Answer the following questions concerned with the alternative procedures for constructing trees in Section 21.4.

(a) Show that the binomial model in Section 21.4 is exactly consistent with the mean and variance of the change in the logarithm of the stock price in time Δt.

(b) Show that the trinomial model in Section 21.4 is consistent with the mean and variance of the change in the logarithm of the stock price in time Δt when terms of order (Δt)2 and higher are ignored.

(c) Construct an alternative to the trinomial model in Section 21.4 so that the probabilities are 1/6, 2/3, and 1/6 on the upper, middle, and lower branches emanating from each node. Assume that the branching is from S to Su, Sm, or Sd with m2 = ud. Match the mean and variance of the change in the logarithm of the stock price exactly.