Consider two binomial assets, each with price equal to 1.

The first is a stock that at the end of a period pays either 3 or 0 with probabilities \(p\) and \(1-p\), respectively. The second asset is risk-free and pays \(R\) at the end of the period.

(a) A portfolio is defined by investing a portion \(\alpha\) of one's wealth in the stock and a portion \(1-\alpha\) in the risk-free asset. Find the log-optimal value of \(\alpha\).

(b) What is the risk-neutral probability for this situation?

(c) What is the value of \(\alpha\) if \(p=q\) ?