Exercise 2 Consider the following discrete-time multiple period economy with a single repre- sentative agent. There is n

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Exercise 2 Consider The Following Discrete Time Multiple Period Economy With A Single Repre Sentative Agent There Is N 1 (410.26 KiB) Viewed 38 times

Exercise 2 Consider the following discrete-time multiple period economy with a single repre- sentative agent. There is no terminal period so the economy continues forever. The agent is endowed with one unit of an asset paying dividends Dt which follows from the recursion In D:+44 = In Dt + pAt +ov Atɛt+at, (1) where pl, o are constants and the noise terms Et+At have a mean of zero, a variance of 1, and are mutually independent for all t, and hence independent of Dt. All other assets are in zero-net supply. The agent's preferences are characterized by the utility function u(c) 1, with 7 > 0 and for y=1 she has log utility. In addition she has time-additive expected utility with a time preference parameter d. (a) Argue that in equilibrium, the agent's optimal consumption must be equal to the dividend of the asset, i.e. Ct = D for all t. (b) Show/argue that the relative state-price deflator induced by the agent's preferences and optimal consumption are given as e-8(st) (2) 2 over any time period (t, s]. (e) Find the equilibrium one-period risk-free rate R. To find the equilibrium price of the risky asset, we conjecture that it is given as P = DA for some constant A >0. (d) Argue that the price of the risky asset must follow the recursive expectation P = E ]z [ Sitat St (Petar + Detar) (3) (e) Show that the constant A > 0 is given as A= 1 (8-(1-7)-1(1-7)20%) At 1 and therefore we must have that 8- (1 - 7M - 4-5(1 – »)?o? > 0
(f) Argue that the one-period return on the risky asset is given as D++A+1+A Rt+At = Dt А (4) 3 Now consider a zero-net supply asset called the variance derivative introduced at time t, with a payoff at time T = t + 4At related to the realized one-period log returns of the risky asset over the following 4 periods, given as VI = Vt+4At = 10 000 t+4AL 1 4 s=t+At (In R) (5) (g) Argue that the time-t price of the variance derivative must be given as Ve = Et Er [vi St+4At St VT (6) (h) Estimate the time-t price of the variance derivative by simulating the 4-quarter economy M = 10000 times with Dų = 100, At = 1, u = 0.02, 0 = 0.05, 8 = 0.02 and for two different y = 2 and y = 10. Explain in detail how you simulate the economy and how you find the final random variable you are using in your Monte-Carlo estimation.