Interest rate barrier options: An interest rate barrier option is a regular option whose payoff at maturity depends on whether or not a certain level of interest rate has been touched during the life of the option. For instance, a down-and-out option is an interest rate option that expires if before maturity the interest rate hits a given barrier. Let the current term structure of interest rates be flat at the continuously compounded 5% rate. Consider an at-the-money, down-and-out call option with $100 million notional and one year to maturity. The barrier is set at r = 3%. You also know that a 6-month regular call option on the 1-month interest rate trades at C(l/12) = 2.6505 (for 100 of notional).

(a) Fit the Ho-Lee interest rate tree with monthly steps to the term structure of interest rates. Make sure that the tree can price (approximately) the simple 6-month option described above.
(b) Use Monte Carlo simulations to compute the value of the down-and-out option. What goes wrong if you try to compute the value of the down-and-out call option by using the backward methodology?
(c) Fit a simple BDT interest rite model and do the same exercise. Is the price the same? Discuss.