Bond Evaluation, Selection, and Management, Second Edition by R. Stafford Johnson Copyright 2010 R. Stafford Johnson PPENDIX H Pricing Interest Rate Options with a Binomial Interest Rate Tree In Chapter 14, we examined how the binomial interest rate model can be used to price bonds with embedded call and put options, sinking fund arrangements, and convertible clauses, and in Chapter 15, we looked at two approaches to estimating the tree. In this appendix, we show how the binomial interest rate tree can be used to price interest rate options. H.1 VLUING T-BILL OPTIONS WITH BINOMIL TREE Figure H.1 shows a two-period binomial tree for an annualized risk-free spot rate (S) and the corresponding prices on a T-bill (B) with a maturity of years and face value of $100 and also a futures contract (f ) on the T-bill, with the futures expiring at the end of period 2. The length of each period is six months (six-month steps); the upward parameter on the spot rate (u) is 1.1 and the downward parameter (d) is 1/1.1 0.9091; the probability of the spot rate increasing in each period is ; and the yield curve is assumed flat. s shown in the figure, given an initial spot rate of 5% (annual), the two possible spot rates after one period (six months) are 5% and 44545%, and the three possible rates after two periods (one year) are 6.05%, 5%, and 4.13223%. t the current spot rate of 5%, the price of the T-bill is B 0 98.79 [ 100/ ]; in period 1, the price is 98.67 when the spot rate is 5% [ 100/(1.055) ] and 98.895 when the rate is 44545% [ 100/(1.0454545) ]. In period 2, the T-bill prices are 984, 98.79, and 99 for spot rates of 6.05%, 5%, and 4.13223%, respectively. The futures prices shown in Figure H.1 are obtained by assuming a risk-neutral market. If the market is risk neutral, then the futures price is an unbiased estimator of the expected spot price: f t E(S T ). The futures prices at each node in the exhibit are therefore equal to their expected price next period. Given the spot T-bill prices in period 2, the futures prices in period 1 are 98.665 [ E(B) (984) + (98.79)] and 98.895 [ E(B) (98.79) + (99)]. Given these prices, the current futures price is f 0 98.78 [ E(f 1 ) (98.665) + (98.895)]. Given the binomial tree of spot rates, prices on the spot T-bill, and prices on the T-bill futures, we can determine the values of call and put options on spot and futures T-bills. For European options, the methodology for determining the price is to start at 753
754 PPENDIX H: PRICING INTEREST RTE OPTIONS WITH BINOMIL INTEREST RTE TREE S nnualized spot rate u 1.1, d 1/1.1.9091, n 2, length of period years B Price of T-bill, M, F $100 f Price on T-bill futures, expiration one year (n 2) u 2 6.05% Buu 100/(1.0605) 984 fuu Buu 984 u 5% Bu 100/(1.055) 98.67 fu (984)+(98.79) 98.665 5.00% Bu 100/ 98.79 f0 (98.67) +(98.895) 98.78 d ud 5.00% Bud 100/ 98.79 fud Bud 98.79 ds 0 44545% Bd 100 /(1.0454545) 98.895 fd (98.79)+(99) 98.895 2 d d 4.13223% Bdd 100/(1.0413223) 99 fdd Bdd 99 FIGURE H.1 Binomial Tree of Spot Rates, T-Bill Prices, and T-Bill Futures Prices expiration where we know the possible option values are equal to their intrinsic values, IVs. Given the option s IVs at expiration, we then move to the preceding period and price the option to equal the present value of its expected cash flows for next period. Given these values, we then roll the tree to the next preceding period and again price the option to equal the present value of its expected cash flows. We continue this recursive process to the current period. If the option is merican, then its early exercise advantage needs to be taken into account by determining at each node whether or not it is more valuable to hold the option or exercise. This is done by starting one period prior to the expiration of the option and constraining the price of the merican option to be the maximum of its binomial value (present value of next period s expected cash flows) or the intrinsic value (i.e., the value from exercising). Those values are then rolled to the next preceding period, and the merican option values for that period are obtained by again constraining the option prices to be the maximum of the binomial value or the IV; this process continues to the current period. Spot T-Bill Call ppose we want to value a European call on a spot T-bill with an exercise price of 98.75 per $100 face value and expiration of one year. To value the call option on the T-bill, we start at the option s expiration, where we know the possible call values are equal to their intrinsic values, IVs. In this case, at spot rates of 5% and 4.13223%, the call is in the money with IVs of.04 and, respectively, and at the spot rate of 6.05% the call is out of the money and thus has an IV of zero (see Figure H.2).
