Closed-form Approximation Formula Research Articles

Best sum-throughput evaluation of cooperative downlink transmission nonorthogonal multiple access system

In cooperative simultaneous wireless information and power transfer (SWIPT) nonorthogonal multiple access (NOMA) downlink situations, the current research investigates the total throughput of users in center and edge of cell. We focus on creating ways to solve these problems because the fair transmission rate of users located in cell edge and outage performance are significant hurdles at NOMA schemes. To enhance the functionality of cell-edge users, we examine a two-user NOMA scheme whereby the cell-center user functions as a SWIPT relay using power splitting (PS) with a multiple-input single-output. We calculated the probability of an outage for both center and edge cell users, using closed-form approximation formulas and evaluate the system efficacy. The usability of cell edge users is maximized by downlink transmission NOMA (CDT-NOMA) employing a SWIPT relay that employs PS. The suggested approach calculates the ideal value of the PS coefficient to optimize the sum throughput. Compared to the noncooperative and single-input single-output NOMA systems, the best SWIPT-NOMA system provides the cell-edge user with a significant throughput gain. Applying SWIPT-based relaying transmission has no impact on the framework’s overall throughput.

Pricing vanilla, barrier, and lookback options under two-scale stochastic volatility driven by two approximate fractional Brownian motions

<p>In this paper, we proposed a stochastic volatility model in which the volatility was given by stochastic processes representing two characteristic time scales of variation driven by approximate fractional Brownian motions with two Hurst exponents. We obtained an approximate closed-form formula for a European vanilla option price and the corresponding implied volatility formula based on singular and regular perturbations and a Mellin transform. The explicit formula for the implied volatility allowed us to find the slope of the implied volatility skew with respect to the Hurst exponent and time-to-maturity. The proposed model allows the market volatility behavior to be captured uniformly in time-to-maturity. We conducted an empirical analysis to find the validity of the proposed model by comparing it with other models and Monte Carlo simulation. Further, we extended the pricing result for the vanilla option to two path-dependent exotic (barrier and lookback) options and obtained the corresponding price formulas explicitly.</p>

On a time-changed Lévy risk model with capital injections and periodic observation

In this paper, an insurance company’s surplus process is modeled by a time-changed Lévy process, where the continuous-time change is realized by integrating the Cox–Ingersoll–Ross (CIR) process. Assume that the surplus level is monitored periodically with constant monitoring frequency, and the insurer makes decisions on capital injections based on the observed surplus levels. Given a critical level b (b>0), whenever the observed surplus level belongs to the interval [0,b), capitals are injected into the insurance company to bring the surplus level back to b; whenever the observed surplus level is below level 0, ruin is declared, and the surplus process is stopped. The finite-time expected present value of operating costs until ruin is studied. Utilizing the Fourier series expansion in cosine form, we derive the closed-form approximation formula for this function and analyze the approximation error. Finally, many numerical examples are given to demonstrate the effectiveness of our method, and the impacts of some parameters are also studied.

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.

Valuation of Annuity Guarantees Under a Self-Exciting Switching Jump Model

This article investigates the valuation of annuity guarantees under a regime-switching model when the dynamics of the underlying stock price follow a self-exciting switching jump-diffusion process. In this framework, we add a jump component to a regime-switching geometric Brownian for large shocks on the stock price. The intensity of shock arrivals is a Hawkes process modulated by a continuous time hidden Markov chain with a finite number of states. The interest rate used for discounting is stochastic and correlated to the stock market. In an incomplete market, we define an equivalent martingale measure to price a variable annuity contract that guarantees a minimum living or death benefit. Under this equivalent martingale measure, we propose closed-form approximation formulas using the inverse Fourier transform technique. A numerical implementation highlights the impact of self-exciting jumps and economic regimes on the valuation of guarantees.

Convergence analysis for continuous-time Markov chain approximation of stochastic local volatility models: Option pricing and Greeks

This paper establishes the precise second order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing options under the general framework of stochastic local volatility (SLV) models. The stochastic local volatility models studied in this paper include Heston model, 4/2 model, α-Hypergeometric model, stochastic alpha beta rho (SABR) model, Heston-SABR model and quadratic SLV model. Using the stochastic flow theorem, the closed-form CTMC approximation formula for the Greeks are obtained and the second order convergence rates are proved. Numerical examples confirm the theoretical findings.

The closed-form approximation to price basket options under stochastic interest rate

The paper presents closed-form approximation formulas for pricing basket options. We assume that the underlying asset prices follow geometric Brownian motions and the interest rate follows the one-factor Hull-White model. Under a given forward measure, we obtain the analytical lower and upper bounds of basket options. By finding a simple random variable to replace the sum of the lognormal random variables, we combine the conditioning and the moment matching approaches, and derive approximation formulas to price basket options. Numerical results illustrate that our results fall in the sharp lower and upper bounds of basket options and are consistent with Monte Carlo simulation results.

