|Title:||A class of two-stage stochastic quadratic programs with applications in oil market|
|Subject:||Hong Kong Polytechnic University -- Dissertations|
|Department:||Department of Applied Mathematics|
|Pages:||xxi, 146 pages : color illustrations|
|Abstract:||In this thesis, two applications in mathematical finance and economics are modelled by a class of two-stage stochastic programs. The focuses are on the modeling of the practical problems as well as their real life implementations with numerical implementations. The first application on portfolio selection is approached by the construction of a relaxed second-order stochastic dominance (SSD) constrained stochastic optimization problem. The second model concerns the equilibrium in the world oil market share, which is treated as a solution of a two-stage stochastic Nash equilibrium problem. General framework of a class of two-stage convex stochastic optimization problems is analyzed in details, with special attentions paid to the constructions of the Lagrangian together with its saddle-point characterization of optimality. The problem of portfolio selection aims to construct optimal assets allocation strategies subject to constraints on risk management. More specifically, a portfolio optimization model with relaxed SSD constraints is proposed and solved, and its solution is the portfolio of choice. The proposed model uses Conditional Value at Risk (CVaR) constraints at probability level ε(0, 1) to relax SSD constraints. The relaxation is justified by theoretical convergence results based on sample average approximation (SAA) method when sample size N -> ∞ and CVaR probability level tends to 1. SAA method is used to reduce infinite number of inequalities of SSD constraints to finite ones and also to calculate the expectation value. The proposed relaxation on the SSD constraints in portfolio optimization problem is achieved when the probability level of CVaR takes value less than but close to 1, and the model can be readily solved by cutting plane method. The performance and characteristics of the constructed portfolios are tested empirically on three sets of real market data, and the results obtained from the numerical experiments are analyzed and discussed in details. It is shown that with appropriate choices of CVaR probability level, the constructed CVaR-SSD portfolios are sparse and outperform both the benchmark portfolios and the portfolios constructed by solving the portfolio optimization problems with SSD constraints. The second application is an attempt to provide explanation and mechanism about the stable patterns observed in market share of world's crude oil trading over the last several decades via a two-stage stochastic model of Nash equilibrium for Cournot competition. To summarize, a convex two-stage non-cooperative multi-agent equilibrium problem under uncertainty is formulated as a two-stage stochastic variational inequality (SVI). Under standard assumptions, sufficient conditions for the existence of solutions of the two-stage SVI are provided. A regularized SAA method is proposed to solve it. The convergence of the method is proved as the regularization parameter tends to zero and the sample size tends to infinity. In order to explain the oil market share observation, a two-stage stochastic production and supply planning problem with homogeneous commodity in an oligopolistic market is constructed under the framework of two-stage SVI. Numerical experiments are performed based on historical data. The data are used in-sample to aid the selection of parameters as well as the modification of the game model structure. Out-ofsample tests are presented to demonstrate the effectiveness of the proposed model in its ability for describing the market share of oil producing agents.|
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