ppendix H: Pricing Interest Rate Options with a Binomial Interest Rate Tree 755 Given the three possible option values at expiration, we next move to period 1 and price the option at the two possible spot rates of 5% and 44545% to equal the present values of their expected cash flows next period. ssuming there is an equal probability of the spot rate increasing or decreasing in one period (q ), the two possible call values in period 1 are.01947 and.1418: C u C d (0) + (.04).01947 (1.055) (.04) + ().1418 (1.0454545) Rolling these call values to the current period and again determining the option s price as the present value of the expected cash flow, we obtain a price on the European T-bill call of.0787: C 0 (.01947) + (.1418).0787 If the call option were merican, its two possible prices in period 1 are constrained to be the maximum of the binomial value (present value of next period s expected cash flows) or the intrinsic value (i.e., the value from exercising): C t Max[C t, IV] In period 1, the IV slightly exceeds the binomial value when the spot rate is 44545%. s a result, the merican call price is equal to its IV of.145 (see Figure H.2. Rolling this price and the upper rate s price of.01947 to the current period yields a price for the merican T-bill call of.08. This price slightly exceeds the European value of.0787, reflecting the early exercise advantage of the merican option. Futures T-Bill Call If the call option were on a European T-bill futures contract instead of a spot T-bill, with the futures and option having the same expiration, then the value of the futures option would be the same as the spot option. That is, at the expiration spot rates of 6.05%, 5%, and 4.13223%, the futures prices on the expiring contract would be equal to the spot prices (984, 98.79, and 99), and the corresponding IVs of the European futures call with an exercise price of 98.75 would be 0,.04, and the same as the spot call s IV. Thus, when we roll these call values back to the present period, we end up with the price on the European futures call of.0787 the same as the European spot. If the futures call option were merican, then the option prices at each node need to be constrained to be the maximum of the binomial value or the futures option s IV. Since the IV of the futures call in period 1 is zero when the spot rate is 5% (IV Max[98.665 98.75, 0] 0) and.145 when the rate is 44545% (IV Max[98.895 97.75, 0].145), the corresponding prices of the merican futures option would therefore be the same as the spot option:.01947 and.145. Rolling
756 PPENDIX H: PRICING INTEREST RTE OPTIONS WITH BINOMIL INTEREST RTE TREE S nnualized spot rate u 1.1, d 1/1.1.9091, n 2, Length of period years B Price of T-bill, M, F $100 C Price on call option on spot T-bill, X 98.75, Expiration, n 2 5%, Bu 98.67 e (0)+(.04) Cu.01947 (1.055) IVu Max[98.67 98.75,0] 0 Cu Max[IV,Cu].01947 5%, B0 98.79 e (.01947) +(.1418) C0.0787 (.01947) +(.145) C0.08 44545%, Bd 98.895 e (.04)+() Cd.1418 (1.0454545) IVd Max[98.895 98.75,0].145 C d Max[IV,Cu].145 S B C 6.05% Buu 100/(1.0605) 984 Cuu Max[984 98.75,0] 0 ud ud ud 5.00% 100/ 98.79 Max[98.79 98.75,0].04 d 4.13223% Bdd 100/(1.0413223) 99 Cdd Max[99 98.75,0] FIGURE H.2 Binomial Tree of Spot Rates and T-Bill Call Prices theses prices to the current period yields a price on the merican T-bill futures call of.08 the same price as the merican spot option. 