Resumen de Wavelet approach in computational finance
In the computational finance world both derivatives pricing and risk management have attracted lots of interest amongst practitioners and academia. This thesis aims to provide wavelets based techniques to enhance some of the methodologies used in the mentioned areas. Wavelets are families of functions that can be arbitrarily translated and dilated in order to generate orthogonal basis of L2(R). In relation to them, a collection of Fourier inversion methods has emerged; they are based on the approximation of functions by projecting on the wavelets basis such that the coefficients of the expansion are expressed by means of the Fourier transform of the function to approximate.
The SWIFT (Shannon wavelet inverse Fourier technique) method for pricing European-style options on one underlying asset was recently published and presented as an accurate, robust and highly efficient technique based on Shannon wavelets. One of the achievements of the thesis is the extension of the method to higher dimensions by pricing exotic option contracts, called rainbow options, whose payoff depends on multiple assets. The multidimensional extension inherits the properties of the one-dimensional method, being the exponential convergence one of them. Thanks to the nature of local Shannon wavelets basis, we do not need to rely on a-priori truncation of the integration range, we have an error bound estimate and we use fast Fourier transform (FFT) algorithms to speed up computations. We test the method for several examples comparing it with state-of-the-art methods found in the literature.
When managing the risk, regulators measure the risk exposure of a financial institution to determine the amount of capital that the institution must hold as a buffer against unexpected losses. The Basel Committee on Banking Supervision (BCBS) is the committee of the world's bank regulators. BCBS has recently set out the revised standards for minimum capital requirements for market risk, it has focused, among other things, on the two key areas of moving risk measures from Value-at-Risk (VaR) to Expected Shortfall (ES) and considering a comprehensive incorporation of the risk of market illiquidity. Another goal of this thesis is the presentation of a novel numerical method based on SWIFT to compute the VaR and ES of a given portfolio within the stochastic holding period framework to take into account liquidity issues. Two approaches are considered: the delta-gamma approximation, for modelling the change in value of the portfolio as a quadratic approximation of the change in value of the risk factors, and some of the state-of-the-art stochastic processes for driving the dynamics of the log-value change of the portfolio like the Merton jump-diffusion model and the Kou model.
Credit risk is the risk of losses from the obligor's failure to honour the contractual agreements and it is usually the main source of risk in a commercial bank. In this thesis, we also investigate the challenging problem of estimating credit risk measures of portfolios with exposure concentration under the multi-factor Gaussian and multi-factor t-copula models. It is well-known that Monte Carlo (MC) methods are highly demanding from the computational point of view in these situations. To overcome this issue, we present efficient and robust numerical techniques based on the Haar wavelets theory for recovering the cumulative distribution function of the loss variable from its characteristic function. The analysis of the approximation error and the results obtained in the numerical experiments section show a reliable and useful machinery for credit risk capital measurement purposes in line with Pillar II of the Basel Accords.
