Exponential time integration for fast finite element solutions of some financial engineering problems

We consider exponential time integration schemes for fast numerical pricing of European, American, barrier and butterfly options when the stock price follows a dynamics described by a jump-diffusion process. The resulting pricing equation which is in the form of a partial integro-differential equation is approximated in space using finite elements. Our methods require the computation of a single matrix exponential and we demonstrate using a wide range of numerical tests that the combination of exponential integrators and finite element discretisations with quadratic basis functions leads to highly accurate algorithms for cases when the jump magnitude is Gaussian. Comparison with other time-stepping methods are also carried out to illustrate the effectiveness of our methods.     

Analysis of algebraic systems arising from fourth‐order compact discretizations of convection‐diffusion equations

We study the properties of coefficient matrices arising from high‐order compact discretizations of convection‐diffusion problems. Asymptotic convergence factors of the convex hull of the spectrum and the field of values of the coefficient matrix for a one‐dimensional problem are derived, and the convergence factor of the convex hull of the spectrum is shown to be inadequate for predicting the convergence rate of GMRES. For a two‐dimensional constant‐coefficient problem, we derive the eigenvalues of the nine‐point matrix, and we show that the matrix is positive definite for all values of the cell‐Reynolds number. Using a recent technique for deriving analytic expressions for discrete solutions produced by the fourth‐order scheme, we show by analyzing the terms in the discrete solutions that they are oscillation‐free for all values of the cell Reynolds number. Our theoretical results support observations made through numerical experiments by other researchers on the non‐oscillatory nature of the discrete solution produced by fourth‐order compact approximations to the convection‐diffusion equation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 155–178, 2002; DOI 10.1002/num.1041     

A new fourth-order numerical scheme for option pricing under the CEV model

The empirically observed negative relationship between a stock price and its return volatility can be captured by the constant elasticity of variance option pricing model. For European options, closed form expressions involve the non-central chi-square distribution whose computation can be slow when the elasticity factor is close to one, volatility is low or time to maturity is small. We present a fast numerical scheme based on a high-order compact discretisation which accurately computes the option price. Various numerical examples indicate that for comparable computational times, the option price computed with the scheme has higher accuracy than the Crank–Nicolson numerical solution. The scheme accurately computes the hedging parameters and is stable for strongly negative values of the elasticity factor.     

A fast high-order finite difference algorithm for pricing American options

We describe an improvement of Han and Wu’s algorithm [H. Han, X.Wu, A fast numerical method for the Black–Scholes equation of American options, SIAM J. Numer. Anal. 41 (6) (2003) 2081–2095] for American options. A high-order optimal compact scheme is used to discretise the transformed Black–Scholes PDE under a singularity separating framework. A more accurate free boundary location based on the smooth pasting condition and the use of a non-uniform grid with a modified tridiagonal solver lead to an efficient implementation of the free boundary value problem. Extensive numerical experiments show that the new finite difference algorithm converges rapidly and numerical solutions with good accuracy are obtained. Comparisons with some recently proposed methods for the American options problem are carried out to show the advantage of our numerical method.     

A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems

Convergence of the implicitly restarted Arnoldi (IRA) method for nonsymmetric eigenvalue problems has often been studied by deriving bounds for the angle between a desired eigenvector and the Krylov projection subspace. Bounds for residual norms of approximate eigenvectors have been less studied and this paper derives a new a-posteriori residual bound for nonsymmetric matrices with simple eigenvalues. The residual vector is shown to be a linear combination of exact eigenvectors and a residual bound is obtained as the sum of the magnitudes of the coefficients of the eigenvectors. We numerically illustrate that the convergence of the residual norm to zero is governed by a scalar term, namely the last element of the wanted eigenvector of the projected matrix. Both cases of convergence and non-convergence are illustrated and this validates our theoretical results. We derive an analogous result for implicitly restarted refined Arnoldi (IRRA) and for this algorithm, we numerically illustrate that convergence is governed by two scalar terms appearing in the linear combination which drives the residual norm to zero. We provide a set of numerical results that validate the residual bounds for both variants of Arnoldi methods.     

Numerical pricing of options using high-order compact finite difference schemes

We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black–Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.     

High-order computational methods for option valuation under multifactor models

Many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second-order discretization schemes. A higher-order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high-order convergence rates for the solution of multi-factor option problems. These new schemes employ Galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two-asset option problems and time integration of the resulting semi-discrete systems requires the computation of a single matrix exponential. The computations indicate that this combination of high-order finite elements and exponential time integration leads to efficient algorithms for multi-factor problems. Highly accurate European prices are obtained with relatively coarse meshes and high-order convergence rates are also observed for options with the American early exercise feature. Various numerical examples are provided for illustrating the accuracy of the option prices for Heston’s and Bates stochastic volatility models and for two-asset problems under Merton’s jump-diffusion model.     

A note on Hermitian splitting induced relaxation methods for convection-diffusion equations

The solution of the linear system Ax = b by iterative methods requires a splitting of the coefficient matrix in the form A = M − N where M is usually chosen to be a diagonal or a triangular matrix. In this article we study relaxation methods induced by the Hermitian and skew-Hermitian splittings for the solution of the linear system arising from a compact fourth order approximation to the one dimensional convection-diffusion equation and compare the convergence rates of these relaxation methods to that of the widely used successive overrelaxation (SOR) method. Optimal convergence parameters are derived for each method and numerical experiments are given to supplement the theoretical estimates. For certain values of the diffusion parameter, a relaxation method based on the Hermitian splitting converges faster than SOR. For two-dimensional problems a block form of the iterative algorithm is presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 581–591, 1998     

Analysis of incomplete factorizations for a nine-point approximation to a convection–diffusion model problem

We study the stability of zero-fill incomplete LU factorizations of a nine-point coefficient matrix arising from a high-order compact discretisation of a two-dimensional constant-coefficient convection–diffusion problem. Nonlinear recurrences for computing entries of the lower and upper triangular matrices are derived and we show that the sequence of diagonal entries of the lower triangular factor is unconditionally convergent. A theoretical estimate of the limiting value is derived and we show that this estimate is a good predictor of the computed value. The unconditional convergence of the diagonal sequence of the lower triangular factor to a positive limit implies that the incomplete factorization process never encounters a zero pivot and that the other diagonal sequences are also convergent. The characteristic polynomials associated with the lower and upper triangular solves that occur during the preconditioning step are studied and conditions for the stability of the triangular solves are derived in terms of the entries of the tridiagonal matrices appearing in the lower and upper subdiagonals of the block triangular system matrix and a triplet of parameters which completely determines the solution of the nonlinear recursions. Results of ILU-preconditioned GMRES iterations and the effects of orderings on their convergence are also described.     

A method for improving the performance of the WENO5 scheme near discontinuities

WENO5 uses a convex combination of the polynomials reconstructed on the three stencils of ENO3 in order to achieve higher accuracy on smooth profiles. However, in some cases WENO5 generates oscillations or smears near discontinuities due to the time scheme used. Here, we present a method to reduce those oscillations without damping and this yields a sharper approximation. Our technique uses smoothness indicators to identify severe shocks and switches from WENO5 to ENO3. Numerical tests show that the behaviour of WENO5 is improved near discontinuities while preserving high accuracy on smooth profiles.