scholarly journals Series Expansion and Fourth-Order Global Padé Approximation for a Rough Heston Solution

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1968
Author(s):  
Siow Woon Jeng ◽  
Adem Kilicman
Keyword(s):  
Riccati Equation ◽  
Series Expansion ◽  
Fourth Order ◽  
Padé Approximation ◽  
Approximation Error ◽  
Heston Model ◽  
Hurst Parameter ◽  
Option Prices ◽  
Pade Approximation ◽  
Adomian Decomposition

The rough Heston model has recently been shown to be extremely consistent with the observed empirical data in the financial market. However, the shortcoming of the model is that the conventional numerical method to compute option prices under it requires great computational effort due to the presence of the fractional Riccati equation in its characteristic function. In this study, we contribute by providing an efficient method while still retaining the quality of the solution under varying Hurst parameter for the fractional Riccati equations in two ways. First, we show that under the Laplace–Adomian-decomposition method, the infinite series expansion of the fractional Riccati equation’s solution corresponds to the existing expansion method from previous work for at least up to the fifth order. Then, we show that the fourth-order Padé approximants can be used to construct an extremely accurate global approximation to the fractional Riccati equation in an unexpected way. The pointwise approximation error of the global Padé approximation to the fractional Riccati equation is also provided. Unlike the existing work of third-order global Padé approximation to the fractional Riccati equation, our work extends the availability of Hurst parameter range without incurring huge errors. Finally, numerical comparisons were conducted to verify that our methods are indeed accurate and better than the existing method for computing both the fractional Riccati equation’s solution and option prices under the rough Heston model.

scholarly journals Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 959 ◽  
Author(s):  
Siow W. Jeng ◽  
Adem Kilicman
Keyword(s):  
Numerical Methods ◽  
Characteristic Function ◽  
Riccati Equation ◽  
Financial Market ◽  
Approximation Method ◽  
Padé Approximation ◽  
Heston Model ◽  
Pade Approximation ◽  
Multipoint Padé Approximation ◽  
Rough Volatility

Rough volatility models are recently popularized by the need of a consistent model for the observed empirical volatility in the financial market. In this case, it has been shown that the empirical volatility in the financial market is extremely consistent with the rough volatility. Currently, fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form and therefore, we must rely on numerical methods to obtain a solution. In this paper, we will be giving a short introduction to option pricing theory (Black–Scholes model, classical Heston model and its characteristic function), an overview of the current advancements on the rough Heston model and numerical methods (fractional Adams–Bashforth–Moulton method and multipoint Padé approximation method) for solving the fractional Riccati equation. In addition, we will investigate on the performance of multipoint Padé approximation method for the small u values in D α h ( u − i / 2 , x ) as it plays a huge role in the computation for the option prices. We further confirm that the solution generated by multipoint Padé (3,3) method for the fractional Riccati equation is incredibly consistent with the solution generated by fractional Adams–Bashforth–Moulton method.

scholarly journals Approximate Solution of Riccati Differential Equation via Modified Greens Decomposition Method

2020 ◽  
Vol 70 (4) ◽  
pp. 419-424
Author(s):  
Amit Ujlayan ◽  
Mohit Arya
Keyword(s):  
Numerical Solution ◽  
Decomposition Method ◽  
Padé Approximation ◽  
Medical Science ◽  
Adomian Decomposition Method ◽  
Specific Class ◽  
Pade Approximation ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Adomian Decomposition

Riccati differential equations (RDEs) plays important role in the various fields of defence, physics, engineering, medical science, and mathematics. A new approach to find the numerical solution of a class of RDEs with quadratic nonlinearity is presented in this paper. In the process of solving the pre-mentioned class of RDEs, we used an ordered combination of Green’s function, Adomian’s polynomials, and Pade` approximation. This technique is named as green decomposition method with Pade` approximation (GDMP). Since, the most contemporary definition of Adomian polynomials has been used in GDMP. Therefore, a specific class of Adomian polynomials is used to advance GDMP to modified green decomposition method with Pade` approximation (MGDMP). Further, MGDMP is applied to solve some special RDEs, belonging to the considered class of RDEs, absolute error of the obtained solution is compared with Adomian decomposition method (ADM) and Laplace decomposition method with Pade` approximation (LADM-Pade`). As well, the impedance of the method emphasised with the comparative error tables of the exact solution and the associated solutions with respect to ADM, LADM-Pade`, and MGDMP. The observation from this comparative study exhibits that MGDMP provides an improved numerical solution in the given interval. In spite of this, generally, some of the particular RDEs (with variable coefficients) cannot be easily solved by some of the existing methods, such as LADM-Pade` or Homotopy perturbation methods. However, under some limitations, MGDMP can be successfully applied to solve such type of RDEs.

