FIN 684 Professor Robert B.H. Hauswald Fixed-Income Analysis Kogod School of Business, AU Assignment 7 Please be reminded that you are expected to use contemporary computer software to solve the following problems. 1. Caplets. One period caps - call options on a spot rate or yield - are also known as caplets. If the caplet is written on a 6 month spot rate and matures in six months its payoff is given by ( ) [ ] 6 1 r 6 k + c := 12 2 1 + r N (1) 6 2 where r 6 r 6,12 is the simple six month interest rate at the option s maturity, k the strike and N the principal with f (x) = [x] + max {x, 0}. Consider the following 6M LIBOR Black-Derman-Toy spot rate lattice that was derived from actual market data so that the equivalent martingale (risk neutral) probabilities are q = 1 2 = 1 q (in %): t = 0 t = 6 12 r 0 = 5.20 r u = 5.96 r d = 5.17 (a) The lattice rates are continuously compounded rates while the payoff formula (1) is defined in terms of simple interest rates. Show that the payoff can be expressed as ( ) ( 6 c = 1 + k ) [ ( )] 6 + K P 12 2 12, 1 N where K := ( 1 + k 2 ) 1 and P ( 6 12, 1) = ( 1 + r 6 2 ) 1 is the price of a one year T-bill in 6 months time. (b) In terms of interpreting caps, what does the preceding derivation show? (c) Calculate the caplet s value at t = 0 for k = 5.40% and N = $10m. 2. Hedging Caps with Futures. The following 1M LIBOR spot rate lattice was generated under the Black-Derman-Toy methodology with equivalent martingale (risk neutral) probabilities q = 1 2 = 1 q (in %): r uu = 12.24 r u = 8.80 r 0 = 6.000 r ud = 8.21 r d = 5.90 r dd = 5.50
(a) Price a two year European interest rate caplet on the 1Y LIBOR struck at 8.70%. Since this is an [ interest ] rate option on a coupon the option s payoff at maturity is defined as r2,3 k + c (2) := 1+r 2,3 100 where r2,3 is the one year spot rate in two years time and k the cap rate. What is the caplet s value today at t = 0? (b) Suppose that there exists a two year futures contract written on a one year T-bill, i.e., the contract will mature in two years time and deliver a (then) one year T-bill. Determine the futures price at t = 0. (c) How would you hedge the cap at t = 0 using the T-bill futures contract and a 1Y T-bill (not the money market!!)? Assuming a face value of $100 derive the appropriate number of T-bills and T-bill futures contracts. (d) Price a 2 year cap on 1Y LIBOR struck at-the-money. 3. Hedging T-Bill Options. The below spot rate lattice was generated under the BDT methodology so that the equivalent martingale probabilities are q = 1 2 = 1 q. r uu = 12.24 r u = 8.80 r 0 = 6.000 r ud = 8.21 r d = 5.90 r dd = 5.50 (a) Calculate the following prices of T-bills and T-note strips with $100 face value at t = 0 : P (0, 1), P (0, 2), P (0, 3). (b) Calculate the value of a two year European call option on a 1Y T-bill for a strike of K = 94.50. (c) Calculate the replicating portfolio using 2 and 3 year T-note strips. Indicate the investments in each strip. 4. Black-Derman-Toy Lattice. Consider the following information reflecting current market conditions T-Strips Price Yield (%) Volatility (%) P (0, 1) 86.9358 14.00 7.14 P (0, 2) 75.5029 14.05 6.75 P (0, 3) 66.8527 13.42 and suppose that the conditions of the Black-Derman-Toy model hold, that time is measured in years and that the equivalent martingale probabilities are q = 1 2 = 1 q. (a) Determine the (recombining) one year spot rate lattice. (b) Calculate the one year futures price of a contract maturing at t = 2 written on a one year T-bill. (c) Calculate the one year forward price on the same T-bill from year 2 to year 3. How does the futures price compare to the forward price? 2
