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2. Lucas tree economy: Consider the one-tree economy we studied in class with equilibrium stochastic discount factor M0,t = etvt, vt N(t,2t), > 0, > 0. Let us assume that the time-0 dividend (fruit) is Y0 = 1.
(a) Consider a so-called digital option, which pays out 1 at the given time, T > 0, if (and only if) the price of the economys stock (tree) pt > Z, for some Z > 0. Thus, if pT > Z, the owner of the option gets 1 at time T, and if pT < Z, the owner gets nothing. Denote the price at time t of the option by Dt. Using what you know about SDFs, derive an expression for the value of the option at time 0, D0.
(b) Now consider an asset that pays pT at time T. What is the value of this asset at time 0?
3. Equity premium puzzle: Consider a one-tree economy with expected log growth rate per year 2
of E[ln(Yt+1) ln(Yt)] = 1%, and annual variance of V ar[ln(Yt+1) ln(Yt)] = (2%) . The representative agent has CRRA utility, with risk aversion coefficient = 5 and personal discount rate = 2% per year.
(a) What is the expected return on the market in this economy?
(b) What is the risk-free rate in this economy?
(c) How well do these numbers match with historical returns in the stock market and bond markets?
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