Small Exponent Research Articles

Volatility and variance swaps and options in the fractional SABR model

Appropriate capturing the nature of financial market volatility is a significant factor for the pricing of volatility derivatives. A recent study by Gatheral, Jaisson and Rosenbaum [2018. “Volatility is Rough.” Quantitative Finance 18 (6): 933–949] has found that log-volatility behaves as a fractional Brownian motion with a small Hurst exponent at any reasonable time scale. Also, there are several empirical works showing that a stochastic volatility model driven by the fractional Brownian motion well approximates at-the-money volatility skew near expiration. In this paper, we choose the log-normal SABR model with fractional stochastic volatility to valuate variance and volatility swaps. We derive a closed-form exact solution for the fair strike price of the variance swap by using fractional Ito calculus, while we obtain an approximate solution for the fair strike price of the volatility swap by exploiting the shifted log-normal approximation. Also, solution formulas for the variance and volatility option prices are derived. Their accuracy is confirmed through numerical studies. Calibration to market variance swap rates demonstrates the strength of fractional SABR model compared to the Heston and SABR models.

Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles

AbstractWe consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.

Inversion of convex ordering in the VIX market

We investigate conditions for the existence of a continuous model on the S&P 500 index (SPX) that jointly calibrates to a full surface of SPX implied volatilities and to the VIX smiles. We present a novel approach based on the SPX smile calibration condition . In the limiting case of instantaneous VIX, a novel application of martingale transport to finance shows that such model exists if and only if, for each time t, the local variance is smaller than the instantaneous variance in convex order. The real case of a 30-day VIX is more involved, as averaging over 30 days and projecting onto a filtration can undo convex ordering. We show that in usual market conditions, and for reasonable smile extrapolations, the distribution of in the market local volatility model is larger than the market-implied distribution of in convex order for short maturities T, and that the two distributions are not rankable in convex order for intermediate maturities. In particular, a necessary condition for continuous models to jointly calibrate to the SPX and VIX markets is the inversion of convex ordering property: the fact that, even though associated local variances are smaller than instantaneous variances in convex order, the VIX squared is larger in convex order in the associated local volatility model than in the original model for short maturities. We argue and numerically demonstrate that, when the (typically negative) spot–vol correlation is large enough in absolute value, (a) traditional stochastic volatility models with large mean reversion, and (b) rough volatility models with small Hurst exponent, satisfy the inversion of convex ordering property, and more generally can reproduce the market term-structure of convex ordering of the local and stochastic squared VIX.

SLOW GROWTH RATE OF THE DIGITS IN ENGEL EXPANSIONS

We are concerned with the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] is the digit of the Engel expansion of [Formula: see text] and [Formula: see text] is a function such that [Formula: see text] as [Formula: see text]. The Hausdorff dimension of [Formula: see text] is studied by Lü and Liu [Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory 185 (2018) 490–498] under the condition that [Formula: see text] grows to infinity. The aim of this paper is to determine the Hausdorff dimension of [Formula: see text] when [Formula: see text] slowly increases to infinity, such as in logarithmic functions and power functions with small exponents. We also provide a detailed analysis of the gaps between consecutive digits. This includes the central limit theorem and law of the iterated logarithm for [Formula: see text] and the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] with the convention [Formula: see text].

Analysis of the RSA-cryptosystem in abstract number rings

Quantum computers can be a real threat to some modern cryptosystems (such as the RSA-cryptosystem). The analogue of the RSA-cryptosystem in abstract number rings is not affected by this threat, as there are currently no factorization algorithms using quantum computing for ideals. In this paper considered an analogue of RSA-cryptosystem in abstract number rings. Proved the analogues of theorems related to its cryptographic strength. In particular, an analogue of Wiener’s theorem on the small secret exponent is proved. The analogue of the re-encryption method is studied. On its basis the necessary restrictions on the parameters of the cryptosystem are obtained. It is also shown that in numerical Dedekind rings the factorization problem is polynomial equivalent to factorization in integers.

