Pasquini, I., Raimondi, D.
Nuovi metodi numerici per la risoluzione dell’EDP di Black & Scholes in grande dimensione
Tuesday 22nd September 2009
major challenge in computational finance is the pricing of options that depend on a large number of assets. A prominent example is basket options, where dozens or even hundreds of stocks constitute the underlying asset and determine the high dimensionality of the Black and Scholes model. In order to avoid the so-called curse of dimensionality, which characterizes the conventional discretisation techniques employed in the resolution of the PDE problem formulation, derivative prices are currently calculated using Monte Carlo methods. The objective of this study is to show how an efficient PDE discretisation can be achieved by the sparse grid approach. Introduced by Zenger in 1990 and based on hierarchical tensor product approximation spaces, sparse grids have turned out to be a very efficient approach in order to improve the ratio of invested storage and computing time to the achieved accuracy. Discretisations on sparse grids involve O(Nlog(N)d−1) degrees of freedom only, where d denotes the underlying problem’s dimensionality and N is the number of grid points in one coordinate direction at the boundary. The accuracy obtained with piecewise linear basis functions, for example, is O(N−2log(N)d−1) with respect to the L2- and L1 -norm, if the solution has bounded second mixed derivatives. This way, the curse of dimensionality, i.e. the exponential dependence O(Nd) of conventional approaches, is overcome to some extent. Furthermore, the interpolation function can be obtained as a result of the combination of numerical solutions on a series of anisotropic, relatively small, full grids. The most important advantage of this approach, introduced by Griebel in 1992 as the « Combination Technique », is that each of these solutions can be computed in parallel. As a result, there are additional benefits in terms of computing time, although the result is slightly less precise than the one computed with the Sparse Grids method.