Gauss Hermite Research Articles

Fast and simple method for pricing exotic options using Gauss–Hermite quadrature on a cubic spline interpolation

There is a vast literature on numerical valuation of exotic options using Monte Carlo (MC), binomial and trinomial trees, and finite difference methods. When transition density of the underlying asset or its moments are known in closed form, it can be convenient and more efficient to utilize direct integration methods to calculate the required option price expectations in a backward time-stepping algorithm. This paper presents a simple, robust and efficient algorithm that can be applied for pricing many exotic options by computing the expectations using Gauss–Hermite integration quadrature applied on a cubic spline interpolation. The algorithm is fully explicit but does not suffer the inherent instability of the explicit finite difference counterpart. A "free" bonus of the algorithm is that it already contains the function for fast and accurate interpolation of multiple solutions required by many discretely monitored path dependent options. For illustrations, we present examples of pricing a series of American options with either Bermudan or continuous exercise features, and a series of exotic path-dependent options of target accumulation redemption note (TARN). Results of the new method are compared with MC and finite difference methods, including some of the most advanced or best known finite difference algorithms in the literature. The comparison shows that, despite its simplicity, the new method can rival with some of the best finite difference algorithms in accuracy and at the same time it is significantly faster. Virtually the same algorithm can be applied to price other path-dependent financial contracts such as Asian options and variable annuities.

A new class of nonlinear Rauch–Tung–Striebel cubature Kalman smoothers

In this paper, a new Rauch–Tung–Striebel type of nonlinear smoothing method is proposed based on a class of high-degree cubature integration rules. This new class of cubature Kalman smoothers generalizes the conventional third-degree cubature Kalman smoother using the combination of Genz׳s or Mysovskikh׳s high-degree spherical rule with the moment matching based arbitrary-degree radial rule, which considerably improves the estimation accuracy. A target tracking problem is utilized to demonstrate the performance of this new smoother and to compare it with other Gaussian approximation smoothers. It will be shown that this new cubature Kalman smoother enhances the filtering accuracy and outperforms the extended Kalman smoother, the unscented Kalman smoother, and the conventional third-degree cubature Kalman smoother. It also maintains close performance to the Gauss–Hermite quadrature smoother with much less computational cost.

The improved Hagedorn wavepacket method for semiclassical Schrödinger equation

The Hagedorn wavepacket method is an important numerical method for solving the semiclassical time-dependent Schrödinger equation. In this paper, a new semi-discretization in space is obtained by wavepacket operator. In a sense, such semi-discretization is equivalent to the Hagedorn wavepacket method, but this discretization is more intuitive to show the advantages of wavepacket methods. Moreover, we apply the multi-time-step method and the Magnus-expansion to obtain the improved algorithms in time-stepping computation. The improved algorithms are of the Gauss–Hermite spectral accuracy to approximate the analytical solution of the semiclassical Schrödinger equation. And for the given accuracy, the larger time stepsize can be used for the higher oscillation in the semiclassical Schrödinger equation. The superiority is shown by the error estimation and numerical experiments.

Bearings-only maneuvering target tracking based on truncated quadrature Kalman filtering

In this paper, a novel bearings-only maneuvering target tracking algorithm based on truncated quadrature Kalman filtering (TQKF) is proposed. In the proposed method, when the target maneuvers, in order to reduce the effect on performance duo to the increasing variance of the prior distribution, a modified prior distribution based on the current measurement is proposed. In the update step, the first two moments of the modified prior distribution is approximately estimated based on the least square estimation method and Gauss–Hermite quadrature rule, and the posterior distribution is jointly updated by using the prior distribution and the modified prior distribution. Moreover, in order to adaptively choose the estimated results obtained by the prior PDF and the truncated prior PDF, a fuzzy logic approach in which a Gaussian membership function is employed is proposed to determine the weight α. Finally, the experiment results show that the proposed algorithm results in more accurate tracking than the existing one, namely, the unscented Kalman filter (UKF), the quadrature Kalman filter (QKF), interact multiple model extended Kalman filter (IMMEKF) and multiple model Rao–Blackwellized particle filter (MMRBPF).

