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

of the payoff is also accountable for the severe spikes in sparse vega solution near the
at-the-money position.
6.4 Discussion and Recommendations.
In financial community, the quality of a computational method is, to a large extent,
determined by its ability to show accurate approximations of risk sensitivities, such as
∆ and V. We can conclude from ∆ and V slice plots of each test case that the impact of
the spikes effect is being magnified as we start operating on smaller ranges of values.
What didn’t show in quality of price estimates, shows distinctly in the values for the
Greeks. This fact makes immediate use of a sparse grid method in a discussed setting
quite problematic. Steps need to be taken to improve the accuracy of approximation.
There are several possible recipes one could follow:
• Iterative Solver. When benchmarking against the results from[Blackham, 2004],
the latter shows smoother convergence in both discretizations, and, moreover,
surprisingly close results on earlier levels. Most likely, such behaviour is inflicted
by the choice of an iterative solver. The author of [Blackham, 2004]
reports to have used second-order finite differences with a multigrid iterative
solver. The multigrid technique is designed to operate on a hierarchy of increasingly
“coarser” meshes where, after desired accuracy has been reached, every
individual mesh represents a part of the common solution.
However, despite smoother convergence, the author of [Blackham, 2004] reports
the presence of spikes in the solution. Thus, similar inaccuracies are to be expected
in the results for the Greeks.
• Order of FD. A possible way of with dealing discretization errors could be refining
spatial discretization to a fourth-order finite differences, as explored in
[Leentvaar and Oosterlee, 2006]. Again, such discretization poses difficulties
for ”narrow” grids and solutions adjacent to boundary conditions, that appear to
rely on ”ghost” outlier points. These issues can be addressed with application of
extrapolation or rewriting of PDE with one-sided differences on the borders of
solution domain.
• Combination technique and sparse grid type. It would be worthwhile investigating
the effect of the choice of an alternative sparse grid combination technique.
Some of the possible alternatives to the “classical” sparse grid and combination
technique are discussed in [Klimke, 2003] and [Leentvaar and Oosterlee,
2006], respectively.
• Quantification Error. In practice, the quantification error of discontinuous payoff
can be significantly reduced by peforming coordinate transformation. As described
in [Travella and Randal, 2000], by rewriting the PDE in different terms,
one can transform the grid in such a way that a larger amount of points is concentrated
in the discontinuity region. Since this procedure does not degrade the
accuracy of discretization, it could be an employed by the sparse grid method to
battle the quantification error in case of digital option.
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