Simple Closed-form Approximation Research Articles

A Simple Closed-Form Approximation for Constant Elasticity of Variance Spread Options

By applying the Lie-Trotter operator splitting method and the idea of the WKB method, we have developed a simple, accurate and efficient analytical approximation for pricing the constant elasticity of variance (CEV) spread options. The derived option price formula bears a striking resemblance to Kirk's formula of the Black-Scholes spread options. Illustrative numerical examples show that the proposed approximation is not only extremely fast and robust, but also it is remarkably accurate for typical volatilities and maturities of up to two years.

A simple closed-form approximation for constant elasticity of variance spread options

By applying the Lie–Trotter operator splitting method and the idea of the WKB method, we have developed a simple, accurate and efficient analytical approximation for pricing the constant elasticity of variance (CEV) spread options. The derived option price formula bears a striking resemblance to Kirk’s formula of the Black–Scholes spread options. Illustrative numerical examples show that the proposed approximation is not only extremely fast and robust, but it is also remarkably accurate for typical volatilities and maturities of up to two years.

CDS Returns

We show that existing metrics of CDS returns poorly approximate cash flow-based CDS returns. Given the complexities involved in computing CDS returns correctly, we provide a simple closed-form approximation that bears a correlation of no less than 99% with the true return series. Our work emphasizes the importance of distinguishing between changes in credit spreads and CDS returns. In addition, it highlights the need to rely on true CDS return metrics to evaluate investment strategies and predictive return regressions that involve the selling or buying of CDS contracts.

Test sensitivity of mammography and mean sojourn time over 40 years of breast cancer screening in Nijmegen (The Netherlands).

ObjectivesWe investigated whether changes in mammographic technique and screening policy have improved mammographic sensitivity, and elongated the mean sojourn time, since the introduction of biennial breast cancer screening in Nijmegen, the Netherlands, in 1975.MethodsMaximum likelihood estimation, non-linear regression, and Markov Chain Monte Carlo simulation were used to estimate test sensitivity, mean sojourn time, and underlying breast cancer incidence in four time periods, covering 40 years of breast cancer screening in Nijmegen (1975–2012).ResultsMaximum likelihood estimation generated an estimated test sensitivity of approximately 90% and a mean sojourn time around three years, while the estimates based on non-linear regression and Markov Chain Monte Carlo simulation were 80% and four years, respectively. All three methods estimated a rise in the underlying breast cancer incidence over time, with approximately one case more per 1000 women per year in the final period compared with the first period.ConclusionsThe three methods showed a slightly higher mammographic sensitivity and a longer mean sojourn time in the last period, after the introduction of digital mammography. Estimates were more realistic for the more sophisticated methods, non-linear regression and Markov Chain Monte Carlo simulation, while the simple closed form approximation of maximum likelihood estimation led to rather high estimates for sensitivity in the early periods.

Approximation of Meta Distribution and its Moments for Poisson Cellular Networks

The notion of meta distribution as the distribution of the conditional coverage probability was introduced in the literature before. In this letter, we show how we can reconstruct the entire meta distribution only from its moments using Fourier–Jacobi expansion. As an example, we specifically consider Poisson cellular networks. We also provide a simple closed-form approximation for its moments, along with its error analysis. Lastly, we apply the approximation to obtain a power scaling law for downlink Poisson cellular networks.

Coverage and Rate Maximization via User Association in Multi-Antenna HetNets

We formulate two statistical optimization frameworks for multi-antenna heterogeneous networks (HetNets). The first one maximizes coverage, via user association, and the second one maximizes a rate utility function by jointly optimizing user association and the spectrum allocated to each tier. Since both performance metrics depend on the coverage probability, we first derive the exact analytical coverage expression for a frequency-partitioned multi-antenna HetNet. We then present a simple closed-form approximation using the Gil-Pelaez inversion theorem, making both optimization frameworks concave in the optimizing variables. We show that to maximize coverage, the user should connect to the tier with the highest ratio of the array gain to the multiplexing gain, highlighting that max-power association becomes far from optimal in terms of coverage. To maximize the rate utility, we show that the optimal resource allocation of a tier is equal to the portion of users connected to that tier. Furthermore, maximizing the rate utility breaks down into maximizing two objectives: coverage and the number of users to be served in a resource block. Several theoretical insights are presented and further validated via simulations, showing that both frameworks provide considerable coverage and rate gains compared to the standard max-power and small-cell range expansion schemes.

