## Question

(a) Assume that the interest rate follows one-period binomial tree process such that the current interest rate can go up to 3.25% or down to 1.50% in 6 months. What will happen to the price of 1-year bond in each scenario?

(b) Consider an interest rate option, which matures in 6 months, with a payoff equal to, where is the strike rate of 2%, and denotes the interest rate in 6 months.

Show a portfolio of zero coupon bonds that replicates the payoffs of the interest rate option.

(c) Consider a swap that pays in 6 months the amount , where is the swap rate, and is 2%. Compute the value of the swap when the interest rate changes, and construct a replicating portfolio of bonds.

(d) Consider a bond option that matures in 6 months, where the underlying is a 6-month bond.

Assume that the strike price is $99.00. Compute the payoff of the bond option in 6 months and form a replicating portfolio of zeros.

(e) Now use the risk neutral probability to compute the value of the interest rate options in the question (b) above.

(f) Now use the risk neutral probability to compute the value of the swap in the question (c) above.

(g) Now use the risk neutral probability to compute the value of the bond option in the question (d) above.

2. Assume that the yields to maturity, continuously compounded, for zero coupon bonds maturing in half a year, one year, and one and half years from now are 0.85%, 0.90%, and 0.97%, respectively.

Assume that the current short rate, i.e. the rate on the 6-month bonds, follows a binomial stochastic recombining process such that it either increases or decreases 20% every six months with a 50% probability in either direction at every node. Now consider a risk-free zero coupon bond that pays a par value of $100 at the end of one year from now.

As a financial engineer, we are about to value a straddle with a strike price at $99.5928, which matures a year hence. Assume that the underlying asset is the bond maturing in one and half years from now. (Choose 1).

(a) Write the payoff function for this straddle.

(b) Based on the interest rate assumptions, value the derivative.

(c) How would you replicate this derivative?

3. THIS IS A MUST QUESTION TO ANSWER, AS ALL OTHER QUESTIONS BELOW WILL USE THE RESULTS FROM THIS QUESTION. Today, the prices of zero coupon bonds maturing in 6 months, 1 year and 1 ½ years are 99.30, 98.45, and 97.50, respectively. The standard deviation for changes in short rates is 1%. Assume that the expected changes in the short rate in 6 months and in one year are 0.6314% and 0.4474%, respectively. Construct the interest rate trees for two future 6-month time periods according to the Ho-Lee model.

4. Choose one of the following:

a. The Ho-Lee model often results in negative interest rate forecasting to be market consistent as in (4). How does the Black-Derman-Toy (BDT) model differ to remedy the negative interest rate problems? Hint: Show how BDT solves the Ho-Lee problem by way of a numerical example.

b. In both Ho-Lee and BDT models, do we allow investors to readjust their expectations about the future changes in short rates? Why or why not. Hint: Write down the interest stochastic model and explain.

5. Choose one of the following:

a. Compute all possible values of the price of one year bond in six months.

b. Verify if your expectation about the short rate, i.e. 0.6314% is in fact consistent with the today’s price of bonds maturing in one year by actually computing the price of the bond using your interest rate tree assumption.

6. Consider a bond that pays a 2% semiannual coupon maturing in one year. (Choose two from the following).

a. How much should the bond be valued today if the bond is not callable?

b. If the bond is callable on any interest date before it matures, what is the value of the option?

c. Value the bond if it is callable.

d. Show how you would replicate this callable bond and value the portfolio. Hint: You must show the exact replication numerically.

7. Answer two from below.

a. Suppose that you currently have a cap agreement at the strike rate of 1.35% on a continuously compounded rate basis. The agreement matures in one year. The interest rates are reset every 6 month whose settlement occurs only after the 6-month interest period. Assuming that your notional principal is $600k, what is the value of the cap?

b. Suppose that you enter into a swap agreement for a period of 1.5 years. Compute the swap rate.

c. Compute the value of the European swaption, if the derivative matures in 1 year.

d. Compute the value of the American swaption, which matures in 1 year but is exercisable in every 6 month.

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