Exercise 3 (1 mark for each part). Suppose a stock pays dividends m times per ye
ID: 3167816 • Letter: E
Question
Exercise 3 (1 mark for each part). 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) = Stat q/m)-m(T-t) Z(t,T) (c) Compute the limit as n oo (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 portfolioExplanation / Answer
(a) q/m*m is the stock units you will have at q/m years later which will come out to be q..
(b) since (T-t) is an integer multiple of 1/m, so we will get an integer and q/m is the stock price dividend so, it will be an increase in the stock price as given in the power of integer which will be deducted and result will be integer which will be negative. Z(t,T)*F(t,T)/St will be achieved this will your forward price or change in price movement from before.
c.) Since infinity is zero divided by zero, so in short there will be no movement in dividend and dividend payment will also tend to zero.
d.) There will be arbitrage but to the extent of the value of q so if dividend paid is 5 of stock price which is some percent of stock, then, we get (1+5)^-.5 which will give earning potential of .40828
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