Dividend yield
On page 132 of McDonald (2006), it is assumed that stock dividends are paid
continuously at a rate proportional to the stock price.
More precisely, for each share of the stock,
the amount of dividends paid between time
t
and
t
+d
t
is assumed to be
S
(
t
)
d
t
, where
S
(
t
)
denotes the price of one share of the stock at time
t
,
t
0.
(Note that McDonald also writes
S
(
t
)
as
S
t
.)
This is not exactly a reasonable assumption for stock dividends, but it is needed to obtain
the important BlackScholes optionpricing formula (12.1).
It is indicated on page 132 that, if all dividends are reinvested immediately, then one
share of the stock at time 0 will grow to
e
t
shares at time
t
,
t
0.
A calculus proof of this fact is
as follows.
Let
n
(
t
) denote the number of shares of the stock at time
t
under this immediate
reinvestment policy.
Thus,
n
(0) = 1.
Because the additional number of shares purchased
between time
t
and
t
+d
t
is
n
(
t
+d
t
)
−
n
(
t
) = d
n
(
t
), we have
n
(
t
)
S
(
t
)
d
t
=
S
(
t
)d
n
(
t
),
or
t
d
d
n
(
t
)
=
n
(
t
)
.
Rewriting the last equation as
t
d
d
ln[
n
(
t
)]
=
integrating, exponentiating, and applying the condition
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 Spring '08
 Tang,Q
 Dividend, McDonald

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