scholarly journals Padé-Sumudu-Adomian Decomposition Method for Nonlinear Schrödinger Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-19
Author(s):  
Metomou Richard ◽  
Weidong Zhao
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Decomposition Method ◽  
Schrodinger Equation ◽  
Padé Approximation ◽  
Adomian Decomposition Method ◽  
Nonlinear Schrodinger Equation ◽  
Pade Approximation ◽  
Nonlinear Schrödinger ◽  
Adomian Decomposition

The main purpose of this paper is to solve the nonlinear Schrödinger equation using some suitable analytical and numerical methods such as Sumudu transform, Adomian Decomposition Method (ADM), and Padé approximation technique. In many literatures, we can see the Sumudu Adomian decomposition method (SADM) and the Laplace Adomian decomposition method (LADM); the SADM and LADM provide similar results. The SADM and LADM methods have been applied to solve nonlinear PDE, but the solution has small convergence radius for some PDE. We perform the SADM solution by using the function P L / M · called double Padé approximation. We will provide the graphical numerical simulations in 3D surface solutions of each application and the absolute error to illustrate the efficiency of the method. In our methods, the nonlinear terms are computed using Adomian polynomials, and the Padé approximation will be used to control the convergence of the series solutions. The suggested technique is successfully applied to nonlinear Schrödinger equations and proved to be highly accurate compared to the Sumudu Adomian decomposition method.

Convergence analysis of Padé approximations for Helmholtz frequency response problems

2018 ◽  
Vol 52 (4) ◽  
pp. 1261-1284 ◽  
Author(s):  
Francesca Bonizzoni ◽  
Fabio Nobile ◽  
Ilaria Perugia
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Padé Approximation ◽  
Approximation Error ◽  
Approximation Technique ◽  
Pade Approximation ◽  
Solution Map ◽  
Type Approximation ◽  
Numerical Tests ◽  
The Laplace Operator

The present work concerns the approximation of the solution map S associated to the parametric Helmholtz boundary value problem, i.e., the map which associates to each (real) wavenumber belonging to a given interval of interest the corresponding solution of the Helmholtz equation. We introduce a least squares rational Padé-type approximation technique applicable to any meromorphic Hilbert space-valued univariate map, and we prove the uniform convergence of the Padé approximation error on any compact subset of the interval of interest that excludes any pole. This general result is then applied to the Helmholtz solution map S, which is proven to be meromorphic in ℂ, with a pole of order one in every (single or multiple) eigenvalue of the Laplace operator with the considered boundary conditions. Numerical tests are provided that confirm the theoretical upper bound on the Padé approximation error for the Helmholtz solution map.

scholarly journals A Few Iterative Methods by Using [1,n]-Order Padé Approximation of Function and the Improvements

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 55 ◽  
Author(s):  
Shengfeng Li ◽  
Xiaobin Liu ◽  
Xiaofang Zhang
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Padé Approximation ◽  
Single Step ◽  
Order Convergence ◽  
Pade Approximation ◽  
Derivatives Of ◽  
Third Derivative ◽  
Approximation Of Function

In this paper, a few single-step iterative methods, including classical Newton’s method and Halley’s method, are suggested by applying [ 1 , n ] -order Padé approximation of function for finding the roots of nonlinear equations at first. In order to avoid the operation of high-order derivatives of function, we modify the presented methods with fourth-order convergence by using the approximants of the second derivative and third derivative, respectively. Thus, several modified two-step iterative methods are obtained for solving nonlinear equations, and the convergence of the variants is then analyzed that they are of the fourth-order convergence. Finally, numerical experiments are given to illustrate the practicability of the suggested variants. Henceforth, the variants with fourth-order convergence have been considered as the imperative improvements to find the roots of nonlinear equations.

