Time Step: Δt 0.083333333 ( = 1/12 )
Mean Reversion: a 5.0000% u(i,j)= j · Δu
Volatility: σ 20.0000%
State Step: Δu 9.9792%
Tree for driver u(t)
6 0.5988 0.5988 0.5988 0.5988 0.5988 0.5988 0.5988
5 0.4990 0.4990 0.4990 0.4990 0.4990 0.4990 0.4990
4 0.3992 0.3992 0.3992 0.3992 0.3992 0.3992 0.3992
3 0.2994 0.2994 0.2994 0.2994 0.2994 0.2994 0.2994
2 0.1996 0.1996 0.1996 0.1996 0.1996 0.1996 0.1996
State ( j )
1 0.0998 0.0998 0.0998 0.0998 0.0998 0.0998 0.0998
0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-1 -0.0998 -0.0998 -0.0998 -0.0998 -0.0998 -0.0998 -0.0998
-2 -0.1996 -0.1996 -0.1996 -0.1996 -0.1996 -0.1996 -0.1996
-3 -0.2994 -0.2994 -0.2994 -0.2994 -0.2994 -0.2994 -0.2994
-4 -0.3992 -0.3992 -0.3992 -0.3992 -0.3992 -0.3992 -0.3992
-5 -0.4990 -0.4990 -0.4990 -0.4990 -0.4990 -0.4990 -0.4990
-6 -0.5988 -0.5988 -0.5988 -0.5988 -0.5988 -0.5988 -0.5988
0 1 2 3 4 5 6
Time ( i )
Time Step: Δt 0.083333333 ( = 1/12 )
Mean Reversion: a 5.0000%
Volatility: σ 20.0000%
j k β p_u
1 1 6 6 -0.0249 0.1545
pu = + ( b 2 + b ), 5 5 -0.0208 0.1565
6 2
4 4 -0.0166 0.1585
2
pm = - b 2 , 3 3 -0.0125 0.1605
3 2 2 -0.0083 0.1625
1 1 -0.0042 0.1646
1 1
pd = + ( b 2 - b ), 0 0 0.0000 0.1667
6 2 -1 -1 0.0042 0.1688
b = je - a D t - k , -2 -2 0.0083 0.1709
-3 -3 0.0125 0.1730
é 1ù -4 -4 0.0166 0.1751
k = int êje - a D t + ú -5 -5 0.0208 0.1773
ê
ë 2ú
û -6 -6 0.0249 0.1795
p_m p_d
0.6660 0.1795
0.6662 0.1773
0.6664 0.1751
0.6665 0.1730
0.6666 0.1709
0.6666 0.1688
0.6667 0.1667
0.6666 0.1646
0.6666 0.1625
0.6665 0.1605
0.6664 0.1585
0.6662 0.1565
0.6660 0.1545
Time Step: Δt 0.083333333 ( = 1/12 )
Mean Reversion: a 5.0000% r(i,j) = r0(i)exp[ j · Δu ]
Volatility: σ 20.0000%
State Step: Δu 9.9792%
Mean Rate: r0(i) 1.8500% 1.8600% 1.8800% 1.9100% 1.9300% 1.9600% 1.9700%
Black-Karasinski short rate tree
6 3.3667% 3.3849% 3.4213% 3.4759% 3.5123% 3.5669% 3.5851%
5 3.0470% 3.0634% 3.0964% 3.1458% 3.1787% 3.2281% 3.2446%
4 2.7576% 2.7725% 2.8023% 2.8470% 2.8768% 2.9215% 2.9365%
3 2.4957% 2.5092% 2.5362% 2.5766% 2.6036% 2.6441% 2.6576%
2 2.2587% 2.2709% 2.2953% 2.3319% 2.3563% 2.3930% 2.4052%
State ( j )
1 2.0441% 2.0552% 2.0773% 2.1104% 2.1325% 2.1657% 2.1767%
0 1.8500% 1.8600% 1.8800% 1.9100% 1.9300% 1.9600% 1.9700%
-1 1.6743% 1.6833% 1.7014% 1.7286% 1.7467% 1.7739% 1.7829%
