Callable Bond/EJ/e

H.-J. Büttler: Callable Bonds: Finite Difference Methods 10
Percentage Error
5
4
3
2
1
0
-1
#4
#2
#3
Percentage Error
5
4
3
2
1
0
-1
#4
#2
#3
(b)
-2
-3
0.0
#1
0.5 1.0
Interest Rate
1.5
(a)
2.0
-2
-3
0.00
#1
0.05
Interest Rate
0.10
0.15
Fig. 4a & b: Percentage Error after 6.811 Years. ‡
0.8
0.8
Percentage Error
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
0.0
#3
#2
#4
0.5 1.0
Interest Rate
#1
1.5
(a)
2.0
Percentage Error
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
0.00
#4
#3
0.05
Interest Rate
Fig. 5a & b: Percentage Error of the Underlying Straight Bond. ‡
#1
#2
0.10
(b)
0.15
Table 2: Five Boundary Schemes.
Scheme # Derivative Denominato F n
F n – 1
F n – 2
F n – 3
F n – 4
r
1 F x ∆x 1 – 1 — — —
F xx (∆x) 2 1 – 2 1 — —
2 F x 2 ∆x 3 – 4 1 — —
F xx (∆x) 2 1 – 2 1 — —
3 F x 2 ∆x 3 – 4 1 — —
F xx (∆x) 2 2 – 5 4 – 1 —
4 F x 6 ∆x 11 – 18 9 – 2 —
F xx (∆x) 2 2 – 5 4 – 1 —
5 F x 12 ∆x 25 – 48 36 – 16 3
F xx 12 (∆x) 2 35 – 104 114 – 56 11
‡ The numbers refer to the boundary schemes of Table 2. The parameters of the Lawson-Morris method are n 1
= 50, r m = 0.15, n 2 = 50, ψ = 200, ∆t = 1/74th of a year, s = 10 and m;^ = 1.
Royal Economic Society Conference University of Exeter, 28 – 31 March 1994