A closed-form approximation formula for pricing European options under a three-factor model

AbstractThe double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

Free vibration of viscoelastically supported beam bridges under moving loads: Closed-form formula for maximum resonant response

In this paper, a closed-form approximate formula for estimating the maximum resonant response of beam bridges on viscoelastic supports (VS) under moving loads is proposed. The methodology is based on the discrete approximation of the fundamental vertical mode of a non-proportionally damped Bernoulli–Euler beam, which allows the derivation of closed-form expressions for the fundamental modal characteristics and maximum amplitude of free vibration at the mid-span of VS beams. Finally, an approximate formula to estimate maximum resonant acceleration of VS beams under passage of articulated trains has been proposed. Verification studies prove that the approximate closed-form formula estimates the resonant peaks with good accuracy and is a useful tool for preliminary assessment of railway beam bridges considering the effect of soil-structure interaction at resonance. In combination with the use of full train signatures through the Residual Influence Line (LIR) method, the proposed solution yields good results also in the lower range of speeds, where resonant sub-harmonics are more intensely reduced by damping.

A Mellin Transform Approach to the Pricing of Options with Default Risk

The stochastic elasticity of variance model introduced by Kim et al. (Appl Stoch Models Bus Ind 30(6):753–765, 2014) is a useful model for forecasting extraordinary volatility behavior which would take place in a financial crisis and high volatility of a market could be linked to default risk of option contracts. So, it is natural to study the pricing of options with default risk under the stochastic elasticity of variance. Based on a framework with two separate scales that could minimize the number of necessary parameters for calibration but reflect the essential characteristics of the underlying asset and the firm value of the option writer, we obtain a closed form approximation formula for the option price via double Mellin transform with singular perturbation. Our formula is explicitly expressed as the Black–Scholes formula plus correction terms. The correction terms are given by the simple derivatives of the Black–Scholes solution so that the model calibration can be done very fast and effectively.

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.

Nonlocal Effective Electromagnetic Wave Characteristics of Composite Media: Beyond the Quasistatic Regime

We derive exact nonlocal expressions for the effective dielectric constant tensor ${\boldsymbol \varepsilon}_e({\bf k}_I, \omega)$ of disordered two-phase composites and metamaterials from first principles. This formalism extends the long-wavelength limitations of conventional homogenization estimates of ${\boldsymbol \varepsilon}_e({\bf k}_I, \omega)$ for arbitrary microstructures so that it can capture spatial dispersion well beyond the quasistatic regime (where $\omega$ and ${\bf k}_I$ are frequency and wavevector of the incident radiation). This is done by deriving nonlocal strong-contrast expansions that exactly account for multiple scattering for the range of wavenumbers for which our extended homogenization theory applies, i.e., $0 \le |{\bf k}_I| \ell \lesssim 1$ (where $\ell$ is a characteristic heterogeneity length scale). Due to the fast-convergence properties of such expansions, their lower-order truncations yield accurate closed-form approximate formulas for ${\varepsilon}_e({\bf k}_I,\omega)$ that incorporate microstructural information via the spectral density, which is easy to compute for any composite. The accuracy of these microstructure-dependent approximations is validated by comparison to full-waveform simulation methods for both 2D and 3D ordered and disordered models of composite media. Thus, our closed-form formulas enable one to predict accurately and efficiently the effective wave characteristics well beyond the quasistatic regime without having to perform full-blown simulations. Among other results, we show that certain disordered hyperuniform particulate composites exhibit novel wave characteristics. Our results demonstrate that one can design the effective wave characteristics of a disordered composite by engineering the microstructure to possess tailored spatial correlations at prescribed length scales.

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.

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.

Physical Layer Security of QSTBC With Power Scaling in MIMO Wiretap Channels

Quasi-orthogonal space time block codes with power scaling (QSTBC-PS) that linearly combines two Alamouti codes with unequal power scaling has been known to achieve full diversity in multi-input multi-output (MIMO) systems. In this correspondence, we evaluate the physical layer security performance of QSTBC-PS in MIMO wiretap channels under Rayleigh fading. We derive the tight closed-form approximate formula of secrecy outage probability for QSTBC-PS by using moment matching approximation technique. Further, we derive the high signal-to-noise ratio (SNR) approximation and the probability of strictly positive secrecy capacity in explicit form. The numerical examples validate that our analytical results match well to the Monte-Carlo simulation results for general number of receive antennas at the desired receiver and eavesdropper.

