Truncation Of Lattices Research Articles

American Option Pricing: Optimal Lattice Models and Multidimensional Efficiency Tests

This paper introduces a set of lattice techniques with a view to accelerating computation time and improving the accuracy of American Option valuation. Estimation speed can be enhanced through developing a parsimonious early exercise boundary search routine combined with reliance on dynamic memory and lattice truncation. Furthermore, Black-Scholes and Richardson extrapolation modifications to the lattices can also be applied individually and/or together to improve the accuracy of lattices. In this paper, we investigate the improvement introduced by obtaining the best combination of varying features. By introducing these techniques to the Leisen-Reimer and Tian binomial model, we can achieve a level of accuracy and efficiency combined that surpass analytical analogues prominent in the literature. Significantly, the Leisen-Reimer and Tian structure can accommodate arbitrary improvements in accuracy by simply increasing the density of their own mesh. Analytical methods generally do not afford much scope for optimising speed and efficiency in a granular fashion. We also compare efficient lattice models with analytical formulae for pricing different groups of options according to the deepness of American quality and the moneyness of the options. The appropriate model is recommended for pricing particular types of the options. Lattices importantly afford an explicit trade-off locus between accuracy and speed that can be navigated according to predetermined precision tolerance levels and option types. This should have practical relevance to trading platforms that require real-time estimates of implied volatility.

A q-Virasoro algebra at roots of unity, free fermions, and Temperley-Lieb hamiltonians

In this work, we consider the q-deformation of the Virasoro algebra [M. Chaichian and P. Presnajder, Phys. Lett. B 277, 109 (1992)] expressed in terms of free fermions, and we then realize this algebra, when the deformation parameter is a root of unity, on the lattice in a truncated form in terms of the Clifford algebra of Γ matrices. For this finite size truncation, the commutation relations of the Deformed algebra hold exactly albeit without central extension term. We then study the relations existing between this lattice truncation of the deformed Virasoro algebra at roots of unity and the tower of commuting Temperley-Lieb hamiltonians introduced in a previous work.

Ab Initio Thermodynamics and First-Principles Microkinetics for Surface Catalysis

Ab initio thermodynamics and first-principles microkinetic simulations have become standard tools in research on model catalysts. Complementing dedicated in situ experiments these techniques contribute to our evolving mechanistic understanding, in particular of a reaction-induced dynamical evolution of the working catalyst surface. This topical review surveys the methodological foundations and ongoing developments of both techniques, and specifically illustrates the type of insights they provide in the context of in situ model catalyst studies. This insight points at substantial deviations from the standard picture that analyzes catalytic function merely in terms of properties of and processes at active sites as they emerge from a crystal lattice truncation of the nominal catalyst bulk material.

Exceptional points and Bloch oscillations in non-Hermitian lattices with unidirectional hopping

The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open) and for the ring lattice geometries the spectrum is complex, lattice truncation makes the spectrum real. However, an exceptional point of order equal to the number of lattice sites emerges. When a homogeneous dc force is applied to the lattice, in all cases an equally-spaced real Wannier-Stark ladder spectrum is obtained, corresponding to periodic oscillatory dynamics in real space. Possible physical realizations of non-Hermitian lattices with unidirectional hopping are briefly discussed.

Dynamic localization in Glauber–Fock lattices

Glauber–Fock lattices refer to a special class of semi-infinite tight-binding lattices with inhomogeneous hopping rates which are found in certain simple solid state, quantum optics and quantum field theoretical models. Here it is shown that dynamic localization, i.e. suppression of quantum diffusion and periodic quantum self-imaging by an external sinusoidal force (Dunlap and Kenkre 1986 Phys. Rev. B 34 3625), can be exactly realized in Glauber–Fock lattices, in spite of inhomogeneity of hopping rates and lattice truncation.

Lattice Quantization of Phases for Equal Gain Transmission

Equal gain transmission (EGT) requires only the feedback of phase information and is known to enjoy several implementational advantages compared to maximum ratio transmission (MRT). This paper investigates the use of lattice quantization for quantizing phases in EGT systems. In practice, the feedback rate is finite and the lattices need to be truncated to have finite code rate. The truncation of lattices results in boundary effect; hence, earlier results developed for untruncated lattices, e.g., fast algorithm and mean-square error (MSE) analysis, cannot be applied directly. Exploiting the fact that the quantized signals are phases, we show how to quantize using truncated lattices so that there is no boundary effect. Moreover, we propose efficient methods to convert between lattice codewords and binary digits so that the lattice quantization can be used in practical systems. Furthermore, the mean-square quantization error is analyzed for several useful truncated lattices. We also show how to analyze the signal-to-noise ratio (SNR) loss when the proposed quantization is applied to an EGT system.

