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 L\'evy processes and 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.

A novel pricing method for european options based on fourier-cosine series expansions

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 underlying asset processes for which the characteristic function is known 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.

/pdf/a-novel-pricing-method-for-european-options-based-on-fourier-2w9rx9dt1j.pdf

A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions

We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Levy asset price models. The error convergence is exponential for processes characterized by very smooth ($${{\rm{C}}^{\infty}[a,b]\in\mathbb {R}}$$) 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 (Fang and Oosterlee in SIAM J Sci Comput 31(2):826–848, 2008) in which we presented the impressive performance of the Fourier-cosine series method for European options.

https://mpra.ub.uni-muenchen.de/9248/1/MPRA_paper_9248.pdf

Pricing early-exercise and discrete barrier options by fourier-cosine series expansions

SUMMARY Handcock & Stein (1993) introduced the Matern family of spatial correlations into statistics as a flexible parametric class with one parameter determining the smoothness of the paths of the underlying spatial field We document the varied history of this family, which includes contributions by eminent physical scientists and statisticians

Studies in the history of probability and statistics XLIX On the Matérn correlation family

We present a historical survey of Gauss-Christoffel quadrature formulae, beginning with Gauss’ discovery of his well-known method of approximate integration and the early contributions of Jacobi and Christoffel, but emphasizing the more recent advances made after the emergence of powerful digital computing machinery. One group of inquiry concerns the development of the quadrature formula itself, e.g. the inclusion of preassigned nodes and the admission of multiple nodes, as well as other generalizations of the quadrature sum. Another is directed towards the widening of the class of integrals made accessible to Gauss-Christoffel quadrature. These include integrals with nonpositive measures of integration and singular principal value integrals. An account of the error and convergence theory will also be given, as well as a discussion of modern methods for generating Gauss-Christoffel formulae, and a survey of numerical tables.

A Survey of Gauss-Christoffel Quadrature Formulae

The Hankel Transform

A new method for the calculation of the integrals

$$I_1 (m) = \int\limits_a^b {f(x)\sin mxdx} andI_2 (m) = \int\limits_a^b {f(x)\cos mxdx}$$

is presented. The functionf(x) is approximated by a sum of Chebyshev polynomials. The Chebyshev coefficients are then used to calculate a Neumann series approximation forI1(m) andI2(m).

A numerical method for the integration of oscillatory functions

Gaussian Quadrature Formulas for the Integration of Oscillating Functions

A Fourier-transform integral can be transformed into an integral of a periodic function over a finite interval. This integral can be calculated efficiently by the trapezium rule, while the periodic function, which is expressed as an infinite series, can be evaluated using Euler's transformation. A FORTRAN program implementing this method is presented.

Robert Piessens

Papers

The Hankel Transform

A numerical method for the integration of oscillatory functions

Gaussian Quadrature Formulas for the Integration of Oscillating Functions

Numerical calculation of Fourier-transform integrals

Tables of Gaussian quadrature formulas