Simple Closed-form Formulas Research Articles

Generalized Moment Estimators Based on Stein Identities

For parameter estimation of continuous and discrete distributions, we propose a generalization of the method of moments (MM), where Stein identities are utilized for improved estimation performance. The construction of these Stein-type MM-estimators makes use of a weight function as implied by an appropriate form of the Stein identity. Our general approach as well as potential benefits thereof are first illustrated by the simple example of the exponential distribution. Afterward, we investigate the more sophisticated two-parameter inverse Gaussian distribution and the two-parameter negative-binomial distribution in great detail, together with illustrative real-world data examples. Given an appropriate choice of the respective weight functions, their Stein-MM estimators, which are defined by simple closed-form formulas and allow for closed-form asymptotic computations, exhibit a better performance regarding bias and mean squared error than competing estimators.

A simple formula for coherent specular reflection loss of underwater sound from rough ocean surfaces

The interaction of underwater sound with rough sea surfaces affects its propagation. The interaction may involve several coupled mechanisms: (1) a rough air-sea interface scatters and spreads the incident sound energy over a range of angles, (2) entrapped sub-surface air bubbles absorb and scatter sound energy, and (3) medium of bubble clouds refract sound energy upwards. In this paper, we ignore the effect of bubbles and consider the effect of rough air-sea interface only. We further limit our consideration to the coherent component in the specular direction. Two simple closed-form formulas are often used to model the coherent reflection loss from rough ocean surfaces. One is derived from perturbation approximation and suitable for smaller grazing angles. The other is from Kirchhoff approximation and suitable for larger grazing angles. More accurate approximations over a wide-range of grazing angles require numerical evaluation of integrals or complex special functions. In this work, we simplify and adapt earlier results by other authors to obtain a simple expression that transits from the perturbation to the Kirchhoff formulas as grazing angle increases. The new simple analytical formula is easy to use and compares well with results from complex numerical methods.

Interference Analysis in Non-Poisson Networks Under Spatially Correlated Shadowing

Spatial correlations that appear in mobile wireless networks, such as the correlations of node locations and shadowing effects, significantly affect the characteristics of interference, which may degrade the performance of various wireless network systems. In this paper, we theoretically analyze the statistical and temporal characteristics of interference in a network where the node locations and shadowing are spatially correlated. We model the correlation of the node locations by two types of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non-Poisson</i> point processes: determinantal point processes (DPPs) for modeling <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">repulsiveness</i> and doubly Poisson cluster processes (DPCPs) for modeling <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">attractiveness</i> . Furthermore, we consider Gudmundson's model for the spatial correlation of shadowing. Using this model and assuming an i.i.d. mobility of nodes, we analyze the variance along with the spatial and temporal correlations of interference. Since the exact expressions are in non-analytical forms, we derive their simple closed-form asymptotic formulas when the variance of the shadowing is large. The results show a readable relationship between the characteristics of interference and various system parameters. This relationship can be used for realistic modeling and better understanding of various wireless network systems under spatially correlated shadowing. Moreover, we discuss various numerical examples and demonstrate that the obtained asymptotic expressions achieve tight approximation.

The role of the cortex in indentation experiments of animal cells

We present a model useful for interpretation of indentation experiments on animal cells. We use finite element modeling for a thorough representation of the complex structure of an animal cell. In our model, the crucial constituent is the cell cortex—a rigid layer of cytoplasmic proteins present on the inner side of the cell membrane. It plays a vital role in the mechanical interactions between cells. The cell cortex is modeled by a three-dimensional solid to reflect its bending stiffness. This approach allows us to interpret the results of the indentation measurements and extract the mechanical properties of the individual elements of the cell structure. During the simulations, we scan a broad range of parameters such as cortex thickness and Young’s modulus, cytoplasm Young’s modulus, and indenter radius, which define cell properties and experimental conditions. Finally, we propose a simple closed-form formula that approximates the simulated results with satisfactory accuracy. Our formula is as easy to use as Hertz's function to extract cell properties from the measurement, yet it considers the cell’s inner structure, including cell cortex, cytoplasm, and nucleus.

