There is a European put option on a stock that expires in two months. The stock price is $$76$ and the standard deviation of the stock returns is $60$ percent. The option has a strike price of $$80$ and the risk$-$free interest rate is an annual percentage rate of $5$ percent. What is the price of the put option today using one$-$month steps?

In the previous problem, assume that the exercise style on the option is American rather than European. What is the price of the option now? $($Hint: How will you find the value of the option if it can be exercised early? When would you exercise the option early?$)$