Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

Quantum mechanics on non commutative spaces and squeezed states: a functional approach

We review here the quantum mechanics of some noncommutative theories in which no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. We show how the difference of structure between the Poisson brackets and the commutators in these theories generically leads to a harmonic oscillator whose positions and momenta mean values are not strictly equal to the ones predicted by classical mechanics. This raises the question of the nature of quasi classical states in these models. We propose an extension based on a variational principle. The action considered is the sum of the absolute values of the expressions associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory i.e we recover the known Gaussian functions. Besides them, we find other states which can be expressed as products of Gaussians with specific hyper geometrics. We illustrate our construction in two models defined on a four dimensional phase space: a model endowed with a minimal length uncertainty and the non commutative plane. Our proposal leads to second order partial differential equations. We find analytical solutions in specific cases. We briefly discuss how our proposal may be applied to the fuzzy sphere and analyze its shortcomings.Comment: 15 pages revtex. The title has been modified,the paper shortened and misprints have been corrected. Version to appear in JHE

On noncommutative spherically symmetric spaces

Two families of noncommutative extensions are given of a general space-time metric with spherical symmetry, both based on the matrix truncation of the functions on the sphere of symmetry. The first family uses the truncation to foliate space as an infinite set of spheres, is of dimension four and necessarily time-dependent; the second can be time-dependent or static, is of dimension five and uses the truncation to foliate the internal space.Comment: 22 page

Field theory on evolving fuzzy two-sphere

I construct field theory on an evolving fuzzy two-sphere, which is based on the idea of evolving non-commutative worlds of the previous paper. The equations of motion are similar to the one that can be obtained by dropping the time-derivative term of the equation derived some time ago by Banks, Peskin and Susskind for pure-into-mixed-state evolutions. The equations do not contain an explicit time, and therefore follow the spirit of the Wheeler-de Witt equation. The basic properties of field theory such as action, gauge invariance and charge and momentum conservation are studied. The continuum limit of the scalar field theory shows that the background geometry of the corresponding continuum theory is given by ds^2 = -dt^2+ t d Omega^2, which saturates locally the cosmic holographic principle.Comment: Typos corrected, minor changes, 23 pages, no figures, LaTe