a) Write the payoff functions for the European and American up-and-in barrier put option with barrier B.

b) Use the 4-step binomial tree to price the up-and-in barrier put options (European style and American style, separately). Assume that the spot price is $105. The strike is $101. The barrier is $106. The time to maturity is 1 year. Risk-free rate is 5% with the annual compounding, the stock price goes up or down by 20%.

c) Is the European up-and-in barrier put option priced higher or lower than the same option wihout a barrier?