Q. 4 (Derivatives pricing, 25 pts). In the lectures, we discussed pricing call and put options by means of the so-called replicating portfolios. The purpose of this exercise is to extend this pricing technique to more general derivative securities. Consider a one-period binomial model with initial stock price So, up factor u, down factor d, initial bond price Ao, bond return r, up probability p, and terminal time T. Assume that the market satisfies the principle of no-arbitrage. In addition to the risky stock and the riskless bond, let us consider a derivative security written on the stock, that is, its payoff Dr is given as a function of the stock price Sr at time T: Dr = f(Sr), where f : [0,+) + R is a function. Hence, if the stock price goes up at time T, then the payoff is Dr = f(uSo), and if the stock price goes down, then Dy = f( do). (For instance, f(x) = (K - x)+ for a put option with strike price K.) a. (5) Does there exist a replicating portfolio for this derivative? If yes, then find all such portfolios. b. (5) Let (x, y) be a replicating portfolio for this derivative. Using the principle of no-arbitrage, show that its price is given by Do = 2S, + y Ap. c. (5) Show that Do can be written as 1 Do 1+r (9f(uS) + (1 9)f(dSo)), for some 0