scholarly journals A Hybrid Finite Difference Method for Pricing Two-Asset Double Barrier Options

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Y. L. Hsiao ◽  
S. Y. Shen ◽  
Andrew M. L. Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Hybrid Method ◽  
Difference Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Barrier Options ◽  
Barrier Option ◽  
Double Barrier ◽  
Numerical Examples

The pricing of the two-asset double barrier option is modeled as an initial-boundary value problem of the two-dimensional Black-Scholes partial differential equation. We use the hybrid finite different method to solve the problem. The hybrid method is a combination of the Laplace transform and a finite difference method. It is more efficient than a traditional finite difference method to obtain a solution without a step-by-step process. The method is implemented on a computer. Two numerical examples are calculated to verify the performance of the hybrid method. In our numerical examples, the convergence rate of the method is approximately two. We conclude that the method is efficient for pricing two-asset barrier options.

scholarly journals Application of Fuzzy Finite Difference Scheme for the Non-homogeneous Fuzzy Heat Equation

2021 ◽  
Author(s):  
Samaneh Zabihi ◽  
reza ezzati ◽  
F Fattahzadeh ◽  
J Rashidinia
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Heat Equation ◽  
Difference Scheme ◽  
Difference Method ◽  
Taylor Expansion ◽  
Truncation Error ◽  
Initial Boundary ◽  
Convergence Conditions ◽  
Numerical Examples

Abstract A numerical framework based on fuzzy finite difference is presented for approximating fuzzy triangular solutions of fuzzy partial differential equations by considering the type of $[gH-p]-$differentiability. The fuzzy triangle functions are expanded using full fuzzy Taylor expansion to develop a new fuzzy finite difference method. By considering the type of gH-differentiability, we approximate the fuzzy derivatives with a new fuzzy finite-difference. In particular, we propose using this method to solve non-homogeneous fuzzy heat equation with triangular initial-boundary conditions. We examine the truncation error and the convergence conditions of the proposed method. Several numerical examples are presented to demonstrate the performance of the methods. The final results demonstrate the efficiency and the ability of the new fuzzy finite difference method to produce triangular fuzzy numerical results which are more consistent with existing reality.

Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 815-825 ◽  
Author(s):  
Chaobao Huang ◽  
Xiaohui Liu ◽  
Xiangyun Meng ◽  
Martin Stynes
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Value Problem ◽  
Difference Method ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Graded Meshes ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is {(0,1)^{d}} for some {d\in\{1,2,3\}}. This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where {d=1} and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.

Convergence in Positive Time for a Finite Difference Method Applied to a Fractional Convection-Diffusion Problem

2018 ◽  
Vol 18 (1) ◽  
pp. 33-42 ◽  
Author(s):  
José Luis Gracia ◽  
Eugene O’Riordan ◽  
Martin Stynes
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Diffusion Problem ◽  
Initial Time ◽  
Uniform Mesh ◽  
Convection Diffusion ◽  
Initial Boundary Value

AbstractA standard finite difference method on a uniform mesh is used to solve a time-fractional convection-diffusion initial-boundary value problem. Such problems typically exhibit a mild singularity at the initial time {t=0}. It is proved that the rate of convergence of the maximum nodal error on any subdomain that is bounded away from {t=0} is higher than the rate obtained when the maximum nodal error is measured over the entire space-time domain. Numerical results are provided to illustrate the theoretical error bounds.

A hybrid method for transonic flow past multi-element aerofoils

1986 ◽  
Vol 170 ◽  
pp. 253-264 ◽  
Author(s):  
M. G. Hill ◽  
N. Riley
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Compressible Flow ◽  
Hybrid Method ◽  
Difference Method ◽  
Transonic Flow ◽  
Potential Flow ◽  
Supercritical Flow ◽  
Subcritical Flow ◽  
Flow Past

A method for calculating transonic potential flow past a multi-element aerofoil configuration is presented. The method is a hybrid method that is based upon a compressible-flow panel method, valid for subcritical flow, and a finite-difference method that is suitable for supercritical flow calculations. The effectiveness of the proposed method is demonstrated, first by application to a single aerofoil and then to a three-element aerofoil.

scholarly journals Numerical Solution of Stochastic Generalized Fractional Diffusion Equation by Finite Difference Method

2018 ◽  
Vol 23 (4) ◽  
pp. 53 ◽  
Author(s):  
Amaneh Sepahvandzadeh ◽  
Bahman Ghazanfari ◽  
Nader Asadian
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Difference Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Mean Square ◽  
Numerical Examples ◽  
Mean Square Convergence

The present study aimed at solving the stochastic generalized fractional diffusion equation (SGFDE) by means of the random finite difference method (FDM). Moreover, the conditions of mean square convergence of the numerical solution are studied and numerical examples are presented to demonstrate the validity and accuracy of the method.

scholarly journals On accurate geoid modeling: derivation of dirichlet problems that govern geoidal undulations and geoid modeling by means of the finite difference method and a hybrid method

2014 ◽  
Vol 20 (2) ◽  
pp. 334-353
Author(s):  
Eduardo Del Rio
Keyword(s):  
Dirichlet Problem ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Hybrid Method ◽  
Difference Method ◽  
Gravity Data ◽  
Dirichlet Problems ◽  
Geoid Model ◽  
The Earth ◽  
Stokes's Integral

The geoid is the reference surface used to measure heights (orthometric). These are used to study any mass variability in the Earth system. As the Earth is represented by an oblate spheroid (Ellipsoid), the geoid is determined by geoidal undulations (N) which are the separation between these surfaces. N is determined from gravity data by Stokes's Integral. However, this approach takes a Spherical rather than an Ellipsoidal Earth. Here it is derived a Partial Differential Equation (PDE) that governs N over the Earth by means of a Dirichlet problem and show a method to solve it which precludes the need for a Spherical Earth. Moreover, Stokes's Integral solves a boundary value problem defined over the whole Earth. It was found that the Dirichlet problem derived here is defined only over the region where a geoid model is to be computed, which is advantageous for local geoid modeling. Moreover, the method eliminates several of the sources of uncertainty in Stokes's Integral. However, estimates indicate that the errors due to discretization are very large in this new method which calls for its modification. So, here it is also proposed an optimal combination of techniques by means of a Hybrid method and shown that it alleviates the uncertainty in Finite Difference Method. Moreover, a rigorous error analysis indicates that the Hybrid method proposed here may well outperform Stokes's Integral.

scholarly journals A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yifan Qin ◽  
Xiaocheng Yang ◽  
Yunzhu Ren ◽  
Yinghong Xu ◽  
Wahidullah Niazi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Scheme ◽  
Difference Method ◽  
Stability And Convergence ◽  
Sobolev Equation ◽  
Numerical Examples ◽  
Convergence Orders ◽  
Theoretical Results ◽  
Crank Nicolson

In this paper, one class of finite difference scheme is proposed to solve nonlinear space fractional Sobolev equation based on the Crank-Nicolson (CN) method. Firstly, a fractional centered finite difference method in space and the CN method in time are utilized to discretize the original equation. Next, the existence, uniqueness, stability, and convergence of the numerical method are analyzed at length, and the convergence orders are proved to be O τ 2 + h 2 in the sense of l 2 -norm, H α / 2 -norm, and l ∞ -norm. Finally, the extensive numerical examples are carried out to verify our theoretical results and show the effectiveness of our algorithm in simulating spatial fractional Sobolev equation.

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