Orthogonal Expansion Research Articles

Analyzing the Risks Embedded in Option Prices with rndfittool

This paper introduces a new computational tool for the analysis of the risks embedded in a set of prices of European-style options. The software enables the estimation of the risk-neutral density (RND) from the observed option prices by means of orthogonal polynomial expansions. Orthogonal polynomials offer a viable alternative to more standard techniques based on interpolation and estimation of the second-order derivatives of option prices. The app rndfittool is available on GitHub and its usage is illustrated with examples based on real data.

Efficient Approach to Description of Heat Transfer and Multicomponent Diffusion in Ionized Gases

The formulas for the heat fluxes of heavy components and electrons as well as the Stefan–Maxwell relations for the diffusion fluxes in amagnetic field are derived for amulticomponent two-temperature plasma with regard to the higher-order approximations in orthogonal expansions of the component distribution functions in Sonine polynomials. For the complex transport coefficients of heavy components and electrons exact formulas are obtained in the significantly simpler form as compared with the standard procedure of the Chapman–Enskog method with the minimum number of minimum-order matrix inversions.

Analytical modeling of oscillatory heat transfer in coated sorption beds

A novel analytical model that considers the thermal contact resistance (TCR) at the interface between the sorbent layer and heat exchanger (HEX) is developed to investigate the oscillatory heat transfer and performance of coated sorption beds. The governing energy equation is solved using an orthogonal expansion technique and closed-form relationships are obtained to calculate the temperature distribution inside the sorbent coating and HEX. In addition, a new gravimetric large pressure jump (GLAP) test bed is designed to measure the uptake of sorption material. Novel graphite coated sorption beds were prepared and tested in the GLAP test bed. The model was successfully validated with the measurements performed in the GLAP test bed. It is found that specific cooling power (SCP) of a sorption cooling system (SCS) enhances by increasing the sorbent thermal diffusivity and decreasing the TCR. For example, SCP of the sorption cooling system (SCS) can be enhanced from 90 to 900 (W/kg) by increasing the sorbent thermal diffusivity from 2.5e−7 to 5.25e−6 (m2/s) and decreasing the TCR from 4 to 0.3 (K/W). Moreover, the results show that SCP increases by reducing the HEX to sorbent thickness ratio (HSTR). Therefore, the proposed graphite coated sorption beds with high thermal diffusivity and low thickness are suitable for sorption cooling applications.

Trigonometric series in orthogonal expansions for density estimates of deep image features

In this paper we study image recognition tasks in which the images are described by high dimensional feature vectors extracted with deep convolutional neural networks and principal component analysis. In particular, we focus on the problem of high computational complexity of a statistical approach with non-parametric estimates of probability density implemented by the probabilistic neural network. We propose a novel statistical classification method based on the density estimators with orthogonal expansions using trigonometric series. It is shown that this approach makes it possible to overcome the drawbacks of the probabilistic neural network caused by the memory-based approach of instance-based learning. Our experimental study with Caltech-101 and CASIA WebFace datasets demonstrates that the proposed approach reduces the error rate by 1–5 % and increases the computational speed by 1.5 – 6 times when compared to the original probabilistic neural network for small samples of reference images.

Dimension-free Lp estimates for vectors of Riesz transforms associated with orthogonal expansions

An explicit Bellman function is used to prove a bilinear embedding theorem for operators associated with general multi-dimensional orthogonal expansions on product spaces. This is then applied to obtain $L^p,$ $1<p<\infty,$ boundedness of appropriate vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates for the $L^p$ norms of these Riesz transforms are both dimension-free and linear in $\max(p,p/(p-1)).$ The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.

Stochastic Collocation Methods via Minimisation of the Transformed L&lt;sub&gt;1&lt;/sub&gt;-Penalty

Abstract The sparse reconstruction of functions via a transformed ℓ1(TL1) minimisation is studied and theoretical results concerning recoverability and accuracy of such reconstruction from undersampled measurements are obtained. To identify the coefficients of sparse orthogonal polynomial expansions in uncertainty quantification, the method is combined with the stochastic collocation approach. The DCA-TL1 algorithm [37] is used in implementing the TL1 minimisation. Various numerical examples demonstrate the recoverability and efficiency of this method.

