Infinite Variation Research Articles

Exercise boundary of the American put near maturity in an exponential Lévy model

We study the behavior of the critical price of an American put option near maturity in an exponential Levy model. In particular, we prove that in situations where the limit of the critical price is equal to the strike price, the rate of convergence to the limit is linear if and only if the underlying Levy process has finite variation. In the case of infinite variation, a variety of rates of convergence can be observed: we prove that when the negative part of the Levy measure exhibits an α-stable density near the origin, with 1<α<2, the convergence rate is ruled by $\theta^{1/\alpha}|\ln \theta|^{1-\frac{1}{\alpha}}$ , where θ is the time until maturity.

Is a pure jump process fitting the high frequency data better than a jump-diffusion process?

Two families of processes: pure jump processes and jump-diffusion processes are widely used in literatures. Recently, empirical findings demonstrate that the underlying processes of high frequency data sets are pure-jump processes of infinite variation in many situations. Statistical tests are also proposed to make the empirical findings theoretically grounded. In this paper, we extend the work of Jing et al. (2012) in two aspects: (1) the jump process in the null hypothesis and the alternative hypothesis could be different; (2) the null hypothesis covers more flexible processes which are more relevant in finance when considering models for asset prices or nominal interest rates. Theoretically, the test is proven to be very powerful and can control the type I error probabilities well under the nominal level.

Affine processes on positive semidefinite [formula omitted] matrices have jumps of finite variation in dimension [formula omitted

The theory of affine processes on the space of positive semidefinite d×d matrices has been established in a joint work with Cuchiero et al. (2011) [4]. We confirm the conjecture stated therein that in dimension d>1 this process class does not exhibit jumps of infinite total variation. This constitutes a geometric phenomenon which is in contrast to the situation on the positive real line (Kawazu and Watanabe, 1971) [8]. As an application we prove that the exponentially affine property of the Laplace transform carries over to the Fourier–Laplace transform if the diffusion coefficient is zero or invertible.

Continuous-time modeling of random searches: statistical properties and inference

Mathematical modeling of random searches is of great relevance in the fields of physics, chemistry, biology or modern ecology. A large number of existing studies record the search movement at equidistant time intervals and model such time series data directly with discrete-time random walks, such as Lévy flights and correlated random walks. Given the increasing availability of high-resolution observation data, statistical inference for search paths based on such high-resolution data has recently become one of the major interests and has raised an important issue of robustness of random walk models to the sampling rate, as estimation results for the discrete observation data are found to be largely different at different sampling rates even when the underlying movement is supposedly independent of scale. To address this issue, in this paper, we propose to model the continuous-time search paths directly with a continuous-time stochastic process for which the observer makes statistical inference based on its discrete observation. As continuous-time counterparts of Lévy flights, we consider two-dimensional Lévy processes and discuss the relevance of those models based upon advantages and limitations in terms of statistical properties and inference. Among the proposed models, the Brownian motion is most tractable in various ways while its Gaussianity and infinite variation of sample paths do not well describe the reality. Such drawbacks in statistical properties may be remedied by employing the stable and tempered stable Lévy motions, while those models are less tractable and cause an issue in statistical inference.

Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations

We investigate transition law between consecutive observations of Ornstein–Uhlenbeck processes of infinite variation with tempered stable stationary distribution. Thanks to the Markov autoregressive structure, the transition law can be written in the exact sense as a convolution of three random components; a compound Poisson distribution and two independent tempered stable distributions, one with stability index in (0, 1) and the other with index in (1, 2). We discuss simulation techniques for those three random elements. With the exact transition law and proposed simulation techniques, sample paths simulation proves significantly more efficient, relative to the known approximative technique based on infinite shot noise series representation of tempered stable Lévy processes.

Hp-DGFEM FOR KOLMOGOROV–FOKKER–PLANCK EQUATIONS OF MULTIVARIATE LÉVY PROCESSES

We analyze the discretization of nonlocal degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes. Such equations arise in option pricing problems when the stochastic dynamics of the markets is modeled by Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes, in particular the discontinuous Galerkin Finite Element Methods (DG-FEM). In the DG-FEM, a new regularization of hypersingular integrals in the Dirichlet form of the pure jump part of infinite variation processes is proposed, allowing in particular a stable DG discretization of hypersingular integral operators. Robustness of the stabilized discretization with respect to various degeneracies in the characteristic triple of the stochastic process is proved. We provide in particular an hp-error analysis of the DG-FEM. Numerical experiments for model equations confirm the theoretical results.

Small-Time Asymptotics of Option Prices and First Absolute Moments

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process S follows a general martingale. This is equivalent to studying the first centered absolute moment of S. We show that if S has a continuous part, the leading term is of order √T in time T and depends only on the initial value of the volatility. Furthermore, the term is linear in T if and only if S is of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation of S so that calculations are necessary only for the class of Lévy processes.