Research on cascading high-dimensional isomorphic chaotic maps

In order to overcome the security weakness of the discrete chaotic sequence caused by small Lyapunov exponent and keyspace, a general chaotic construction method by cascading multiple high-dimensional isomorphic maps is presented in this paper. Compared with the original map, the parameter space of the resulting chaotic map is enlarged many times. Moreover, the cascaded system has larger chaotic domain and bigger Lyapunov exponents with proper parameters. In order to evaluate the effectiveness of the presented method, the generalized 3-D Hénon map is utilized as an example to analyze the dynamical behaviors under various cascade modes. Diverse maps are obtained by cascading 3-D Hénon maps with different parameters or different permutations. It is worth noting that some new dynamical behaviors, such as coexisting attractors and hyperchaotic attractors are also discovered in cascaded systems. Finally, an application of image encryption is delivered to demonstrate the excellent performance of the obtained chaotic sequences.

Nucleation Theory for Yielding of Nearly Defect-Free Crystals: Understanding Rate Dependent Yield Points.

Experiments and simulations show that when an initially defect-free rigid crystal is subjected to deformation at a constant rate, irreversible plastic flow commences at the so-called yield point. The yield point is a weak function of the deformation rate, which is usually expressed as a power law with an extremely small nonuniversal exponent. We reanalyze a representative set of published data on nanometer sized, mostly defect-free Cu, Ni, and Au crystals in light of a recently proposed theory of yielding based on nucleation of stable stress-free regions inside the metastable rigid solid. The single relation derived here, which is not a power law, explains data covering 15 orders of magnitude in timescales.

Room temperature deformation of 6H–SiC single crystals investigated by micropillar compression

The room-temperature plastic deformation behavior of 6H–SiC single crystals has been investigated by uniaxial compression of micropillar specimens as a function of crystal orientation and specimen size. Plastic flow is observed even at room temperature by basal and prism slip, latter of which have never been observed in the bulk. The CRSS values for basal and prism slip are as high as above 5 and 6 GPa at the specimen size of 5 μm, respectively, each of which increases with decreasing specimen size, following an inverse power-law relationship with a relatively small power-law exponent of ∼0.10 and ∼0.21, respectively. The CRSS values for basal slip are not virtually affected by the existence of basal dislocations introduced at 1300 °C prior to micropillar compression tests at room temperature. The majority of basal dislocations observed after micropillar compression are perfect (undissociated) screw dislocations, and they are considered to be introduced in the shuffle-set plane during micropillar testing, unlike widely dissociated dislocations introduced in the glide-set plane in the bulk during high-temperature deformation. Prism dislocations are observed also to glide as perfect (undissociated) dislocations and tend to align strongly along their screw orientation. The fracture toughness values are estimated to be 1.37 ± 0.13 and 1.57 ± 0.13 MPa m1/2 by three-point bend tests for chevron-notched single crystalline specimens with a notch plane being parallel to (0001) and {011¯0} planes, respectively.

Bloch functions, asymptotic variance, and geometric zero packing

We study a new type of extremal problem in complex analysis, referred to as "geometric zero packing", which is the hyperbolic analogue of a problem considered by Abrikosov in the 1950s concerning Bose-Einstein condensates. We relate the corresponding minimal discrepancy density with the asymptotic variance for Bloch functions of the form "Bergman projection of bounded functions" and obtain a corresponding identity. Together with related work of Ivrii, this gives the asymptotic behavior of the universal quasiconformal integral means spectrum for small values of quasiconformality k and small exponents t. In particular, the conjectured behavior is shown to be smaller than conjectured by Prause and Smirnov, which also shows that there are no quasidisks with dimension 1+k^2, at least for small k.

Imaginary quadratic number fields with class groups of small exponent

Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor of $8$. We compute all $D$ with $|D|\leq 3.1\cdot 10^{20}$ such that $E(D)\leq 8$.

