit of dividend yield again by applying that in your stocks.

Hi I know that, but the problem is I can mathematically derive the result by using share repurchase, but I can not derive the cost of the position (S0 * exp((r - q)* T)) (q is dividend yield) so I do not understand how that (the yield being subtracted from the interest) happens?

 

[darth]

where did you get the figure of 3.5% that you have used for calculating the FV of money borrowed.

Current Share Price (St) * Number of stocks (Nt) - Money Borrowed ($20 million * exp(0.035 * 40/365) = $439 * 45485 - $20.0769

 

[angelo]

where did you get the figure of 3.5% that you have used for calculating the FV of money borrowed.

Sorry, the figure in the question should be 3.5% not 3.2%. And I figured it out:

Divide one year to m intervals with 1/m year long for each. At the first interval the money you owed grows to S0 * (1 + r/m) but you use the dividend to pay back part of the debt so you only owed S0 * (1 + r / m - q / m). After one year, you will owe S0 * (1 + (r-q)/m)^m. When m is infinity the money you owed for one year grows to S0 * exp(r-q).

Is is correct?
Thanks.

 

[darth]

yes , you are right .... that's the limiting case .