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

tion of spot-rate model to the observed market data is far from straight-forward, since
such rates do not exist in practice [Pelsser, 2000].
In the third chapter we turn our attention to market rate models, namely the LIBOR
Market Model (LMM), where LIBOR stands for London Interbank Offered Rate. Published
as [Brace et al., 1997], it has been widely accepted and became one of the most
popular models used in practice. In contrast to spot-rate models, it takes the process
for a real market rate as the basic modelling unit. This fact simplifies fitting the model
to market parameters to a great extent. On the other hand, the number of interest rates
underlying the derivative has a direct impact on the dimensionality of the problem.
In fact, without sophisticated extensions, e.g. [Pietersz et al., 2005], the number of
dimensions is equal to the number of underlying forward rates.
Since the analytical valuation formulae are rarely available in a multi-dimensional
setting, one needs to approximate the solution with a computational method of choice.
The fourth chapter contains the discussion of Monte Carlo simulation (MC) and Finite
Difference Method (FDM) from the perspective of LMM valuation. As of this moment,
valuation of interest-rate derivatives lies in the domain of Monte Carlo simulation,
since it is weakly dependent on the dimensionality of the problem and allows to price
claims contigent on a large (up to 30) number of rates. Yet, as discussed in Chapter 4,
FDM has an number of important advantages over MC and the main reason for it to be
underfavoured is its ’curse of dimensionality’. The exponential growth in the number
of mesh points with the increase in the dimensionality makes it unusable in more than
three dimensions. The challenge of breaking the curse motivated researchers to seek
for alternative discretisation methods. One such method, known as the Sparse Grid
method was introduced in [Zenger, 1991] and has been able to show promising results
in application to high-dimensional problems in many areas of science. It is discussed
in the fifth chapter of this text.
The idea of application of sparse grid technique to the field of option pricing is not
new [Reisinger, 2004]; nor is it such with respect to the derivatives in the LIBOR Market
Model [Blackham, 2004]. The author of the latter reports price convergence of full
and sparse grid solutions for products contingent on 2,3,4 and possibly 5 forward rates.
The current master thesis was conceived as an extension of the work of [Blackham,
2004] in lower (2D and 3D) dimensions, with the threefold objective for the practical
part:
1. Independent verification of the findings regarding the option valuation for 2D
and 3D cases of [Blackham, 2004] and exploration of a larger parameter space.
2. Investigation of the behaviour for a discontinuous payoff in case of a sparse grid
method.
3. A more rigorous study of the quality of the sparse ’Greeks’. Essential to risk
management, ’Greeks’ are the quantities that show sensitivity of the computed
price to the changes in the input parameters.
The results and due observations are presented in the final chapter, followed by the
summary of recommendations for future work.
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