3. For an option on a non-dividend paying stock, we know that in a one-period binomial...

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3. For an option on a non-dividend paying stock, we know that in a one-period binomial model, the price of the option is given by where f" and fd P* = r-d u-d' price = (1+r)(p.f" + (1 p.) fd) are the option payoffs on the up and down nodes, respectively, and Suppose now that the stock pays dividends continuously at rate & per period and that the dividends are reinvested to purchase additional shares of stock. Assume that the risk-free rate of interest is r per period with continuous compound- ing. (We usually use periodic compounding in the binomial and other discrete-time models, but when dividends are paid continuously, it's convenient to use continuous compounding.) In this problem, you will show that when the stock pays dividends continuously, the option price is given by price e (p.f" + (1 p.) d) - where f" and fd are defined as above and p. is the risk-neutral probability modified to incorporate the dividends. 2 (a) (10 points) Let x be the number of shares of stock purchased at time 0 and let y be the amount invested in the risk-free asset. Compute x and y to replicate the option payoff. Your expressions for x and y should be in terms of u, d, r, 8, fu, fd, and S(0). Hint: Recall that when dividends are paid continuously and reinvested, one share of stock at time 0 will grow to e shares at time 1. (b) (5 points) Compute the time 0 price of the replicating portfolio. This will be the price of the option. (c) (5 points) If you did not already do so, rearrange your expression in Problem 3b so that it's in the form price e (p.f" + (1 p.) ). - Write an expression for p. in terms of r, 8, u, and d. 3. For an option on a non-dividend paying stock, we know that in a one-period binomial model, the price of the option is given by where f" and fd P* = r-d u-d' price = (1+r)(p.f" + (1 p.) fd) are the option payoffs on the up and down nodes, respectively, and Suppose now that the stock pays dividends continuously at rate & per period and that the dividends are reinvested to purchase additional shares of stock. Assume that the risk-free rate of interest is r per period with continuous compound- ing. (We usually use periodic compounding in the binomial and other discrete-time models, but when dividends are paid continuously, it's convenient to use continuous compounding.) In this problem, you will show that when the stock pays dividends continuously, the option price is given by price e (p.f" + (1 p.) d) - where f" and fd are defined as above and p. is the risk-neutral probability modified to incorporate the dividends. 2 (a) (10 points) Let x be the number of shares of stock purchased at time 0 and let y be the amount invested in the risk-free asset. Compute x and y to replicate the option payoff. Your expressions for x and y should be in terms of u, d, r, 8, fu, fd, and S(0). Hint: Recall that when dividends are paid continuously and reinvested, one share of stock at time 0 will grow to e shares at time 1. (b) (5 points) Compute the time 0 price of the replicating portfolio. This will be the price of the option. (c) (5 points) If you did not already do so, rearrange your expression in Problem 3b so that it's in the form price e (p.f" + (1 p.) ). - Write an expression for p. in terms of r, 8, u, and d.