Buffelhead's stock price is $$238$ and could halve or double in each six$-$month period. Assume that you own a one$-$year American put option on Buffelhead stock with an exercise price of $$238.$ The interest rate is $19\%$ a year.

Would you ever want to exercise an American put early?

Calculate the value of the Buffelhead American put.

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

Calculate the value of the put if it had been European$-$style.

what is the value put