Valuation of volatility derivatives with time-varying volatility: An analytical probabilistic approach using a mixture distribution for pricing nonlinear payoff volatility derivatives in discrete observation case

In this paper, we present an analytical probabilistic approach for pricing nonlinear payoff volatility derivatives with discrete sampling by assuming that the underlying asset price evolves according to the Black–Scholes model with time-varying volatility. A major difficulty to solve the pricing problem analytically is the volatility of the underlying asset price is time-varying, resulting the realized variance is distributed according to a mixture distribution, the probability density function of which is unknown. By utilizing the properties of a linear combination of noncentral chi-square random variables, we can calculate the expectation of square root of the realized variance analytically and provide formulas for pricing volatility swaps, volatility options, and variance options, including put–call parity relationships. Furthermore, we demonstrate an interesting application of our formulas by constructing simple closed-form approximate formulas for pricing the volatility derivatives for the constant elasticity of variance model. Finally, Monte Carlo simulations are conducted to illustrate the performance our approach, and effects of price volatility on the fair strike prices of the volatility derivatives are investigated and analyzed through several numerical experiments.

Approximating Displacements in ${\mathbb {R}}^3$ by Rotations in ${\mathbb {R}}^4$ and Its Application to Pointcloud Registration

No proper norm exists to measure the distance between two object poses essentially because a general pose is defined by a rotation and a translation, and thus, it involves magnitudes with different units. As a means to solve this dimensional-inhomogeneity problem, the concept of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">characteristic length</i> has been put forward in the area of kinematics. The idea consists of scaling translations according to this characteristic length and then approximating the corresponding displacement defining the object pose in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathbb {R}}^3$</tex-math></inline-formula> by a rotation in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathbb {R}}^4$</tex-math></inline-formula> , for which a norm exists. This article sheds new light on this kind of approximations which permits simplifying optimization problem whose cost functions involve translations and rotations simultaneously. A good example of this kind of problems is the pointcloud registration problem in which the optimal rotation and translation between two sets of corresponding 3-D point data, so that they are aligned/registered, have to be found. As a result, a simple closed-form formula for solving this problem is presented which is shown to be an attractive alternative to the previous approaches.

Simple Closed-Form Formulas for Conditional Moments of Inhomogeneous Nonlinear Drift Constant Elasticity of Variance Process

The stochastic differential equation (SDE) has been used to model various phenomena and investigate their properties. Conditional moments of stochastic processes can be used to price financial derivatives whose payoffs depend on conditional moments of underlying assets. In general, the transition probability density function (PDF) of a stochastic process is often unavailable in closed form. Thus, the conditional moments, which can be directly computed by applying the transition PDFs, may be unavailable in closed form. In this work, we studied an inhomogeneous nonlinear drift constant elasticity of variance (IND-CEV) process, which is a class of diffusions that have time-dependent parameter functions; therefore, their sample paths are asymmetric. The closed-form formulas for conditional moments of the IND-CEV process were derived without having a condition on eigenfunctions or the transition PDF. The analytical results were examined through Monte Carlo simulations.

Analytically Pricing Formula for Contingent Claim with Polynomial Payoff under ECIR Process

Contingent claims, such as bonds, swaps, and options, are financial derivatives whose payoffs depend on uncertain future real values of underlying assets which emphasize various real-world applications. In general, valuations for contingent claims can be derived from the conditional expectations of underlying assets. For a simple process, the moments are usually directly obtained from its transition probability density function (PDF). However, if the transition PDF is unavailable in simple form, the derivations of the moments and the contingent claim prices will not be accessible in closed forms. This paper provides a closed-form formula for pricing contingent claims with nonlinear payoff under a no-arbitrage principle when underlying assets follow the extended Cox–Ingersoll–Ross (ECIR) process with the symmetry properties of the Brownian motion. The formula proposed here is a consequence of successfully solving an explicit solution for a system of recurrence partial differential equations in which its solution subtly depends on the conditional moments. For the particular CIR process, we obtain simple closed-form formulas by solving the Riccati differential equation. Furthermore, we carry out a complete investigation of the convergent case for those formulas. In case such as that of the unsolvable Riccati differential equation, ECIR case, a numerical method for numerically evaluating the mentioned analytical formulas and numerical validations for the formulas are examined. The validity and efficiency of the formulas are numerically demonstrated by comparison with results from Monte Carlo simulations for various modeling parameters. Finally, the proposed formula is applied to the value contingent claims such as coupon bonds, interest rate swaps, and arrears swaps.

