Circulant preconditioning technique for barrier options pricing under fractional diffusion models

Abstract

In recent years, considerable literature has proposed the more general class of exponential Lévy processes as the underlying model for prices of financial quantities, which thus better explain many important empirical facts of financial markets. Finite moment log stable (FMLS), CGMY and KoBoL models are chosen from those above-mentioned models as the dynamics of underlying equity prices in this paper. With such models pricing barrier options, one kind of financial derivatives, is transformed to solve specific fractional partial differential equations (FPDEs). This study focuses on numerically solving these FPDEs via the fully implicit scheme, with the shifted Grünwald approximation. The circulant preconditioned generalized minimal residual method which converges very fast with theoretical proof is incorporated for solving resultant linear systems. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show the accuracy of our method compared with that done by the Fourier cosine expansion method as a benchmark.

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions

“…We aim at constructing efficient finite difference schemes for fractional ordinary differential equations (FODEs) with non-smooth solutions. In recent decades, due to the increasing interest in problems with anomalous transport dynamics, fractional differential equations have become significant mathematical models in many fields of science and engineering, such as viscoelastic models in blood flow [33], underground transport [20], options pricing model in financial markets [41], etc. Though some fractional differential equations (FDEs) with special form, e.g., linear equations, can be solved by analytical methods, e.g., the Fourier transform method or the Laplace transform method [34], the analytical solutions of many generalized FDEs (e.g.…”

Fast solution algorithms for exponentially tempered fractional diffusion equations

“…The clustering property of the normalized preconditioned matrix is proven via a matrix‐based approach, which yields the superlinear convergence of the preconditioned Krylov subspace method. It is remarked that the coefficient matrix generated from the second order scheme in this article, which involves more parameters, is quite different from the matrices in some previous papers such as , thus the proof of the clustering spectrum here is different.…”

“…To accelerate the convergence rate of the fast Krylov subspace method for 1D TFDE, a circulant‐and‐skew‐circulant splitting preconditioner is constructed in and all eigenvalues of the preconditioned matrix are proven to be inside the circle centered at with radius strictly less than 1. In , circulant preconditioner is proposed for solving a first order finite difference scheme for some options pricing models, which are variations of the 1D TFDE. The preconditioned Krylov subspace method is shown to converge superlinearly so that the total complexity is .…”

“…Tempered fractional diffusion equation (TFDE), which is an exponentially tempered extension of fractional diffusion equation (FDE), is recently shown to be useful in geophysics [1][2][3] and finance [4][5][6][7]. However, analytical solutions for TFDEs and FDEs are usually not available.…”

“…Numerical methods for solving FDEs have been extensively studied in [8][9][10][11][12][13][14][15][16] in recent decades. For tempered fractional derivatives, if the same difference approximation for fractional derivatives is directly adopted, the stability of the resulted finite difference scheme can only be guaranteed when the spatial step size is small enough, see for example [6]. With a new shifted approximation to tempered fractional derivatives proposed in [17][18][19], unconditionally stable numerical schemes for one-dimensional (1D) TFDEs with constant coefficients are developed successfully.…”