Answer the following questions concerned with the alternative procedures for construct- ing trees in Section 20.4:

(a) Show that the binomial model in Section 20.4 is exactly consistent with the mean and variance of the change in the logarithm of the stock price in time At.

(b) Show that the trinomial model in Section 20.4 is consistent with the mean and variance of the change in the logarithm of the stock price in time At when terms of order (At) and higher are ignored.

(c) Construct an alternative to the trinomial model in Section 20.4 so that the prob- abilities 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 m = ud. Match the mean and variance of the change in the logarithm of the stock price exactly.