Please solve the questions step by step with equations and answers.A stock price is currently $$200.$ During each $4-$month period for the next $8$ months it will increase by $12\%$ or reduce by $10\%.$ The risk$-$free interest rate is $10\%$ per

annum with continuous compounding. Consider evaluating derivatives that pay off $\frac{1}{4}|S_t-200|$ at the end of $t$ months, where $S_t$ is the stock price in $t$ months.

$(1)$ The probability of an up movement in a risk$-$neutral world for a $4-$month time step is

$(2)$ The value of a European derivative with a maturity of $4$ months given by a binomial tree is

$(3)$ The value of a European derivative with a maturity of $8$ months given by a two$-$step binomial tree is

$(4)$ The value of an American derivative with a maturity of $8$ months given by a two$-$step binomial tree is