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Financial Mathematics Old FM Forum

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Old 01-11-2009, 05:31 PM
Marid Audran's Avatar
Marid Audran Marid Audran is offline
 
Join Date: Nov 2008
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Question Reconciling two "prepaid forward" formulas; understanding "cost of carry"

Hi everyone --- I have two questions from Chapter 5 of McDonald.

The two "prepaid forward" formulas
Here are the two formulas for pricing prepaid forwards with dividends (McDonald, pp. 131-32):
With discrete dividends:

With continuous dividends:
I understand each of these formulas on its own, but I am having a hard time reconciling them. I did find a forum discussion about why the risk-free rate is omitted from the continuous-dividend formula, but that's not the reconciliation I'm looking for.

Here's as far as I got: I did some figuring using the approximation
(this seems to be the gist of the "daily dividend" formula on Page 132), which led me to
and ultimately (after substituting and simplifying) to
. Is this accurate/meaningful? Is there some other way of understanding the two prepaid-forward price formulas simultaneously? Am I barking up the wrong tree?

Understanding "cost of carry"
I don't understand McDonald's discussion of this concept on Page 141:
Quote:
Suppose you buy a unit of the index that costs S and fund the position by borrowing at the risk-free rate. You will pay rS on the borrowed amount, but the dividend yield will provide offsetting income of . You will have to pay the difference, , on an ongoing basis. This difference is the net cost of carying a long position in the asset; hence, it is called the "cost of carry."
This discussion seems to ignore the fact that the stock price changes over time. Also, further down, I'm a little confused about Formula (5.13):
Forward price = Spot price + Interest to carry the asset - Asset lease rate,
where the last two terms on the right-hand side are grouped together as "cost of carry." This formula seems to apply directly to the discrete-dividends case, but I don't see how to apply it directly to the continuous-dividends case. Is there a clearer interpretation?

I will sip coffee while you discuss TIA
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