A New Closed Form Approximation for BER for Optical Wireless Systems in Weak Atmospheric Turbulence

Abstract Weak atmospheric turbulence condition in an optical wireless communication (OWC) is captured by log-normal distribution. The analytical evaluation of average bit error rate (BER) of an OWC system under weak turbulence is intractable as it involves the statistical averaging of Gaussian Q-function over log-normal distribution. In this paper, a simple closed form approximation for BER of OWC system under weak turbulence is given. Computation of BER for various modulation schemes is carried out using proposed expression. The results obtained using proposed expression compare favorably with those obtained using Gauss-Hermite quadrature approximation and Monte Carlo Simulations.

Closed-form approximation of the axial force-bending moment interaction diagram for hollow circular reinforced concrete cross-sections

This paper proposes a simple closed-form approximation of the axial force-bending moment interaction diagram for solid and hollow circular reinforced concrete (RC) cross-sections with an arbitrary number of layers of longitudinal reinforcement. The implemented nonlinear material models are those of the European building code. Specifically, the parabola-rectangular strain-stress relation and the stress-block model are considered for the concrete in compression. On the other hand, the elastic-perfectly plastic behavior is assumed for the reinforcing steel bars. The proposed semi-exact closed-form approximation involves one parameter only, which is calibrated by means of a numerical optimization procedure taking into account a large database of cross-sections. Moreover, an efficient procedure is developed for the design of the outer layer of longitudinal reinforcement in solid or hollow circular RC cross-sections under combined axial force and bending moment (the mechanical ratio of any other layers of longitudinal reinforcement is considered as known function of the mechanical ratio of the outer layer of longitudinal reinforcement). Finally, several numerical examples are included to demonstrate the accuracy of the approximated axial force-bending moment interaction diagram and the effectiveness of the derived design procedure.

A Note on CDS Returns

We show that commonly used metrics of CDS returns poorly approximate cash-flow-based CDS returns, and that they may erroneously represent the relationship between changes in prices of equity and bonds. Given the complexities involved in computing CDS returns correctly, we provide a simple closed-form approximation that bears a correlation of no less than 99% with the true return series. Our work emphasizes the importance of distinguishing between changes in credit spreads and CDS returns. In addition, it highlights the need to rely on true CDS return metrics to evaluate investment strategies and predictive return regressions that involve the selling or buying of CDS contracts.

Asymptotic Approximation of the Standard Condition Number Detector for Large Multi-Antenna Cognitive Radio Systems

Standard condition number (SCN) detector is a promising detector that can work efficiently in uncertain environments. In this paper, we consider a Cognitive Radio (CR) system with large number of antennas (eg. Massive MIMO) and we provide an accurate and simple closed form approximation for the SCN distribution using the generalized extreme value (GEV) distribution. The approximation framework is based on the moment-matching method where the expressions of the moments are approximated using bi-variate Taylor expansion and results from random matrix theory. In addition, the performance probabilities and the decision threshold are considered. Since the number of antennas and/or the number of samples used in the sensing process may frequently change, this paper provides simple form decision threshold and performance probabilities offering dynamic and real-time computations. Simulation results show that the provided approximations are tightly matched to relative empirical ones.

