Fourier Cosine Series Expansion Research Articles

Moment Lyapunov exponent for three-dimensional system under real noise excitation

The pth moment Lyapunov exponent of a two-codimension bifurcation system excited parametrically by a real noise is investigated. By a linear stochastic transformation, the differential operator of the system is obtained. In order to evaluate the asymptotic expansion of the moment Lyapunov exponent, via a perturbation method, a ralevant eigenvalue problem is obtained. The eigenvalue problem is then solved by a Fourier cosine series expansion, and an infinite matrix is thus obtained, whose leading eigenvalue is the second-order of the asymptotic expansion of the moment Lyapunov exponent. Finally, the convergence of procedure is numerically illustrated, and the effects of the system and the noise parameters on the moment Lyapunov exponent are discussed.

A Fourier Cosine Method for an Efficient Computation of Solutions to BSDEs

We develop a Fourier method to solve backward stochastic differential equations (BSDEs). A general theta-discretization of the time-integrands leads to an induction scheme with conditional expectations. These are approximated by using Fourier cosine series expansions, relying on the availability of a characteristic function. The method is applied to BSDEs with jumps. Numerical experiments demonstrate the applicability of BSDEs in financial and economic problems and show fast convergence of our efficient probabilistic numerical method.

An efficient pricing method for rainbow options based on two-dimensional modified sine–sine series expansions

An efficient option pricing method based on the Fourier cosine series expansion was proposed by Fang and Oosterlee [A novel option pricing method based on Fourier-cosine series expansions, SIAM J. Sci. Comput. 31 (2008), pp. 826–848], and recently it has been used in two-dimensional cases by Ruijter and Oosterlee [Two-dimensional Fourier cosine series expansion method for pricing financial options, SIAM J. Sci. Comput. 34 (2012), pp. B642–B671]. In this paper, we consider a modification of the Fourier sine–sine expansion, whereby in [−1, 1] the sin(π nx) functions are replaced by sin(π(n−)x), n≥1, for pricing rainbow options. Feasible approaches for obtaining the pricing formulae are presented. A numerical comparison and a flexible and general error analysis are considered to prove the effectiveness of the method. Practical applications to basket options and correlation options under different models are also given. They indicate satisfactory convergence and efficiency as expected.

On the Fourier cosine series expansion method for stochastic control problems

SUMMARYWe develop a method for solving stochastic control problems under one‐dimensional Lévy processes. The method is based on the dynamic programming principle and a Fourier cosine expansion method. Local errors in the vicinity of the domain boundaries may disrupt the algorithm. For efficient computation of matrix–vector products with Hankel and Toeplitz structures, we use a fast Fourier transform algorithm. An extensive error analysis provides new insights based on which we develop an extrapolation method to deal with the propagation of local errors. Copyright © 2013 John Wiley & Sons, Ltd.

A Fourier-Cosine Method for an Efficient Computation of Solutions to BSDEs

We develop a Fourier method to solve backward stochastic differential equations (BSDEs). General theta-discretization of the time-integrands leads to an induction scheme with conditional expectations. These are approximated by using Fourier-cosine series expansions, relying on the availability of a characteristic function. The method is applied to BSDEs with jumps. Numerical experiments demonstrate the applicability of BSDEs in financial and economic problems and show fast convergence of our efficient probabilistic numerical method.

Acceleration of option pricing technique on graphics processing units

SUMMARYThe acceleration of an option pricing technique based on Fourier cosine expansions on the graphics processing unit (GPU) is reported. European options, in particular with multiple strikes, and Bermudan options will be discussed. The influence of the number of terms in the Fourier cosine series expansion, the number of strikes, as well as the number of exercise dates for Bermudan options is explored. We also give details about the different ways of implementing on a GPU. Numerical examples include asset price processes on the basis of a Lévy process of infinite activity and the stochastic volatility Heston model. Furthermore, we discuss the issue of precision on the present GPU systems. Copyright © 2012 John Wiley & Sons, Ltd.

Two-Dimensional Fourier Cosine Series Expansion Method for Pricing Financial Options

The COS method for pricing European and Bermudan options with one underlying asset was developed in [F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826--848] and [F. Fang and C. W. Oosterlee, Numer. Math., 114 (2009), pp. 27--62]. In this paper, we extend the method to higher dimensions, with a multidimensional asset price process. The algorithm can be applied to, for example, pricing two-color rainbow options but also to pricing under the popular Heston stochastic volatility model. For smooth density functions, the resulting method converges exponentially in the number of terms in the Fourier cosine series summations; otherwise we achieve algebraic convergence. The use of an FFT algorithm, for asset prices modeled by Lévy processes, makes the algorithm highly efficient. We perform extensive numerical experiments.