Elliptic averaging of optical transfer functions for estimating astigmatism and defocus

We elaborate an elliptic averaging procedure to estimate astigmatism and defocus of optical transfer functions that allows to reduce the problem to a simpler one of estimating defocus only. Based on approximate closed-form formulas for the zeros of defocus optical transfer functions, we reparametrise the frequency domain and derive a class of nonlinear transformations that turn elliptic null-curves of astigmatic optical transfer functions into circles. Our results make astigmatic images amenable to restoration.

Dyadic analysis of a cylindrical wire consisting of a cover with fully-populated surface conductivity tensor.

Dyadic Green's function for a monolayer cylinder consisting of a cover with fully-populated surface conductivity tensor is formulated in this paper. Cylindrically wrapped densely packed graphene strips behaving as a hyperbolic meta-surface and arbitrary shaped graphene patterns supporting magneto-plasmons are investigated as two special cases. Closed-form approximate formulas based on effective medium theory (EMT) are available for the surface conductivity of the former and the latter can be analyzed by the effective tensor surface conductivity extraction by means of the finite element method (FEM). Both cases are treated under locally flat consideration of the conformal structures. Our formulation is based on the scattering superposition method by imposing the discontinuity boundary condition on tangential magnetic fields. Numerical examples are demonstrated to further validate the method for both guided and radiated modes.

A general model of demand-responsive transportation services: From taxi to ridesharing to dial-a-ride

The paper presents a general analytic framework to model transit systems that provide door-to-door service. The model includes as special cases non-shared taxi and demand responsive transportation (DRT). In the latter we include both, paratransit services such as dial-a-ride (DAR), and the form of ridesharing (shared taxi) currently being used by crowd-sourced taxi companies like Lyft and Uber. The framework yields somewhat optimistic results because, among other things, it is deterministic and does not track vehicles across space. By virtue of its simplicity, however, the framework yields approximate closed form formulas for many cases of interest.

Pricing Discretely Monitored Barrier Options Under Markov Processes Using a Markov Chain Approximation

We propose an explicit closed-form approximation formula for the price of discretely monitored single or double barrier options whose underlying asset evolves according to a generic one-dimensional Markov process. This set of stochastic processes includes, but is not limited to, diffusion and jump diffusion processes commonly used in derivative pricing applications. The formula’s derivation combines the integral equation method, the Z−transform technique, and a continuous-time Markov chain approximation of the underlying Markov process. It does not require one to perform an intermediate numerical quadrature or related potentially runtime-intensive, error-prone, or otherwise complicated numerical procedure that may require a high degree of tuning to ensure appropriate accuracy (e.g. an inverse Laplace transform or inverse Z−transform). Rather, the price and Greeks of a discretely-monitored double barrier option may be 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. Examples including time-dependent barriers and non-uniform monitoring time intervals, may be seamlessly incorporated into this framework. In addition, by limiting the monitoring frequency to a large value, we also obtain an accurate closed-form formula for the price and Greeks of continuously-monitored double barrier options with time-dependent barriers under general Markov processes. Finally, we provide many 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.

Regularized Zero-Forcing Precoding-Aided Adaptive Coding and Modulation for Large-Scale Antenna Array-Based Air-to-Air Communications

We propose a regularized zero-forcing transmit precoding (RZF-TPC)-aided and distance-based adaptive coding and modulation (ACM) scheme to support aeronautical communication applications, by exploiting the high spectral efficiency of the large-scale antenna arrays and link adaption. Our RZF-TPC-aided and distance-based ACM scheme switches its mode according to the distance between the communicating aircraft. We derive the closed-form asymptotic signal-to-interference-plus-noise ratio (SINR) expression of the RZF-TPC for the aeronautical channel, which is Rician, relying on a non-centered channel matrix that is dominated by the deterministic line-of-sight component. The effects of both realistic channel estimation errors and of the co-channel interference are considered in the derivation of this approximate closed-form SINR formula. Furthermore, we derive the analytical expression of the optimal regularization parameter that minimizes the mean square detection error. The achievable throughput expression based on our asymptotic approximate SINR formula is then utilized as the design metric for the proposed RZF-TPC-aided and distance-based ACM scheme. Monte-Carlo simulation results are presented for validating our theoretical analysis as well as for investigating the impact of the key system parameters. The simulation results closely match the theoretical results. In the specific example that two communicating aircrafts fly at a typical cruising speed of 920km/h, heading in opposite direction over the distance up to 740km taking a period of about 24 min, the RZF-TPC-aided and distance-based ACM is capable of transmitting a total of 77 GB of data with the aid of 64 transmit antennas and four receive antennas, which is significantly higher than that of our previous eigen-beamforming transmit precoding-aided and distance-based ACM benchmark.