1 T-Bill Put In the case of a spot or futures T-bill put, their prices can be determined given a binomial tree of spot rates and their corresponding spot and futures prices. Figure H.3 shows the binomial valuation of a European T-bill futures put contract with an exercise price of 98.75 and expiration of one year (two periods). t the expiration spot rate of 6.05%, the put is in the money with an IV of.21, and at the spot rates of 5% and 4.13223% the put is out of the money. In period 1, the two possible values for the European put are.1022 and 0. Since these values exceed or equal their IV, they would also be the prices of the put if it were merican. Rolling these values to the current period, we obtain the price for the futures put of.05. H.2 VLUING CPLET ND FLOORLET WITH BINOMIL TREE The price of a caplet or floorlet can also be valued using a binomial tree of the option s reference rate. For example, consider an interest rate call on the spot rate defined by our binomial tree, with an exercise rate of 5%, time period applied to the payoff of φ, and notional principal of NP 100. s shown in Figure H.4,
ppendix H: Pricing Interest Rate Options with a Binomial Interest Rate Tree 757 S nnualized spot rate u 1.1, d 1/1.1.9091, n 2, Length of period years f Price of T-bill future, Expiration, n 2 P Price on put option on futures T-bill, X 98.75, Expiration, n 2 6.05% f uu Buu 984 Puu Max[98.75 984,0].21 5%, B0 98.79, f 0 98.78 e (.1022) + (0) P0 P0.05 5%, Bu 98.67, f u 98.665 e (.21) + (0) Pu.1022 (1.055) IVu Max[98.775 98.665,0] 0 Pu Max[IV, Pu ].1022 S ud 5.00% B ud f ud 98.79 Pud Max[98.75 98.79,0] 0 44545%, Bd 98.895, f d 98.895 e (0) + (0) Pd 0 (1.0454545) IVd Max[98.75 98.895,0] 0 Pd Max[IV,Cu] 0 d 4.13223% Bdd f dd 99 Pdd Max[98.75 99,0] 0 FIGURE H.3 Binomial Tree of Spot Rates and T-Bill Futures Put Prices S nnualized rate spot S uu 6.05% u 1.1, d 1/1.1.9091, n 2, Length of period years Caplet Max[.0605.05,0]()(100) Caplet and floorlet: Exercise rate.05, Reference rate Spot rate,.2625 NP 100, φ, Expiration, n 2. Floorlet Max[.05.0605,0]()(100) S 5% 0 u (.2625) + (0) Caplet.1278 (1.055) (0) + (0) Floorlet 0 (1.055) 5% S ud 5.00% (.1278) + (0) Caplet.06236 Caplet Max[.05.05,0]()(100) 0 (0) + (.1061) Floorlet.05177 Floorlet Max[.05.05,0]()(100) 0 44545% (0) + (0) Caplet 0 (1.0454545) (0) + (.2169) Floorlet.1061 (1.0454545) S dd 4.13223% Caplet Max[.0413223.05,0]()(100) 0 Floorlet Max[.05.0413223,0]()(100).2169 FIGURE H.4 Binomial Tree: Caplets and Floorlets
758 PPENDIX H: PRICING INTEREST RTE OPTIONS WITH BINOMIL INTEREST RTE TREE the interest rate call is in the money at expiration only at the spot rate of 6.05%. t this rate, the caplet s payoff is.2625 [ (.0605.05)()(100)]. In period 1, the value of the caplet is.1278 ( [(.2625) + (0)]/(1.055) ) at spot rate 5% and 0 at spot rate 44545%. Rolling theses values to the current period, in turn, yields a price on the interest rate call of.06236 ( [(.1278) + (0)]/ ). In contrast, an interest rate put with similar features would be in the money at expiration at the spot rate of 4.13223%, with a payoff of.2169 [ (.05.0413223)()(100) and out of the money at spot rates 5% and 6.05%. In period 1, the floorlet s values would be.1061 ( [(0) + (.2169)]/(1.0454545) ) at spot rate 4.454545% and 0 at spot rate 6.05%. Rolling these values to the present period, we obtain a price on the floorlet of.05177 ( [(0) + (.1061)]/ ). Since a cap is a series of caplets, its price is simply equal to the sum of the values of the individual caplets making up the cap. To price a cap, we can use a binomial tree to price each caplet and then aggregate the caplet values to obtain the value of the cap. Similarly, the value of a floor can be found by summing the values