Shannon wavelets in computational finance

Derivative securities, when used correctly, allow investors to increase their expected profits and minimize their exposure to risk. Options offer leverage and insurance for risk-averse investors while they can be used as ways of speculation for the more risky investors. When an option is issued, we face the problem of determining the price of a product at the same time we must make sure to eliminate arbitrage opportunities. In this thesis, we introduce a robust, accurate, and highly efficient financial option valuation technique, the so-called SWIFT method (Shannon wavelets inverse Fourier technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. This method is adapted to the pricing of European options and Discrete Lookback options. Numerical experiments show exponential convergence and confirm the robustness, efficiency and versatility of the method

Transformada de Fourier: aplicacions a la resolució d'equacions en derivades parcials

The objective of this thesis is to make a theoretical and formal study of the Fourier Transform and to introduce some of its many applications. We start studying the Fourier Transform in L^1 and L^2, its behavior respect to the convolution and the multidimensional generalization. This study will allow us to solve, analyze and understand more two of the most well-known and important Partial Differential Equations: the Heat equation and the Wave equation. Finally, we will introduce and study the most relevant properties of filters. In order to give the most general results and exploit the full potential of the Fourier Transform, we will introduce the distributions, their basic properties and the theory of the Fourier Transform for distributions

Transformada de Fourier: aplicacions a la resolució d'equacions en derivades parcials

The objective of this thesis is to make a theoretical and formal study of the Fourier Transform and to introduce some of its many applications. We start studying the Fourier Transform in L^1 and L^2, its behavior respect to the convolution and the multidimensional generalization. This study will allow us to solve, analyze and understand more two of the most well-known and important Partial Differential Equations: the Heat equation and the Wave equation. Finally, we will introduce and study the most relevant properties of filters. In order to give the most general results and exploit the full potential of the Fourier Transform, we will introduce the distributions, their basic properties and the theory of the Fourier Transform for distributions

Shannon wavelets in computational finance

Derivative securities, when used correctly, allow investors to increase their expected profits and minimize their exposure to risk. Options offer leverage and insurance for risk-averse investors while they can be used as ways of speculation for the more risky investors. When an option is issued, we face the problem of determining the price of a product at the same time we must make sure to eliminate arbitrage opportunities. In this thesis, we introduce a robust, accurate, and highly efficient financial option valuation technique, the so-called SWIFT method (Shannon wavelets inverse Fourier technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. This method is adapted to the pricing of European options and Discrete Lookback options. Numerical experiments show exponential convergence and confirm the robustness, efficiency and versatility of the method