http://bura.brunel.ac.uk/handle/2438/235 Fri, 01 May 2026 12:10:02 GMT 2026-05-01T12:10:02Z http://bura.brunel.ac.uk/handle/2438/33226 Title: Mixed topics on geometry of varieties of Fano type Authors: Jiao, Dongchen Abstract: In this thesis, we investigate the deformation properties of Fano threefolds and the birational ge-ometry of foliations. First, we try to find compactification of several families of Fano threefolds. Then we give a description of the connections between foliated minimal models. Finally, we will discuss geometric properties of Fano foliations. This thesis contains results of (1), (26), (19) and some recent independent work. Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London Wed, 01 Jan 2025 00:00:00 GMT http://bura.brunel.ac.uk/handle/2438/33226 2025-01-01T00:00:00Z http://bura.brunel.ac.uk/handle/2438/32341 Title: Application of factor models to risk premium estimation Authors: Penco, Vittorio Abstract: We provide a rigorous mathematical approach to the beta-pricing model, starting from the standard two-step cross-sectional regression, through Nonlinear Seemingly Unrelated Regression (NSUR) and Generalized Method of Moments (GMM), and finally compare the results with several linear approximation methods. The use of the linear approximation applied to a single-factor nonlinear system of equations is new in the literature and is one of the major contributions of this work. Our results show that, in the presence of heavy-tailed distributions, the L1-norm methods proposed in this study are more appropriate (exhibiting lower bias and variance) for risk price estimation than traditional L2-norm approaches. It is also the first time that the Capital Asset Pricing Model (CAPM) is applied systematically to compare the integration and segmentation between different markets and a given portfolio set. Our study, Penco and Lucas (2024), applies a two-factor integration model to the economies of Asia, Europe, Japan and North America, showing integration between the European and North American economies. We also extend the integration model to commodity markets. To capture more accurately the cross-sectional pricing of commodity risk we use the Cochrane factor mimicking approach and compare the results with alternative dependence-based integration measures using copulas. We show how the copula correlation between the Stochastic Discount Factor (SDF) and returns differentiates the contribution of joint dependence from the contribution of the risk prices. Finally, we introduce a penalised p-value Fama-MacBeth Generalized Least Squares (GLS) regularisation, which provides several advantages over other methods as it ensures that retained factors contribute not only to statistical fit but also to risk pricing. Unlike other approaches, this method regularises the pricing kernel directly. Factors that lack significance or explanatory power are penalised and removed, while priced and relevant sources of risk are preserved. Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London Wed, 01 Jan 2025 00:00:00 GMT http://bura.brunel.ac.uk/handle/2438/32341 2025-01-01T00:00:00Z http://bura.brunel.ac.uk/handle/2438/31718 Title: Echo state networks in forecasting chaotic dynamics and emergent universalities Authors: Grublys, Mykolas Abstract: Deep learning has become a popular way to accurately recognise, classify, generate and forecast data in various complex scenarios. It is commonly based on artificial neural networks, a type of machine learning architecture inspired by the human brain, that consist of multiple layers of neurons with non-linear activations. Their inside workings can be considered as a ‘black box’ because it is difficult, if not impossible, to truly understand “how” or “why” such networks work. Furthermore, the training relies on gradient decent methods which are inherently difficult and computationally expensive. Reservoir Computing has recently emerged as a new paradigm aimed to alleviate such difficulties. In this thesis, we study its particular realisation, known as Echo State Networks (ESNs), which show promise in many tasks, particularly in forecasting the dynamics of chaotic systems. The thesis will provide a detailed discussion of the ESN architecture, hyperparameters, and implementation. We introduce a new performance metric that highlights the networks maintaining small errors for the longest duration and use it in GridSearch to optimise ESN hyperparameters. We study the statistical properties of the correlation matrix that is formed during training, offering a novel approach not applied to ESNs or other types of recurrent networks. Our extensive numerical studies reveal universal results, consistent across tests and different chaotic training signals. Although analytical studies are limited, we introduce a simple ‘toy model’ to qualitatively describe some properties. We conclude our work with study of the statistics of trained output weights, which also exhibit universal characteristics. These universalities provide deeper insights into the inner workings and behaviour of ESNs, enhancing our understanding of how information spreads throughout the network. Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London Mon, 01 Jan 2024 00:00:00 GMT http://bura.brunel.ac.uk/handle/2438/31718 2024-01-01T00:00:00Z http://bura.brunel.ac.uk/handle/2438/31502 Title: New control variates for pricing basket and Asian options Authors: Jipreze, Kam Abstract: In this thesis, we investigate new control variates for simulation-based pricing of options where the option price is a function of the sum of (or integral of) lognormal random variables. We use two different approaches: one is the use of Hermite polynomial approximation of the relevant function and another is the use of upper and lower bounds on the option prices obtained using the properties of Brownian motion. We provide detailed numerical experiments to illustrate the use of these approaches for accurate and low variance pricing basket and Asian options. First order Hermite polynomial approximation also gives a reasonable direct approximation to the basket or Asian option price for at the money and in-the-money options. Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London Mon, 01 Jan 2024 00:00:00 GMT http://bura.brunel.ac.uk/handle/2438/31502 2024-01-01T00:00:00Z