scholarly journals Finite Difference Methods for Option Pricing under Lévy Processes: Wiener-Hopf Factorization Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Oleg Kudryavtsev
Keyword(s):  
Finite Difference ◽  
Lévy Processes ◽  
Finite Difference Methods ◽  
Factorization Method ◽  
Levy Processes ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Pricing Options ◽  
Black Scholes ◽  
Difference Methods

In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments.

On the Solution of the Diffusion Equations by Numerical Methods

1966 ◽  
Vol 88 (4) ◽  
pp. 421-427 ◽  
Author(s):  
H. Z. Barakat ◽  
J. A. Clark
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Present Method ◽  
Finite Difference Methods ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Procedure ◽  
Implicit Methods ◽  
Difference Methods ◽  
Explicit Finite Difference

An explicit-finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, nonhomogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems are compared with the exact solution and with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in closer agreement with the exact solution than are those obtained by the other methods.

SPECTRALLY ACCURATE OPTION PRICING UNDER THE TIME-FRACTIONAL BLACK–SCHOLES MODEL

2021 ◽  
pp. 1-21
Author(s):  
GERALDINE TOUR ◽  
NAWDHA THAKOOR ◽  
DÉSIRÉ YANNICK TANGMAN
Keyword(s):  
Finite Difference ◽  
Option Pricing ◽  
Time Discretization ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Spectral Approximation ◽  
Barrier Option ◽  
Order Accuracy ◽  
Black Scholes ◽  
Richardson Extrapolation Method

Abstract We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular $L1$ finite difference approximation, which converges with order $\mathcal {O}((\Delta \tau )^{2-\alpha })$ for functions which are twice continuously differentiable. However, when using the $L1$ scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

scholarly journals A Comparative Study between Implicit and Crank-Nicolson Finite Difference Method for Option Pricing

2020 ◽  
Vol 40 (1) ◽  
pp. 13-27
Author(s):  
Tanmoy Kumar Debnath ◽  
ABM Shahadat Hossain
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Option Pricing ◽  
Difference Method ◽  
Finite Difference Methods ◽  
Put Option ◽  
Black Scholes ◽  
Implicit Finite Difference ◽  
Difference Methods ◽  
Crank Nicolson

In this paper, we have applied the finite difference methods (FDMs) for the valuation of European put option (EPO). We have mainly focused the application of Implicit finite difference method (IFDM) and Crank-Nicolson finite difference method (CNFDM) for option pricing. Both these techniques are used to discretized Black-Scholes (BS) partial differential equation (PDE). We have also compared the convergence of the IFDM and CNFDM to the analytic BS price of the option. This turns out a conclusion that both these techniques are fairly fruitful and excellent for option pricing. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 13-27

Spectrally accurate option pricing under the time fractional Black-Scholes model

2021 ◽  
Vol 63 ◽  
pp. 228-248
Author(s):  
Geraldine Tour ◽  
Nawdha Thakoor ◽  
Désiré Yannick Tangman
Keyword(s):  
Finite Difference ◽  
Option Pricing ◽  
Time Discretization ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Spectral Approximation ◽  
Barrier Option ◽  
Order Accuracy ◽  
Black Scholes ◽  
Richardson Extrapolation Method

We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular \(L1\) finite difference approximation, which converges with order \(\mathcal{O}((\Delta \tau)^{2-\alpha})\) for functions which are twice continuously differentiable. However, when using the \(L1\) scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.   doi:10.1017/S1446181121000286  

Compact Crank–Nicolson and Du Fort–Frankel Method for the Solution of the Time Fractional Diffusion Equation

2015 ◽  
Vol 12 (06) ◽  
pp. 1550041 ◽  
Author(s):  
Faoziya Al-Shibani ◽  
Ahmad Ismail
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Finite Difference Methods ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Time Fractional Diffusion Equation ◽  
The Stability ◽  
The One

In this paper, two compact implicit finite difference methods are developed and analyzed for solving the one-dimensional time fractional diffusion equation. The temporal derivative is approximated by using Grünwald–Letnikov formula. Compact finite difference approximation is used for the second-order derivative in space. The local truncation errors are discussed. The stability analysis and the convergence of the proposed methods are investigated by means of Fourier series method. A comparison between the results of these methods and the exact solution is made. Numerical tests are given to verify the feasibility and accuracy of the methods.

Sign in / Sign up

Export Citation Format

Share Document