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published_or_final_version / abstract / Industrial and Manufacturing Systems Engineering / Doctoral / Doctor of Philosophy

Li Chin Pang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 64-66). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Structural Models of Credit Risk --- p.7 / Chapter 2.1 --- Barrier-Independent Models --- p.8 / Chapter 2.2 --- Barrier-Dependent Models --- p.9 / Chapter 2.3 --- Empirical Literature --- p.10 / Chapter 3 --- Jump-Diffusion Models --- p.13 / Chapter 3.1 --- Analytical Option Pricing Formula --- p.14 / Chapter 3.1.1 --- The Jump-Diffusion Model of Merton --- p.14 / Chapter 3.1.2 --- The Jump-Diffusion Model of Kou --- p.15 / Chapter 3.2 --- Simulation for Options --- p.19 / Chapter 3.2.1 --- Simulation for Barrier-Independent Options --- p.19 / Chapter 3.2.2 --- Brownian Bridge Simulation for DOC Option --- p.20 / Chapter 4 --- Likelihood Function for Equity Returns --- p.24 / Chapter 4.1 --- Likelihood Function on Equity Return --- p.26 / Chapter 4.2 --- Degeneracy Problem of Likelihood Function --- p.27 / Chapter 5 --- The Proposed Framework --- p.31 / Chapter 5.1 --- Penalized Likelihood Estimation --- p.31 / Chapter 5.2 --- Expectation-Maximization Algorithm --- p.36 / Chapter 5.3 --- The MJD Structural Model --- p.41 / Chapter 5.4 --- The K<JD Structural Model --- p.43 / Chapter 5.5 --- Computation of the E-step --- p.47 / Chapter 6 --- Performance of Estimation --- p.49 / Chapter 6.1 --- Simulation Checks --- p.49 / Chapter 6.2 --- Empirical Performance --- p.55 / Chapter 6.2.1 --- Bond Selection --- p.55 / Chapter 6.2.2 --- Empirical Results --- p.57 / Chapter 7 --- Conclusion --- p.62 / Bibliography --- p.64

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 / Industrial and Manufacturing Systems Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Industrial and Manufacturing Systems Engineering / Doctoral / Doctor of Philosophy

The procurement of commodity materials for production is an important issue in supply chain management. Effective procurement should consider both uncertain customer demand and fluctuating commodity price which, when act together, give rise to the procurement risk. To protect the bottom line, a manufacturer has to plan its procurement activities with special attention given to such procurement risk. Existing research has studied the use of exchange market-traded commodities in mitigating procurement risk. This study addresses the case of a manufacturer with long-term procurement commitments who wishes to hedge against the risk exposure by using long-dated futures contracts. In the commodities markets, however, long-dated futures are often illiquid or even unavailable, thus making the hedge ineffective. Alternatively, in a stack-and-roll hedge, the hedging positions are rolled forward in actively traded short-dated futures contracts of equal maturity until the procurement is executed. This in effect replicates the long-term futures contract in performing a hedge. This study therefore aims at developing a dynamic stack-and-roll approach that can effectively manage the long maturity procurement risk. The proposed dynamic stack-and-roll approach is inherently a discrete-time hedging strategy that divides the procurement planning horizon into multiple decision stages. The nearby futures are adopted as the short-dated futures as they are typically liquid. The hedging positions are adjusted periodically in response to the commodity price behaviour and updated information about the forward customer demand. For a manufacturer who wishes to mitigate the procurement risk as well as maximise the terminal revenue after the procurement, the mean-variance objective function is employed to model the manufacturer’s risk aversion behaviour. Then, a dynamic program formulation of the approach is presented for determining a closed-form expression of the optimal hedging positions. Notice that the hedging policy is a time-consistent mean-variance policy in discrete-time, in contrast to the existing discrete hedging approaches that employ minimum-variance policies. In this study, the commodity prices are modelled by a fractal nonlinear regression process that employs a recurrent wavelet neural network as the nonlinear function. The purpose of this arrangement is to incorporate the fractal properties discovered in commodity prices series. In the wavelet transform domain, fractal self-similarity and self-affinity information of the price series over a certain time scale can be extracted. The Extended Kalman Filter (EKF) algorithm is applied to train the neural network for its lower training error comparing with classical gradient descent algorithms. Monthly returns and volatility of commodity prices are estimated by daily returns data in order to increase the estimation accuracy and facilitate effective hedging. The demand information is updated stage by stage using Bayesian inference. The updating process are defined and adapted to a filtration, which can be regarded as the information received at the beginning of each decision stage. Numerical experiments are carried out to evaluate the performance of the proposed stack-and-roll approach. The results show that the proposed approach robustly outperforms other hedging strategies that employ minimum-variance or naïve policies, and effectively mitigate the procurement risk. / published_or_final_version / Industrial and Manufacturing Systems Engineering / Doctoral / Doctor of Philosophy

Over the past decades, researchers in economics, financial mathematics and actuarial science have introduced results to the concept of comonotonicity in their respective fields of interest. Comonotonicity is a very strong dependence structure and is very often mistaken as a dependence structure that is too extreme and unrealistic. However, the concept of comonotonicity is actually a useful tool for solving several research and practical problems in capital allocation, risk sharing and optimal allocation. The first topic of this thesis is focused on the application of comonotonicity in optimal capital allocation. The Enterprise Risk Management process of a financial institution usually contains a procedure to allocate the total risk capital of the company into its different business units. Dhaene et al. (2012) proposed a unifying capital allocation framework by considering some general deviation measures. This general framework is extended to a more general optimization problem of minimizing separable convex function with a linear constraint and box constraints. A new approach of solving this constrained minimization problem explicitly by the concept of comonotonicity is developed. Instead of the traditional Kuhn-Tucker theory, a method of expressing each convex function as the expected stop-loss of some suitable random variable is used to solve the optimization problem. Then, some results in convex analysis with infimum-convolution are derived using the result of this new approach. Next, Borch's theorem is revisited from the perspective of comonotonicity. The optimal solution to the Pareto optimal risk-sharing problem can be obtained by the Lagrangian method or variational arguments. Here, I propose a new method, which is based on a Breeden-Litzanbeger type integral representation formula for increasing convex functions. It enables the transform of the objective function into a sum of mixtures of stop-losses. Necessary conditions for the existence of optimal solution are then discussed. The explicit solution obtained allows us to show that the risk-sharing problem is indeed a “point-wise” problem, and hence the value function can be obtained immediately using the notion of supremum-convolution in convex analysis. In addition to the above classical risk-sharing and capital allocation problems, the problem of minimizing a separable convex objective subject to an ordering restriction is then studied. Best et al. (2000) proposed a pool adjacent violators algorithm to compute the optimal solution. Instead, we show that using the concept of comonotonicity and the technique of dynamic programming the solution can be derived in a recursive manner. By identifying the right-hand derivative of the convex functions with distribution functions of some suitable random variables, we rewrite the objective function into a sum of expected deviations. This transformation and the fact that the expected deviation is a convex function enable us to solve the minimizing problem. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

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