Exercise 1:

You have estimated the risk neutral interest rate tree for the continuously compounded interest rate as in the table above. There is equal risk neutral probability to move up or down the tree and each interval time represents 6 months, that is, = 0.5.
Part I:
1. Compute the discount factors for all maturities.
2. Compute the price of a security that pays \$100 at time i = 2 if the continuously
compounded interest rate at that time is more than 7.00% and zero otherwise.
3. Compute the price of a 1.5-year cap, with semi-annual payment, strike rate rK = 5%,
and notional N = 100.
Part II:
Answer the questions of Part I using the Monte-Carlo Simulations approach. You have to precisely explain your reasoning and to show the standard error, as well as the confidence interval. Compare the results obtained in both parts.
Exercise 2:
EF5157
2

You have estimated the risk neutral interest rate tree for the continuously compounded interest rate as in the table above. There is equal risk neutral probability to move up or down the tree and each interval time represents 1 year, that is, = 1.
1. a) Compute the value of a non-callable bond with principal N = 100, maturity i = 3, and annual coupon rate of 5.00%.
b) Compute the value of a similar bond that is callable starting i = 1. c) Which of the two bonds is more expensive for investors? Why?
2. Consider a 2-year mortgage with annual payment, \$30, 000 face value, and a contin- uously compounded mortgage rate of 5.25%.
a) If there is no prepayment, what is the annual payment?
b) Given the annual payment, compute the principal and interest payment each period (i.e. i = 1 and i = 2). Also, compute the value of the mortgage without prepayment option along the tree.
c) Compare the value of the mortgage at each node with the principal outstanding. What is the option value? When is it optimal to exercise the option?
N.B.: You cannot exercise at i = 0.
d) Compute the value of the mortgage with prepayment. What is the homeowner really paying?
e) How would you calculate the duration of the mortgage? Give its value.
Exercise 3:
Suppose you are an investor who invested in callable bonds. In particular, your bond portfolio is long in 3.5%, 10-year AAA rated corporate coupon bonds (par = 500 million), which will become callable in exactly three years.
The discount factors are the following:

T Z(t,T) T Z(t,T) T Z(t,T) T Z(t,T) 0.5 0.9738 3 0.8598 5.5 0.7574 8 0.6646 1 0.9484 3.5 0.8385 6 0.7380 8.5 0.6473 1.5 0.9250 4 0.8178 6.5 0.7190 9 0.6303 2 0.9031 4.5 0.7973 7 0.7005 9.5 0.6137 2.5 0.8811 5 0.7772 7.5 0.6823 10 0.5975
1. Assume = 18.22%, and use the discount curve to fit a simple BDT tree with semi- annual steps up to T = 10. You should report the interest rate tree and explain the methodology used to construct it.
N.B.: You can use the file BDT BinomialTree Macro available on Canvas