Comparison of algorithms for Doppler broadening pointwise tabulated cross sections

In this paper, three algorithms for Doppler broadening pointwise tabulated cross sections are compared on the basis of speed and accuracy. The use of a Gauss–Legendre quadrature for numerical integration on subintervals of the convolution integral for which the cross section is smooth was found to reduce run time by 48% compared to the exact method used in NJOY and other cross section processing codes. However, a comparable decrease in run time was also achieved simply by using a more efficient implementation of the erfc function. The use of a Gauss–Hermite quadrature for performing the convolution integral resulted in an order of magnitude decrease in run time for broadening. The accuracy of the Gauss–Hermite quadrature was also studied, and it was found that quadrature orders greater than 15 reach an asymptotic level of error. Lastly, an on-the-fly implementation of Doppler broadening in OpenMC using a Gauss–Hermite quadrature resulted in at least a factor of 15 increase in run time relative to a simulation without Doppler broadening for the BEAVRS benchmark problem, underscoring the need for advanced methods for on-the-fly temperature treatment for Monte Carlo codes.

The large homogeneous portfolio approximation with a two-factor Gaussian copula and random recovery rate

In this paper we consider the large homogeneous portfolio (LHP) approximation with a two-factor Gaussian copula and random recovery rate. In addition, we assume that the earlier the default occurs, the less the asset recovers; in other words, random recovery rate and individual default times have a positive rank correlation. Under the LHP assumption, the conditional cumulative loss of the reference portfolio is approximated by the product of loss given default and conditional default probability. In order to derive semi-analytic formulas for the loss distribution and the expected tranche loss, we use a Gaussian two-factor model and assume that the recovery rate depends on one systematic factor. In addition, we consider stochastic correlation for a better fit to credit default swap index tranches. The derived semi-analytic formula only involves integration with respect to the standard normal density and can be computed by Gauss–Hermite quadrature. Numerical tests show that the two-factor model with stochastic correlation and random recovery fits iTraxx tranche premiums better than other correlation or recovery assumptions under the Gaussian LHP framework. We also apply our model to credit risk assessment such as value-at-risk of the loss distribution.

Integration in Hermite spaces of analytic functions

We study integration in a class of Hilbert spaces of analytic functions defined on the Rs. The functions are characterized by the property that their Hermite coefficients decay exponentially fast. We use Gauss–Hermite integration rules and show that the errors of our algorithms decay exponentially fast. Furthermore, we study tractability in terms of s and logε−1 and give necessary and sufficient conditions under which we achieve exponential convergence with EC-weak, EC-polynomial, and EC-strong polynomial tractability.

MDGHM-SURF: A robust local image descriptor based on modified discrete Gaussian–Hermite moment

This paper proposes a novel family of local feature descriptors, a variant of the speed up robust features (SURF) descriptor, which is capable of demonstrably better performance. The conventional SURF descriptor is an efficient implementation of the SIFT descriptor. Although the SURF descriptor can represent the nature of the underlying image pattern, it is still sensitive to more complicated deformations such as large viewpoint and rotation changes. To solve this problem, our family of descriptors, called MDGHM-SURF, is based on the modified discrete Gaussian–Hermite moment (MDGHM), which devises a movable mask to represent the local feature information of non-square images. Whereas conventional SURF uses first-order derivatives, MDGHM-SURF uses MDGHM, which offers more feature information than first-order derivative-based local descriptors such as SURF and SIFT. Consequently, by redefining the conventional SURF descriptor using MDGHM, MDGHM-SURF can extract more distinctive features than conventional SURF. The results of evaluations conducted with six types of deformations indicate that our proposed method outperforms the matching accuracy of other SURF related algorithms.

Quadrature Kalman filter–based state of charge estimation for lithium-ion battery

This article presents an approach to lithium-ion battery state of charge estimation based on the quadrature Kalman filter. Among the existing state of charge estimation approaches, the extended Kalman filter–based state of charge and unscented filter–based state of charge algorithms are influenced by the linearization or the solution of sigma points. The proposed quadrature Kalman filter–based state of charge algorithm avoids these problems. Specifically, the battery system equations are built based on the second-order resistance–capacitance equivalent circuit model, and the parameters are identified according to the hybrid pulse power characterization discharging test. Then, the quadrature points and corresponding weights are defined by the Gauss–Hermite quadrature rule, and the Kronecker tensor product is adopted to solve the points of multivariate. In addition, the stability of quadrature Kalman filter–based state of charge is verified. Finally, the simulation is carried out under the discharging and urban dynamometer driving schedule condition, which demonstrates that the quadrature Kalman filter–based state of charge algorithm has a better performance compared with extended Kalman filter–based state of charge and unscented filter–based state of charge.

