Valuation on a Multiplicative Binomial Lattice

This problem reviews some of the main ideas of valuation on a binomial lattice and the properties of put and call options. You may wish to review the relevant lecture material and readings.

Suppose that the price of a share of KAF stock is S(0) = 120 in period 0. At the beginning of period 1, the price of a share can either move upward to S(1) = u S(0) or downward to S(1) = d S(0). Suppose that u = 4/3 = 1.333 and d = 3/4 = .75, so that S(1) = u S(0) = 160 after an up move and S(1) = d S(0) = 90 after a down move. Suppose that the probability of an up move is p = 0.5.

Similarly, suppose that, at the beginning of period 2, the share price either moves up or down by the same multiplicative factors and with the same probability (0.5) of an up move. (If the probability of an up move in a period is 0.5, then the probability of a down move in a period is also 0.5.) Hence, if the share price in period 1 is S(1), then the share price at the beginning of period 2 is either S(2) = u S(1) = 4/3 S(1) or S(2) = d S(1) = 3/4 S(1).

For simplicity, suppose that a period is a year, and let the riskless interest rate be r = .12, that is, 12% per period.

3. (i) Using the replicating portfolio you determined in question 3, calculate the numerical value of d, the 1-period-ahead down state price for the state where the price of a share of KAF stock makes a down move from period 0 to period 1. Also calculate the numerical value of u, the 1-period-ahead up state price for the state where the price of a share of KAF stock makes an up move from period 0 to period 1. Explain your reasoning.

(ii) Replicating portfolios, risk neutral probabilities, and state prices can each be used to calculate the market value in period 0 of any asset or portfolio of assets with payoffs in period 1 that are determined by the movement in the price of a share of KAF stock. Write down the formulas that connect the relevant risk neutral probabilities for period 1, pu and pd, and the 1-periodahead state prices, u and d. What are the numerical values of the risk neutral probabilities, pu and pd ?