Consider a two-period arbitrage-free economy where the resolution of uncertainty is illustrated in the following binomial tree.

t = 0 t = 1 t = 2 1

0.9 0.95 0.95

(2; 8)

(8; 2)

(6; 10)

(10; 4) (10; 6)

(4; 10) 0.85 0.9 Each branch in the tree has a conditional probability of 1 2 . Assets in the economy are priced by a state-price deflator ζ = (ζt)t∈{0,1,2}. The numbers along the branches show the possible values of the state-price deflator over that period, that is ζ1/ζ0 over the first period and ζ2/ζ1 over the second period. The pair of numbers written at each node shows the dividend payments of asset 1 and asset 2, respectively, if that node is reached. For example, if the up-branch is realized in both periods, then asset 1 will pay a dividend of 10 and asset 2 a dividend of 6 at time 2.

(a) For each of the two assets compute the following quantities in both the up-node and the down-node at time 1: (i) the conditional expectation of the dividend received at time 2, (ii) the ex-dividend price, and (iii) the expected net rate of return over the second period.

(b) For each of the two assets compute the following quantities at time 0: (i) the expectation of the dividend received at time 1, (ii) the price, and (iii) the expected net rate of return over the first period.

(c) Compare the prices of the two assets. Compare the expected returns of the two assets.
Explain the differences.

(d) Is it always possible in this economy to construct a portfolio with a risk-free dividend over the next period? If so, find the one-period risk-free return at time 0 and in each of the two nodes at time 1.

(e) Is the market complete? Explain!