Vibration and power flow analysis of periodically reinforced plates

In this paper an analytical method is proposed to investigate the vibration and power flows of periodically reinforced plate with general boundary conditions. Both the plate and stiffening beams are modeled as 3D structural components, and the couplings at the interfaces are specified in terms of 3D elastic joints. The displacement function for each stiffening beam is expressed as a modified Fourier cosine series, and the transverse and in-plane displacements for the plate are similarly expressed as the 2D versions of the modified Fourier cosine series expansions. The unknown Fourier coefficients are calculated using the Rayleigh-Ritz technique. The key advantages of the proposed method include: 1) it is capable of dealing with arbitrary boundary and coupling conditions, 2) it allows modeling any number of reinforcing beams with arbitrary lengths, and 3) the structural intensity, power flows, and kinetic energy distributions are readily calculated analytically from the displacement functions through appropriate mathematical (differential) operations, to name a few. The power flow characteristics of periodically reinforced plates are studied against various influencing factors, such as, plate and beam boundary conditions, coupling conditions, excitation locations, and dislocations resulting from minor misplacement of a reinforcing beam.

A Fourier-Based Valuation Method for Bermudan and Barrier Options under Heston's Model

We develop an efficient Fourier-based numerical method for pricing Bermudan and discretely monitored barrier options under the Heston stochastic volatility model. The two-dimensional pricing problem is dealt with by a combination of a Fourier cosine series expansion, as in [F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826-848, F. Fang and C. W. Oosterlee, Numer. Math., 114 (2009), pp. 27-62], and high-order quadrature rules in the other dimension. Error analysis and experiments confirm a fast error convergence.

Fast Valuation and Calibration of Credit Default Swaps Under Levy Dynamics

In this paper we address the issue of finding an efficient and flexible numerical approach for calculating survival/default probabilities and pricing Credit Default Swaps under advanced jump dynamics. We have chosen to use the firm’s value approach, modeling the firm’s value by an exponential Levy model. For this approach the default event is defined as a first passage of a barrier and it is therefore possible to exploit a numerical technique developed to price barrier options under Levy models to calculate the default probabilities. The method presented is based on the Fourier-cosine series expansion of the underlying model’s density function.

Fast Valuation and Calibration of Credit Default Swaps Under Levy Dynamics

In this paper we address the issue of finding an efficient and flexible numerical approach for calculating survival/default probabilities and pricing Credit Default Swaps under advanced jump dynamics. We have chosen to use the firm's value approach, modeling the firm's value by an exponential\levy\model. For this approach the default event is defined as a first passage of a barrier and it is therefore possible to exploit a numerical technique developed to price barrier options under\levy\models to calculate the default probabilities. The method presented is based on the Fourier-cosine series expansion of the underlying model's density function.With this method we calibrate two well known\levy\models, the Normal Inverse Gaussian and the CGMY, to the market quotes of the iTraxx Series 7 and 8 constituents. In the calibration study we look at the root mean square error of the fit and find that both models give a very good fit to market data. Furthermore, we investigate how the out of calibration given parameters evolve over the year covered by the two series. Finally, we look at the changes in the default probability term structure over the year.

Pricing early-exercise and discrete barrier options by 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.

Vibrations of rectangular plates with arbitrary non-uniform elastic edge restraints

Arbitrary non-uniform elastic edge restraints represent the most general class of boundary conditions for plate problems, and are encountered in many real-world applications. The vibrations of plates with this kind of boundary conditions, however, are rarely studied in the literature perhaps because there is a lack of suitable analytical or numerical techniques. In this investigation, a general analytical method is derived for the vibration analysis of rectangular plates with elastic edge restraints of varying stiffness. Both rotational and translational restraints can be arbitrarily applied to an edge, and their stiffness distributions are generally described in terms of a set of invariants, cosine functions. The displacement solution is sought simply as a linear combination of several one- and two-dimensional Fourier cosine series expansions. All the unknown Fourier coefficients are treated equally as a set of independent generalized coordinates and solved directly from the Rayleigh–Ritz formulation. Unlike the existing techniques, the current method does not require any special procedures or schemes to deal with different boundary conditions. A few “classical” problems involving non-uniform rotational restraints are first solved and used to check the current solution against some of the existing techniques. The modal results are also presented for plates with more complicated boundary conditions in which an edge is no longer completely restrained in the translational direction. The accuracy and reliability of the current method are repeatedly demonstrated through all these examples.