Multiple sensor estimation using a new fifth-degree cubature information filter

In this paper, a new class of cubature information filters (CIF) is proposed for multiple sensor estimation. This new CIF generalizes the conventional third-degree CIF to attain higher estimation accuracy using a class of higher-degree cubature integration rules including the fifth-degree Mysovskikh’s spherical rule and the arbitrary degree radial rule. The statistical linear error propagation method is utilized to incorporate the high-degree cubature rule into the extended information filtering framework such that more accurate estimation can be achieved than the extended information filter. It also outperforms the unscented information filter as well as the particle filter. In addition, the high-degree CIF maintains close performance to the Gauss–Hermite information filter but uses significantly fewer quadrature points. As a result, the curse of the dimensionality problem existing in the tensor product-based Gauss–Hermite information filter can be greatly alleviated. Besides the improved estimation accuracy and computation efficiency, the high-degree CIF also exhibits the desirable robustness under unknown noise statistics. The proposed CIF is compared with other information filters via the state estimation of a two-phase permanent magnet synchronous motor and a target tracking problem, and demonstrates the best performance.

Multidimensional Item Response Model for Nominal Variables

This article describes a multidimensional generalization of the nominal categories model that serves to estimate factorial models from nominal and ordinal observed responses, and includes a structural model for latent variables that distinguishes between endogenous and exogenous factors. The model includes a scale parameter for each response category in each factor. Item parameters relate the logit between categories to the vector of latent variables. The inferential framework is marginal maximum likelihood, implemented via static and adaptive Gauss–Hermite quadrature and Monte Carlo EM. The properties of estimators are investigated in a simulation study. An example with real data illustrates the utility of the model in analyzing local dependencies in testlets composed of multiple-choice items that are clustered in several groups around a common theme.

Static Hedging of Geometric Average Asian Options with Standard Options

In this article, we first establish a theorem that represents the price of an Asian option in terms of standard European options with a shorter term and different strikes. Then using Gauss–Hermite numerical integration, we discretize our theorem so as to use Monte Carlo simulation to examine the error of the static hedging under the Black–Scholes model and the Merton jump-diffusion model. For ease of comparison, we also provide the error of the dynamic hedging. The numerical results show that the static hedging strategy performs better than the dynamic one under both models.

A Frequency-Domain Approach for the $$\hbox {P}_1$$ P 1 Approximation of Time-Dependent Radiative Transfer

We propose a new frequency-domain method to solve the simplified \(\text {P}_1\) approximation of time-dependent radiative transfer equations. The method employs the Fourier transform and consists of two stages. In the first stage the equations are transformed into an elliptic problem for the frequency variables. The numerical solutions of this problem are approximated using a Galerkin projection method based on the tensor-product B-spline interpolants. In the second stage a Gauss–Hermite quadrature procedure is proposed for the computation of the inverse Fourier transform to recover the numerical solutions of the original simplified \(\text {P}_1\) problem. The method avoids the discretization of the time variable in the considered system and it accurately resolves all time scales in radiative transfer regimes. Several test examples are used to verify high accuracy, effectiveness and good resolution properties for smooth and discontinuous solutions.

Method for estimating capacity and predicting remaining useful life of lithium-ion battery

Reliability of lithium-ion (Li-ion) rechargeable batteries used in implantable medical devices has been recognized as of high importance from a broad range of stakeholders, including medical device manufacturers, regulatory agencies, physicians, and patients. To ensure Li-ion batteries in these devices operate reliably, it is important to be able to assess the capacity of Li-ion battery and predict the remaining useful life (RUL) throughout the whole life-time. This paper presents an integrated method for the capacity estimation and RUL prediction of Li-ion battery used in implantable medical devices. A state projection scheme from the author’s previous study is used for the capacity estimation. Then, based on the capacity estimates, the Gauss–Hermite particle filter technique is used to project the capacity fade to the end-of-service (EOS) value (or the failure limit) for the RUL prediction. Results of 10years’ continuous cycling test on Li-ion prismatic cells in the lab suggest that the proposed method achieves good accuracy in the capacity estimation and captures the uncertainty in the RUL prediction. Post-explant weekly cycling data obtained from field cells with 4–7 implant years further verify the effectiveness of the proposed method in the capacity estimation.

Multilevel mixed effects parametric survival models using adaptive Gauss-Hermite quadrature with application to recurrent events and individual participant data meta-analysis.

Multilevel mixed effects survival models are used in the analysis of clustered survival data, such as repeated events, multicenter clinical trials, and individual participant data (IPD) meta-analyses, to investigate heterogeneity in baseline risk and covariate effects. In this paper, we extend parametric frailty models including the exponential, Weibull and Gompertz proportional hazards (PH) models and the log logistic, log normal, and generalized gamma accelerated failure time models to allow any number of normally distributed random effects. Furthermore, we extend the flexible parametric survival model of Royston and Parmar, modeled on the log-cumulative hazard scale using restricted cubic splines, to include random effects while also allowing for non-PH (time-dependent effects). Maximum likelihood is used to estimate the models utilizing adaptive or nonadaptive Gauss-Hermite quadrature. The methods are evaluated through simulation studies representing clinically plausible scenarios of a multicenter trial and IPD meta-analysis, showing good performance of the estimation method. The flexible parametric mixed effects model is illustrated using a dataset of patients with kidney disease and repeated times to infection and an IPD meta-analysis of prognostic factor studies in patients with breast cancer. User-friendly Stata software is provided to implement the methods.

