Abstract. Here we develop an option pricing method for European options based on the Fourier-cosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fourier-cosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including Lévy processes and the Heston stochastic volatility model, and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a follow-up paper we will present its application to options with early-exercise features.Key words. option pricing, European options, Fourier-cosine expansion AMS subject classifications. 65T40, 42A10, 60E10, 62P05, 91B28 Preferred short title : Cosine expansions for option pricing 1. Introduction. Efficient numerical methods are required to rapidly price complex contracts and calibrate various financial models.In option pricing, it is the famous Feynman-Kac theorem that relates the conditional expectation of the value of a contract payoff function under the risk-neutral measure to the solution of a partial differential equation. In the research areas covered by this theorem, various numerical pricing techniques can be developed. In brief, existing numerical methods can be classified into three major groups: partial-(integro) differential equation (PIDE) methods, monte Carlo simulation and numerical integration methods. Each of them has its merits and demerits for specific applications in finance, but the methods from the latter class are often used for calibration purposes. An important aspect of research in computational finance is to further increase the performance of the pricing methods.State-of-the-art numerical integration techniques have in common that they rely on a transformation to the Fourier domain [8,21]. The Carr-Madan method [8] is one of the best known examples of this class. The probability density function appears in the integration in the original pricing domain, which is not known for many relevant pricing processes. However, its Fourier transform, the characteristic function, is often available, for example from the Lévy-Khinchine theorem for underlying Lévy processes or by other means, as for the Heston model. In the Fourier domain it is possible to solve various derivative contracts, as long as the characteristic function is available. By means of the Fast Fourier Transform (FFT), integration can be performed with a computational complexity of O(N log 2 N ), where N represents the number of integration points. The computational speed, especially for plain vanilla options, makes the integration methods state-of-the-art for calibration at financial institutions.Quadrature rule based techniques are, however, not of the highest efficiency when solving Fourier transformed in...

Abstract. An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz-type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner. 1. Introduction. In this paper we present a novel preconditioner for high wavenumber Helmholtz problems in heterogeneous media. The preconditioner is based on the Helmholtz operator, where an imaginary term is added. This preconditioner can be handled by multigrid. This is somewhat surprising as multigrid, without enhancements, has convergence troubles for the original Helmholtz operator at high wavenumbers.A part of this paper is therefore reserved for the analysis of the multigrid method for Helmholtz problems with a complex zeroth order term. This is done, for constant wavenumbers, by means of Fourier analysis. The preconditioned system leads to a favorably clustered spectrum for a Krylov subspace convergence acceleration. As the preconditioner is not based on a regular splitting of the original Helmholtz problem, it must be used in the setting of Krylov subspace methods. The particular example presented can be viewed as a generalization of the work by Bayliss, Goldstein, and Turkel [3] from the 1980s, where the Laplacian was used as a preconditioner for Helmholtz problems. This work has been generalized by Laird and Giles [17], proposing a Helmholtz preconditioner with a positive sign in front of the Helmholtz term. In [13] we have proposed a preconditioner with a purely imaginary shift added to the Laplacian. The method here is an improvement of that method.In this paper we benefit from Fourier analysis in several ways. First of all, for idealized (homogeneous boundary conditions, constant coefficients) versions of the preconditioned system it is possible to visualize its spectrum for different values of the wavenumber, as Fourier analysis provides all eigenvalues. Second, for analyzing multigrid algorithms quantitatively, Fourier smoothing, two-, and three-grid analysis [6,7,23,24,30] are the tools of choice.

We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Lévy asset price models. The error convergence is exponential for processes characterized by very smooth (C ∞ [a, b] ∈ R) transitional probability density functions. The computational complexity is O((M − 1)N log N ) with N a (small) number of terms from the series expansion, and M , the number of early-exercise/monitoring dates. This paper is the follow-up of [22] in which we presented the impressive performance of the Fourier-cosine series method for European options.

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called "shifted Laplace" preconditioners of the form ∆φ − αk 2 φ with α ∈ C. Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.

Abstract. A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within many regular affine models, among which the class of exponential Lévy models. For an M -times exercisable Bermudan option, the overall complexity is O(M N log 2 (N )) with N grid points used to discretise the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.