5. Replicating Floaters. Suppose that the evolution of the yield curve {r t,t : t = 0, 1, 2; t + 1 T 4} (in %) is as follows r uu (2, T ) : T = 3, 4 15.42 14.53 r u (1, T ) : T = 2,..., 4 12.38 11.84 11.37 r (0, T ) : T = 1,..., 4 9.42 9.22 9.03 8.85 r d (1, T ) : T = 2,..., 4 6.53 6.65 6.74 r ud (2, T ) : T = 3, 4 9.42 9.22 r dd (2, T ) : T = 3, 4 3.72 4.15 where all spot rates are annualized and time is measured in years. Consider a 2Y floating rate note maturing in t = 2 where the variable coupon rate in year t is equal to the annually compounded two year spot rate in year t 1. Notice that you do not need (and, given the information, actually can not) assume that the lattice s equivalent martingale probabilities are q = 1 2 = 1 q. Instead, one calculates the value of this floater as of t = 0 by replicating its cashflows with the zeros implied by the above tree. To solve this problem, construct a synthetic floater from the zeros that replicates the actual floater s cashflows at each node. You should proceed by using our usual replication argument working backward through the floater s tree toward t = 0. This is in complete analogy with our options calculations. Hence, you first calculate the floater s value at each node for t = 2, then you move to the t = 1 nodes (do not forget to discount!) and finally to t = 0. Since you can not appeal to the LEH you need to explicitly replicate the cashflows with 1Y and 2Y zeros, say. In order to do so, set up our usual two equations d u (1, 1) n 1 + d u (1, 2) n 2 = V u (1) d d (1, 1) n 1 + d d (1, 2) n 2 = V d (1) (2) where d s (1, 1) 1, s {u, d} and solve for the quantities of 1Y and 2Y strips n 1, n 2 to find V (0) as V (0) = d (0, 1) n 1 + d (0, 2) n 2. From this one then obtains V F RN (0) = d 1 n 1 + d 2 n 2! (a) Calculate the floater s value at t = 0. (b) What is the floater s sensitivity to interest rate changes, i.e., F RN? Here, you are asked to calculate a in complete analogy with an option s. 3
6. Inverse Floaters. With the spectacular success of FRNs several issuers such as the Student Loan Marketing Association ( Sallie Mae ) have taken the concept a step further and issued inverse floating rate notes better known as inverse floaters. Here, the variable coupon is inversely linked to a spot rate benchmark such as Treasuries or LIBOR: coupon payments are made in the form k r s for time period s. Hence, coupon rate rise when interest rates fall and vice versa. Notice, how one can naturally use caps and floors to limit interest rate risk on inverse floaters. Consider a two year inverse floater where k = 17% so that the variable coupon payments are k r t 1,t, t = 1, 2 where r t 1,t is the one year spot rate at t 1. (a) Value the inverse floater assuming that the current term structure is (in %) r 0,1 r 0,2 r 0,3 7.25 7.59 7.87 (b) Verify your answer to (a) by using the synthetic replication method sketched in the preceding question together with the following spot rate lattice: r uu (2, 3) 13.13 r u (1, T ) : T = 2, 3 10.16 10.18 r (0, T ) : T = 1, 2, 3 7.25 7.59 7.87 (c) Calculate the inverse floater s IF RN. r d (1, T ) : T = 2, 3 4.43 5.07 r ud (2, 3) 7.25 r dd (2, T ) 1.68 7. Inverse Floaters and Swaps. The relationship between floaters and swaps is quite obvious: a floater is one of the two underlying instruments of a swap. Hence, it should be quite intuitive that a similar relationship exists between inverse floaters and swaps. Recall that inverse floaters make variable coupon payments N (k r t 1,t ) at t for some fixed rate k. Suppose that P (0, 3) = 89.00 and that the k of a 3Y LIBOR inverse floater with annual resetting is set at k = 3.94265%. Also, suppose that a newly negotiated 3Y plain-vanilla 1Y LIBOR swap pays a fixed par swap rate of c = 3.94265%. What is the value of the inverse floater at t = 0? You do not need to use a replication argument that synthetically creates the inverse floater from the swap and the 3Y zero similar to equations (2): a simple cashflow diagram should be sufficient. 8. Protected Inverse Floaters. One of the problems of inverse floaters is that coupon rates could fall below zero so that the holder (investor) pays the issuer! In order to circumvent this unpleasantness, investment banks came up with the concept of protected inverse floating rate notes ( protected inverse floaters ) where the coupon never falls below zero so that the instrument pays N [k r t 1,t ] + at t for some fixed rate k. 4
