Question $2(20$ points$)$

A binomial model is used to explain the changes in the price of a stock. The current

spot price is $$100.$ The $u$ factor is $1\mathrm{.}12$ and $d$ is $0\mathrm{.}87$

Calculate all possible values of the Stock price at the end of $t=3$

How many times each value $($from $1)$ will show up at the end of $t=3?$

Calculate the probability for each value in $(1).$ Explain the calculation.

If the binomial tree is extended to $12$ periods, what is the minimum value of

up movements to exercise a call option in the stock with a Strike value of

$$181\mathrm{.}6?$