Complex derivatives valuation: applying the Least-Squares Monte Carlo Simulation Method with several polynomial basis

Background:This article investigates the Least-Squares Monte Carlo Method by using different polynomial basis in American Asian Options pricing.The standard approach in the option pricing literature is to choose the basis arbitrarily.By comparing four different polynomial basis we show that the choice of basis interferes in the option's price.Methods:We assess Least-Squares Method performance in pricing four different American Asian Options by using four polynomial basis:Power,Laguerre,Legendre and Hermite A.To every American Asian Option priced,three sets of parameters are used in order to evaluate it properly.Results:We show that the choice of the basis interferes in the option's price by showing that one of them converges to the option's value faster than any other by using fewer simulated paths.In the case of an Amerasian call option,for example,we find that the preferable polynomial basis is Hermite A.For an Amerasian put option,the Power polynomial basis is recommended.Such empirical outcome is theoretically unpredictable,since in principle all basis can be indistinctly used when pricing the derivative.Conclusion:In this article The Least-Squares Monte Carlo Method performance is assessed in pricing four different types of American Asian Options by using four different polynomial basis through three different sets of parameters.Our results suggest that one polynomial basis is best suited to perform the method when pricing an American Asian option.Theoretically all basis can be indistinctly used when pricing the derivative.However,our results does not confirm these.We find that when pricing an American Asian put option,Power A is better than the other basis we have studied here whereas when pricing an American Asian call,Hermite A is better.

A Novel Accurate Method forMulti-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains

Anovel accuratemethod is proposed to solve a broad variety of linear and nonlinear(1+1)-dimensional and(2+1)-dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity.For(1+1)-dimensional problems,analytical solutions that satisfy the boundary requirements are derived.Such solutions are numerically calculated using the trigonometric basis approximation for(2+1)-dimensional problems.With the aid of these analytical or numerical approximations,the original problems can be converted into the fractional ordinary differential equations,and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients.An efficient backward substitution strategy that was previously provided for a single fractional ordinary differential equation is then used to solve the corresponding systems.The straightforward quasilinearization technique is applied to handle nonlinear issues.Numerical experiments demonstrate the suggested algorithm’s superior accuracy and efficiency.

A Novel Method for Linear Systems of Fractional Ordinary Differential Equations with Applications to Time-Fractional PDEs

This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations(FODEs)which have been widely used in modeling various phenomena in engineering and science.An approximate solution of the system is sought in the formof the finite series over the Müntz polynomials.By using the collocation procedure in the time interval,one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure.This technique also serves as the basis for solving the time-fractional partial differential equations(PDEs).The modified radial basis functions are used for spatial approximation of the solution.The collocation in the solution domain transforms the equation into a system of fractional ordinary differential equations similar to the one mentioned above.Several examples have verified the performance of the proposed novel technique with high accuracy and efficiency.

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Based on the nonlocal theory and Mindlin plate theory,the governing equations(i.e.,a system of partial differential equations(PDEs)for bending problem)of magnetoelectroelastic(MEE)nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle.The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions(MPS)to solve the governing equations numerically.It is confirmed that for the present bending model,the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points.The effects of different boundary conditions,applied loads,and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method.Some important conclusions are drawn,which should be helpful for the design and applications of electromagnetic nanoplate structures.

An improved element-free Galerkin method for solving the generalized fifth-order Korteweg-de Vries equation

In this paper, an improved element-free Galerkin （IEFG） method is proposed to solve the generalized fifth-order Korteweg-de Vries （gfKdV） equation. When the traditional element-free Galerkin （EFG） method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares （MLS） shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares （IMLS） approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.

HIGH-ORDER RUNGE-KUTTA DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR 2-D RESONATOR PROBLEM

The Runge-Kutta discontinuous Galerkin finite element method （RK-DGFEM） is introduced to solve the classical resonator problem in the time domain. DGFEM uses unstructured grid discretization in the space domain and it is explicit in the time domain. Consequently it is a best mixture of FEM and finite volume method （FVM）. RK-DGFEM can obtain local high-order accuracy by using high-order polynomial basis. Numerical experiments of transverse magnetic （TM） wave propagation in a 2-D resonator are performed. A high-order Lagrange polynomial basis is adopted. Numerical results agree well with analytical solution. And different order Lagrange interpolation polynomial basis impacts on simulation result accuracy are discussed. Computational results indicate that the accuracy is evidently improved when the order of interpolation basis is increased. Finally, L^2 errors of different order polynomial basis in RK-DGFEM are presented. Computational results show that L^2 error declines exponentially as the order of basis increases.

Ensemble kernel method：SVM classification based on game theory

With the development of the support vector machine（SVM）,the kernel function has become one of the cores of the research on SVM.To a large extent,the kernel function determines the generalization ability of the classifier,but there is still no general theory to guide the choice and structure of the kernel function.An ensemble kernel function model based on the game theory is proposed,which is used for the SVM classification algorithm.The model can effectively integrate the advantages of the local kernel and the global kernel to get a better classification result,and can provide a feasible way for structuring the kernel function.By making experiments on some standard datasets,it is verified that the new method can significantly improve the accuracy of classification.

ON APPROXIMATION BY SPHERICAL REPRODUCING KERNEL HILBERT SPACES

The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.

A radial basis function for reconstructing complex immersed boundaries in ghost cell method

It is important to track and reconstruct the complex immersed boundaries for simulating fluid structure interaction problems in an immersed boundary method（IBM）. In this paper, a polynomial radial basis function（PRBF） method is introduced to the ghost cell immersed boundary method for tracking and reconstructing the complex moving boundaries. The body surfaces are fitted with a finite set of sampling points by the PRBF, which is flexible and accurate. The complex or multiple boundaries could be easily represented. A simple treatment is used for identifying the position information about the interfaces on the background grid. Our solver and interface reconstruction method are validated by the case of a cylinder oscillating in the fluid. The accuracy of the present PRBF method is comparable to the analytic function method. In ta flow around an airfoil, the capacity of the proposed method for complex geometries is well demonstrated.

On Higher Order Pyramidal Finite Elements

In this paper we first prove a theorem on the nonexistence of pyramidal polynomial basis functions.Then we present a new symmetric composite pyramidal finite element which yields a better convergence than the nonsymmetric one.It has fourteen degrees of freedom and its basis functions are incomplete piecewise triquadratic polynomials.The space of ansatz functions contains all quadratic functions on each of four subtetrahedra that form a given pyramidal element.