Suppose a stock pays dividends m times per year at evenly spaced times with annual yield q. (So each dividend payment is equal to q/m of the stock price and the payments are made at times t + 1/m, t + 2/m, . . . , where t is the current time.) Suppose the dividends are automatically reinvested in the stock.

(a) If you have 1 unit of stock at time t, how many units will you have 1/m years later when the first dividend is paid?

(b) If T t is an integer multiple of 1/m, use a replication argument to show that the forward price for the stock is F(t, T) = St(1 + q/m) m(T t) Z(t, T) . ()

(c) Compute the limit as m .

(d) Suppose m = 1 and T t = 0.5 (so T t is not an integer multiple of 1/m). Show that if () holds, then you can build an arbitrage portfolio. Verify the portfolio is an arbitrage portfolio.