A Simple Closed form Approximation of Average Channel Capacity for Weakly Turbulent Optical Wireless Links

Performance of optical wireless link is degraded by the presence of atmospheric turbulence induced fading. In this paper, a simple tight closed form expression for average channel capacity of optical wireless communication (OWC) system in weak atmospheric turbulence is given. The channel capacity of OWC systems employing spatial diversity reception with maximum ratio combining and equal gain combining over independent and identical turbulence induced fading channels is also computed using proposed expression. Results computed by employing our expressions are in excellent agreement with results based on numerical Monte Carlo simulations over permissible turbulence strength for a wide range of signal to noise ratio.

A Closed-Form Approximation for the CDF of the Sum of Independent Random Variables

In this letter, we use the Berry–Esseen theorem and the method of tilted distributions to derive a simple tight closed-form approximation for the tail probabilities of a sum of independent but not necessarily identically distributed random variables. We also provide lower and upper bounds. The expression can also be used for computing the cumulative distribution function. We illustrate the accuracy of the method by analyzing some convergence properties of the theoretical approximation and comparing it with previous results in the literature when available and/or numerical results.

Nonlinear electroelastic deformations of dielectric elastomer composites: I—Ideal elastic dielectrics

This paper puts forth homogenization solutions for the macroscopic elastic dielectric response—under finite deformations and finite electric fields—of ideal elastic dielectric composites with two-phase isotropic particulate microstructures. Specifically, solutions are presented for three classes of microstructures: (i) an isotropic iterative microstructure wherein the particles are infinitely polydisperse in size, (ii) an isotropic distribution of polydisperse spherical particles of a finite number of different sizes, and (iii) an isotropic distribution of monodisperse spherical particles. The solution for the iterative microstructure, which corresponds to the viscosity solution of a Hamilton–Jacobi equation in five “space” variables, is constructed by means of a novel high-order WENO finite-difference scheme. On the other hand, the solutions for the microstructures with spherical particles are constructed by means of hybrid finite elements.Prompted by the functional features shared by all three obtained solutions, a simple closed-form approximation is proposed for the macroscopic elastic dielectric response of ideal elastic dielectric composites with any type of (non-percolative) isotropic particulate microstructure. As elaborated in a companion paper, the proposed approximate solution proves particularly useful as a fundamental building block to generate approximate solutions for the macroscopic elastic dielectric response of dielectric elastomer composites made up of non-Gaussian dielectric elastomers filled with nonlinear elastic dielectric particles.

On the Performance of Non-orthogonal Multiple Access Systems With Partial Channel Information

In this paper, a downlink single-cell non-orthogonal multiple access (NOMA) network with uniformly deployed users is considered and an analytical framework to evaluate its performance is developed. Particularly, the performance of NOMA is studied by assuming two types of partial channel state information (CSI). For the first one, which is based on imperfect CSI , we present a simple closed-form approximation for the outage probability and the average sum rate, as well as their high signal-to-noise ratio (SNR) expressions. For the second type of CSI, which is based on second order statistics (SOS) , we derive a closed-form expression for the outage probability and an approximate expression for the average sum rate for the special case two users. For the addressed scenario with the two types of partial CSI, the results demonstrate that NOMA can achieve superior performance compared to the traditional orthogonal multiple access (OMA). Moreover, SOS-based NOMA always achieves better performance than that with imperfect CSI, while it can achieve similar performance to the NOMA with perfect CSI at the low SNR region. The provided numerical results confirm that the derived expressions for the outage probability and the average sum rate match well with the Monte Carlo simulations.

An Improved Short-Beam Method for Measuring Multiple Constitutive Properties for Composites

Abstract The short beam shear (SBS) test method combined with digital image correlation (DIC) deformation measurement is used in this work to study constitutive properties of the IM7/8552 carbon/epoxy composite material system. Dependence of the shear stress approximation on the SBS specimen span-to-thickness (s/t) ratio has been investigated. A stress model developed in a previous study is expanded to account for this dependency. In addition, the effect of nonzero transverse normal stresses on a simple closed-form approximation for tensile and compressive axial moduli has been assessed. Results from the expanded closed-form solution verify that transverse normal stress has little influence on the measurement of the axial moduli. Also, the SBS test setup has been refined as a result of the sensitivity of the measured axial strain distribution to the specimen alignment in the test fixture. The test results obtained in this work are compared with previously published results to demonstrate the ability of a single unidirectional SBS test to capture multiple constitutive properties including axial tensile and compressive moduli, the nonlinear shear stress–strain response in the plane of loading, and to show that measured material properties do not depend on the test specimen size.

