Financial Engineering – IC302
Autumn Term 2020/1
Seminar 6: Interest Rate Options Questions
1. Consider a company that borrows $100 million in a 5-year loan on which they pay interest each quarter at an annual rate equal to 3-month USD LIBOR plus 20 basis points. They decide to protect themselves against an increase in interest rates by purchasing a 5-year cap on 3-month USD LIBOR.
(a) On what notional amount will they purchase the cap?
(b) How many caplets will the cap contain?
(c) Suppose that the cap strike is 2%. If 3-month USD LIBOR for one of the forward periods covered by the cap fixes at 2.5%, approximately what payoff will the company receive for the caplet that covers that period?
[For the purpose of this question, you may assume that the length of the forward period is exactly one-quarter of one year.]
(d) Suppose that 3-month USD LIBOR for a different forward period covered by the cap fixes at 1.5%. Approximately what payoff will the company receive for the caplet that covers that period?
[For the purpose of this question, you may assume that the length of the forward period is exactly one-quarter of one year.]
(e) Suppose that the amortized cost of the cap is 50 basis points per year (i.e. the upfront cost of the cap is equivalent to 50 basis points per year when spread out over the 5-year life of the cap). What is the maximum annual interest rate the company will pay in any quarter after taking into account the interest its pays on the loan, any payoffs it receives from the cap, and the amortized cost of the cap?
2. Use the extension to Black’s model described in the lecture (see equation (20) in lecture 6) to value a European swaption that gives the holder the right to enter into a 3-year annual-pay swap in four years’ time in which a fixed rate of 5% is paid and LIBOR is received. Assume that the LIBOR/swap yield curve is used for discounting, that this curve is flat at 5% per annum with annual compounding, and that the volatility of the swap rate is 20%. The swaption notional amount is $10 million.
3. How would the value of the swaption in the previous question change if discounting were done at OIS rates, with all swap rates 5% and all OIS rates 4.7%, all with annual compounding?
4. Would you expect a swaption to be cheaper than a cap (for payer swaptions) or floor (for receiver swaptions) that covers the same set of forward LIBORs? Explain your answer.
5. The Refinitiv Eikon screen image below shows pricing information for an at-the- money 2y5y European payer swaption:
(a) Suppose that the 2y5y forward starting swap rate were to increase to 1.4110%. At this new level of the swap rate, will the at-the-money 2y5y swaption with strike rate 1.3110% shown here be in the money, at the money, or out of the money?
(b) What effect would you expect the change in the swap rate in part (a) to have on the mark-to-market value of a long position in the swaption?
(c) Suppose that the 2y5y swaption shown in the Refinitiv Eikon screen image were trading at a price of 200 bp rather than 177.6357 bp. Other things remaining equal, would its implied normal volatility be higher or lower than 66.98 bp?
(d) As we have seen, the ATM swaption was trading at an implied volatility of approximately 67 bp. The Refinitiv Eikon screen image below shows part of the implied volatility cube for European swaptions on this date. Based on
this information, at roughly what implied normal volatility would you expect a 2y5y swaption with a strike 50 bp lower to trade?