On the efficacy of Fourier series approximations for pricing European options

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the halfrange cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together,these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

Lacunary Fourier series and a qualitative uncertainty principle for compact Lie groups

We define lacunary Fourier series on a compact connected semisimple Lie group G. If f is an element of L-1 (G) has lacunary Fourier series and f vanishes on a non empty open subset of G, then we prove that f vanishes identically. This result can be viewed as a qualitative uncertainty principle.

Lq estimates for rearrangements of Fourier series

This work is concerned with estimating the upper envelopes S* of the absolute values of the partial sums of rearranged trigonometric sums. A.M. Garsia [Annals of Math. 79 (1964), 634-9] gave an estimate for the L₂ norms of the S*, averaged over all rearrangements of the original (finite) sum. This estimate enabled him to prove that the Fourier series of any function in L₂ can be rearranged so that it converges a.e. The main result of this thesis is a similar estimate of the L_q norms of the S*, for all even integers q. This holds for finite linear combinations of functions which satisfy a condition which is a generalization of orthonormality in the L₂ case. This estimate for finite sums is extended to Fourier series of L_q functions; it is shown that there are functions to which the Men’shov-Paley Theorem does not apply, but whose Fourier series can nevertheless be rearranged so that the S* of the rearranged series is in L_q.

Transient thermal analysis of power devices based on Fourier-series thermal model

A new thermal model based on Fourier series expansion method has been presented for dynamic thermal analysis on power devices. The thermal model based on the Fourier series method has been programmed in MATLAB SIMULINK and integrated with a physics-based electrical model previously reported. The model was verified for accuracy using a two-dimensional Fourier model and a two-dimensional finite difference model for comparison. To validate this thermal model, experiments using a 600V 50A IGBT module switching an inductive load, has been completed under high frequency operation. The result of the thermal measurement shows an excellent match with the simulated temperature variations and temperature time-response within the power module. ©2008 IEEE.

Fourier series expansion method for gain measurement from amplified spontaneous emission spectra of Fabry-Perot semiconductor lasers

A gain measurement technique, based on Fourier series expansion of periodically extended single fringe of the amplified spontaneous emission spectrum, is proposed for Fabry-Perot semiconductor lasers. The underestimation of gain due to the limited resolution of the measurement system is corrected by a factor related to the system response function. The standard deviations of the gain-reflectivity product under low noise conditions are analyzed for the Fourier series expansion method and compared with those of the Hakki-Paoli method and Cassidy's method. The results show that the Fourier series expansion method is the least sensitive to noise among the three methods. The experiment results obtained by the three methods are also presented and compared.

Algorithmic Computation of Formal Fourier Series

KAAD (Katholischer Akademischer Ausländer-Dienst)

Fourier Series

Lecture notes on Fourier Series

Modelling of surface identifying characteristics using Fourier series

Texture and small-scale surface details are widely recognised as playing an important role in the haptic identification of objects. In order to simulate realistic textures in haptic virtual environments, it has become increasingly necessary to identify a robust technique for modelling of surface profiles. This paper describes a method whereby Fourier series spectral analysis is employed in order to describe the measured surface profiles of several characteristic surfaces. The results presented suggest that a bandlimited Fourier series can be used to provide a realistic approximation to surface amplitude profiles.

Correlation between Fourier series expansion and Hückel orbital theory

The Fourier series can be used to describe periodic phenomena such as the one-dimensional crystal wave function. By the trigonometric treatements in Hückel theory it is shown that Hückel theory is a special case of Fourier series theory. Thus, the conjugated π system is in fact a periodic system. Therefore, it can be explained why such a simple theorem as Hückel theory can be so powerful in organic chemistry. Although it only considers the immediate neighboring interactions, it implicitly takes account of the periodicity in the complete picture where all the interactions are considered. Furthermore, the success of the trigonometric methods in Hückel theory is not accidental, as it based on the fact that Hückel theory is a specific example of the more general method of Fourier series expansion. It is also important for education purposes to expand a specific approach such as Hückel theory into a more general method such as Fourier series expansion.

