Key Words: continuous-time model, mean-variance, transaction costs, stochastic singular control, Lagrange multiplier method, parabolic free-boundary problem, double-obstacle problem, Skorokhod problem. / We study continuous-time Markowitz's mean-variance portfolio selection problem in a market with one stock, one bond and proportional transaction costs. The presence of transaction costs makes the problem being a singular control problem in a finite time horizon, which is very hard to deal with from the point view of control theory. Using a partial differential equation approach, we formulate the problem as a double obstacle problem, and prove the smoothness of the value function. Then we give the necessary and sufficient conditions for the existence of an optimal solution and completely characterize the optimal strategy when the problem is feasible. We show three critical distinctive features of the Markowitz model under the presence of transaction costs. First, the expected return on the portfolio could be too high to achieve if the time to maturity is not long enough, while without transaction costs, any expected return can be reached in an arbitrary short time. Second, instead of keeping the investment ratio between stock and bond to be a constant, there exists time-dependent upper and lower boundaries, transaction is carried out only if the investment ratio is on the boundaries. Third, there exists a critical time, which only depends on the market parameters, such that beyond the time no more investment will be added to stock holding. Our result is closer to real investment practice where people tend not to invest on risky assets towards the end of the investment horizon. / Xu Zuoquan. / "January 2007." / Adviser: Xunyu Zhou. / Source: Dissertation Abstracts International, Volume: 68-08, Section: B, page: 5290. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 118-123). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Kam, Kwok Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 100-103). / Abstracts in English and Chinese. / Abstract Page --- p.11 / Abstract (Chinese) --- p.12 / Acknowledgment Page --- p.13 / Table of Contents --- p.1 / Table of Figures --- p.1 / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Utility and Value Function --- p.5 / Chapter 1.2.1 --- Expected utility theory --- p.5 / Chapter 1.2.2 --- Prospect Theory --- p.9 / Chapter 1.3 --- Mental Accounting --- p.14 / Chapter 1.3.1 --- Segregation vs Aggregation --- p.17 / Chapter 2 --- Moving reference point with loss aversion --- p.21 / Chapter 2.1 --- Model Setup --- p.21 / Chapter 2.2 --- Simulation Results --- p.27 / Chapter 3 --- Constant Rebalancing Portfolio with Additive Utility --- p.30 / Chapter 3.1 --- Model setting --- p.31 / Chapter 3.1.1 --- Additive Utility Theory (AUT) --- p.33 / Chapter 3.2 --- Analysis --- p.34 / Chapter 3.3 --- Results --- p.35 / Chapter 3.4 --- Summary --- p.38 / Chapter 4 --- Revision of Gomes´ة Work --- p.40 / Chapter 4.1 --- Background --- p.40 / Chapter 4.2 --- Portfolio Allocation with zero surplus wealth --- p.44 / Chapter 4.3 --- Portfolio Allocation with Negative Surplus --- p.46 / Chapter 4.4 --- Portfolio Allocation with Positive Surplus --- p.50 / Chapter 4.5 --- Numerical Results --- p.51 / Chapter 4.5.1 --- Gomes´ة Work --- p.56 / Chapter 4.6 --- Summary --- p.57 / Chapter 5 --- Mental Accounting under Value Function in the Prospect Theory --- p.59 / Chapter 5.1 --- Cognitive dissonance --- p.59 / Chapter 5.2 --- Market Setting --- p.60 / Chapter 5.3 --- Single Mental Account --- p.61 / Chapter 5.4 --- Two Mental Accounts --- p.63 / Chapter 5.5 --- Numerical results --- p.67 / Chapter 5.5.1 --- Pessimistic View --- p.71 / Chapter 5.6 --- Summary --- p.72 / Chapter 6 --- Mental Accounting under Friedman-Savage Value Function --- p.74 / Chapter 6.1 --- Two Assets with Single mental account --- p.76 / Chapter 6.1.1 --- Different Sharpe ratios --- p.78 / Chapter 6.1.2 --- Same Sharpe ratio --- p.82 / Chapter 6.2 --- Two Assets with two mental accounts --- p.85 / Chapter 6.2.1 --- Segregation or Aggregation --- p.86 / Chapter 6.2.2 --- Numerical results --- p.90 / Chapter 6.3 --- Summary --- p.93 / Chapter 7 --- Conclusion --- p.96 / Bibliography --- p.100

