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 tand t+dtis assumed to be S(t)dt, where S(t) denotes the price of one share of the stock at time t, t0. (Note that McDonald also writes S(t) as St.) 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 etshares at time t, t0. A calculus proof of this fact is as follows. Let n(t) denote the number of shares of the stock at time tunder this immediate reinvestment policy. Thus, n(0) = 1. Because the additional number of shares purchased between time tand t+dtis n(t+dt) −n(t) = dn(t), we have n(t)S(t)dt= S(t)dn(t), or tddn(t) = n(t). Rewriting the last equation as tddln[n(t)] = integrating, exponentiating, and applying the condition
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