Exponential Time Integration Scheme Research Articles

Convection experiments with the exponential time integration scheme

We present the construction and preliminary tests of the exponential time integration schemes for the Euler equations. The equations are written in terms of potential temperature and Exner function. The time stepping is accomplished with the EPI2 and EPI3 schemes which have been used in the past for very accurate, long time step integration of the shallow water equations on the sphere. The stability of exponential integrators of this class is ensured by approximation of the oscillatory term using an absolutely stable exponential formula. Consequently, the schemes are not subject to the CFL condition imposed by the linear oscillatory part of the system and they also eliminate the phase errors associated with all the known implicit time stepping algorithms. All calculations are performed using the phipm algorithm based on the Krylov space methods. It is shown that the spectral radius of the Jacobian and consequently the Krylov space dimension are very sensitive to the presence of spatial noise. Efficiency and accuracy increase after the Shapiro filter is applied every two time steps. The method is investigated using several standardized and well accepted benchmark tests, including convective bubbles and a cold density current proposed by Straka and collaborators. Numerical experiments show that a stable integration is obtained for the wave Courant number of the order of 60 to 250. We present a detailed discussion of the development of a cold density current capturing the spiraling Prandtl vortex and the subsequent appearance of the Kelvin–Helmholtz billows. Our results from the exponential method compare very well with those obtained from the accurate schemes reported in the recent literature. The algorithm accurately handles both advection as well as acoustic and gravity waves. These results warrant the further development of the scheme for modeling atmospheric processes on a wide range of scales.

A Spectral Element Method for Option Pricing Under Regime-Switching with Jumps

In this paper, we propose the spectral element method to price European, digital, butterfly, American, discrete and continuous barrier options in a Markovian jump-diffusion regime-switching economy. The spectral element method discretisation is considered for the approximation of the spatial derivatives in a system of partial integro-differential equations and is chosen because it possesses spectral accuracy such that highly accurate option prices can be obtained using a small number of grid discretisation nodes. Essentially, the spectral element method consists of splitting the computational domain into as many elements as needed and approximating the basis functions by high-order orthogonal polynomials within each element. In order to sustain the high-order convergence in time, we also use an exponential time integration scheme to solve the semi-discrete system. Our numerical examples support our error analysis and indicate that the spectral element method converges exponentially for the values and the hedging parameters of the regime-dependent options. Therefore, the proposed scheme provides a viable alternative to the finite difference or finite element methods which usually converge only algebraically.

Multi‐field formulation of large deformation ductile fracture

AbstractIn the present contribution we focus on a novel computational framework for the simulation of ductile fracture in elastoplastic solids. In particular, a fully non‐linear formulation of gradient plasticity is applied together with a higher order phase‐field approach to fracture to account for large elastic and plastic deformations along with three dimensional crack propagation. The proposed formulation is based on a multiplicative decomposition of the deformation gradient into elastic and plastic contributions and a subsequent multiplicative decomposition of the elastic part into a fracture sensitive and a fracture insensitive contribution, see [1]. Postulating that fracture requires a local state of tensile/shear deformation, the latter decomposition is used to formulate the isochoric and tensile contributions to the elastic strain energy, whereas the compressive contribution remains unaffected by the crack phase‐field and vice versa. The modification of the Griffith brittle fracture theory to ductile fracture is most crucial and has to be adjusted carefully. Within our framework, this adaptation is considered as dependency of the critical fracture energy density by the equivalent plastic strain. Concerning the elastic/plastic decomposition, we assume an isochoric plastic flow which requires special attention for the update procedure of the plastic deformation. To be specific, we apply an exponential time integration scheme such that we preserve the deviatoric state of the plastic deformation and obtain a correct evaluation of the phase‐field driving force. Eventually, a number of numerical examples will demonstrate the accuracy and capability of the proposed multi‐field approach to ductile fracture.

Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models

A new barycentric spectral domain decomposition methods algorithm for solving partial integro-differential models is described. The method is applied to European and butterfly call option pricing problems under a class of infinite activity Levy models. It is based on the barycentric spectral domain decomposition methods which allows the implementation of the boundary conditions in an efficient way. After the approximation of the spatial derivatives, we obtained the semi-discrete equations. The computation of these equations is performed by using the barycentric spectral domain decomposition method. This is achieved with the implementation of an exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that Greek options, such as Delta and Gamma sensitivity, are computed with no spurious oscillation.

A high-order finite difference method for option valuation

In this paper, we propose the use of an efficient high-order finite difference algorithm to price options under several pricing models including the Black–Scholes model, the Merton’s jump–diffusion model, the Heston’s stochastic volatility model and the nonlinear transaction costs or illiquidity models. We apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non-uniform high-order five-point stencil finite difference scheme. For linear models, the time-stepping is dealt with by using an exponential time integration scheme with Carathéodory–Fejér approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices. Nonlinear Black–Scholes equations are solved using an efficient iterative scheme coupled with a Richardson extrapolation. Our numerical experiments clearly demonstrate the high-order accuracy of the proposed finite difference method, making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems.