Small-Time Asymptotics of Option Prices and First Absolute Moments

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price processSfollows a general martingale. This is equivalent to studying the first centered absolute moment ofS. We show that ifShas a continuous part, the leading term is of order √Tin timeTand depends only on the initial value of the volatility. Furthermore, the term is linear inTif and only ifSis of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation ofSso that calculations are necessary only for the class of Lévy processes.

Modification of fracture toughness parameter K1c following CO2 laser surface treatment of Si3N4 engineering ceramic

Surface treatment of a Si3N4 engineering ceramic assisted by a CO2 laser was conducted to identify changes in the fracture toughness parameter K1c. The K1c was determined by using a Vickers macrohardness indentation method before and after the CO2 laser surface treatment. Optical microscopy and infinite focus variation techniques were then adopted to measure the crack length and to investigate the surface morphology as well as the crack geometry within the engineering ceramic. Thereafter, computational and analytical methods were employed to calculate the K1c. Chemical analysis was further conducted to elucidate the change in composition as a result of the CO2 laser surface irradiation. The results showed that a decrease in the near surface hardness of 7 and 44% in the resulting crack lengths was found with the Si3N4 engineering ceramic following the CO2 laser surface treatment. This inherently led to an increase in the K1c. A rise in the K1c of 64% for the Si3N4 engineering ceramics was found (under the applied conditions) in comparison to the as received surface. This occurred due to the Si3N4 engineering ceramic being oxidised and further forming a new surface layer, which was somewhat softer than that of the as received or laser unaffected surface. Compositional analysis showed that the formation of the new surface layer as a result of the CO2 laser surface treatment was found to be SiO2.

Volatility Jumps

The article undertakes a nonparametric analysis of the high-frequency movements in stock market volatility using very finely sampled data on the VIX volatility index compiled from options data by the CBOE. We derive theoretically the link between pathwise properties of the latent spot volatility and the VIX index, such as presence of continuous martingale and/or jumps, and further show how to make statistical inference about them from the observed data. Our empirical results suggest that volatility is a pure jump process with jumps of infinite variation and activity close to that of a continuous martingale. Additional empirical work shows that jumps in volatility and price level in most cases occur together, are strongly dependent, and have opposite sign. The latter suggests that jumps are an important channel for generating leverage effect.

Nonparametric tests for pathwise properties of semimartingales

We propose two nonparametric tests for investigating the pathwise properties of a signal modeled as the sum of a L\'{e}vy process and a Brownian semimartingale. Using a nonparametric threshold estimator for the continuous component of the quadratic variation, we design a test for the presence of a continuous martingale component in the process and a test for establishing whether the jumps have finite or infinite variation, based on observations on a discrete-time grid. We evaluate the performance of our tests using simulations of various stochastic models and use the tests to investigate the fine structure of the DM/USD exchange rate fluctuations and SPX futures prices. In both cases, our tests reveal the presence of a non-zero Brownian component and a finite variation jump component.

Engineering Application of Time-changed Lévy Process to Capture Jumps in Stock Market

In financial engineering, volatility in the stock return processes is one of crucial factors when we deal with asset pricing and risk management. Besides the continuous part and the big jumps, there are a great amount of small jumps in stock prices. In this paper, under the continuous-time financial framework, we use the time-changed Lévy process with infinite activity and infinite variation to construct the SVNIG model, which can capture small jumps. This model can describe the continuous volatility component and the jump component simultaneously. MCMC approach is then employed to estimate parameters and identify latent variables. Using Hushen300 composite index in China and Hang Seng index in Hong Kong, the empirical results show that there are massive small jumps in both markets, and SVNIG model can describe jump behavior more accurately than other models.

The speed of convergence of the Threshold estimator of integrated variance

In this paper we consider a semimartingale model for the evolution of the price of a financial asset, driven by a Brownian motion (plus drift) and possibly infinite activity jumps. Given discrete observations, the Threshold estimator is able to separate the integrated variance IV from the sum of the squared jumps. This has importance in measuring and forecasting the asset risks. In this paper we provide the exact speed of convergence of I V ˆ h , a result which was known in the literature only in the case of jumps with finite variation. This has practical relevance since many models used have jumps of infinite variation (see e.g. Carr et al. (2002) [4]).

Numerical pricing of American options under infinite activity Lévy processes

AbstractUnder infinite activity Lévy models, American option prices can be obtained by solving a partial integro‐differential equation (PIDE), which has a singular kernel. With increasing degree of singularity, standard time‐stepping techniques may encounter difficulties. This study examines exponential time integration (ETI) for solving this problem and the performance of this scheme is compared with the Crank–Nicolson (CN) method and an implicit–explicit method in conjunction with an extrapolation (IMEX‐Extrap), in terms of computational speed and convergence orders. These findings indicate that ETI is faster and more accurate among PIDE‐based methods for solving the system of ordinary differential equations resulting from spatial discretization of the PIDE. For very singular problems, it is shown that the IMEX‐Extrap scheme becomes unfavorable compared with the other schemes as it is relatively more time consuming and the global convergence deteriorates from quadratic to linear, whereas the ETI scheme yields both point‐wise and global quadratic convergence. For illustration, under the infinite variation process, the IMEX‐Extrap achieves a precision of the order of 10−4 in 663.016 s, whereas for the same set of parameters, the CN method and the ETI scheme reach an accuracy of the order of 10−5 in 237.891 s and 22.772 s, respectively. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:809–829, 2011