Attractors with large complex structure for one-parameter families of Calabi-Yau manifolds

The attractor equations for an arbitrary one-parameter family of Calabi-Yau manifolds are studied in the large complex structure region. These equations are solved iteratively, generating what we term an N-expansion, which is a power series in the Gromov-Witten invariants of the manifold. The coefficients of this series are associated with integer partitions. In important cases we are able to find closed-form expressions for the general term of this expansion. To our knowledge, these are the first generic solutions to attractor equations that incorporate instanton contributions. In particular, we find a simple closed-form formula for the entropy associated to rank two attractor points, including those recently discovered. The applications of our solutions are briefly discussed. Most importantly, we are able to give an expression for the Wald entropy of black holes that includes all genus 0 instanton corrections.

Glulam – closed form solution using statistical smearing, truncating, calibrating

ABSTRACT This paper describes a relatively simple closed form formula for predicting the strength of glulam assemblies in bending about the major axis based on statistical engineering mechanics using the tension strength of finger jointed of bonded lamination pairs. It has been calibrated against the strengths of full-size glulam beams ranging in depths from 150 mm to 1200 mm over spans of 2.7 m up to 19.5 m. Comparisons are made at both the mean strength and 5th percentile strength levels with the latter drawing attention to the fact that all test series invariably involve observation of minimum values that means the statistical distributions are invariably truncated. For Weibull distributions, this has an impact on the shape factor and, therefore, the local coefficient of variation of the lower tail, which plays an important role in size effects in glulam and potentially has wider implications.

Large deviations of the entropy production rate for a class of Gaussian processes

We prove a large deviation principle (LDP) and a fluctuation theorem for the entropy production rate (EPR) of the following d dimensional stochastic differential equation dXt=AXtdt+QdBt, where A is a real normal stable matrix, Q is positive definite, and the matrices A and Q commute. The rate function for the EPR takes the following explicit form: I(x)=x1+ℓ0(x)−12+12∑k=1dαk2−βk2ℓ0(x)+αk for x ≥ 0 and I(x)=−x1+ℓ0(x)+12+12∑k=1dαk2−βk2ℓ0(x)+αk for x &amp;lt; 0, where αk ±iβk are the eigenvalues of A and ℓ0(x) is the unique solution of the equation x=1+ℓ×∑k=1dβk2αk2−ℓβk2,−1≤ℓ&amp;lt;mink=1,…,dαk2βk2. Simple closed form formulas for rate functions are rare, and our work identifies an important class of large deviation problems where such formulas are available. The logarithmic moment generating function (the fluctuation function) Λ associated with the LDP is given as Λ(λ)=−12∑k=1dαk2−4λ(1+λ)βk2+αk for λ∈D and Λ(λ) = ∞ for λ∉D, where D is the domain of Λ. The functions Λ(λ) and I(x) satisfy the Cohen–Gallavotti symmetry properties: Λ(x)=Λ(−(1+x)),I(x)=I(−x)−x, for all x∈R. In particular, the functions I and Λ do not depend on the diffusion matrix Q and are determined completely by the real and imaginary parts of the eigenvalues of A. Formally, the deterministic system with Q = 0 has zero EPR, and thus, the model exhibits a phase transition in that the EPR changes discontinuously at Q = 0.