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.

A harmonic approach to global ocean tide analysis based on TOPEX/Poseidon satellite

The constant and harmonic parts of the global ocean tide are modeled by up to nine major tidal constituents, namely, S2, M2, N2, K1, P1, O1, Mf, Mm, and Ssa. Our computations start with the Fourier sine and cosine series expansion for the tidal constituents, including the constant Mean Sea Level (MSL). Although the frequencies of the tidal constituents are considered known, the coefficients of the sine and cosine functions are assumed to be unknown. Subsequently, the coefficients of the sine and cosine functions, as well as the constant part of the Fourier expansion, were expanded into spherical harmonics up to degree and order n, where n corresponds to the number of linearly independent spherical harmonic base functions needed to model the tidal constituents, determined via independent columns of the Gram matrix. The unknown coefficients of the spherical harmonic expansions are computed using sea level observations within cycles #1–#350 of the TOPEX/Poseidon satellite altimetry over 11 years of its mission. A set of orthonormal base functions was generated for the marine areas covered by TOPEX/Poseidon observations from the spherical harmonics using a Gram-Schmidt orthogonalization process. These were used for modeling the dominant tidal constituents. The computed models based on orthonormal base functions for the nine tidal constituents and the constant part of the Fourier expansion, were tested numerically for their validity and accuracy, proving centimeter accuracy.

Electric Field Dependence of the Competition Between Permanent and Induced Dipole Orientation in Suspensions of Large Anisotropic Particles

ABSTRACT Accurate orientational probability distribution functions are derived which describe the dynamical behavior of permanent and induced dipole moments in an external field. The method involves a transformation of the Smoluchowski equation into a Schrödinger–like equation with real eigenvalues. The method is applied here for the first time to the case of suspensions of large noninteracting anisotropic particles which are of interest as artificial Kerr materials for nonlinear electrooptical applications. A matrix presentation is made in which the effects of any combination of permanent and induced dipole moments are described by a single matrix. The eigenfunctions and eigenvalues of this matrix when specific material parameters are substituted provide a Fourier cosine series expansion for the angular distribution function under conditions of cylindrical symmetry which is valid for aligning fields of any magnitude. The only restriction is that larger fields require larger matrices for more accurate results. The method is used to analyze the competition between the aligning contributions of permanent and induced dipole moments and the effects of permanent dipole moment and field magnitude on alignment times are obtained.

The torsional-wagging tunneling problem and the torsional-wagging-rotational problem in methylamine

A theoretical formalism is presented for fitting rotational energy levels in isolated (unperturbed) vibrational states of methylamine. This formalism is obtained by recasting and extending theoretical studies in the earlier literature, which were undertaken to help analyze the methylamine microwave spectrum. The present formalism is applicable when both the NH 2 umbrella (wagging) motion and the CH 3 internal rotation (torsion) motion take place near the high-barrier limit and leads to the usual Fourier sine and cosine series expansions for molecular energy levels. The derivation is separated into two parts, one treating the large-amplitude vibrational problem (the torsional-wagging problem) by itself, the other treating the torsional-wagging-rotational problem. In both treatments, permutation-inversion group and extended group ideas are used to determine the allowed terms in an effective rotational-tunneling Hamiltonian operator and to block diagonalize the matrix representation of this operator for a near-prolate symmetric top. The resulting energy levels and selection rules are discussed, but application of the method in detail to the methylamine spectrum is planned for a later paper.

Program for Calculation of Augmented Jacobi Polynomials

The augmented Jacobi polynomials, designated Zℓmn by Roe, or the closely related generalized Legendre functions, Pℓmn(cos⁡ϕ) of Bunge, are required for series expansions of the crystallite orientation distribution. The problem of publishing extensive tabular information has limited compilation of these functions to certain crystal and physical symmetries. This paper gives a brief description and listing of a computer program which permits calculation of Fourier sine (m–n odd) or cosine (m–n even) series expansions of Zℓmn(ξ) for ℓ=0,1,…,32; m=0,1,…,ℓ; n=0,1,…,ℓ.