A stock price is currently $$80.$ Over each of the next two six$-$month periods, it will either increase by $15\%$ or decrease by $10\%.$ The risk$-$free interest rate is $5\%$ per annum with continuous compounding.

$1)$ To use the $2-$step binomial tree model to calculate a option price on the stock, calculate p $($the risk$-$neutral probability$)$ for each step $($a six$-$month period$).2)$ What is the value of a one$-$year European call option with a strike price of $$80?$ Use no$-$arbitrage arguments $3)$ What is the value of a one$-$year American put option with a strike price of $$82?$ Use no$-$arbitrage arguments and the two$-$step binomial model.