Mathematical Methods in Finance: Modeling and Numerical Analysis
This thesis concerns the study of strategies and the development of mathematical methods to deal with three specific problems in quantitative finance. In the first problem, we address the use of Fourier methods for derivative pricing. We present a novel method to compute options prices, which extends the existing literature of Fourier methods in finance. The method makes it possible to price several payoffs not treated in the literature and also a portfolio of derivatives with different maturities. We study the approximation of Fourier operators in different frameworks, having the financial application as a particular case. We also present a non-uniform fast Fourier transform (NUFFT) for the approximations used here and several numerical results. The second problem concerns commodity pricing. We present a model for the liquefied natural gas (LNG) market based on a multidimensional stochastic process. Several derivatives over LNG are also presented with a numerical method to evaluate them. In the third problem, we present a way of using interest rate derivatives to recover market expectation regarding future decisions by the monetary authority. This chapter describes a model that has monetary decisions as input and it proposes some regularization techniques in order to recover interest rate expectations from real data.