現代風險管理需要對金融產品的相關結構做出刻畫，而在實際生活中，我們通常使用相關係數和自相關係數去刻畫這種結構。然而，越來越多的人意識到自相關函數在度量相關結構上面被高估了，特別是在風險管理中我們更關心極端事件。同樣的，偏自相關函數也有這樣的短板。在這篇論文中，我們在有限維分佈服從有正尾係數的正則變差的嚴平穩過程上定義了Partial Extremogram。 這個指標僅僅依賴於隨機過程中的極端值。我們給出了它的一個估計并且研究了這個估計的漸進性質。此外，为了刻畫时间序列的負相關結構，我們把 Inverse Tail Dependence 的想法推廣到了隨機過程上面並且引入了Inverse Extremogram 的概念。我們給出了Inverse Extremogram 在ARMA模型中的顯示表達式。理論推導和數據模擬都說明這個指標可以很好的刻畫出一個隨機過程的尾部的負相關結構。 / Modern risk management calls for deeper understanding of the dependence structure of financial products, which is usually measured by correlation or autocorrelation functions. More and more people realized that autocorrelation function is overvalued as a tool to measure dependence, especially when one has to deal with extremal events in risk management. Likewise, partial autocorrelation function also suffers similar shortcomings as autocorrelation function. In this thesis, an analog of the partial autocorrelation function for a strictly stationary sequence of random variables whose finite-dimensional distributions are jointly regularly varying with positive index, the partial extremogram, is introduced. This function only depends on the extremal events of the underlying process. A natural estimator of the partial extremogram is also proposed and its asymptotic properties are studied. Furthermore, to measure the negative dependence of a time series, the idea of inverse tail dependence is extended to a stochastic process and the notion of inverse extremogram is proposed. A closed form of the inverse extremogram for an ARMA model is deduced. The theoretical and simulation results show that the inverse extremogram is a useful tool for measuring the negative tail dependence of a process. / Detailed summary in vernacular field only. / Chen, Pengcheng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 53-56). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Tail Dependence --- p.2 / Chapter 1.2 --- Extremogram --- p.4 / Chapter 1.2.1 --- Regularly Varying Time Series --- p.4 / Chapter 1.2.2 --- Extremogram for Regularly Varying Time Series --- p.7 / Chapter 1.3 --- Motivation and Organization --- p.8 / Chapter 2 --- Partial Extremogram --- p.9 / Chapter 2.1 --- Definition of Partial Extremogram --- p.9 / Chapter 2.2 --- Applications of Partial Extremogram --- p.15 / Chapter 2.2.1 --- AR(1) Process --- p.15 / Chapter 2.2.2 --- MA(1) process --- p.17 / Chapter 2.2.3 --- Stochastic Volatility Model --- p.19 / Chapter 2.3 --- Estimation of Partial Extremogram --- p.19 / Chapter 2.4 --- Simulation Study --- p.22 / Chapter 3 --- Inverse Extremogram --- p.28 / Chapter 3.1 --- Definition of Inverse Extremogram --- p.28 / Chapter 3.2 --- Applications of Inverse Extremogram --- p.29 / Chapter 3.2.1 --- MA(q) Model --- p.29 / Chapter 3.2.2 --- MA(∞) Model --- p.35 / Chapter 3.2.3 --- ARMA Model --- p.40 / Chapter 3.2.4 --- GARCH Model and SV Model --- p.41 / Chapter 3.3 --- Simulation Study --- p.42 / Chapter 4 --- Conclusions and Further Research --- p.50 / Bibliography --- p.53