Extended biologically inspired model for object recognition based on oriented Gaussian–Hermite moment

Hierarchical Model and X (HMAX) presents a biologically inspired model for robust object recognition. The HMAX model, based on the mechanisms of the visual cortex, can be described as a four-layer structure. Although the performance of HMAX in object recognition is robust, it has been shown to be sensitive to rotation, which limits the model׳s performance. To alleviate this limitation, we propose an Oriented Gaussian–Hermite Moment-based HMAX (OGHM-HMAX). In contrast to HMAX which uses a Gabor filter for local feature representation, OGHM-HMAX employs the Oriented Gaussian–Hermite Moment (OGHM), which is a local representation method that represents features and is robust against distortions. OGHM is an extension of the modified discrete Gaussian–Hermite moment (MDGHM). To show the effectiveness of the proposed method, experimental studies on object categorization are conducted on the CalTech101, CalTech5, Scene13 and GRAZ01 databases. Experimental results demonstrate that the performance of OGHM-HMAX is a significant improvement on that of the conventional HMAX.

Generalized Linear Mixed Models for Binary Data: Are Matching Results from Penalized Quasi-Likelihood and Numerical Integration Less Biased?

BackgroundOver time, adaptive Gaussian Hermite quadrature (QUAD) has become the preferred method for estimating generalized linear mixed models with binary outcomes. However, penalized quasi-likelihood (PQL) is still used frequently. In this work, we systematically evaluated whether matching results from PQL and QUAD indicate less bias in estimated regression coefficients and variance parameters via simulation.MethodsWe performed a simulation study in which we varied the size of the data set, probability of the outcome, variance of the random effect, number of clusters and number of subjects per cluster, etc. We estimated bias in the regression coefficients, odds ratios and variance parameters as estimated via PQL and QUAD. We ascertained if similarity of estimated regression coefficients, odds ratios and variance parameters predicted less bias.ResultsOverall, we found that the absolute percent bias of the odds ratio estimated via PQL or QUAD increased as the PQL- and QUAD-estimated odds ratios became more discrepant, though results varied markedly depending on the characteristics of the datasetConclusionsGiven how markedly results varied depending on data set characteristics, specifying a rule above which indicated biased results proved impossible.This work suggests that comparing results from generalized linear mixed models estimated via PQL and QUAD is a worthwhile exercise for regression coefficients and variance components obtained via QUAD, in situations where PQL is known to give reasonable results.

GAUSS QUADRATURES – THE KEYSTONE OF LATTICE BOLTZMANN MODELS

In this paper, we compare two families of Lattice Boltzmann (LB) models derived by means of Gauss quadratures in the momentum space. The first one is the HLB (N;Qx,Qy,Qz) family, derived by using the Cartesian coordinate system and the Gauss–Hermite quadrature. The second one is the SLB (N;K,L,M) family, derived by using the spherical coordinate system and the Gauss–Laguerre, as well as the Gauss–Legendre quadratures. These models order themselves according to the maximum order N of the moments of the equilibrium distribution function that are exactly recovered. Microfluidics effects (slip velocity, temperature jump, as well as the longitudinal heat flux that is not driven by a temperature gradient) are accurately captured during the simulation of Couette flow for Knudsen number (kn) up to 0.25.

Simplified Gauss Hermite Filter Based on Sparse Grid Gauss Hermite Quadrature

In order to improve estimation accuracy of nonliear system with linear measurement model, simplifed gauss hermite filter based on sparse grid gauss hermite quadrature (SGHF) is proposed. Comparing to conventional Gauss-Hermite filter (GHF) based on tensor product gauss quadrature rule, simplified SGHF not only maintains GHF's advantage of precission controllable, high estimation accuracy, but also relieves the curse of dimension problem by reduce the number of gaussian intrgration points to the number of sigma points that scaled unscented transform uses. Theoretical analysis and experimental results show that estimation base on new filter performs significantly beter than extended kalman filter (EKF), and slightly better than unscented kalman filter (UKF) on estimation accuracy and convergence speed, and computational burden is significantly reduced comparing with traditional GHKF. DOI: http://dx.doi.org/10.11591/telkomnika.v11i12.3693 Full Text: PDF

A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss–Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed of hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying subsonic slightly rarefied gas flows.