Analytically pricing volatility swaps and volatility options with discrete sampling: Nonlinear payoff volatility derivatives

This paper presents the first analytical pricing formulas for volatility swaps and volatility options with discrete sampling under the Black-Scholes model with time varying risk-free interest rate. Despite numerous analytical works on the pricing of variance swaps with discrete sampling under different models of asset prices, an analytical pricing formula for volatility swaps as well as volatility options had not been well addressed until now. The main challenge in pricing volatility swaps and volatility options is that payoff functions contain a square root operator, making their expectations nonlinear. By utilizing properties of noncentral chi random variables, we can compute expectations of payoff functions analytically and obtain formulas for pricing volatility swaps and volatility options, including variance swaps and variance options. Furthermore, we investigate the accuracy of the well-known convexity correction formula. Most interestingly, we extend our results to the Black-Scholes model with time varying parameters and to the Heston stochastic volatility model in which the variance process is assumed to follow the extended Cox-Ingersoll-Ross process by constructing simple closed-form approximate formulas for pricing volatility swaps and demonstrate the accuracy and efficiency of this approach by comparing the approximated prices against those obtained with Monte Carlo simulations.

Pricing options on a mean-reverting asset by the analytical operator splitting method

In this paper, we propose an operator splitting method to valuate options on the inhomogeneous geometric Brownian motion. By exploiting the approximate dynamical symmetry of the pricing equation, we derive a simple closed-form approximate price formula for a European call option which resembles closely the Black–Scholes price formula for a European vanilla call option. Numerical tests show that the proposed method is able to provide very accurate estimates and tight bounds of the exact option prices. The method is very efficient and robust as well.

Simple closed-form expressions for the effective properties of multilaminated flexoelectric composites

A multilayered rectangular flexoelectric structure is considered. This composite has a unit cell of N-constituents which belong to the cubic crystal symmetry. Using the two-scale asymptotic homogenization method, explicit expressions of the local problems are derived. Simple closed-form formulas of the effective stiffness, piezoelectric, dielectric, and flexoelectric tensors are given, based on the solutions of local problems of stratified multilayered composites with perfect contact at the interface. These formulas provide information about the symmetry of the homogenized structure. As a numerical example, a bilaminate composite where the layers are perpendicular to the $$x_3$$ axis is studied, and the constituents are Barium Titanate and Gallium Arsenide. The composite after the homogenization process exhibits flexoelectric properties with tetragonal $$\bar{4}2$$ m crystal symmetry. The effective properties are computed for several constituents of volume fractions.

Mathematical Relations Between Measures of Brain Connectivity Estimated From Electrophysiological Recordings for Gaussian Distributed Data.

A large variety of methods exist to estimate brain coupling in the frequency domain from electrophysiological data measured, e.g., by EEG and MEG. Those data are to reasonable approximation, though certainly not perfectly, Gaussian distributed. This work is based on the well-known fact that for Gaussian distributed data, the cross-spectrum completely determines all statistical properties. In particular, for an infinite number of data, all normalized coupling measures at a given frequency are a function of complex coherency. However, it is largely unknown what the functional relations are. We here present those functional relations for six different measures: the weighted phase lag index, the phase lag index, the absolute value and imaginary part of the phase locking value (PLV), power envelope correlation, and power envelope correlation with correction for artifacts of volume conduction. With the exception of PLV, the final results are simple closed form formulas. In an excursion we also discuss differences between short time Fourier transformation and Hilbert transformation for estimations in the frequency domain. We tested in simulations of linear and non-linear dynamical systems and for empirical resting state EEG on sensor level to what extent a model, namely the respective function of coherency, can explain the observed couplings. For empirical data we found that for measures of phase-phase coupling deviations from the model are in general minor, while power envelope correlations systematically deviate from the model for all frequencies. For power envelope correlation with correction for artifacts of volume conduction the model cannot explain the observed couplings at all. We also analyzed power envelope correlation as a function of time and frequency in an event related experiment using a stroop reaction task and found significant event related deviations mostly in the alpha range.

