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

6.1.1 Convergence of MC, Full and Sparse Grid solutions.
MC convergence and its standard error are shown in Figure (6.1). Simulation was
performed for the maximum of 4, 000, 000 trials, reaching minimum standard error of
0.022. FD solution for full grid discretization was computed for up to discretization
level L = 9. Figure (6.1) shows convincing convergence behavior of simulated solution
Level of Discretization
6 6.5 7 7.5 8 8.5 9
10.2
MC Simulation Value+Std.Error
Full-Grid FD value
10.15
10.1
Option price
10.05
10
9.95
9.9
9.85
500000 1e+006 1.5e+006 2e+006 2.5e+006 3e+006 3.5e+006 4e+006
Number of MC simulations
Figure 6.1: 2D Chooser Option. MC vs. FD Convergence.
and full-grid FD, providing accurate reference values for further tests. For our sparse
approximation experiments we have used a classical sparse grid (a.k.a ”maximumnorm-based”
type) and a basic combination technique, both described in the previous
chapter. Table (6.3) and Figure (6.2) give idea of FD convergence on full and sparse
grids 2 :
Level
Full grid Execution Sparse Grid Execution Full grid Sparse grid
solution time(sec.) solution time(sec.) points count points count
5 10.5969 1.5 11.351 0.36 1089 257
6 10.1755 6.6 11.477 0.88 4225 577
7 10.0964 30.1 9.8731 2.16 16641 1281
8 10.0762 176.4 10.005 5.18 66049 2817
9 10.0762 651.4 10.165 12.89 263169 6145
10 10.153 31.64 1050625 13313
11 9.998 74.86 4198401 28673
Table 6.3: 2D Chooser. Convergence values for full and sparse grid solutions.
2 The execution time for sparse grid technique does not include interpolation procedure
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