Size Of Jumps Research Articles

Valuing Options Under Nonlognormality Using Relaxed Lattices

Option pricing models based on an underlying lognormal distribution typically exhibit volatility smiles or smirks where the implied volatility varies by strike price. To adequately model the underlying distribution, a less restrictive model is needed. A relaxed binomial model is developed here that can account for the skewness of the underlying distribution and a relaxed trinomial model is developed that can account for the skewness and kurtosis of the underlying distribution. The new model incorporates the usual binomial and trinomial tree models as restricted special cases. Unlike previous flexible tree models, the size and probability of jumps are held constant at each node so only minor modifications in existing code for lattice models are needed to implement the new approach. Also, the new approach allows calculating implied skewness and implied kurtosis. Numerical results show that the relaxed binomial and trinomial tree models developed in this study are at least as accurate as tree models based on lognormality when the true underlying distribution is lognormal and substantially more accurate when the underlying distribution is not lognormal.

The Sequential Occupancy Problem through Group Throwing of Indistinguishable Balls

The occupancy problem is generalized to the case where instead of throwing one ball at a time, a fixed size group of indistinguishable balls are distributed sequentially into cells. Bose-Einstein statistics is used for analyzing the distribution of the waiting time until each cell is occupied by at least one ball. Each trial is classified according to its jump size, i.e. the number of newly occupied cells. We propose an approach to decompose the occupancy and filling processes in terms of the jumps sizes using a multi-dimensional representation. A set of recursive equations is built in order to obtain the joint generating probability function of a series of random variables, each of which denotes the number of trials for a given jump size that occurred during the filling process. As a special case, the joint probability function of these random variables is obtained.

Inhomogeneity of deformation of solids at the nanometer level

A new method for processing interferometrically recorded deformation data has been implemented for studying an inhomogeneity in the rate and parameters of deformation jumps at the nanostructure level, which provides detection of deformation jumps of less than 300 nm. It is shown that the lower limit for deformation jumps lies in the range 10–30 nm for aluminum and is 130 nm for amorphous polymer (poly(methyl methacrylate)). It is assumed that the sizes of jumps correspond to scales of ordered structures, as was previously established for higher level structures. The results obtained make it possible to investigate more thoroughly the multilevel character of deformation and to evaluate the sizes of the nanostructural units, their evolution during deformation and under the effect of external fields, as well as their relation to the microscopic and macroscopic inhomogeneities of deformation.

Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

Abstract. We consider a stochastic process driven by diffusions and jumps. Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred. This allows us to determine a consistent non‐parametric estimator of the integrated volatility when the infinite activity jump component is Lévy. Jump size estimation and central limit results are proved in the case of finite activity jumps. Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data.

Stretched-Gaussian asymptotics of the truncated Lévy flights for the diffusion in nonhomogeneous media

The Lévy, jumping process, defined in terms of the jumping size distribution and the waiting time distribution, is considered. The jumping rate depends on the process value. The fractional diffusion equation, which contains the variable diffusion coefficient, is solved in the diffusion limit. That solution resolves itself to the stretched Gaussian when the order parameter μ → 2 . The truncation of the Lévy flights, in the exponential and power-law form, is introduced and the corresponding random walk process is simulated by the Monte Carlo method. The stretched Gaussian tails are found in both cases. The time which is needed to reach the limiting distribution strongly depends on the jumping rate parameter. When the cutoff function falls slowly, the tail of the distribution appears to be algebraic.

Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements

This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted $L^2$ projection onto $H(\mathrm{div})$ conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps.

Jumps, Cojumps and Macro Announcements

We use recently proposed tests to extract jumps and cojumps from three types of assets: stock index futures, bond futures, and exchange rates. We then characterize the dynamics of these discontinuities and informally relate them to U.S. macroeconomic releases before using limited dependent variable models to formally model how news surprises explain (co)jumps. Nonfarm payroll and federal funds target announcements are the most important news across asset classes. Trade balance shocks are important for foreign exchange jumps. We relate the size, frequency and timing of jumps across asset classes to the likely sources of shocks and the relation of asset prices to fundamentals in the respective classes.

Anthropic prediction for a large multi-jump landscape

The assumption of a flat prior distribution plays a critical role in the anthropicprediction of the cosmological constant. In a previous paper we analytically calculatedthe distribution for the cosmological constant, including the prior and anthropicselection effects, in a large toy ‘single-jump’ landscape model. We showed thatit is possible for the fractal prior distribution that we found to behave asan effectively flat distribution in a wide class of landscapes, but only if thesingle-jump size is large enough. We extend this work here by investigating a large(N∼10500) toy‘multi-jump’ landscape model. The jump sizes range over three orders of magnitude and an overall freeparameter c determines the absolute size of the jumps. We will show that for ‘large’c the distribution of probabilities of vacua in the anthropic range is effectively flat, and thusthe successful anthropic prediction is validated. However, we argue that for smallc, the distribution may not be smooth.