Inversion of Convex Ordering in the VIX Market

We investigate conditions for the existence of a continuous model on the S&P 500 index (SPX) that jointly calibrates to a full surface of SPX implied volatilities and to the VIX smiles. We present a novel approach based on the SPX smile calibration condition (the fact that the conditional expectation of the instantaneous variance given the spot equals the market local variance). In the limiting case of instantaneous VIX, a novel application of martingale transport to finance shows that such model exists if and only if, for each time t, the local variance is smaller than the instantaneous variance in convex order. The real case of a 30 day VIX is more involved, as averaging over 30 days and projecting onto a filtration can undo convex ordering. We show that in usual market conditions, and for reasonable smile extrapolations, the distribution of the VIX squared in the market local volatility model is larger than the market-implied distribution of the VIX squared in convex order for short maturities T, and that the two distributions are not rankable in convex order for intermediate maturities. In particular, a necessary condition for continuous models to jointly calibrate to the SPX and VIX markets is the inversion of convex ordering property: the fact that, even though associated local variances are smaller than instantaneous variances in convex order, the VIX squared is larger in convex order in the associated local volatility model than in the original model for short maturities. We argue and numerically demonstrate that, when the (typically negative) spot-vol correlation is large enough in absolute value, (a) traditional stochastic volatility models with large mean reversion, and (b) rough volatility models with small Hurst exponent, satisfy the inversion of convex ordering property, and more generally can reproduce the market term-structure of convex ordering of the local and stochastic squared VIX.

Fredholm Toeplitz operators with VMO symbols and the duality of generalized Fock spaces with small exponents

AbstractWe characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 &lt; p &lt; 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.

Interactive Effects of Climatic Factors on Seasonal Vegetation Dynamics in the Central Loess Plateau, China

The interactive effects of climatic factors (precipitation and temperature) on vegetation growth can be characterized by their effect on vegetation seasonal dynamics. The interactive effects, seasonal trend of vegetation growth, and its future consistency (potential for future trend) have not been adequately studied in the literature. In this work, using the Enhanced Vegetation Index (EVI) and gridded climate data at a resolution of 250 m in the central Loess Plateau region, we examined seasonal vegetation dynamics with climate changes and the interactive effects of climatic factors on vegetation growth at the pixel and regional scales from the period 2000 to 2015. Vegetation cover in the Central Loess Plateau in China has dramatically changed due to the Grain-for-Green (GFG) ecological restoration program, which was designed to convert cropland to forestland or grassland since 1999. Our results show that the EVI increased significantly during the 16 year period and is likely to continue to increase in the near future. Relatively small Hurst exponents for forestland suggests that the potential for a future increased trend will be weak for the forest. Large Hurst exponents for grassland indicate its strong potential of further increase. Significant increases in spring precipitation have promoted vegetation growth, while significant decreases in summer temperature have had negative effects on vegetation growth. For temperatures between 10 to 20 °C, the impact of temperature on vegetation growth has a clear positive relationship with the moderator variable precipitation. For precipitation &lt; 200 mm in the growing season, the impact of precipitation on vegetation growth has a clearly positive relationship with the moderator variable temperature. Results of this study will provide useful and important guidelines for designing forestland and grassland restoration plans in arid, semiarid and sub-humid regions.

Stress sensitivity of tight reservoirs and its effect on oil saturation: A case study of Lower Cretaceous tight clastic reservoirs in the Hailar Basin, Northeast China