-2 1.5153% 1.5235% 1.5399% 1.5644% 1.5808% 1.6054% 1.6136%
-3 1.3714% 1.3788% 1.3936% 1.4158% 1.4307% 1.4529% 1.4603%
-4 1.2411% 1.2478% 1.2613% 1.2814% 1.2948% 1.3149% 1.3216%
-5 1.1232% 1.1293% 1.1415% 1.1597% 1.1718% 1.1900% 1.1961%
-6 1.0166% 1.0221% 1.0331% 1.0495% 1.0605% 1.0770% 1.0825%
0 1 2 3 4 5 6
Time ( i )
Time Step: Δt 0.083333333 ( = 1/12 )
Mean Reversion: a 5.0000%
Volatility: σ 20.0000%
State Step: Δu 9.9792%
Mean Rate: r0(i) 1.8500% 1.8600% 1.8800% 1.9100% 1.9300% 1.9600% 1.9700%
Black-Karasinski short rate tree
6 3.3667% 3.3849% 3.4213% 3.4759% 3.5123% 3.5669% 3.5851%
5 3.0470% 3.0634% 3.0964% 3.1458% 3.1787% 3.2281% 3.2446%
4 2.7576% 2.7725% 2.8023% 2.8470% 2.8768% 2.9215% 2.9365%
3 2.4957% 2.5092% 2.5362% 2.5766% 2.6036% 2.6441% 2.6576%
2 2.2587% 2.2709% 2.2953% 2.3319% 2.3563% 2.3930% 2.4052%
1 2.0441% 2.0552% 2.0773% 2.1104% 2.1325% 2.1657% 2.1767%
r(i,j) 0 1.8500% 1.8600% 1.8800% 1.9100% 1.9300% 1.9600% 1.9700%
-1 1.6743% 1.6833% 1.7014% 1.7286% 1.7467% 1.7739% 1.7829%
-2 1.5153% 1.5235% 1.5399% 1.5644% 1.5808% 1.6054% 1.6136%
-3 1.3714% 1.3788% 1.3936% 1.4158% 1.4307% 1.4529% 1.4603%
-4 1.2411% 1.2478% 1.2613% 1.2814% 1.2948% 1.3149% 1.3216%
-5 1.1232% 1.1293% 1.1415% 1.1597% 1.1718% 1.1900% 1.1961%
-6 1.0166% 1.0221% 1.0331% 1.0495% 1.0605% 1.0770% 1.0825%
0 1 2 3 4 5 6
Lattice Implied Term Structure
Cont. 1M Rate: R(i) 1.8500% 1.8631% 1.8862% 1.9194% 1.9426% 1.9759%
Zero Prices: Z(i) 1.000000 0.998460 0.996911 0.995345 0.993754 0.992147 0.990514
0 1 2 3 4 5 6
Arrow-Debreu Prices
6 0.0000
5 0.0001 0.0004
4 0.0007 0.0024 0.0048
3 0.0044 0.0118 0.0203 0.0286
2 0.0273 0.0546 0.0757 0.0908 0.1011
1 0.1664 0.2215 0.2351 0.2335 0.2266 0.2181
AD(i, j) 0
-1
1.0000 0.6656
0.1664
0.4991
0.2215
0.4069
0.2352
0.3501
0.2336
0.3117
0.2267
0.2838
0.2183
-2 0.0274 0.0546 0.0758 0.0909 0.1013
-3 0.0044 0.0118 0.0203 0.0287
-4 0.0007 0.0024 0.0049
-5 0.0001 0.0005
-6 0.0000
0 1 2 3 4 5 6
Probabilities
j k β p_u p_m p_d
6 6 -0.0249 0.1545 0.6660 0.1795
5 5 -0.0208 0.1565 0.6662 0.1773
4 4 -0.0166 0.1585 0.6664 0.1751
3 3 -0.0125 0.1605 0.6665 0.1730
2 2 -0.0083 0.1625 0.6666 0.1709
1 1 -0.0042 0.1646 0.6666 0.1688
0 0 0.0000 0.1667 0.6667 0.1667
-1 -1 0.0042 0.1688 0.6666 0.1646
-2 -2 0.0083 0.1709 0.6666 0.1625
