Asymptotic Terms Research Articles

PRICES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS IN LÉVY-DRIVEN MODELS, NEAR BARRIER

We calculate the leading term of asymptotics of the prices of barrier options and first-touch digitals near the barrier for wide classes of Lévy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carr's randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Lévy processes commonly used in option pricing.

Generalized kernel regression estimator for dependent size-biased data

This paper considers nonparametric regression estimation in the context of dependent biased nonnegative data using a generalized asymmetric kernel. It may be applied to a wider variety of practical situations, such as the length and size biased data. We derive theoretical results using a deep asymptotic analysis of the behavior of the estimator that provides consistency and asymptotic normality in addition to the evaluation of the asymptotic bias term. The asymptotic mean squared error is also derived in order to obtain the optimal value of smoothing parameters required in the proposed estimator. The results are stated under a stationary ergodic assumption, without assuming any traditional mixing conditions. A simulation study is carried out to compare the proposed estimator with the local linear regression estimate.

Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson’s integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd, or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.

Leading asymptotic term of the solution to the hamer equation

Leading asymptotic term of the solution to the hamer equation

Scattering and radiation of waves by a thin toroidal tube with semi-transparent walls: an asymptotic model

The work deals with a scalar model of wave scattering and radiation by a thin toroidal tube with varying longitudinal ‘conductivity’. The walls of the toroidal tube are assumed to be semi-transparent for the waves. The corresponding boundary conditions represent a direct scalar analog of the electromagnetic boundary conditions describing electromagnetic properties of carbon nanotubes recently discussed in the literature. The method of matching of asymptotic expansions is applied to study the model problem provided that the length of the tube is much greater than its thickness and the ‘conductivity’ is high. The leading term of the formal asymptotics is developed and analyzed. The scattering amplitude is then considered and approximate resonances are discussed.

An extension of the Hardy-Ramanujan circle method and applications to partitions without sequences

We develop a generalized version of the Hardy-Ramanujan "circle method" in order to derive asymptotic series expansions for the products of modular forms and mock theta functions. Classical asymptotic methods (including the circle method) do not work in this situation because such products are not modular, and in fact, the "error integrals" that occur in the transformations of the mock theta functions can (and often do) make a significant contribution to the asymptotic series. The resulting series include principal part integrals of Bessel functions, whereby the main asymptotic term can also be identified. To illustrate the application of our method, we calculate the asymptotic series expansion for the number of partitions without sequences. Andrews showed that the generating function for such partitions is the product of the third order mock theta function $\chi$ and a (modular) infinite product series. The resulting asymptotic expansion for this example is particularly interesting because the error integrals in the modular transformation of the mock theta function component appear in the exponential main term.

Mellin and Green operators of the corner calculus

We study spaces of parameter-dependent Mellin plus Green operators occurring as operator-valued symbols of the pseudo-differential calculus on a manifold with corners. Here we mainly focus on holomorphic families in the complex Mellin covariable belonging to the corner axis variable \({t\in\mathbb{R}_{+},}\) while we impose the full asymptotic information with respect to the subordinate cone axis variable \({r\in\mathbb{R}_{+},}\) here in terms of continuous asymptotics.

Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials

We consider the unperturbed operator H_0 : = (-i \nabla - A)^2 + W , self-adjoint in L^2(\mathbb{R}^2) . Here A is a magnetic potential which generates a constant magnetic field b>0 , and the edge potential W is a non-decreasing non constant bounded function depending only on the first coordinate x \in \mathbb{R} of (x,y) \in \mathbb{R}^2 . Then the spectrum of H_0 has a band structure and is absolutely continuous; moreover, the assumption \lim_{x \to \infty}(W(x) - W(-x)) < 2b implies the existence of infinitely many spectral gaps for H_0 . We consider the perturbed operators H_{\pm} = H_0 \pm V where the electric potential V \in L^{\infty}(\mathbb{R}^2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H_\pm in the spectral gaps of H_0 . We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to V . Further, we restrict our attention on perturbations V of compact support and constant sign. We establish a geometric condition on the support of V which guarantees the finiteness of the eigenvalues of H_{\pm} in any spectral gap of H_0 . In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of H_+ (resp. H_- ) to the left (resp. right) edge of a given spectral gap, is Gaussian.

Spectral Asymptotics of Self-Adjoint Fourth Order Differential Operators with Eigenvalue Parameter Dependent Boundary Conditions

A fourth-order regular ordinary differential operator with eigenvalue dependent boundary conditions is considered. This problem is realized by a quadratic operator pencil with self-adjoint operators. The location of the eigenvalues is discussed and the first four terms of the eigenvalue asymptotics are evaluated explicitly.

