Here are some of my papers that relate to phynance:
The present report contains an introduction to some elementary concepts in noncommutative differential geometry. The material extends upon ideas first presented by Dimakis and Mueller-Hoissen. In particular, stochastic calculus and the Ito formula are shown to arise naturally from introducing noncommutativity of functions (0-forms) and differentials (1-forms). The abstract construction allows for the straightforward generalization to lattice theories for the direct implementation of numerical models. As an elementary demonstration of the formalism, the standard Black-Scholes model for option pricing is reformulated.
In the present report, a review of discrete calculus on directed graphs is presented. It is found that the binary tree is a special directed graph that contains both the exterior calculus and stochastic calculus as different continuum limits are taken. In the latter case, we arrive at something that may be referred to as “discrete stochastic calculus.” The resulting discrete stochastic calculus may be applied to any stochastic financial model and is guaranteed to produce results that converge in the continuum limit. Discrete stochastic calculus is applied to the Black-Scholes model for an illustration. The resulting algorithm generated by discrete stochastic calculus agrees with that of the Cox-Ross-Rubinstein model, as it should. The results presented here are preliminary and are intended to encourage others to learn discrete stochastic calculus and apply it to more complicated financial models.
This paper represents an overview of the fully geometric approach to performance attribution developed by Menchero of Vestek. In addition to providing further insights into the subtleties involved with geometric attribution, two new fully geometric approaches to attributions are provided. The first represents a slight extension of that provided by Menchero. The second constitutes a slight extension of the exponential approach developed by Carino of the Frank Russell Company. The advantages of the extended exponential approach include absolute transparency in interpretation of the results as well as a direct parallel to the algebraic aproach.
Here are some of my non-phynance related papers:
Differential Geometry in Computational Electromagnetics
Dissertation, University of Illinois at Urbana-Champaign, 2002
This dissertation consists of two parts. Part I constitutes an introduction to some of the elementary concepts of the classical theory of differential geometry that are of relevance to computational electromagnetics. The motivation here is to aid researchers in overcoming the significant investment in mathematics that is typically required in order to learn and apply the theory to problems of practical interest in engineering. Another motivation for introducing the mathematical background in such detail is to provide a springboard to the algebraic model that takes up the second part of the dissertation.
The algebraic model of Part II represents a somewhat radical approach to computation. In this approach, rather than take the classical theory based on the continuum as a given and constructing approximate numerical techniques from there, work is done toward constructing an alternative to the continuum theory that is built up from scratch in a discrete setting. The goal is to take each mathematical objects that is necessary in order to write down the classical electromagnetic theory and develop a corresponding discrete analog. The theory builds upon standard concepts in algebraic topology and is motivated by recent progress in noncommutative differential geometry.
Discrete Differential Geometry on Causal Graphs
Eric Forgy and Urs Schreiber, July 2004
Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric is encoded in a deformation of the inner product on that space, which is the crucial technique of this paper. We study the general case but find that drastic and possibly vital simplifications occur when the underlying lattice is topologically hypercubic, in which case we explicitly construct mimetic analogues of the volume form, the Hodge star operator, and the Hodge inner product for arbitrary discrete geometries. It turns out that the formalism singles out a pseudo-Riemannian metric on topologically hypercubic graphs with respect to which all edges are lightlike. We study such causal graph complexes in detail and consider some of their possible physical applications, such as lattice Yang-Mills theory and lattice fermions.
A Time-Domain Method With Isotropic Dispersion and Increased Stability on an Overlapped Lattice
Eric Forgy and Weng Chew
IEEE Transactions on Antennas and Propagation, July 2002
(Submitted in December 1998!)
Note: This research won 1st Prize in the highly-competitive Student Paper Competition at the 1999 IEEE Antennas and Propagation Society International Symposium. It was essentially my MS thesis.
A time-domain method on an overlapped lattice is presented for the accurate and efficient simulation of electromagnetic wave propagation through inhomogeneous media. The method comprises a superposition of complementary approximations to electromagnetic theory on a lattice. The discrete space–time (DST) method, is set on a pair of dual lattices whose field components are collocated on their respective lattice sites. The other, the time-domain element (TDE) method, is set on overlapping dual lattices whose field components are noncollocated. The TDE method is shown to be a generalization and reinterpretation of the Yee algorithm. The benefits of the combined algorithm over comparable methods include: 1) increased accuracy over larger bandwidths; 2) increased stability allowing larger time steps; 3) local stencil-satisfying boundary conditions on interfaces; 4) self-contained mathematical framework; and 5) it is physically intuitive.
A Discrete Exterior Calculus and Electromagnetic Theory on a Lattice
Eric Forgy and Weng Chew, 2000
A highly accurate FDTD formulation was developed on an overlapped cubic grid that greatly reduces numerical dispersion errors. However, errors in in the FDTD method arise not only from numerical dispersion, but from geometrical modelling as well. Although representing a significant progress in addressing the numerical dispersion problem, it is still confined to a cubic grid with the subsequent “stair-casing” geometric approximations that it entails. The material presented represents a fundamentally new paradigm for finite-difference methods which hopes to address both issues of numerical dispersion and geometrical modelling. It involves a rigorous mathematical framework based on concepts from topology and differential geometry. Particularly, it involves a construction of a discrete analog to the calculus of differential forms. It should be noted that the use of differential forms, and their lattice counterparts, is well known within the field of algebraic topology. However, the original contribution here lies in the introduction of a metric onto the lattice. It is with the metric that the adjoint exterior derivative may be defined, which is required for most physical systems not the least of importance being Maxwell’s equations.