What is put option, European put option, American put option and fair present value?

Problem 2 The AIA stock is currently at $40 per share. At the end of each 1-month period for the coming 2 months, the stock price will either increase by 8% or decrease by 10%. This risk-free interest rate is 2% per period. (a) Consider a put option with strike price 941 and expiration date at 2 months from now. (1) If the put option is a European put option, what is its fair present value? (ii) If the put option is an American put option, what is its fair present value? (b) Consider a European-style call option called the binary option, with strike price $35 and expiration date at 2 months from now. On the expiration date, if 1 the stock price happens to be larger than the strike price, then the owner of the binary option will receive $10. Otherwise, on the expiration date, if the stock price happens to be equal or lower than the strike price, then the owner of such an option will receive nothing. Compute the fair present value of this binary option