Topics for Computational Finance
Topic 4
Finite Differences: Non-Equidistant Grids
Equidistant grids can waste many nodes in areas where the solution is
rather flat. Frequently this happens with options far out of the money.
This situation calls for flexible grids having large mesh length for regions
with V ≈ 0 and small mesh length where the solution V has significant
gradients. This is the ideal scenario for finite element methods. But also
for finite differences the effort of introducing variable grid sizes improves
the efficiency. This holds in particular in the two-asset case.
The figure shows V (S1 , S2 , 0) for a butterfly option, calculated with nonequidistant grid. Notice the area where both S1 and S2 are large, and the
grid is very different from the areas where either S1 or S2 are small.
0.2
V(S1,S2,T)
0.15
0.1
0.05
0
−0.05
8
0
7
1
6
2
5
3
S
2
4
4
3
5
1
2
6
1
7
8
www.compfin.de
S
0
Michael Tenvenne
RS
May 2012