To analyze stress sensitivity of tight reservoirs and the boundary conditions for oil filling, core samples of tight clastic reservoirs in the Lower Cretaceous Nantun Formation in the Hailar Basin were selected. The measurements include optical observation of thin sections, changes of porosity and permeability under overburden pressure, together with physical simulation experiments of oil filling. The results show that the stress sensitivity coefficient of permeability is a product of two parameters including the power exponent of porosity-permeability and pore compressibility, which reflect the geometric and mechanical characteristics of pores. The stress sensitivity and pore geometry of tight reservoirs of different lithologies vary greatly. Medium-coarse sandstones-conglomerates have relatively small stress sensitivity coefficient and large power exponent of porosity-permeability, indicating stronger anti-compaction, weaker stress sensitivity and dominated fissure-type pores. Fine sandstone with large stress sensitivity coefficient and small power exponent of porosity-permeability, indicating that stress sensitivity is high, and matrix circular/elliptical pores predominate. Oil filling in tight reservoirs displays two pressure gradient points with threshold characteristics (i.e., starting pressure gradient and critical pressure gradient) and two flow characteristics (i.e., low-speed nonlinear seepage and quasi-linear seepage). Only when the actual pressure gradient breaks through the critical pressure gradient into the quasi-linear seepage zone can oil saturation in the tight reservoir reach the lower limit of 30%, attaining higher oil saturation levels gradually during the subsequent filling process. The favorable tight reservoir is characterized by a low stress sensitivity coefficient and a high power exponent of porosity-permeability, and oil exhibits rapid filling characteristics with quasi-linear seepage. Another type of tight reservoir is characterized by a high stress sensitivity coefficient and a low power exponent of porosity-permeability, and oil exhibits slow filling rate with nonlinear seepage. It is therefore difficult in actual geological conditions for such reservoirs to achieve high oil saturation. Increase in oil saturation during the filling process is controlled by a combination of initial permeability, pore geometry, stress sensitivity and the displacement pressure gradient. These factors affect and compensate each other, comprehensively controlling the accumulation of tight oil, and could be used to classify the reservoir types.

Determining the Optimal Random-Padding Size for Rabin Cryptosystems

Rabin encryption and a secure ownership transfer protocol based on the difficulty of factorization of a public key use a small public exponent. Such encryption requires random number padding. The Coppersmith's shortpad attack works effectively on short padding, thereby allowing an adversary to extract the secret message. However, the criteria for determining the appropriate padding size remains unclear. In this paper, we derived the processing-time formula for the shortpad attack and determined the optimal random-padding size in order to achieve the desired security.

Generalized cryptanalysis of small CRT-exponent RSA

There have been several works for studying the security of CRT-RSA with small CRT exponents dp and dq by using lattice-based Coppersmith's method. Thus far, two attack scenarios have been mainly studied: (1) dq is small with unbalanced prime factors p≪q. (2) Both dp and dq are small for balanced p≈q. The best attacks for the both scenarios were proposed by Takayasu-Lu-Peng (Eurocrypt'17, Journal of Cryptology'19) and the attack conditions are much better than the other known attacks. Although the attacks have been very useful for studying the security of CRT-RSA, the structures of their proposed lattices are not well understood. In this paper, to further study the security of CRT-RSA, we first define a generalized attack scenario to unify the previous ones. Specifically, all p,q,dp, and dq can be of arbitrary sizes. Furthermore, we propose improved attacks in this paper when dp and/or p is sufficiently small. Technically, we construct a lattice whose basis vectors are chosen flexibly depending on the sizes of p,q,dp, and dq. Since the attack scenarios (1) and (2) are simpler than our general scenario, the previous Takayasu-Lu-Peng's lattices are simple special cases of ours. We are able to achieve the flexible lattice constructions by exploiting implicit but essential structures of Takayasu-Lu-Peng's lattices. We check the validity of our proposed attacks by computer experiments. We believe that the deeper understanding of the lattice structures will be useful for studying the security of CRT-RSA even in other scenarios.

GOOD GOVERNANCE IN MUNICIPAL AND RURAL MUNICIPAL OFFICES OF LOCAL ADMINISTRATION IN POLAND

Public governance constitutes an answer to the limitations regarding the possibilities of reforming public administration in the stream of new public management. One of the detailed concepts of public governance was originally formulated by the World Bank. Its use in practice is limited to operationalization through formulating a variety of indices which are usually dedicated to countries and regions. Scientific discourse lacks studies indicating the possibility of using good governance in public administration at the level of its smaller exponents such as the units of local self-government. Equally, there are no available results of research regarding the influence of the functioning of units of public administration according to the principles of good governance on their perception on the part of citizens. This study aims at operationalizing the concept of good governance through the formulation of a model of activity of local government administration according to the principles of good governance, as well as proposes a tool for evaluating&nbsp; the level of maturity of management solutions based on the concept of good governance. The research results regarding the level of compatibility of these offices with the rules of good governance are compared with the results of surveys examining opinions and levels of satisfaction among inhabitants of selected boroughs. &nbsp;The aim of this comparison was to find the answer to the question of dependence between the activities of local administration in accordance with good governance and satisfaction on the part of local inhabitants.The publication was the outcome of the project entitled ”Adaptation of the management system of local government office to the good governance concept” which was financed by The National Science Centre, based on the decision no. DEC-2013/09/B/HS4/01143.

