7,300 research outputs found
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
Physical states in the canonical tensor model from the perspective of random tensor networks
Tensor models, generalization of matrix models, are studied aiming for
quantum gravity in dimensions larger than two. Among them, the canonical tensor
model is formulated as a totally constrained system with first-class
constraints, the algebra of which resembles the Dirac algebra of general
relativity. When quantized, the physical states are defined to be vanished by
the quantized constraints. In explicit representations, the constraint
equations are a set of partial differential equations for the physical
wave-functions, which do not seem straightforward to be solved due to their
non-linear character. In this paper, after providing some explicit solutions
for , we show that certain scale-free integration of partition functions
of statistical systems on random networks (or random tensor networks more
generally) provides a series of solutions for general . Then, by
generalizing this form, we also obtain various solutions for general .
Moreover, we show that the solutions for the cases with a cosmological constant
can be obtained from those with no cosmological constant for increased .
This would imply the interesting possibility that a cosmological constant can
always be absorbed into the dynamics and is not an input parameter in the
canonical tensor model. We also observe the possibility of symmetry enhancement
in , and comment on an extension of Airy function related to the
solutions.Comment: 41 pages, 1 figure; typos correcte
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Trefftz Difference Schemes on Irregular Stencils
The recently developed Flexible Local Approximation MEthod (FLAME) produces
accurate difference schemes by replacing the usual Taylor expansion with
Trefftz functions -- local solutions of the underlying differential equation.
This paper advances and casts in a general form a significant modification of
FLAME proposed recently by Pinheiro & Webb: a least-squares fit instead of the
exact match of the approximate solution at the stencil nodes. As a consequence
of that, FLAME schemes can now be generated on irregular stencils with the
number of nodes substantially greater than the number of approximating
functions. The accuracy of the method is preserved but its robustness is
improved. For demonstration, the paper presents a number of numerical examples
in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering
of electromagnetic (acoustic) waves, and wave propagation in a photonic
crystal. The examples explore the role of the grid and stencil size, of the
number of approximating functions, and of the irregularity of the stencils.Comment: 28 pages, 12 figures; to be published in J Comp Phy
Theoretical Interpretations and Applications of Radial Basis Function Networks
Medical applications usually used Radial Basis Function Networks just as Artificial Neural Networks. However, RBFNs are Knowledge-Based Networks that can be interpreted in several way: Artificial Neural Networks, Regularization Networks, Support Vector Machines, Wavelet Networks, Fuzzy Controllers, Kernel Estimators, Instanced-Based Learners. A survey of their interpretations and of their corresponding learning algorithms is provided as well as a brief survey on dynamic learning algorithms. RBFNs' interpretations can suggest applications that are particularly interesting in medical domains
Renormalization procedure for random tensor networks and the canonical tensor model
We discuss a renormalization procedure for random tensor networks, and show
that the corresponding renormalization-group flow is given by the Hamiltonian
vector flow of the canonical tensor model, which is a discretized model of
quantum gravity. The result is the generalization of the previous one
concerning the relation between the Ising model on random networks and the
canonical tensor model with N=2. We also prove a general theorem which relates
discontinuity of the renormalization-group flow and the phase transitions of
random tensor networks.Comment: 23 pages, 5 figures; Comments on first order transitions and
discontinuity of RG added, and minor correction
Motion in Quantum Gravity
We tackle the question of motion in Quantum Gravity: what does motion mean at
the Planck scale? Although we are still far from a complete answer we consider
here a toy model in which the problem can be formulated and resolved precisely.
The setting of the toy model is three dimensional Euclidean gravity. Before
studying the model in detail, we argue that Loop Quantum Gravity may provide a
very useful approach when discussing the question of motion in Quantum Gravity.Comment: 30 pages, to appear in the book "Mass and Motion in General
Relativity", proceedings of the C.N.R.S. School in Orleans, France, eds. L.
Blanchet, A. Spallicci and B. Whitin
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