Lam, Ka Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 51-55). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Levy Processes --- p.6 / Chapter 2.1 --- Definition --- p.6 / Chapter 2.2 --- Levy-Khinchine formula --- p.7 / Chapter 2.3 --- Applications of Levy Processes in Finance --- p.10 / Chapter 2.4 --- Option pricing under Levy Processes --- p.12 / Chapter 2.4.1 --- Black-Scholes Formula with Characteristic Function --- p.12 / Chapter 2.4.2 --- Fast Fourier Transform --- p.14 / Chapter 2.4.3 --- Other Payoff Functions --- p.16 / Chapter 3 --- Dynamic Fund Protection --- p.19 / Chapter 3.1 --- Discrete Dynamic Fund Protection --- p.20 / Chapter 3.2 --- Link DFP to Discrete Lookback Options --- p.22 / Chapter 4 --- Spitzer´ةs Identity --- p.25 / Chapter 4.1 --- Applications of Spitzer's Identity --- p.25 / Chapter 4.2 --- Discrete Lookback Options --- p.29 / Chapter 5 --- Pricing Discrete DFP --- p.32 / Chapter 5.1 --- Girsanov´ةs Theorem --- p.32 / Chapter 5.2 --- Equivalent Martingale Measure in DFP --- p.34 / Chapter 5.3 --- Pricing DFP at any Time Points --- p.36 / Chapter 5.4 --- The Main Algorithm --- p.38 / Chapter 6 --- Numerical Results --- p.40 / Chapter 6.1 --- Simulation of Discrete DFP --- p.40 / Chapter 6.2 --- Numerical Implementation --- p.42 / Chapter 7 --- Conclusion --- p.50 / Bibliography --- p.51

Jiang, Kening. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 90-93). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Portfolio Selection Using Clustering Methodology --- p.7 / Chapter 2.1 --- Heuristic algorithm --- p.8 / Chapter 2.1.1 --- Step 1: Security transformation by factor model --- p.8 / Chapter 2.1.2 --- Step 2: Clustering algorithm --- p.10 / Chapter 2.1.3 --- Step 3: Representative selection by t he Sliarpe ratio --- p.16 / Chapter 2.2 --- Numerical results --- p.17 / Chapter 3 --- Modified Portfolio Selection Using Clustering Methodology --- p.22 / Chapter 3.1 --- Analysis of artificial factors --- p.23 / Chapter 3.2 --- Problem reformulation --- p.27 / Chapter 3.3 --- Numerical results --- p.29 / Chapter 4 --- Minimum Variance Point --- p.70 / Chapter 4.1 --- Iterative elimination scheme I --- p.72 / Chapter 4.2 --- Iterative elimination scheme II --- p.74 / Chapter 4.3 --- Orthogonal matrix mapping --- p.76 / Chapter 4.4 --- Condition to solve diagonal dominant problem --- p.77 / Chapter 4.5 --- L1 formulation --- p.82 / Chapter 4.6 --- Numerical results --- p.85 / Chapter 5 --- Summary and Future work --- p.88

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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

Transportation procurement problem (TPP) is the problem of setting transportation service prices, delivery timing and quantity, and controlling costs and capacity to reduce empty movements and improve market efficiency. The purchase of transportation service is traditionally achieved using a request for proposal and long-term contracts. However, as business relationships become ever more flexible and dynamic, there has been an increasing need to hedge the risks of traditional transportation procurement such as entrance of new carriers and sudden drop in fuel price. This thesis proposes a holistic aution-based solution for the TPP. Four typical scenarios are investigated. The first scenario incorporates bilateral bidding into auction mechanism design for multi-unit TPP. This scenario considers one-sided Vickrey-Clarke-Groves (O-VCG) combinatorial auctions for a complex transportation marketplace with multiple lanes. This scenario then designs three alternative multi-unit trade reduction (MTR) mechanisms for the bilateral exchange transportation marketplace where all the lanes are partitioned into distinct markets. Proposed mechanisms ensure incentive compatibility, individual rationality, budget balance and asymptotical efficiency. The second scenario presents a double auction model for the TPP in a dynamic single-lane transportation environment. This scenario first addresses the TPP in a transportation spot market with stochastic but balanced or “symmetric” demand and supply. A periodic sealed double auction (PSDA) is proposed. This scenario then devises a modified PSDA (M-PSDA) to address the TPP with “asymmetric” demand and supply. The auctioneer is likely to gain higher profits from setting a relatively short auction length. However, it is optimal to run the auction (either PSDA or MPSDA) with a relatively large auction length, when maximizing either the social welfare or the utility of shippers and carriers (agents). When the degree of supply-demand imbalance is low, the auctioneer’s myopic optimal expected profit under supply-demand imbalance is larger than that under symmetric demand and supply. This third scenario presents an auction-based model for the TPP in make-toorder systems. The optimality of dynamic base-stock type (S(x)-like policy) is established. The optimal allocation can be achieved by running an O-VCG auction or a first-price auction with closed-form reserve prices. By mild technical modifications, the results derived in the infinite horizon case can all be extended to the finite horizon case. The fourth scenario proposes allocatively efficient auction mechanisms for the distributed transportation procurement problem (DTPP), which is generally the problem of matching demands and supplies over a transportation network. This scenario constructs an O-VCG combinatorial auction for the DTPP where carriers are allowed to bid on bundles of lanes. To simplify the execution of auction, this scenario next proposes a primal-dual Vickrey (PDV) auction based on insights from the known Ausubel auctions and the primal-dual algorithm. The PDV auction realizes VCG payments and truthful bidding under the condition of seller-submodularity, which implies that the effect of each individual carrier is decreasing when the coalition increases. / 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