Consider an American Put option, having exercise price $10$ euro and maturity $1$ year.

The underlying asset price follows a Binomial model, at each semester the price moves up by

$40\%$ or moves down by $60\%,$ with historical probability equal to $50\%$ in each semester. The

current price is S$0=10$ euro. The risk free rate is $4\%.$

i$)$ Determine if the early exercise is convenient.

ii$)$ Compute the current price of the American Put option.

iii$)$ If the $($historical$)$ probability of increasing $($resp$.$ decreasing$)$ is equal to $80\%($resp$.20\%),$

which is the option price? Justify your answer.