Probing the Local Site Environments in Mn:CdSe Quantum Dots

A quantum dot (QD) contains a well-defined surface passivated by ligands and a bulklike core. The effect of surface passivation and lattice truncation on local structural and electronic microenvironments within a CdSe QD is an area of active research. Selectively probing the local microenvironments that exist at the surface and core of the QD is difficult but can be achieved by use of a Mn(II) impurity ion doped into a CdSe QD using electron paramagnetic resonance (EPR). By use of high frequency EPR (HF-EPR) spectroscopy, the site-dependent perturbation experienced for Mn(II) incorporated as a guest ion into CdSe QDs allows the distinguishing of two unique microenvironments within the QD, namely, an unperturbed core and an electronically distorted surface. Analysis of the Landé g-factor, hyperfine constant (A), and the distribution of g and D (Δg, ΔD) allows the local microenvironments within CdSe to be probed as a function of size, ligand passivation, and site of Mn(II) incorporation.

A Unified FDTD Lattice Truncation Method for Dispersive Media Based on Periodic Boundary Conditions

A unified treatment for the truncation of finite-difference time-domain lattices, applicable to dispersive and conductive media alike, is proposed. The method is based on periodic boundary conditions, hence necessitating that the medium under study be periodic along the direction of truncation. When this condition (which is satisfied in many practical cases) is met, a much simpler but equally effective alternative to the PML is provided by the combination of periodic boundaries with an array-scanning method. The proposed formulation does not need any additional auxiliary variables when applied to dispersive media, unlike the PML. Applications include a Bragg filter and a negative-refractive-index super lens.

An efficient time‐domain method analysis of quasiperiodic structures by a finite‐element tearing and interconnecting algorithm

AbstractBased on the finite‐element approximation and nonoverlapping domain decomposition, an efficient tearing and interconnecting algorithm of the finite‐element time‐domain method is presented. Using edge‐based finite element, the method divides the computation domain into several nonoverlapping subdomains and computes electric fields in each subdomain by solving the vector curl–curl wave equation. Adjacent subdomains are related to each other by Dirichlet and Neumann boundary condition. By use of the repetition of the quasiperiod structures, the method further reduces the memory requirement and CPU time. The time‐dependent formulation employs an unconditional stable time‐integration method based on the Newmark‐beta method. An anisotropic perfectly matched layers absorbing material terminated by the first‐order boundary condition is applied for the truncation of lattices. Numerical results demonstrate that our proposed method is extremely efficient for the time‐domain analysis of the quasiperiodic structures, such as electromagnetic band gap structures.© 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1072–1078, 2010; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/mop.25100

Bound states in the continuum in a single-level Fano-Anderson model

Bound states in the continuum (BIC) are shown to exist in a single-level Fano-Anderson model with a colored interaction between the discrete state and a tight-binding continuum, which may describe mesoscopic electron or photon transport in a semi-infinite one-dimensional lattice. The existence of BIC is explained in the lattice realization as a boundary effect induced by lattice truncation.

A Reflectionless Sponge Layer Absorbing Boundary Condition for the Solution of Maxwell's Equations with High-Order Staggered Finite Difference Schemes

We develop, implement, and demonstrate a reflectionless sponge layer for truncating computational domains in which the time-dependent Maxwell equations are discretized with high-order staggered nondissipative finite difference schemes. The well-posedness of the Cauchy problem for the sponge layer equations is proved, and the stability and accuracy of their discretization is analyzed. With numerical experiments we compare our approach to classical techniques for domain truncation that are based on second- and third-order physically accurate local approximations of the true radiation condition. These experiments indicate that our sponge layer results in a greater than three orders of magnitude reduction of the lattice truncation error over that afforded by such classical techniques. We also show that our strongly well-posed sponge layer performs as well as the ill-posed split-field Berenger PML absorbing boundary condition. Being an unsplit-field approach, our sponge layer results in ∼25% savings in computational effort over that required by a split-field approach.

Three-dimensional computer modeling of electromagnetic fields: A global lookback lattice truncation scheme

A new lattice truncation scheme for the finite difference time domain approach to the solution of Maxwell's equations has been developed. The problem space is truncated near the sources and the field components on its boundary are generated from those field values known at retarded times on an interior surface one cell from it with an integral representation of the electromagnetic field. The numerical implementation of this global lookback scheme is discussed. Examples which have been used to determine its characteristics and its validity are given.

Application of the Finite-Difference Time-Domain Method to Sinusoidal Steady-State Electromagnetic-Penetration Problems

A numerical method for predicting the sinusoidal steady-state electromagnetic fields penetrating an arbitrary dielectric or conducting body is described here. The method employs the finite-difference time-domain (FD-TD) solution of Maxwell's curl equations implemented on a cubic-unit-cell space lattice. Small air-dielectric loss factors are introduced to improve the lattice truncation conditions and to accelerate convergence of cavity interior fields to the sinusoidal steady state. This method is evaluated with comparison to classical theory, method-of-moment frequency-domain numerical theory, and experimental results via application to a dielectric sphere and acylindrical metal cavity with an aperture. Results are also given for a missile-like cavity with two different types of apertures illuminated by an axial-incidence plane wave.