A dispersive regularization of the modulational instability of stratified gravity waves

We derive high-order corrections to a modulation theory for the propagation of internal gravity waves in a density-stratified fluid with coupling to the mean flow. The methodology we use allows for strong modulations of wavenumber and mean flow, extending previous approaches developed for the quasi-monochromatic regime. The wave mean flow modulation equations consist of a system of nonlinear conservation laws that may be hyperbolic, elliptic or of mixed type. We investigate the regularizing properties of the asymptotic correction terms in the case when the system becomes unstable and ill-posed due to a change of type (loss of hyperbolicity). A linear analysis reveals that the regularization by the added correction terms does so by introducing a short-wave cut-off of the unstable wavenumbers. We perform various numerical experiments that confirm the regularizing properties of the correction terms, and show that the growth of unstable modes is tempered by nonlinearity. We also find an excellent agreement between the solution of the corrected modulation system and the modulation variables extracted from the numerical solution of the nonlinear Boussinesq equations.

Contact with stick zone between an indenter and a thin incompressible layer

Abstract The dominant asymptotic term for the indentation of a thin elastic incompressible layer by an axisymmetric rigid indenter is considered. Complete adhesion is supposed everywhere in the contact area or else in a given inner region surrounded by an annular frictionless zone. Both the problems are formulated in the form of systems of coupled dual integral equations. Using operators transforming kernels of the Hankel transform into kernels of the Weber–Orr transform, the dual integral equations are reduced to systems of Fredholm integral equations of the second kind whose structures permit deriving asymptotic solutions. Simple expressions for the contact stresses, the penetration depth, and the contact radius in the case of an unknown contact area are obtained. Explicit formulae, derived for the flat and power law indenter profiles, allow us to analyze how stick and frictionless zones affect mechanical characteristics. Results manifest that the punch penetration exhibits strong sensitivity to contact conditions inspite of the fact that the radial traction is small. A conical indenter is less sensitive than flat-ended and spherical indenters.

Singular Perturbations of Curved Boundaries in Three Dimensions. The Spectrum of the Neumann Laplacian

We calculate the main asymptotic terms for eigenvalues, both simple and multiple, and eigenfunctions of the Neumann Laplacian in a three-dimensional domain \Omega(h) perturbed by a small (with diameter O(h) ) Lipschitz cavern \overline{\omega_h} in a smooth boundary \partial\Omega=\partial\Omega(0) . The case of the hole \overline{\omega_h} inside the domain but very close to the boundary \partial\Omega is under consideration as well. It is proven that the main correction term in the asymptotics of eigenvalues does not depend on the curvature of \partial\Omega while terms in the asymptotics of eigenfunctions do. The influence of the shape of the cavern to the eigenvalue asymptotics relies mainly upon a certain matrix integral characteristics like the tensor of virtual masses. Asymptotically exact estimates of the remainders are derived in weighted norms.

Remark on the phase shift in the Kuzmak-Whitham ansatz

We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for the nonlinear Klein-Gordon equation and for the Korteweg-de Vries equation. In this case, the leading asymptotic expansion term has the form X(S(x, t)/h+Φ(x, t), I(x, t), x, t) +O(h), where h ≪ 1 is a small parameter and the phase S}(x, t) and slowly changing parameters I(x, t) are to be found from the system of “averaged” Whitham equations. We obtain the equations for the phase shift Φ(x, t) by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially follows from papers by Haberman and collaborators, is that if we incorporate the phase shift Φ into the phase and adjust the parameter Ĩ by setting \( \tilde S \) = S +hΦ+O(h 2),Ĩ = I + hI 1 + O(h 2), then the functions \( \tilde S \)(x, t, h) and Ĩ(x, t, h) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term, which is X(\( \tilde S \)(x, t, h)/h, Ĩ(x, t, h), x, t) + O(h).

Finite-time relaxation of the solution of a nonlinear pseudoparabolic equation

A model equation is considered that describes the relaxation of an initial perturbation in a crystalline semiconductor in the case when its electrical conductivity depends nonlocally on the field. For certain initial parameters, the effect of finite-time “cooling” is proved to occur. For other parameters, the first term of the long-time asymptotics is found and the remainder of the asymptotic expansion is estimated.

