1. A company buys a caplet today (time 0) with maturity T = 1 year and a strike rate of k = 3 percent on a notional of LN = $200 million. Suppose the six-month bbalibor rate realized after one year is 4.5 percent. Assume that there are 181 days in this six-month period and the year has 365 days. what is the caplet's payoff after 1.5 years?
2. A company buys a caplet today (time 0) with maturity T = 1 year and a strike rate of k = 3 percent on a notional of LN = $200 million. Suppose the six-month bbalibor rate realized after one year is 4.5 percent. Assume that there are 181 days in this six-month period and the year has 365 days. Suppose that after one year, the price of a zero-coupon bond that matures after six months is worth $0.9780. What is the caplet's payoff after one year?
3. Calculate the price of an option that caps the three-month rate, starting in 15 months' time, at 13% (quoted with quarterly compounding) on a principal amount of $1,000. The forward interest rate for the period in question is 12% per annum (quoted with quarterly compounding), the 18-month risk-free interest rate (continuously compounded) is 11.5% per annum, and the volatility of the forward rate is 12% per annum.
4. Suppose that the LIBOR yield curve is flat at 8% with annual compounding. A swaption gives the holder the right to receive 7.6% in a five-year swap starting in four years. Payments are made annually. The volatility of the forward swap rate is 25% per annum and the principal is $1 million. Use Black's model to price the swaption.
5. Suppose that the short rate is currently 4% and its standard deviation is 1% per annum. What will standard deviation be if the short rate increases to 8% in assuming the interest rate follows the Cox, Ingersoll, and Ross model.
6. Suppose that kappa = theta = 0.1 and in both the Vasicek and the Cox, Ingersoll, Ross model. In both models, the initial short rate is 10% and the initial volatiltiy of the short rate change 0.02 (i.e. in time t the s.d. of the short rate is 0.02 sqrt(t).
Calculate the prices given by the models for a zero-coupon bond that matures in year 10 and give the absolute difference (i.e. a positive number) of the two prices.