(a) How would you replicate such a security using inverse floaters and spot rate (yield) options? (b) An investment bank offers a protected floater with k = 7.50%. At the same time, you could buy a 3Y arrears cap with a strike rate of k = 7.50% and annual reset for $1.075 per $100 face value. An arrears cap makes payments in arrears so that this one will only pay in t = 1, 2. Decomposing the protected inverse floater into its constituent parts (what are they?) derive its fair value. 9. Options on Swaps. A very widely used and traded instrument is an option on a swap called, surprise, surprise, a swaption. The swaption is in a new class of derivative instruments since it is a derivative of a derivative: the underlying instrument, a plain-vanilla swap, is itself a derivative! Otherwise, the option on a swap is priced with the same methods as before: set up a swap tree derived from the spot rate or zero price lattice, use a cashflow replication or equivalent martingale probability argument and calculate the option s value at each node recursively going backward in the lattice. However, since swaps have no value at inception by convention (this is how we calculate rates or prices!) the strike K 0, too. Consider a 5Y plain-vanilla interest rate swap where the variable coupon rate at t is equal to the annualized one year spot rate at t 1. Suppose A bought the swap (pays fixed). The evolution of the zero price curve is given by (a) Calculate the swap rate. P uu (2, T ) 88.69 78.16 68.51 P u (1, T ) 90.48 81.14 72.22 63.87 P (0, T ) 92.31 84.24 76.13 68.22 60.70 P d (1, T ) 94.18 87.46 80.25 72.87 P ud (2, T ) 92.31 84.24 76.13 P dd (2, T ) 96.08 90.80 84.59 (b) A swaption is nothing else but the option (right) but not the obligation to take over the position of A: it is the (call) option to pay fixed and receive floating for years t S + 1 where S is the option s expiration date. 5
Suppose the swaption is a one year European call to pay floating on the preceding swap. Calculate the swaption s value at t = 0. Derive the swap s value in the up and down states at t = 1 and then apply a synthetic replication argument for the swaption s value such as in (2) since you are not given the equivalent martingale probabilities of the zero lattice. (c) If the swaption were an American option, i.e., you could exercise at t = 0 or t = 1, would you pay more for this option than its European analog (think about exercise strategies based on your swaption value lattice in (b))? 10. Bonds with Embedded Options. Many bonds come with early repayment options for the issuer ( callables ) or the investor ( putables ) or other assorted options features. Consider a callable bond with two years to maturity remaining, a 15% annual coupon and a call schedule indicating call prices (strikes) of K 0 = 107.00 and K 1 = 106.00 in years t = 0, 1. If the issuer decides to call the bond she has to pay the call price and the coupon on the date the bond is called. Suppose that the evolution of the 1Y spot rate (in %) and the ex-coupon price 1 of a 2Y non-callable bond with a 15% annual coupon is described by the lattice below with no further information on the lattice s type available. Using a standard synthetic replication argument price the callable on an ex-coupon basis. First set up the ex-coupon bond price lattice, then adjust prices at nodes where the bond would have been called and finally use cashflow replication similar to (2) to price the bond. r uu = 13.01 P (2, 2) = 100 r u = 11.00 P u (1, 2) = 103.017 r 0 = 9.00 P (0, 2) = 109.90 r d = 7.00 P d (1, 2) = 107.23 r ud = 9.00 P (2, 2) = 100 r dd = 4.99 P (2, 2) = 100 1 Ex-coupon is the price the buyer (original issuer) pays for the bond buying it back; obviously, the seller (investor) will receive the ex-coupon price plus the accrued coupon. 6