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 Fractional Riccati Equation and Its Applications to Rough Heston Model

2020 ◽  
Author(s):  
Siow W. Jeng ◽  
Adem Kilicman
Keyword(s):  
Numerical Methods ◽  
Characteristic Function ◽  
Riccati Equation ◽  
Financial Market ◽  
Heston Model ◽  
Short Introduction ◽  
Volatility Models ◽  
Rough Volatility ◽  
Pricing Theory ◽  
Option Pricing Theory

Rough volatility models are popularized by \cite{gatheral2018volatility}, where they have shown that the empirical volatility in the financial market is extremely consistent with rough volatility. Fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form as of now and therefore, we must rely on numerical methods to obtain a solution. In this paper, we give a short introduction to option pricing theory and an overview of the current advancements on the rough Heston model.

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 A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Jingjing Feng ◽  
Qichang Zhang ◽  
Wei Wang ◽  
Shuying Hao
Keyword(s):  
Dynamic Systems ◽  
Approximation Method ◽  
Padé Approximation ◽  
Global Bifurcation ◽  
Heteroclinic Orbits ◽  
Symmetric System ◽  
Pade Approximation ◽  
New Approach ◽  
Homoclinic And Heteroclinic Orbits ◽  
Single Orbit

In dynamic systems, some nonlinearities generate special connection problems of non-Z2symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2symmetric nonlinear quintic systems (orbit with one cusp); and Z2symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.

Improved bounds on the conductivity of composites by interpolation

1994 ◽  
Vol 444 (1921) ◽  
pp. 363-374 ◽  
Keyword(s):  
Composite Material ◽  
Lower Bounds ◽  
Approximation Method ◽  
Interpolation Method ◽  
Padé Approximation ◽  
Upper Bounds ◽  
Pade Approximation ◽  
Two Component ◽  
Null Lagrangian

Different bounds on the conductivity of a composite material may improve on each other in different conductivity régimes. If so, the question arises of how to efficiently interpolate between the bounds. In this paper I show how to do an interpolation with a two-point Padé approximation method. For bounds on two-component composites the interpolation method is shown to be, in a sense, the best possible. The method is discussed in the context of equiaxed polycrystals where the classic Hashin-Shtrikman bounds and the more recent null-lagrangian bounds, partly improve on each other. Denoting the principal conductivities of the crystallite σ 1 ≼ σ 2 ≼ σ 3 , the method gives improved lower bounds for equiaxed polycrystals which have σ 2 (0.77σ 1 + 0.23σ 3 ) ≽ σ 1 σ 3 . The method also gives improved upper bounds.

scholarly journals Solution of nonlinear equations via Pade approximation. A Computer Algebra approach

2021 ◽  
Vol 66 (2) ◽  
pp. 321-328
Author(s):  
Radu T. Trimbitas
Keyword(s):  
Numerical Methods ◽  
Nonlinear Equations ◽  
Computer Algebra ◽  
Padé Approximation ◽  
High Order ◽  
Pade Approximation ◽  
Algebra Approach

"We generate automatically several high order numerical methods for the solution of nonlinear equations using Pad e approximation and Maple CAS."

scholarly journals New Refinements for the Error Function with Applications in Diffusion Theory

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2017
Author(s):  
Gabriel Bercu
Keyword(s):  
Fourier Series ◽  
Approximation Method ◽  
Diffusion Theory ◽  
Error Function ◽  
Padé Approximation ◽  
Absolute Error ◽  
Series Method ◽  
Pade Approximation ◽  
Fourier Series Method ◽  
The Absolute

In this paper we provide approximations for the error function using the Padé approximation method and the Fourier series method. These approximations have simple forms and acceptable bounds for the absolute error. Then we use them in diffusion theory.

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