5 10.5 Assume ABC stock price follows a binomial process, is trading at \(S_{0}=\) \(\$ 100\), has \(u=1.10, d=0.95\), and probability of its price increasing in one period is \(0.5(q=0.5)\).

a. Show with a binomial tree ABC's possible stock prices, logarithmic returns, and probabilities after one period, two periods, and three periods.

b. What are the stock's expected logarithmic return and variance for each period?

c. Define the properties of a binomial distribution.

d. Verify that the \(u\) and \(d\) formulas yield the \(u\) and \(d\) values of 1.10 and 0.95 , given the logarithmic return's mean and variance after three periods:

\[\begin{aligned}
& u=e^{\sqrt{V_{e} / n}+\mu_{e} / n} \\
& d=e^{-\sqrt{V_{c} / n}+\mu_{e} / n}
\end{aligned}\]