Semiparametric Single-Index Predictive Regression

This paper studies a semi-parametric single-index predictive regression model with multiple nonstationary predictors that exhibit co-movement behaviour. Orthogonal series expansion is employed to approximate the unknown link function in the model and the estimator is derived from an optimization under the constraint of identification condition for index parameter. The main finding includes two types of super-consistency rates of the estimators of the index parameter along two orthogonal directions in a new coordinate system. The central limit theorem is established for a plug-in estimator of the unknown link function. In the empirical studies, we provide evidence in favour of nonlinear predictability of the stock return using long term yield and treasury bill rate.

An Analytical Solution for Transient Heat Conduction in a Composite Slab with Time-Dependent Heat Transfer Coefficient

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of n layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. The composite slab, which has thermal contact resistance at n-1 interfaces, as well as an arbitrary initial temperature distribution and internal heat generation, convectively exchanges heat at the external boundaries with two different time-varying surroundings. To obtain the analytical solution, the shifting function method is first used, which yields new partial differential equations under conventional types of external boundary conditions. The solution for the derived differential equations is then obtained by means of an orthogonal expansion technique. Numerical calculations are performed for two composite slabs, whose heat transfer coefficient on the heated surface is either an exponential or a trigonometric function of time. The numerical results demonstrate the effects of temporal variations in the heat transfer coefficient on the transient temperature field of composite slabs.

Algebraic differentiators through orthogonal polynomials series expansions

ABSTRACTNumerical differentiation is undoubtedly a fundamental problem in signal processing and control engineering, due to its countless applications. The goal of this paper is to address this question within an algebraic framework. More precisely, we consider a noisy signal and its orthogonal polynomial series expansion. Through the algebraic identification of the series coefficients, we then propose algebraic differentiators for the signal. Examples based on Hermite and Laguerre polynomials illustrate these algebraic differentiators.

Option Pricing with Orthogonal Polynomial Expansions

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein–Stein, and Hull–White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier‐transform‐based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.

A general scheme for log-determinant computation of matrices via stochastic polynomial approximation

We study the approximation of determinant for large scale matrices with low computational complexity. This paper develops a generalized stochastic polynomial approximation frame as well as a stochastic Legendre approximation algorithm to calculate log-determinants of large-scale positive definite matrices based on the prior eigenvalue distributions. The generalized frame is implemented by weighted L2 orthogonal polynomial expansions with an efficient recursion formula and matrix–vector multiplications. So the proposed scheme is efficient both in computational complexity and data storage. Respective error bounds are given in theory which guarantee the convergence of the proposed algorithms. We illustrate the effectiveness of our method by numerical experiments on both synthetic matrices and counting spanning trees.

Fast polynomial transforms based on Toeplitz and Hankel matrices

Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive algorithms with an observed complexity ofO(Nlog2N)\mathcal {O}(N\log ^2 \! N), based on the fast Fourier transform, for converting coefficients of a degreeNNpolynomial in one polynomial basis to coefficients in another. Numerical results show that this approach is competitive with state-of-the-art techniques, requires no precomputational cost, can be implemented in a handful of lines of code, and is easily adapted to extended precision arithmetic.

Convergence of the Spectral Decomposition of a Function from the Class W p , m 1 G , p &gt; 1 $$ {W}_{p,m}^1(G),p&gt;1 $$ , in the Vector Eigenfunctions of a Differential Operator of the Third Order

We consider a third-order differential operator with matrix coefficients. The absolute and uniform convergence of the orthogonal expansion of a vector function from the class $$ {W}_{p,m}^1(G),p>1 $$ , in the vector eigenfunctions of this operator is studied and the rate of uniform convergence of this expansion on $$ \overline{G}=\left[0,1\right] $$ is estimated.

Marcinkiewicz–Zygmund type results in multivariate domains

We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in $${\mathbb{R}^d}$$ . These inequalities provide a basic tool for the discretization of the L p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains.

