Q$.$ Consider a two$-$period binomial model.

The stock's price $S$ is $$100.$ Every three months, it either goes up and gets multiplied by the factor $u=1\mathrm{.}2,$ or it goes down and gets multiplied by the factor $d=0\mathrm{.}8.$

The continuously compounded risk$-$free interest rate $r$ is $2$ percent per quarter.

There is an American put option with an exercise price of $$115,$ maturing in six months.

a$.$ What are the risk neutral probabilities?

b$.$ What is the price of the put option?