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2011Dividend_yield - Dividend yield On page 132 of...

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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 Black-Scholes option-pricing formula (12.1). It is indicated on page 132 that, if all dividends are re-invested 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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