Lévy flights in nonhomogeneous media: Distributed-order fractional equation approach

A jumping process, defined in terms of the Lévi distributed jumping size and the Poissonian, position-dependent waiting time with the algebraic jumping rate, is discussed on the assumption that parameters of both distributions are themselves random variables which are determined from given probability distributions. The fractional equation for the distributed Lévy order parameter mu is derived and solved. The solution is of the form of a combination of the Fox functions and simple scaling is lacking. The problem of accelerated diffusion is also discussed. The case of the distributed waiting time parameter theta is similarly solved and the solution offers a possibility to manage processes which are characterized by more general forms of the jumping rate, not only algebraic. Moreover, we mention a possibility that the parameters mu and theta are mutually dependent.

Jumplike microdeformation of nanostructured metals

The parameters of microdeformation jumps for copper, aluminum, titanium, and Armco iron with the initial (annealed) structure and after equal-channel angular pressing are investigated in a creep mode under low compressive stresses. The strain rate is measured with a laser interferometer in 0.15-μm linear displacements. It is demonstrated that the values of the microstrain rate and the mean sizes of jumps for the annealed metals are larger than those for the metals subjected to severe deformation. It is revealed that there is a correlation between the jumps of microplastic deformation and the size of nanometal grains. The inference is made that, for nanostructured metals, as for other materials, the structural heterogeneity is one of the factors responsible for the jumplike deformation.

Effect of the magnetic field on nanometer-scale deformation jumps in polymers

The effect of a constant magnetic field on the rate of jumplike creep under compression is investigated for vitreous polymers with a globular structure. The interferometric method used for recording the creep makes it possible to measure deformation jumps from 300 nm and larger. It is demonstrated that the sizes of deformation jumps in polyester and epoxy resins decrease in the magnetic field (B = 0.2 T). Taking into account that the deformation jump size corresponds to the size of structural inhomogeneities, it is assumed that macroglobules under the action of a constant magnetic field are separated into smaller structural units on the nanometer level.

Analysis of filtering and smoothing algorithms for Lévy-driven stochastic volatility models

Filtering and smoothing algorithms that estimate the integrated variance in Lévy-driven stochastic volatility models are analyzed. Particle filters are algorithms designed for nonlinear, non-Gaussian models while the Kalman filter remains the best linear predictor if the model is linear but non-Gaussian. Monte Carlo experiments are performed to compare these algorithms across different specifications of the model including different marginal distributions and degrees of persistence for the instantaneous variance. The use of realized variance as an observed variable in the state space model is also evaluated. Finally, the particle filter's ability to identify the timing and size of jumps is assessed relative to popular nonparametric estimators.

A General Benchmark Model for Stochastic Jump Sizes

Under few technical assumptions and allowing for the absence of an equivalent martingale measure, we show how to price and hedge in a sequence of incomplete markets driven by Wiener noise and a marked point process. We investigate the structure of market prices of risk as markets become approximately complete and consider the limits of traded securities, characterizing explicitly the growth optimal portfolio and investigating arbitrage and diversification in such markets.

Size of flux jumps in superconducting films

Magneto-optical imaging is used to visualize vortex avalanches in $\mathrm{Mg}{\mathrm{B}}_{2}$ films at $4\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. Avalanches ranging from 50 to 50 000 vortices were detected. The size distribution function has a clear peak whose position moves towards larger sizes as the applied field increases. This field dependence as well as variation of flux density profile during an avalanche is well described by a proposed model assuming a thermal origin of the avalanches. The model is based on the adiabatic approach and takes into account nonlocal electrodynamics in thin superconductors. The threshold field for thermal avalanches is predicted to be much smaller than that for thick superconductors, in agreement with the experiment.

Three Regimes of Non-Newtonian Rimming Flow

The present study is related to the rimming flow of non-Newtonian fluid on the inner surface of a horizontal rotating cylinder. Using a scale analysis, the main characteristic scales and nondimensional parameters, which describe the principal features of the process, are found. Exploiting the fact that one of the parameters is very small, an approximate asymptotic mathematical model of the process is developed and justified. For a wide range of fluids, a general constitutive law can be presented by a single function relating shear stress and shear rate that corresponds to a generalized Newtonian model. For this case, the run-off condition for rimming flow is derived. Provided the run-off condition is satisfied, the existence of a steady-state solution is proved. Within the bounds stipulated by this condition, film thickness admits a continuous solution, which corresponds to subcritical and critical flow regimes. It is proved that for the critical regime the solution has a corner on the rising wall of the cylinder. In the supercritical flow regime, a discontinuous solution is possible and a hydraulic jump may occur. It is shown that straightforward leading order steady-state theory can work well to study the shock location and height. For the particular case of a power-law model, the analytical solution of a steady-state equation for the fluid film thickness is found in explicit form. More complex rheological models, which show linear Newtonian behavior at low shear rates with transition to power law at moderate shear rates, are also considered. In particular, numerical computations were carried out for the Ellis model. For this model, some analytical asymptotic solutions have also been obtained in explicit form and compared with the results of numerical computations. Based on these solutions, the optimal values of parameters, which should be used in the Ellis equation for the correct simulation of the coating flows, are determined; the criteria that guarantee the steady-state continuous solutions are defined; and the size and location of the stationary hydraulic jumps, which form when the flow is in the supercritical state, are obtained for the different flow parameters.

Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control

In this study the optimal control of fed-batch glycerol fermentation is investigated based on an impulsive dynamical system. Considering the sudden increase of the glycerol and alkali in fed-batch culture of biodissimilation of glycerol to 1,3-propanediol, this paper proposes a nonlinear impulsive system of fed-batch culture. The existence, uniqueness and regularity properties of piecewise solution for the system are proved. In view of the controllability of volumes of glycerol added to the reactor instantaneously, the paper constructs an optimal control model based on the nonlinear impulsive system and the existence of the optimal control is obtained. The control variables here are the moments and the sizes of jumps in the states at the discrete instants and the objective is to maximize the productivity of 1,3-propanediol over one cycle.

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part I

First-order system least squares (FOSLS) is a recently developed methodology for solving partial differential equations. Among its advantages are that the finite element spaces are not restricted by the inf-sup condition imposed, for example, on mixed methods and that the least-squares functional itself serves as an appropriate error measure. This paper studies the FOSLS approach for scalar second-order elliptic boundary value problems with discontinuous coefficients, irregular boundaries, and mixed boundary conditions. A least-squares functional is defined, and ellipticity is established in a natural norm of an appropriately scaled least-squares bilinear form. For some geometries, this ellipticity is independent of the size of the jumps in the coefficients. The occurrence of singularities at interface corners, cross points, reentrant corners, and irregular boundary points is discussed, and a basis of singular functions with local support around singular points is established. A companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques, also independent of the size of coefficient jumps for some geometries.

Jumps in Financial Markets: A New Nonparametric Test and Jump Clustering

We introduce a new nonparametric jump test for continuous-time asset pricing models. It distinguishes actual arrivals and different sizes of jumps. Asymptotic distribution of the test statistics is provided, and we demonstrate that it is desirable to use high-frequency data. We explore dynamic jump intensity structure through the test, and find empirical evidence of jump clustering in foreign currency exchange markets. Its implication for financial risk management, in particular value at risk, is also discussed.

An Optimal Adaptive Finite Element Method

Although existing adaptive finite element methods for solving second order elliptic equations often perform better in practical computations than nonadaptive ones, usually they are not even proven to converge. Only recently in the work of Dörfler [SIAM J. Numer. Anal., 33 (1996), pp. 1106--1124] and that of Morin, Nochetto, and Siebert [SIAM J. Numer. Anal., 38 (2000), pp. 466--488], adaptive methods were constructed for which convergence could be demonstrated. However, convergence alone does not imply that the method is more efficient than its nonadaptive counterpart. In [ Numer. Math.}, 97 (2004), pp. 219--268], Binev, Dahmen, and DeVore added a coarsening step to the routine of Morin, Nochetto, and Siebert, and proved that the resulting method is quasi-optimal in the following sense: If the solution is such that for some s > 0, the error in energy norm of the best continuous piecewise linear approximations subordinate to any partition with n triangles is $\mathcal{O}(n^{-s})$, then given an $\eps>0$, the adaptive method produces an approximation with an error less than $\eps$ subordinate to a partition with $\mathcal{O}(\eps^{-1/s})$ triangles, in only $\mathcal{O}(\eps^{-1/s})$ operations. In this paper, employing a different type of adaptive partition, we develop an adaptive method with properties similar to those of Binev, Dahmen, and DeVore's method, but unlike their method, our coarsening routine will be based on a transformation to a wavelet basis, and we expect it to have better quantitative properties. Furthermore, all our results are valid uniformly in the size of possible jumps of the diffusion coefficients. Since the algorithm uses solely approximations of the right-hand side, we can even allow right-hand sides in $H^{-1}(\Omega)$ that lie outside $L_2(\Omega)$, at least when they can be sufficiently well approximated by piecewise constants. In our final adaptive algorithm, all tolerances depend on an a posteriori estimate of the current error instead of an a priori one; this can be expected to provide quantitative advantages.

On the Risk of Extinction for a Population Subject to a Random Markov Evolution with Jumps

We consider a one-dimensional population whose evolution is described by a jump-diffusion equation and we study the effects of changing the coefficients of the equation on the extinction time, that is the instant at which the population becomes arbitrarily small. It is shown that, under the same diffusion coefficient, if one reduces the drift and the size of jumps, the speed of extinction increases; moreover, the probability of reaching a higher population state than the present one before reaching a lower population size decreases. If the diffusion coefficient is state-independent, the speed of extinction increases with it. Furthermore, if no jumps are allowed (i.e. for a simple-diffusion equation), then under certain conditions on the coefficients of the equation both large and small values of the diffusion coefficient result in a higher extinction risk.