-3 -3 0.0125 0.1730 0.6665 0.1605
-4 -4 0.0166 0.1751 0.6664 0.1585
-5 -5 0.0208 0.1773 0.6662 0.1565
-6 -6 0.0249 0.1795 0.6660 0.1545
Time Step: Δt 0.083333333 ( = 1/12 )
Mean Reversion: a 5.0000%
Volatility: σ 20.0000%
State Step: Δu 9.9792%
Mean Rate: r0(i) 4.0000% 4.0000% 4.0000% 4.0000% 4.0000% 4.0000%
BK: Coninuously Compounded Short Rate Rate Lattice
6 7.2794% 7.2794% 7.2794% 7.2794% 7.2794% 7.2794%
5 6.5880% 6.5880% 6.5880% 6.5880% 6.5880% 6.5880%
4 5.9623% 5.9623% 5.9623% 5.9623% 5.9623% 5.9623%
3 5.3961% 5.3961% 5.3961% 5.3961% 5.3961% 5.3961%
2 4.8836% 4.8836% 4.8836% 4.8836% 4.8836% 4.8836%
1 4.4198% 4.4198% 4.4198% 4.4198% 4.4198% 4.4198%
r(i, j) 0 4.0000% 4.0000% 4.0000% 4.0000% 4.0000% 4.0000%
-1 3.6201% 3.6201% 3.6201% 3.6201% 3.6201% 3.6201%
-2 3.2763% 3.2763% 3.2763% 3.2763% 3.2763% 3.2763%
-3 2.9651% 2.9651% 2.9651% 2.9651% 2.9651% 2.9651%
-4 2.6835% 2.6835% 2.6835% 2.6835% 2.6835% 2.6835%
-5 2.4286% 2.4286% 2.4286% 2.4286% 2.4286% 2.4286%
-6 2.1980% 2.1980% 2.1980% 2.1980% 2.1980% 2.1980%
0 1 2 3 4 5
Instrument Price: Price $98.0119
Instrument Future Cash Flows
6
5
4
3
2
1
CF(i, j) 0
-1
-2
-3
-4
-5
-6
0 1 2 3 4 5
Lattice Values of Future Cash Flows
6
5 99.4525
4 99.0112 99.5044
3 98.6595 99.1045 99.5513
2 98.3834 98.7855 99.1890 99.5939
1 98.1709 98.5349 98.8998 99.2656 99.6324
V(i, j)
V(i, j) 0
-1
98.0119 98.3418
98.4968
98.6723
98.7969
99.0034
99.0972
99.3350
99.3978
99.6672
99.6988
-2 98.9098 99.1823 99.4548 99.7273
-3 99.2593 99.5063 99.7532
-4 99.5530 99.7766
-5 99.7978
-6
0 1 2 3 4 5
4.0000%
Probabilities
ate Lattice j k β p_u p_m p_d
7.2794% 6 6 -0.0249 0.1545 0.6660 0.1795
6.5880% 5 5 -0.0208 0.1565 0.6662 0.1773
5.9623% 4 4 -0.0166 0.1585 0.6664 0.1751
5.3961% 3 3 -0.0125 0.1605 0.6665 0.1730
4.8836% 2 2 -0.0083 0.1625 0.6666 0.1709
4.4198% 1 1 -0.0042 0.1646 0.6666 0.1688
4.0000% 0 0 0.0000 0.1667 0.6667 0.1667
3.6201% -1 -1 0.0042 0.1688 0.6666 0.1646
3.2763% -2 -2 0.0083 0.1709 0.6666 0.1625
2.9651% -3 -3 0.0125 0.1730 0.6665 0.1605
2.6835% -4 -4 0.0166 0.1751 0.6664 0.1585
2.4286% -5 -5 0.0208 0.1773 0.6662 0.1565
2.1980% -6 -6 0.0249 0.1795 0.6660 0.1545
6
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
100.0000
6
s
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
6