Orthogonal Eigenfunction Expansion Method for One-Dimensional Dual-Phase Lag Heat Conduction Problem With Time-Dependent Boundary Conditions

The separation of variables (SOV) can be used for all Fourier, single-phase lag (SPL), and dual-phase lag (DPL) heat conduction problems with time-independent source and/or boundary conditions (BCs). The Laplace transform (LT) can be used for problems with time-dependent BCs and sources but requires large computational time for inverse LT. In this work, the orthogonal eigenfunction expansion (OEEM) has been proposed as an alternate method for non-Fourier (SPL and DPL) heat conduction problem. However, the OEEM is applicable only for cases where BCs are homogeneous. Therefore, BCs of the original problem are homogenized by subtracting an auxiliary function from the temperature to get a modified problem in terms of a modified temperature. It is shown that the auxiliary function has to satisfy a set of conditions. However, these conditions do not lead to a unique auxiliary function. Therefore, an additional condition, which simplifies the modified problem, is proposed to evaluate the auxiliary function. The methodology is verified with SOV for time-independent BCs. The implementation of the methodology is demonstrated with illustrative example, which shows that this approach leads to an accurate solution with reasonable number of terms in the expansion.

Orthogonal expansions for VIX options under affine jump diffusions

In this work we derive new closed-form pricing formulas for VIX options in the jump-diffusion SVJJ model proposed by Duffie et al. [Econometrica, 2000, 68, 1343–1376]. Our approach is based on the classic methodology of approximating a density function with an orthogonal expansion of polynomials weighted by a kernel. Orthogonal expansions based on the Gaussian distribution, such as Edgeworth or Gram–Charlier expansions, have been successfully employed by a number of authors in the context of equity options. However, these expansions are not quite suitable for volatility or variance densities as they inherently assign positive mass to the negative real line. Here we approximate option prices via expansions that instead are based on kernels defined on the positive real line. Specifically, we consider a flexible family of distributions, which generalizes the gamma kernel associated with the classic Laguerre expansions. The method can be employed whenever the moments of the underlying variance distribution are known. It provides fast and accurate price computations, and therefore it represents a valid and possibly more robust alternative to pricing techniques based on Fourier transform inversions.

Rapidity dependence of transverse-momentum multiplicity correlations

Following previous work [1], we propose to analyze the rapidity dependence of transverse momentum and transverse-momentum multiplicity correlations. We demonstrate that the orthogonal polynomial expansion of the latter has the potential to discriminate between models of particle production.

Sparse Recovery via ℓq-Minimization for Polynomial Chaos Expansions

AbstractIn this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ1 and reweighted ℓ1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ1 and reweighted ℓ1 minimization.

Local orthogonal polynomial expansion for density estimation

ABSTRACTA local orthogonal polynomial expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localised expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalises straightforwardly to multivariate settings. By manner of construction, it is similar to local likelihood density estimation (LLDE). In the limit of small bandwidths, LOrPE functions as kernel density estimation (KDE) with high-order (effective) kernels inherently free of boundary bias, a natural consequence of kernel reshaping to accommodate endpoints. Consistency and faster asymptotic convergence rates follow. In the limit of large bandwidths LOrPE is equivalent to orthogonal series density estimation (OSDE) with Legendre polynomials, thereby inheriting its consistency. We compare the performance of LOrPE to KDE, LLDE, and OSDE, in a number of simulation studies. In terms of mean integrated squared error, the results suggest that with a proper balance of the two tuning parameters, bandwidth and degree, LOrPE generally outperforms these competitors when estimating densities with sharply truncated supports.

Efficient circuit representation of eddy-current fields

PurposeThis paper aims to discuss the relationship between the continued fraction form of the analytical solution in the frequency domain, the orthogonal function expansion and their circuit realization to derive an efficient representation of the eddy-current field in the conducting sheet and wire/cylinder. Effective frequency ranges of representations are analytically derived.Design/methodology/approachThe Cauer circuit representation is derived from the continued fraction form of analytical solution and from the orthogonal polynomial expansion. Simple circuit calculations give the upper frequency bounds where the truncated circuit and orthogonal expansion are applicable.FindingsThe Cauer circuit representation and the orthogonal polynomial expansions for the magnetic sheet in the E-mode and for the wire in the axial H-mode are derived. The upper frequency bound for the Cauer circuit is roughly proportional to N4 with N inductive elements, whereas the frequency bound for the finite element eddy-current analysis with uniform N elements is roughly proportional to N2.Practical implicationsThe Cauer circuit representation is expected to provide an efficient homogenization method because it requires only several elements to describe the eddy-current field over a wide frequency range.Originality/valueThe applicable frequency ranges are analytically derived depending on the conductor geometry and on the truncation types.