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

This thesis focuses on construction, properties and estimation of asymmetric heavy-tailed distributions, as well as on their applications to financial modeling and risk measurement. First of all, we suggest a general procedure to construct a fully asymmetric distribution based on a symmetrically parametric distribution, and establish some natural relationships between the symmetric and asymmetric distributions. Then, three new classes of asymmetric distributions are proposed by using the procedure: the Asymmetric Exponential Power Distributions (AEPD), the Asymmetric Student-t Distributions (ASTD) and the Asymmetric Generalized t Distribution (AGTD). For the first two distributions, we give an interpretation of their parameters and explore basic properties of them, including moments, expected shortfall, characterization by the maximum entropy property, and the stochastic representation. Although neither distribution satisfies the regularity conditions under which the ML estimators have the usual asymptotics, due to a non-differentiable likelihood function, we nonetheless establish asymptotics for the full MLE of the parameters. A closed-form expression for the Fisher information matrix is derived, and Monte Carlo studies are provided. We also illustrate the usefulness of the GARCH-type models with the AEPD and ASTD innovations in the context of predicting downside market risk of financial assets and demonstrate their superiority over skew-normal and skew-Student's t GARCH models. Finally, two new classes of generalized extreme value distributions, which include Jenkinson's GEV (Generalized Extreme Value) distribution (Jenkinson, 1955) as special cases, are proposed by using the maximum entropy principle, and their properties are investigated in detail.

No description available.

published_or_final_version / abstract / Statistics and Actuarial Science / Master / Master of Philosophy

Lim Sze Wah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 71-73). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- A brief review on the basic minority game --- p.6 / Chapter 2.1 --- Model --- p.7 / Chapter 2.2 --- Results --- p.9 / Chapter 2.3 --- Discussion --- p.11 / Chapter 2.3.1 --- Bit-string Statistics and Market Efficiency --- p.11 / Chapter 2.3.2 --- Crowds and Anticrowds Effect --- p.14 / Chapter 2.3.3 --- Hamming Distance and Reduced Strategies Space --- p.15 / Chapter 3 --- A brief review on existing variations on the minority game --- p.17 / Chapter 3.1 --- Darwinism process and MG --- p.17 / Chapter 3.2 --- Evolutionary MG (EMG) --- p.17 / Chapter 3.3 --- Modified EMG (MEMG) --- p.18 / Chapter 3.4 --- MG with arbitrary cutoff --- p.18 / Chapter 3.5 --- Thermal MG (TMG) --- p.19 / Chapter 3.6 --- Three-Sided MG --- p.19 / Chapter 3.7 --- MG with variable payoffs --- p.19 / Chapter 4 --- Minority game with varying number of participants --- p.21 / Chapter 4.1 --- The modified MG --- p.22 / Chapter 4.1.1 --- Model --- p.22 / Chapter 4.1.2 --- Results --- p.23 / Chapter 4.2 --- Mixed-population --- p.33 / Chapter 4.2.1 --- Model --- p.33 / Chapter 4.2.2 --- Results --- p.33 / Chapter 4.2.3 --- Discussions --- p.37 / Chapter 5 --- Minority game considering recent performance of strategies --- p.39 / Chapter 5.1 --- The modified MG --- p.40 / Chapter 5.1.1 --- Model --- p.40 / Chapter 5.1.2 --- Results --- p.40 / Chapter 5.2 --- Mixed-population --- p.46 / Chapter 5.2.1 --- Model --- p.46 / Chapter 5.2.2 --- Results --- p.47 / Chapter 5.2.3 --- Discussions --- p.50 / Chapter 6 --- Minority game combining both modifications --- p.52 / Chapter 6.1 --- Model --- p.52 / Chapter 6.2 --- Results --- p.53 / Chapter 7 --- Conclusion --- p.68 / Bibliography --- p.71

The thesis consists of three independent topics, each of which is discussed in an individual chapter. The first chapter considers a multiperiod optimal transport problem where distributions μ0, . . . , μn are prescribed and a transport corresponds to a scalar martingale X with marginals Xt ∼ μt. We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property of their support, as simultaneous optimizers for a class of bivariate transport cost functions with a Spence–Mirrlees property, and by an order-theoretic minimality property. Left-monotone transports are unique if μ0 is atomless, but not in general. In the one-period case n = 1, these transports reduce to the Left-Curtain coupling of Beiglbo ̈ck and Juillet. In the multiperiod case, the bivariate marginals for dates (0,t) are of Left-Curtain type, if and only if μ0, . . . , μn have a specific order property. The general analysis of the transport problem also gives rise to a strong duality result and a description of its polar sets. Finally, we study a variant where the intermediate marginals μ1,...,μn−1 are not prescribed. The second chapter studies the convergence of Nash equilibria in a game of optimal stopping. If the associated mean field game has a unique equilibrium, any sequence of n-player equilibria converges to it as n → ∞. However, both the finite and infinite player versions of the game often admit multiple equilibria. We show that mean field equilibria satisfying a transversality condition are limit points of n-player equilibria, but we also exhibit a remarkable class of mean field equilibria that are not limits, thus questioning their interpretation as “large n” equilibria. The third chapter studies the equilibrium price of an asset that is traded in continuous time between N agents who have heterogeneous beliefs about the state process underlying the asset’s payoff. We propose a tractable model where agents maximize expected returns under quadratic costs on inventories and trading rates. The unique equilibrium price is characterized by a weakly coupled system of linear parabolic equations which shows that holding and liquidity costs play dual roles. We derive the leading-order asymptotics for small transaction and holding costs which give further insight into the equilibrium and the consequences of illiquidity.