Fourier series for quaternions and the square of the error theorem

In this paper we introduce a type of Hypercomplex Fourier Series based on Quaternions, and discuss on a Hypercomplex version of the Square of the Error Theorem. Since their discovery by Hamilton (Sinegre [1]), quaternions have provided beautifully insights either on the structure of different areas of Mathematics or in the connections of Mathematics with other fields. For instance: I) Pauli spin matrices used in Physics can be easily explained through quaternions analysis (Lan [2]); II) Fundamental theorem of Algebra (Eilenberg [3]), which asserts that the polynomial analysis in quaternions maps into itself the four dimensional sphere of all real quaternions, with the point infinity added, and the degree of this map is n. Motivated on earlier works by two of us on Power Series (Pendeza et al. [4]), and in a recent paper on Liouville’s Theorem (Borges and Mar˜o [5]), we obtain an Hypercomplex version of the Fourier Series, which hopefully can be used for the treatment of hypergeometric partial differential equations such as the dumped harmonic oscillation.

Square of the Error Octonionic Theorem and Hypercomplex Fourier Series

The focus of this paper is to address some classical results for a class of hypercomplex numbers. More specifically we present an extension of the Square of the Error Theorem and a Bessel inequality for octonions.

Hybrid Water Demand Forecasting Model Associating Artificial Neural Network with Fourier Series

This paper addressed the problem of water-demand forecasting for real-time operation of water supply systems. The present study was conducted to identify the best fit model using hourly consumption data from the water supply system of Araraquara, Sa approximate to o Paulo, Brazil. Artificial neural networks (ANNs) were used in view of their enhanced capability to match or even improve on the regression model forecasts. The ANNs used were the multilayer perceptron with the back-propagation algorithm (MLP-BP), the dynamic neural network (DAN2), and two hybrid ANNs. The hybrid models used the error produced by the Fourier series forecasting as input to the MLP-BP and DAN2, called ANN-H and DAN2-H, respectively. The tested inputs for the neural network were selected literature and correlation analysis. The results from the hybrid models were promising, DAN2 performing better than the tested MLP-BP models. DAN2-H, identified as the best model, produced a mean absolute error (MAE) of 3.3 L/s and 2.8 L/s for training and test set, respectively, for the prediction of the next hour, which represented about 12% of the average consumption. The best forecasting model for the next 24 hours was again DAN2-H, which outperformed other compared models, and produced a MAE of 3.1 L/s and 3.0 L/s for training and test set respectively, which represented about 12% of average consumption. DOI: 10.1061/(ASCE)WR.1943-5452.0000177. (C) 2012 American Society of Civil Engineers.

RAPID SPACE TRAJECTORY GENERATION USING A FOURIER SERIES SHAPE-BASED APPROACH

With the insatiable curiosity of human beings to explore the universe and our solar system, it is essential to benefit from larger propulsion capabilities to execute efficient transfers and carry more scientific equipment. In the field of space trajectory optimization the fundamental advances in using low-thrust propulsion and exploiting the multi-body dynamics has played pivotal role in designing efficient space mission trajectories. The former provides larger cumulative momentum change in comparison with the conventional chemical propulsion whereas the latter results in almost ballistic trajectories with negligible amount of propellant. However, the problem of space trajectory design translates into an optimal control problem which is, in general, time-consuming and very difficult to solve. Therefore, the goal of the thesis is to address the above problem by developing a methodology to simplify and facilitate the process of finding initial low-thrust trajectories in both two-body and multi-body environments. This initial solution will not only provide mission designers with a better understanding of the problem and solution but also serves as a good initial guess for high-fidelity optimal control solvers and increases their convergence rate. Almost all of the high-fidelity solvers enjoy the existence of an initial guess that already satisfies the equations of motion and some of the most important constraints. Despite the nonlinear nature of the problem, it is sought to find a robust technique for a wide range of typical low-thrust transfers with reduced computational intensity. Another important aspect of our developed methodology is the representation of low-thrust trajectories by Fourier series with which the number of design variables reduces significantly. Emphasis is given on simplifying the equations of motion to the possible extent and avoid approximating the controls. These facts contribute to speeding up the solution finding procedure. Several example applications of two and three-dimensional two-body low-thrust transfers are considered. In addition, in the multi-body dynamic, and in particular the restricted-three-body dynamic, several Earth-to-Moon low-thrust transfers are investigated.

Fourier Series by Least Square Minimization

An introduction to Fourier Series based on the minimization of the least square error between an approximate series representation and the exact function.