$2.$ Derive the single$-$period binomial model for a put option. Include a single$-$period example where: u $=1\mathrm{.}10,$ d $=0\mathrm{.}95,$ Rf $=0\mathrm{.}05,$ S$0=$ $$100,$ X $=$ $$100.$

$3.$ Assume ABC stock$$s price follows a binomial process, is trading at S$0=$ $$100,$ has u $=1\mathrm{.}10,$ d $=0\mathrm{.}95,$ and probability of its price increasing in one period is $0\mathrm{.}5($q $=0\mathrm{.}5).$

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

b$.$ What are the stock's expected logarithmic return and variance for $2$ periods and $3$ periods?

c$.$ Define the properties of a binomial distribution.