A method to compute the Hall-geometry factor at weak magnetic field in closed analytical form

The Hall-geometry factor describes how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts. Closed analytical formulae are known only for degenerate or highly symmetric shapes of Hall plates. We present an approach which splits up the computation into a superposition of two electrostatic problems with simple scalar conductivity. This procedure is suited to prove various symmetry properties of Hall plates and to derive a number of formulae for the Hall-geometry factor. We apply it to rectangular Hall plates at weak magnetic field with arbitrarily large contacts having different input and output resistances. From this we deduce the case of Hall plates with \(90^{\circ }\) symmetry, for which a simple closed form approximation with an error of less than 0.02 % is given.

On the Outage Capacity of Orthogonal Space-Time Block Codes Over Multi-Cluster Scattering MIMO Channels

Multiple cluster scattering MIMO channel is a useful model for pico-cellular MIMO networks. In this paper, orthogonal space-time block coded transmission over such a channel is considered, where the effective channel equals the product of n complex Gaussian matrices. A simple and accurate closed-form approximation to the channel outage capacity has been derived in this setting. The result is valid for an arbitrary number of clusters n-1 of scatterers and an arbitrary antenna configuration. Numerical results are provided to study the relative outage performance between the multi-cluster and the Rayleigh-fading MIMO channels for which n=1.

On the short-maturity behaviour of the implied volatility skew for random strike options and applications to option pricing approximation

In this paper, we propose a general technique to develop first- and second-order closed-form approximation formulas for short-maturity options with random strikes. Our method is based on a change of numeraire and on Malliavin calculus techniques, which allow us to study the corresponding short-maturity implied volatility skew and to obtain simple closed-form approximation formulas depending on the derivative operator. The numerical analysis shows that these formulas are extremely accurate and improve some previous approaches for two-asset and three-asset spread options such as Kirk’s formula or the decomposition method presented in Alòs et al. [Energy Risk, 2011, 9, 52–57]. This methodology is not model-dependent, and it can be applied to the case of random interest rates and volatilities.

Optimal Backlight Modulation With Crosstalk Control in Stereoscopic Display

Crosstalk between the left-eye and right-eye images is one of the main artifacts affecting the visual quality of stereoscopic liquid crystal display (LCD) systems. In this paper, a novel technique, called Optimal Backlight Modulation (OBM), is proposed to reduce crosstalk by taking the advantage of newly emerged light-emitting diode (LED) backlight panel. Unlike most existing techniques which solely aim at eliminating the effect of crosstalk, ignoring image brightness and contrast, OBM strikes a better balance by maximizing the output luminance while confining the crosstalk within a given level. A simple closed-form approximation of the optimization problem can be easily employed and solved in real time on LCD control hardware. Simulation results show that the proposed OBM algorithm provides the same or higher luminance while reducing the crosstalk by 60% compared with the other tested methods.

A Simple Closed-Form Approximation for the Packet Loss Rate of a TCP Connection Over Wireless Links

This letter presents a new and simple model for a TCP flow experiencing random packet losses due to both transmission errors and congestion events. From this model, we will derive a straightforward expression of a unified loss probability (ULP). This ULP gives the opportunity to reuse classical analytic models to analyze the performance of TCP and to size the buffer to optimize the wireless link utilization. Extensive simulations using TCP Reno in ns-2 demonstrate that our model is valid not only for the extreme cases where either transmission errors or congestion losses dominate but also in the situations where both types of losses are significant.