All-electron fully relativistic Kohn-Sham theory for solids based on the Dirac-Coulomb Hamiltonian and Gaussian-type functions

We present the first full-potential method that solves the fully relativistic 4-component Dirac-Kohn-Sham equation for materials in the solid state within the framework of atom-centered Gaussian-type orbitals (GTOs). Our GTO-based method treats one-, two-, and three-dimensional periodic systems on an equal footing, and allows for a seamless transition to the methodology commonly used in studies of molecules with heavy elements. The scalar relativistic effects and spin-orbit interaction are handled variationally. The full description of the electron-nuclear potential in the core region of heavy nuclei is straightforward due to the local nature of the GTOs and does not pose any computational difficulties. We show how the time-reversal symmetry and a quaternion algebra-based formalism can be exploited to significantly reduce the increased methodological complexity and computational cost associated with multiple wave-function components coupled by the spin-orbit interaction. We provide a description of how to employ the matrix form of the multipole expansion and an iterative renormalization procedure to evaluate the conditionally convergent lattice sums arising for periodic systems. We investigate the problem of inverse variational collapse that arises if the Dirac operator containing a repulsive periodic potential is expressed in a basis containing diffuse functions, and suggest a possible solution. We demonstrate the validity of the method on silver halide crystals with large relativistic effects, and two-dimensional honeycomb structures (silicene and germanene) exhibiting the spin-orbit-driven quantum spin Hall effect. Our results are well-converged with respect to the basis limit using standard bases developed for molecular calculations, and indicate that the rule of removing basis functions with small exponents should not be applied when transferring the molecular basis to solids.

Accelerating spin-space sampling by auxiliary spin dynamics and temperature-dependent spin-cluster expansion

Atomistic simulations of thermodynamic properties of magnetic materials rely on an accurate modelling of magnetic interactions and an efficient sampling of the high-dimensional spin space. Recent years have seen significant progress with a clear trend from model systems to material specific simulations that are usually based on electronic-structure methods. Here we develop a Hamiltonian Monte Carlo framework that makes use of auxiliary spin-dynamics and an auxiliary effective model, the temperature-dependent spin-cluster expansion, in order to efficiently sample the spin space. Our method does not require a specific form of the model and is suitable for simulations based on electronic-structure methods. We demonstrate fast warm-up and a reasonably small dynamical critical exponent of our sampler for the classical Heisenberg model. We further present an application of our method to the magnetic phase transition in bcc iron using magnetic bond-order potentials.

Dead-Core Solutions to Simple Catalytic Reaction Problems in Chemical Engineering

The catalytic chemical reaction is usually carried out in a pellet where the catalyst is distributed throughout its porous structure. The selectivity, yield and productivity of the catalytic reactor often depend on the rates of chemical reactions and the rates of diffusion of species involved in the reactions in the pellet porous space. In such systems, the fast reaction can lead to the consumption of reactants close to the external pellet surface and creation of the dead core where no reaction occurs. This will result in an inefficient use of expensive catalyst. In the discussed simplified diffusion-reaction problems a nonlinear reaction term is of power-law type with a small positive reaction exponent. Such reaction term represents the kinetics of catalytic reaction accompanied by a strong adsorption of the reactant. The ways to calculate the exact solutions possessing dead cores are presented. It was also proved analytically that the exact solution of the nonlinear two-point boundary value problem satisfies physical a-priori bounds. Furthermore, the approximate solutions were obtained using the orthogonal collocation method for pellets of planar, spherical and cylindrical geometries. Numerical results confirmed that the length of the dead core increases for the more active catalysts due to the larger values of the reaction rate constant. The dead core length also depends on the pellet geometry.