Pricing Discretely Monitored Barrier Options under Markov Processes through Markov Chain Approximation

The authors propose an explicit closed-form approximation formula for the price of discretely monitored single or double barrier options with an underlying asset that evolves according to a one-dimensional Markov process, which includes diffusion and jump-diffusion processes. The prices and Greeks of a discretely monitored double barrier option are explicitly expressed in terms of rudimentary matrix operations. In addition, this framework may be extended to include additional features of barrier options often encountered in practice—for example, time-dependent barriers and nonuniform monitoring time intervals. They provide numerical examples to demonstrate the accuracy and efficiency of the proposed formula as well as its ability to reproduce existing benchmark results in the relevant literature in a unified framework. TOPICS:Derivatives, options Key Findings ▪ Markov chain approximations to stochastic processes are used to develop a simple closed-form pricing formula for discretely monitored barrier options. ▪ This article explores price and Greek computations within the constant elasticity of a variance model and Merton’s and Kou’s jump-diffusions models. ▪ Numerical examples are provided to demonstrate the robustness and accuracy of the pricing technique, both to recreate specialized results in related barrier option literature and to extend to multiple novel settings.

HeSRPT: Parallel scheduling to minimize mean slowdown

Modern data centers serve workloads which are capable of exploiting parallelism. When a job parallelizes across multiple servers it will complete more quickly, but jobs receive diminishing returns from being allocated additional servers. Because allocating multiple servers to a single job is inefficient, it is unclear how best to allocate a fixed number of servers between many parallelizable jobs.This paper provides the first optimal allocation policy for minimizing the mean slowdown of parallelizable jobs of known size when all jobs are present at time 0. Our policy provides a simple closed form formula for the optimal allocations at every moment in time. Minimizing mean slowdown usually requires favoring short jobs over long ones (as in the SRPT policy). However, because parallelizable jobs have sublinear speedup functions, system efficiency is also an issue. System efficiency is maximized by giving equal allocations to all jobs and thus competes with the goal of prioritizing small jobs. Our optimal policy, high-efficiency SRPT (heSRPT), balances these competing goals. heSRPT completes jobs according to their size order, but maintains overall system efficiency by allocating some servers to each job at every moment in time. Our results generalize to also provide the optimal allocation policy with respect to mean flow time.Finally, we consider the online case where jobs arrive to the system over time. While optimizing mean slowdown in the online setting is even more difficult, we find that heSRPT provides an excellent heuristic policy for the online setting. In fact, our simulations show that heSRPT significantly outperforms state-of-the-art allocation policies for parallelizable jobs.

Risk Management and the Optimal Combination of Equity Market Factors

Managing the intertemporal risk of optimally constructed multifactor portfolios adds to performance. The increases in Sharpe ratios are in addition to the utility that investors gain from controlling how much active risk they are exposed to over time. We derive a simple closed-form formula for security weights in optimal multifactor portfolios with an active-risk target. We test the risk control of five well-known factors—value, momentum, small size, low beta, and profitability—and the optimal multifactor portfolio. Our empirical research was carried out on the large-capitalization US equity market for 1966 through 2019. We conclude that for the equity market, more active factors are better than fewer if each subportfolio is “pure” as to factor, anchored to the benchmark, and combined on the basis of forecastable risks. Our portfolio construction methodology allows for transparent performance attribution and replication of the process in other markets and time periods.

Dynamic Risk Management of Equity Market Factors

Managing the intertemporal risk of optimally constructed multifactor portfolios adds to performance. The increases in Sharpe ratios are in addition to the utility that investors gain from controlling how much active risk they are exposed to over time. We derive a simple closed-form formula for security weights in optimal multifactor portfolios with an active-risk target. We test the risk control of five well-known factors—value, momentum, small size, low beta, and profitability—and the optimal multifactor portfolio. Our empirical research was carried out on the large-capitalization US equity market for 1966 through 2019. We conclude that for the equity market, more active factors are better than fewer if each factor subportfolio is “pure” as to factor, anchored to the benchmark, and combined on the basis of forecastable risks. Our portfolio construction methodology allows for transparent performance attribution and replication of the process in other markets and time periods.

Pricing options under simultaneous stochastic volatility and jumps: A simple closed-form formula without numerical/computational methods

We overcome the limitations of the previous literature in the European options pricing. In doing so, we provide a closed-form formula that does not require any numerical/computational methods. The formula is as simple as the classical Black–Scholes pricing formula. In addition, we simultaneously include jumps and stochastic volatility. Our approach implies the introduction of a new class of stochastic processes that are based on Clifford algebras. The approach can be easily generalized to higher dimensional problems.