Two extensions of Thurston's spectral theorem for surface diffeomorphisms

Thurston obtained a classification of individual surface homeomorphisms via the dynamics of the corresponding mapping class elements on Teichm\"uller space. In this paper we present certain extended versions of this, first, to random products of homeomorphisms and second, to holomorphic self-maps of Teichm\"uller spaces.Comment: 11 page

Model categories in deformation theory

The aim is the formalization of Deformation Theory in an abstract model category, in order to study several geometric deformation problems from a unified point of view. The main geometric application is the description of the DG-Lie algebra controlling infinitesimal deformations of a separated scheme over a field of characteristic 0

Strategic capital budgeting: asset replacement under market uncertainty

In this paper the impact of product market uncertainty on the optimal replacement timing of a production facility is studied. The existing production facility can be replaced by a technologically more advanced and thus more cost-effective one. We take into account strategic interactions among the firms competing in the product market by analyzing the problem in a duopolistic setting. We calculate the value of each firm and show that i) a preemptive (simultaneous) replacement occurs when the associated sunk cost is low (high), ii) despite the preemption effect uncertainty always raises the expected time to replace, and iii) the relationship between the probability of optimal replacement within a given time interval and uncertainty is decreasing for long time intervals and humped for short time intervals. Furthermore it is shown that result ii) carries over to the case where firms have to decide about starting production rather than about replacing existing facilities

Fast Cross-Validation via Sequential Testing

With the increasing size of today's data sets, finding the right parameter configuration in model selection via cross-validation can be an extremely time-consuming task. In this paper we propose an improved cross-validation procedure which uses nonparametric testing coupled with sequential analysis to determine the best parameter set on linearly increasing subsets of the data. By eliminating underperforming candidates quickly and keeping promising candidates as long as possible, the method speeds up the computation while preserving the capability of the full cross-validation. Theoretical considerations underline the statistical power of our procedure. The experimental evaluation shows that our method reduces the computation time by a factor of up to 120 compared to a full cross-validation with a negligible impact on the accuracy

Dynamic asset (and liability) management under market and credit risk

We introduce a modelling paradigm which integrates credit risk and market risk in describing the random dynamical behaviour of the underlying fixed income assets. We then consider an asset and liability management (ALM) problem and develop a mul- tistage stochastic programming model which focuses on optimum risk decisions. These models exploit the dynamical multiperiod structure of credit risk and provide insight into the corrective recourse decisions whereby issues such as the timing risk of default is appropriately taken into consideration. We also present a index tracking model in which risk is measured (and optimised) by the CVaR of the tracking portfolio in relation to the index. Both in- and out-of-sample (backtesting) experiments are undertaken to validate our approach. In this way we are able to demonstrate the feasibility and flexibility of the chosen framework

Backtesting Parametric Value-at-Risk Estimates in the S&P500 Index

Thanks to its wide diffusion in the industry, Value-at-Risk (VaR) manages to became a cornerstone in the growing and complex regulation of capital requirements (Basel Accords). For this reason, despite the theoretical limitations of VaR, the study of how improve the performance of such risk measure is still fundamental. This thesis concerns the parametric method used to estimate Value-at-Risk and the evaluation of such estimates. The accuracy in predicting future risks, strictly depends on how such measure is calculated. The chosen method for the calculation is the parametric approach based on various extensions of the ARCH-GARCH models, combined with different assumed distributions for the returns. The ARCHGARCH models should be able to fit time series which show a time-varying volatility (heteroskedasticity), while more leptokurtic distributions (such as Student’s t and GED) than the Normal one, and their relative skew version, should provide better tail forecast and hence better VaR estimates. The primary objective of this work is the evaluation of the estimates obtained from the models described above. For this purposes, several backtesting methods were performed and their results compared. Backtesting is a statistical procedure where actual profits and losses are systematically compared to corresponding VaR estimates. Backtesting methods here considered can be broadly divide in two categories. Those tests that evaluate only a single VaR level (i.e. 1% or 5%) and those tests that evaluate a multiple VaR levels (hence they evaluate the entire density forecast). To the first group belong test such as: Kupiec’s Unconditional Coverage test, Christoffersen’s Conditional Coverage test, Mixed Kupiec test and Duration test. While to the second group belongs the Crnkovic-Drachman test, the Q-test and the Berkowitz test. The results are then compared in the light of the strengths and the weaknesses of each approach. It emerged a substantial heterogeneity among the outcomes of these tests, especially between backtesting methods base on a single VaR level and those based on a multiple VaR levels. This empirical work is built on the framework of Angelidis, Benos and Degiannakis (2003). However, different volatility models, distributions and backtesting methods were employed. For these reasons, a comparison between the results of the two study is also provided