$1.$ A stock price is currently $$50.$ Over each of the next two $3-$month periods it is expected to go up by$6\%$ or down by $5\%.$ The risk$-$free interest rate is $5\%$ per annum with continuous compounding.What is the value of a $6-$month European call option with a strike price of $$51?2.$ The volatility of a non$-$dividend$-$paying stock whose price is $$78,$ is $30\%.$ The risk$-$free rate is $3\%$ per annum $($continuously compounded$)$ for all maturities. Calculate the following when a $2-$month time step is used:a$.$ What is the percentage up movement?b$.$ What is the percentage down movement?c$.$ What is the probability of an up movement in a risk$-$neutral world?d$.$ What is the probability of a down movement in a risk$-$neutral world?e$.$ What is the value a $4-$month European call option with a strike price of $$80$ given by a two$-$step binomial tree? Draw your binomial tree