of the floorlets comprising the floor. H.3 VLUING T-BOND OPTIONS WITH BINOMIL TREE The T-bill underlying the spot or futures T-bill option is a fixed-deliverable bill; that is, the features of the bill (maturity of 91 days and principal of $1 million) do not change during the life of the option. In contrast, the T-bond or T-note underlying a T-bond or T-note option or futures option is a specified T-bond or note or the bond from an eligible group that is most likely to be delivered. Because of the specified bond clause on a T-bond or note option or futures option, the first step in valuing the option is to determine the values of the specified T-bond (or bond most likely to be delivered) at the various nodes on the binomial tree, using the same methodology we used in Chapter 14 to value a coupon bond. s an example, consider an OTC spot option on a T-bond with a 6% annual coupon, face value of $100, and with three years left to maturity. In valuing the bond, suppose we have a two-period binomial tree of risk-free spot rates, with the length of each period being one year, the estimated upward and downward parameters being u 1.2 and d.8333, and the current spot rate being 6% (see Figure H). To value the T-bond, we start at the bond s maturity (end of period 3) where the bond s value is equal to the principal plus the coupon, 106. We next determine the three possible values in period 2 given the three possible spot rates. s shown in Figure H, the three possible values of the T-bond in period 2 are 977 ( 106/1.084), 100 ( 106/1.06), and 101.760 ( 106/1.0416667). Given these values, we next roll the tree to the first period and determine the two possible values. The values in that period are equal to the present values of the T-bond s expected cash flows in period 2; that is: B u B d [977 + 6] + [100 + 6] 97.747 1.072 [100 + 6] + [101.760 + 6] 101.79 1.05
ppendix H: Pricing Interest Rate Options with a Binomial Interest Rate Tree 759 S nnualized spot rate u 1.2, d 1/1.2.8333, n 2, Length of period 1 year B Price of T-bond, M 3 years, F $100 (n 3) f Price on T-bond futures, Expiration two years (n 2) u 7.2% (977+6) + (100+6) Bu 97.747 (1.072 ) fu (977) + (100) 98.785 2 u 8.64% Buu 106/(1.0864) 977 fuu Buu 977 106 106 6% (97.747+6)+(101.790+6) B0 99.78 (1.06) f 0 (98.785)+(100.88) 99.83 d ud 6.00% Bud 106/(1.06) 100 fud Bud 100 d 5% (100+6)+(101.760+6) Bd 101.790 fd (100)+(101.760) 100.88 106 2 d d 4.16667 % Bdd 106 /(1.0416667 ) 101.760 fdd Bdd 101.760 106 FIGURE H Binomial Tree: Spot Rates, T-Bond Prices, and T-Bond Futures Prices Finally, using the bond values in period 1, we roll the tree to the current period where we determine the value of the T-bond to be 99.78: B 0 [97.747 + 6] + [101.79 + 6] 1.06 99.78 Figure H also shows the prices on a two-year futures contract on the threeyear, 6% T-bond. The prices are generated by assuming a risk-neutral market. s shown, at expiration (period 2) the three possible futures prices are equal to their spot prices: 977, 100, 101,76; in period 1, the two futures prices are equal to their expected spot prices: f u E(B T ) (977) + (100) 98.875 and f d E(B T ) (100) + (101.76) 100.88; in the current period, the futures price is f 0 E(f 1 ) (98.785) + (100.88) 99.83. Spot T-Bond Call ppose we want to value a European call on the T-bond, with the call having an exercise price of 98 and expiration of two years. t the option s expiration, the underlying T-bond has three possible values: 977, 100, and 101.76. The 98 T-bond call s respective IVs are therefore 0, 2, and 3.76 (see Figure H.6).