scholarly journals Adomian Decomposition Method Combined with Padé Approximation and Laplace Transform for Solving a Model of HIV Infection of CD4+T Cells

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Fang Chen ◽  
Qing-Quan Liu
Keyword(s):  
T Cells ◽  
Approximate Solution ◽  
Hiv Infection ◽  
Laplace Transform ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Padé Approximation ◽  
Adomian Decomposition Method ◽  
Pade Approximation ◽  
Adomian Decomposition

The classical Adomian decomposition method (ADM) is implemented to solve a model of HIV infection of CD4+T cells. The results indicate that the approximate solution by using the ADM is the same as that by using the Laplace ADM, but it can be obtained in a more efficient way. We also use Padé approximation and Laplace transform as a posttreatment technique to obtain the result of the ADM. The advantage of the posttreatment is illustrated by numerical experiments.

scholarly journals Behaviour Analysis of the Padé Sumudu Adomian Decomposition Method Solution

2021 ◽  
pp. 39-45
Author(s):  
Richard Metonou ◽  
Zhao Weidong
Keyword(s):  
Decomposition Method ◽  
Schrodinger Equation ◽  
Padé Approximation ◽  
Adomian Decomposition Method ◽  
Decomposition Methods ◽  
Pade Approximation ◽  
Approximation Solution ◽  
Behaviour Analysis ◽  
The Past ◽  
Adomian Decomposition

Researchers in the past investigate the Sumudu Adomian Decomposition Method (SADM), the Laplace Adomian Decomposition Method (LADM), the Padé Sumudu Adomian Decomposition Methods (PSADM). In this paper we analyse the behaviour of the function P[L/M][.] called double Padé approximation using in the Padé Sumudu Adomian Decomposition Method (PSADM), and provide some criteriums for chosing L and M to obtain the best Padé approximation solution in the case of nonlinear Schrödinger equation and nonlinear KdV Burger's equation.

scholarly journals On the Study of Oscillating Viscous Flows by Using the Adomian-Padé Approximation

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Chi-Min Liu
Keyword(s):  
Fourier Analysis ◽  
Padé Approximation ◽  
Adomian Decomposition Method ◽  
Viscous Flows ◽  
Pulsating Flow ◽  
Pade Approximation ◽  
Oscillating Flows ◽  
Approximation Solution ◽  
Adomian Decomposition ◽  
Stokes Second Problem

The Adomian-Padé technique is applied to examine two oscillating viscous flows, the Stokes’ second problem and the pressure-driven pulsating flow. Main purposes for studying oscillating flows are not only to verify the accuracy of the approximation solution, but also to provide a basis for analyzing more problems by the present method with the help of Fourier analysis. Results show that the Adomian-Padé approximation presents a very excellent behavior in comparison with the exact solution of Stokes’ second problem. For the pulsating flow, only the Adomian decomposition method is required to perform the calculation as the fluid domain is finite where the Padé approximant may not provide a better solution. Based on present results, more problems can be mathematically solved by using the Adomian-Padé technique, the Fourier analysis, and powerful computers.

APPROXIMATE SOLUTION FOR TIME-DELAYED CONVECTIVE FISHER EQUATION BY ADM-PADÉ TECHNIQUE

2010 ◽  
Vol 03 (02) ◽  
pp. 221-233 ◽  
Author(s):  
A. Alharbi ◽  
E. S. Fahmy
Keyword(s):  
Approximate Solution ◽  
Decomposition Method ◽  
Padé Approximation ◽  
Adomian Decomposition Method ◽  
Fisher Equation ◽  
Pade Approximation ◽  
Adomian Decomposition

We present an approximate solution to the time-delayed convective Fisher equation using ADM-Padé technique which is a combination of Adomian decomposition method and Padé approximation. This technique gives the approximate solution with faster convergence and higher accuracy than using ADM alone.

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