sparse grid method in the libor market model. option valuation and the

11.8
11.6
11.4
Black-Scholes Boundary
Fixed Boundary
Full Grid, Level 9 Value
11.2
Option price
11
10.8
10.6
10.4
10.2
10
9.8
6 7 8 9 10 11
Discretization Level
Figure 6.2: 2D Chooser. Convergence of Full and Sparse Grid.
Level
L 1 =L 2 =5.5%, K=5.5% K = 3.9% L 1 =L 2 =7.5%, K=5.5%
Full Sparse Full Sparse Full Sparse
5 83.669 71.9602 49.325 37.3126 251.755 262.1945
6 82.947 83.0537 48.869 37.9287 251.535 258.5324
7 82.547 81.1169 48.818 41.3625 251.449 253.7469
8 82.551 79.0752 48.790 46.9370 251.438 253.1529
9 82.4895 48.0288 251.9835
10 82.5755 48.0717 251.8426
MC
Price Std.error Price Std.error Price Std.error
82.399 0.01298 48.653 0.0089 251.468 0.0218
Table 6.4: 2D Chooser. At-the-money and in-the-money option values.
Level
L 1 =L 2 =2%, K=5.5% L 1 =L 2 =2%, K=2%
Full Sparse Full Sparse
5 0.0572 0.5587 32.7686 23.4880
6 0.0269 0.0239 31.2000 28.7051
7 0.0202 0.0701 30.1360 31.3129
8 0.0193 0.1163 30.1523 28.5730
9 0.0383 28.5550
10 0.0246 29.8075
MC
Price Std.error Price Std.error
0.0194 0.000153 30.0825 0.00472
Table 6.5: 2D Chooser. Deep out-of-the-money and at-the-money with low strike.
In addition to the out-of-the money case from [Blackham, 2004], we were interested
in the convergence of sparse solution in case of different strike prices and different
values of spot forward rates. The results are presented in Tables (6.4) and (6.5), where
MC prices have been calculated for 500000 simulation trials.
46