Radial basis function partition of unity methods for pricing vanilla basket options

Abstract

PostprintThis is the accepted version of a paper published in Computers and Mathematics with Applications. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.Citation for the original published paper (version of record):Shcherbakov, V., Larsson, E. (2016) Radial basis function partition of unity methods for pricing vanilla basket options. Abstract Mesh-free methods based on radial basis function (RBF) approximation are becoming widely used for solving PDE problems. They are flexible with respect to the problem geometry and highly accurate. A disadvantage of these methods is that the linear system to be solved becomes dense for globally supported RBFs. A remedy is to introduce localisation techniques such as partition of unity. RBF partition of unity methods (RBF-PUM) allow for a significant sparsification of the linear system and lower the computational effort. In this work we apply a global RBF method as well as RBF-PUM to problems in option pricing. We consider one-and two-dimensional vanilla options. In order to price American options we employ a penalty approach. A penalty term, suitable for American multi-asset call options, has been designed. RBF-PUM is shown to be competitive compared with a finite difference method and a global RBF method. It is as accurate as the global RBF method, but significantly faster. The results for RBF-PUM look promising for extension to higher-dimensional problems. Computers and Mathematics with

Radial basis function partition of unity operator splitting method for pricing multi-asset American options

“…The error introduced by the penalty is expected to be O(e). This was discussed in [13] for finite difference methods, and also experimentally shown for RBF methods in [18], although the analytically derived bound in the same work was O(e 1/2 ). The modified equation takes the form…”

“…The penalty approach has been implemented in three versions: implicit, implicit-explicit and explicit. Previously the implicit-explicit formulation has been commonly used with radial basis functions [3,17,18], because it allowed to avoid solving a non-linear problem, while enforcing a "rather mild" constraint on the time step size, which still led to unnecessary extra calculations in the time integration. This work shows that, if the partition of unity technique is employed, the implicit version should be preferred.…”

“…We use the type of penalty function for American put options that was introduced by Nielsen et al [13] and has been used later by several authors together with finite differences [14] as well as radial basis functions [3,17,18]. The penalty function is given by…”

“…The form of the penalty term has also been generalised for application to American call options in [18].…”

“…Experiments for RBFs seem to give nearly the same bound [18]. We also implement an entirely implicit BDF-2 scheme, where the penalty is defined at the new time level.…”