760 PPENDIX H: PRICING INTEREST RTE OPTIONS WITH BINOMIL INTEREST RTE TREE S nnualized spot rate u 1.2, d 1/1.2.8333, n 2, Length of period 1 year B Price of T-bond, M 3 years, F $100 (n 3) C Price on spot T-bond call, X 98, Expiration two years (n 2) u 7.2%, Bu 97.747 e (0) + (2) Cu.9328 (1.072) IVu Max[97.747 98,0] 0 Cu Max[.9328,0].9328 8.64 %, B uu 97 7 Cuu Max [97 7 98,0] 0 S 0 6%, B0 99.78 e (.9328) + (2.743) C0 1.734 (1.06) (.9328) + (3.79) C0 2.228 (1.06) 5%, Bd 101.79 e (2) + (3.76) Bd 2.743 IVd Max[101.79 98,0] 3.79 Cd Max[3.79,2.743] 3.79 d 6.00 %, Bud 100 Cud Max [100 98,0 ] 2 d 4.16667 %, B dd 101.760 Cdd Max [101.76 98,0] 3.76 FIGURE H.6 Binomial Tree: T-Bond Call Prices Given these values, the call s possible values in period 1 are.9328 ( [(0) + (2)]/1.072) and 2.743 ( [(2) + (3.76)]/1.05). Rolling these values to the current period, we obtain the price on the European T-bond call of 1.734 ( [(.9328) + (2.743)]/1.06). If the call option were merican, then its value at each node is the greater of the value of holding the call or the value from exercising. s we did in valuing merican T-bill options, this valuation requires constraining the merican price to be the maximum of the binomial value or the IV. In this example, if the T-bond option were merican, then in period 1 the option s price would be equal to its IV of 3.79 at the lower rate. Rolling this price and the upper rate s price of.9328 to the current period yields a price of 2.228. Futures T-Bond Call If the European call were an option on a futures contract on the three-year, 6% T-bond (or if that bond were the most-likely-to-be-delivered bond on the futures contract), with the futures contract expiring at the same time as the option (end of period 2), then the value of the futures option will be the same as the spot. That is, at expiration the futures prices on the expiring contract would be equal to the spot prices, and the corresponding IVs of the European futures call would be the same as the spot call s IV. Thus, when we roll these call values back to the present period, we end up with the price on the European futures call being the same as the European spot: 1.734. If the futures call were merican, then at the spot rate of 5% in period
ppendix H: Pricing Interest Rate Options with a Binomial Interest Rate Tree 761 1, its IV would be 2.88 ( Max[100.88 98,0]), exceeding the binomial value of 2.743. Rolling the 2.88 value to the current period yields a price on the merican futures option of 1.798 [ ((.9328) + (2.88)/1.06] this price differs from the merican spot option price of 2.228. T-Bond Put ppose we want to value a European put on a spot or futures T-bond with the put having similar terms as the call. Given the bond s possible prices at expiration of 977, 100, and 101.76, the corresponding IVs of the put are.43, 0, and 0. In period 1, the put s two possible values are.2006 ( [(.43) + (0)]/1.072) and 0. Rolling these values to the current period yields a price on the European put of.0946 ( [(.2006) + (0)]/1.06). Note that if the spot put were merican, then its possible prices in period 1 would be 3 and 0, and its current price would be.119 ( [(3) + (0)]/1.06); if the futures put were merican, there would be no exercise advantage in period 1 and thus the price would be equal to its European value of.0946. Realism In the above examples, the binomial trees six-month and one-year steps were too simplistic. Ideally, binomial trees need to be subdivided into a number of periods. In pricing a 5-year T-bond, for example, we might use one-month steps and a five-year horizon. This would, in turn, translate into a 60-period tree with the length of each period being one month. By doing this, we would have a distribution of 60 possible T-note values one period from maturity. NOTE 1. It should be noted that if the futures option expired in one period while the T-bill futures expire in two, then the value of the futures option would be.071: C 0 (IV u) + (IV d ) Max(98.665 98.75,0) + Max(98.895 98.75,0).071