A novel partial integrodifferential equation-based framework for pricing interest rate derivatives under jump-extended short-rate models

Interest rate derivatives under jump-extended short-rate models have commonly been valued using lattice methods. Unfortunately, lattice methods have pitfalls, mainly in terms of accuracy, efficiency and ease of programming. This paper proposes a much faster and more accurate valuation method based on partial integrodifferential equations (PIDEs). Spatial differential and integral terms are discretized by fourthorder finite differences and Simpson's rule, respectively. For the time-stepping, we investigate a Crank-Nicolson scheme, an implicit-explicit (IMEX) scheme and an exponential time integration scheme. We demonstrate our method by pricing bonds as well as European and American options on both zero-coupon and coupon bonds under jump-extended constant-elasticity-of-variance processes. Through a variety of test problems, we demonstrate that our PIDE-based approach is remarkably superior to lattice methods in terms of accuracy, speed and ease of implementation. By way of example, pricing a one-year zero-coupon bond under the Vasicek model extended with exponential jumps, our method attains an accuracy of 104 in 0:09 seconds with the IMEX scheme, whereas a truncated tree takes 78 seconds to reach the same accuracy.

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

AbstractThe stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

Discrete Singular Convolution Method for Numerical Solutions of Fifth Order Korteweg-De Vries Equations

A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.

A new radial basis functions method for pricing American options under Merton's jump-diffusion model

A new radial basis functions (RBFs) algorithm for pricing financial options under Merton's jump-diffusion model is described. The method is based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way. The semi-discrete equations obtained after approximation of the spatial derivatives, using RBFs based on differential quadrature are solved, using an exponential time integration scheme and we provide several numerical tests which show the superiority of this method over the popular Crank–Nicolson method. Various numerical results for the pricing of European, American and barrier options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are efficiently computed to high accuracy.

Spectral-element discontinuous Galerkin lattice Boltzmann simulation of flow past two cylinders in tandem with an exponential time integrator

In this paper, a spectral-element discontinuous Galerkin (SEDG) lattice Boltzmann discretization and an exponential time-marching scheme are used to study the flow field past two circular cylinders in tandem arrangement. The basic idea is to discretize the streaming step of the lattice Boltzmann equation by using the SEDG method to get a system of ordinary differential equations (ODEs) whose exact solutions are expressed by using a large matrix exponential. The approximate solution of the resulting ODEs are obtained from a projection method based on a Krylov subspace approximation. This approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension. The exponential time integration scheme is useful especially when computations are carried out at high Courant–Friedrichs–Lewy (CFL) numbers, where most explicit time-marching schemes are inaccurate. Simulations of flow were carried out for a circular cylinder at Re=20 and for two circular cylinders in tandem at Re=40 and a spacing of 2.5D, where D is the diameter of the cylinders. We compare our results with those from a fourth-order Runge–Kutta scheme that is restricted by the CFL number. In addition, important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme. These results are compared with results from our simulation using the RK scheme and with existing benchmark results.

Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation

We apply an exponential time integration scheme combined with a central difference scheme on a piecewise uniform mesh with respect to the spatial variable to evaluate a generalized Black-Scholes equation. We show that the scheme is second-order convergent for both time and spatial variables. It is proved that the scheme is unconditionally stable. Numerical results support the theoretical results.

Fast simplified approaches to Asian option pricing

In this paper a general and simple framework is proposed for pricing Asian options under a variety of pricing models, including the Merton model and the Carr- Geman-Madan-Yor model. The approach uses the exponential time integration scheme in combination with a dimensional splitting technique. For discretely observed Asian options, a simple splitting technique is applied, while a Strang splitting strategy is employed to improve the temporal accuracy for continuously observed averages. For European Asian options it is shown that spectral accuracy can be achieved using a Chebyshev discretization. Barrier features are easily enforced and an operator-splitting technique is used to price Amerasians. For the Black-Scholes model we show that employing best rational approximation via Carathéodory-Fejér approximation leads to algorithms with linear complexity in each spatial dimension.

Fast approximations of bond option prices under CKLS models

A new computational method for approximating prices of zero-coupon bonds and bond option prices under general Chan–Karolyi–Longstaff–Schwartz models is proposed. The pricing partial differential equations are discretized using second-order finite difference approximations and an exponential time integration scheme combined with best rational approximations based on the Carathéodory–Fejér procedure is employed for solving the resulting semi-discrete equations. The algorithm has a linear computational complexity and provides accurate bond and European bond option prices. We give several numerical results which illustrate the computational efficiency of the algorithm and uniform second-order convergence rates for the computed bond and bond option prices.

Fast exponential time integration scheme for option pricing with jumps

SUMMARYA fast exponential time integration scheme is considered for pricing European and double barrier options in jump‐diffusion models. After spatial discretization, the option pricing problem is transformed into the product of a matrix exponential and a vector, while the matrix bears a Toeplitz structure. The shift‐and‐invert Arnoldi method is then employed for fast approximation to such operation. Owing to the Toeplitz structure, the computational cost can be reduced by the fast Fourier transform. Furthermore, the discretized form of option pricing problem satisfies a given condition such that the error bound of the shift‐and‐invert Arnoldi approximation is unrelated to the norm of the matrix. Numerical tests are carried out to compare the proposed method with other option pricing methods. Copyright © 2010 John Wiley & Sons, Ltd.