EKTOPARASIT ACARINA PADA KULIT SISIK ULAR PUTIH (MICROPECIS IKAHEKA) DARI MANOKWARI

&lt;p&gt;&lt;em&gt;Ectoparasites&lt;/em&gt;&lt;em&gt; are species of parasites that live on the outer surface of the body of an animal. They grow and are very dependent on the life of the host. The acarina complex includes predaceous mites of infinite variation, mites associated with plants and other animals and using a variety of organic substrates as food. The result from the identification showed that acarina parasite on white snake comes from the order Parasitiformes. The acarina encountered in snake’s skin inhabit the dorsal scales in lanes or take advantage of the gaps between scales from the posterior head to the scales near the cloaca.&lt;/em&gt;&lt;/p&gt;

On some boundary value problems for linear generalized differential systems with singularities

For a linear Kurzweil generalized differential system with singularities, we consider two-point boundary value problems of two forms. The singularity is understood in the sense that the matrix and vector functions defining the system may have infinite total variation.

Infinite etch selectivity and line edge roughness variation during etching of silicon oxynitride with an extreme ultraviolet resist pattern in dual-frequency CH2F2∕H2∕Ar capacitively coupled plasmas

The process window for infinitely high etch selectivity of silicon oxynitride (SiON) layers to extreme ultraviolet (EUV) resist and the variation in line edge roughness (LER) of etched EUV resist were investigated in a CH2F2∕H2∕Ar dual-frequency superimposed capacitively coupled plasma under various process parameters including the gas flow ratio and low-frequency source power (PLF). The CH2F2∕H2 gas flow ratio was found to play a critical role in determining the process window for infinite selectivity of the SiON/EUV resist due to the change in the degree of polymerization. The preferential chemical reaction of hydrogen and carbon species in the hydrofluorocarbon (CHxFy) layer with nitrogen on the SiON surface led to the formation of HCN etch by-products, resulting in a thinner steady-state CHxFy layer. During continuous SiON etching, the thinner steady-state CHxFy layer was due to enhanced SiF4 formation, while the CHxFy layer was deposited on the EUV resist surface. In addition, the critical dimension and LER tended to increase with increasing Q(CH2F2) flow ratio.

Selection of sustainable technologies for combustion of Bosnian coals

This paper deals with optimization of coal combustion conditions to support selection a sustainable combustion technology and an optimal furnace and boiler design. A methodology for optimization of coal combustion conditions is proposed and demonstrated on the example of Bosnian coals. The properties of Bosnian coals vary widely from one coal basin to the next, even between coal mines within the same basin. Very high percentage of ash (particularly in Bosnian brown coal) makes clear certain differences between Bosnian coal types and other world coal types, providing a strong argument for investigating specific problems related to the combustion of Bosnian coals, as well as ways to improve their combustion behavior. In this work, options of the referent energy system (boiler) with different process temperatures, corresponding to the different combustion technologies; pulverised fuel combustion (slag tap or dry bottom furnace) and fluidized bed combustion, are under consideration for the coals tested. Sustainability assessment, based on calculation economic and environment indicators, in combination with common low cost planning method, is used for the optimization. The total costs in the lifetime are presented by General index of total costs, calculated on the base of agglomeration of basic economic indicators and the economic indicators derived from environmental indicators. So, proposed methodology is based on identification of those combustion technologies and combustion conditions for coals tested for which the total costs in lifetime of the system under consideration are lowest, provided that all environmental issues of the energy system is fulfilled during the lifetime. Inputs for calculation of the sustainability indicators are provided by the measurements on an experimental furnace with possibility of infinite variation of process temperature, supported by good praxis from the power plants which use the fuels tested and by thermal calculations of the different options (different temperature in the boiler furnace) of the referent energy system.

Prices of Barrier and First-Touch Digital Options in Levy-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 Levy processes with exponential jump densities, including Variance Gamma model, 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. In many cases, we calculate the second term of asymptotics as well.

Rational prescribing: the principles of drug selection

Prescribing is the most important tool used by physicians to cure illness, relieve symptoms and prevent future disease. It is also a complex intellectual task that requires formulation of an appropriate treatment regimen from the many thousands available, taking into account the infinite variation in the patients they encounter. Unfortunately, the selection of a medicine and dosage regimen is sometimes suboptimal, leading to poor patient outcomes (eg treatment failure, avoidable adverse reactions). This article will highlight some of the common prescribing errors and will develop a rational approach that includes making a diagnosis, estimating prognosis, establishing the goals of therapy, selecting the most appropriate treatment and monitoring the effects of the treatment.