Please solve the following questions step by step with equations and answers. A stock price is currently $$300.$ Over each of the next two $5-$month periods it is expected to go up by $11\%$ or down by $5\%.$ The risk$-$free interest rate is $7\%$ per

annum with continuous compounding.

$(1)$ The probability of an up movement in a risk$-$neutral world for a $5-$month time step is

$(2)$ The value of a $5-$month European call option with an exercise price of $$293$ using a binomial tree is

$(3)$ The value of a $10-$month European call option with an exercise price of $$293$ using a two$-$step binomial tree is

$(4)$ The value of a $10-$month American call option with an exercise price of $$293$ using a two$-$step binomial tree is