On the Estimation of Mean Squared Prediction Error in Small Area Estimation

We critically examine the exact mean squared prediction error (MSPE) of an empirical best linear unbiased predictor (EBLUP) for certain special cases of the well-known Fay-Herriot model widely used in small area estimation. This study is useful in understanding the impact of the neglected higher order terms in standard second-order aymptotics in small area estimation problems. We argue that in certain cases the naive MSPE estimator, which is generally considered to have a serious underestimation problem as it ignores the uncertainty due to estimation of the variance component, may be adequate and may even overestimate the true MSPE. We propose a new weighted jackknife MSPE estimator as an alternative to the Taylor linearization method. Our proposed method is equivalent to other second-order unbiased MSPE estimators in terms of the standard second-order asymptotics. However, in our simulation study the proposed jackknife MSPE estimators appear to be better than the rival Taylor linearization method in terms of bias, especially when the regularity conditions that govern the second-order unbiasedness property are violated. We observe that for a special case of the Fay-Herriot model, our jackknife MSPE estimator is equivalent, in a higher order asymptotic sense, to a hierarchical Bayes solution under at priors on the hyperparameters.

Sharpening Wald-type inference in robust regression for small samples

The datasets used in statistical analyses are often small in the sense that the number of observations n is less than 5 times the number of parameters p to be estimated. In contrast, methods of robust regression are usually optimized in terms of asymptotics with an emphasis on efficiency and maximal bias of estimated coefficients. Inference, i.e., determination of confidence and prediction intervals, is proposed as complementary criteria. An analysis of MM-estimators leads to the development of a new scale estimate, the Design Adaptive Scale Estimate, and to an extension of the MM-estimate, the SMDM-estimate, as well as a suitable ψ -function. A simulation study shows and a real data example illustrates that the SMDM-estimate has better performance for small n / p and that the use the new scale estimate and of a slowly redescending ψ -function is crucial for adequate inference.

Asymptotics for rank and crank moments

Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to Bringmann and Mahlburg that refined a conjecture of Garvan. Garvan's conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The proof uses the Hardy-Ramanujan method to provide precise asymptotic estimates for rank and crank moments and their differences.

Long-time behavior of the solution to the mKdV equation with step-like initial data

We consider the modified Korteweg–de Vries equation on the line. The initial data is the pure step function, i.e. q(x, 0) = cr for x ⩾ 0 and q(x, 0) = cl for x < 0, where cl > cr > 0 are arbitrary real numbers. Long-time behavior of the solution to the mKdV equation in the case cl > cr = 0 and t → ∞ was studied recently in Kotlyarov and Minakov (2010 J. Math. Phys. 51 093506) where the structure of the compression wave was obtained in the form of a modulated elliptic wave. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t → −∞, i.e. we study the long-time dynamics of the rarefaction wave. Using the steepest descent method and the so-called g-function mechanism we deform the original oscillatory matrix Riemann–Hilbert problem to the explicitly solvable model forms and show that the solution of the initial-value problem has a different asymptotic behavior in different regions of the xt-plane. In the regions x < 6c2lt and x > 6c2rt the main term of asymptotics of the solution is equal to cl and cr, respectively. In the region the asymptotics of the solution tends to . An influence of the dispersion is also studied: the second term of the asymptotics is obtained in the region x > −6c2rt, where the background constant cr is perturbed by the self-similar vanishing wave.

A variant of importance splitting for rare event estimation

Importance splitting is a simulation technique to estimate very small entrance probabilities for Markov processes by splitting sample paths at various stages before reaching the set of interest. This can be done in many ways, yielding different variants of the method. In this context, we propose a new one, called fixed number of successes. We prove unbiasedness for the new and some known variants, because in many papers, the proof is based on an incorrect argument. Further, we analyze its behavior in a simplified setting in terms of efficiency and asymptotics in comparison to the standard variant. The main difference is that it controls the imprecision of the estimator rather than the computational effort. Our analysis and simulation examples show that it is rather robust in terms of parameter choice and we present a two-stage procedure which also yields confidence intervals.

Convergence of Price and Sensitivities in Carr's Randomization Approximation Globally and Near Barrier

Barrier options under wide classes of Lévy processes with exponentially decaying jump densities, including the variance gamma model, KoBoL and CGMY models, normal inverse Gaussian processes, and $\beta$-class, are studied. The leading term of asymptotics of the option price and the leading term of asymptotics in Carr's randomization approximation to the price are calculated as the price of the underlying approaches the barrier. We prove that the order of asymptotics is the same in both cases and that the asymptotic coefficient in the asymptotic formula for Carr's randomization approximation converges to the asymptotic coefficient for the price as the number of time steps $N\to+\infty$. Also, we justify Richardson extrapolation of arbitrary order. Similar results are derived for sensitivities and the leading terms of their asymptotics in Carr's randomization approximation. The convergence of prices and sensitivities is proved in appropriate weighted Hölder spaces.