consider the 3-period binomial model, and assume that r=5% per year. t=1,2,3,4,5(years) t = 0, 1, 2, 3, S0 = 8, St+1 = 2Stor1/2St etc

(1) compute a price of a European call option with its maturity T=1 and strike price K=5

(2) derive a risk neutral probability q

(3) compute a price of a European put option with its maturity T=3, and strike price K=10.

(4) compute a price of an American put option with its maturity T=3, and strike price K=10.

(5) compute a price of a European up-and-out call option with its maturity T=3, strike price k=10,abd barrier B=32.

(6) generally, the following relationship holds among European(E), American(A), and European barrier(EB) options with the same underlying asset, maturity and strike price. A>=E>=EB Prove this relationship with backward inductions.