No description available.

by Lo Ting Shek = 複雜適應系統中的個體為本模型 / 盧庭碩. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 105-107). / Text in English; abstracts in English and Chinese. / by Lo Ting Shek = Fu za shi ying xi tong zhong de ge ti wei ben mo xing / Lu Tingshuo. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Minority game --- p.9 / Chapter 2.1 --- The model --- p.9 / Chapter 2.2 --- Review on selected work on MG --- p.13 / Chapter 2.2.1 --- Market efficiency and Phase transition --- p.13 / Chapter 2.2.2 --- Crowd effect in MG --- p.17 / Chapter 2.2.3 --- Hamming distance between strategies --- p.19 / Chapter 2.2.4 --- Statistical mechanics of systems with heterogeneous agents --- p.21 / Chapter 3 --- Theory of the minority game --- p.23 / Chapter 3.1 --- Formalism --- p.23 / Chapter 3.2 --- Discussion --- p.31 / Chapter 4 --- Evolutionary Minority Game --- p.33 / Chapter 4.1 --- Model --- p.33 / Chapter 4.2 --- Results --- p.36 / Chapter 4.3 --- Discussion --- p.38 / Chapter 5 --- Theory of the Evolutionary Minority game --- p.43 / Chapter 5.1 --- The theory of D'hulst and Rodgers [1] --- p.44 / Chapter 5.1.1 --- Discussion on the D'hulst and Rodgers's theory --- p.51 / Chapter 5.2 --- Theory of EMG [2] --- p.54 / Chapter 5.2.1 --- Formalism --- p.55 / Chapter 5.2.2 --- Results --- p.60 / Chapter 5.2.3 --- Discussion --- p.66 / Chapter 6 --- Evolutionary Minority Game with arbitrary cutoffs --- p.68 / Chapter 6.1 --- Model --- p.68 / Chapter 6.2 --- Results --- p.69 / Chapter 6.3 --- Theory --- p.76 / Chapter 6.4 --- Discussion --- p.85 / Chapter 7 --- Evolutionary minority game with heterogeneous strategy distribution --- p.88 / Chapter 7.1 --- Model --- p.89 / Chapter 7.2 --- Results --- p.90 / Chapter 7.3 --- Discussion --- p.99 / Chapter 8 --- Conclusion --- p.103

Following the recent financial crisis, financial regulators have placed a strong emphasis on reducing expectations of government support for banks, and on better managing and assessing risks in the banking system. This thesis considers three current topics in credit risk and the statistical problems that arise there. The first of these topics is expectations of government support in distressed banks. We utilize unique features of the European credit default swap market to find that market expectations of European government support for distressed banks have decreased -- an important development in the credibility of financial reforms. The second topic we treat is the estimation of covariance matrices from the perspective of market risk management. This problem arises, for example, in the central clearing of credit default swaps. We propose several specialized loss functions, and a simple but effective visualization tool to assess estimators. We find that proper regularization significantly improves the performance of dynamic covariance models in estimating portfolio variance. The third topic we consider is estimation risk in the pricing of financial products. When parameters are not known with certainty, a better informed counterparty may strategically pick mispriced products. We discuss how total estimation risk can be minimized approximately. We show how a premium for remaining estimation risk may be determined when one counterparty is better informed than the other, but a market collapse is to be avoided, using a simple example from loan pricing. We illustrate the approach with credit bureau data.