In this thesis, we develop partial differential equation (PDE) based numerical methods to solve certain optimal stochastic control problems in finance. The value of a stochastic control problem is normally identical to the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation or an HJB variational inequality. The HJB equation corresponds to the case when the controls are bounded while the HJB variational inequality corresponds to the unbounded control case. As a result, the solution to the stochastic control problem can be computed by solving the corresponding HJB equation/variational inequality as long as the convergence to the viscosity solution is guaranteed. We develop a unified numerical scheme based on a semi-Lagrangian timestepping for solving both the bounded and unbounded stochastic control problems as well as the discrete cases where the controls are allowed only at discrete times. Our scheme has the following useful properties: it is unconditionally stable; it can be shown rigorously to converge to the viscosity solution; it can easily handle various stochastic models such as jump diffusion and regime-switching models; it avoids Policy type iterations at each mesh node at each timestep which is required by the standard implicit finite difference methods. In this thesis, we demonstrate the properties of our scheme by valuing natural gas storage facilities---a bounded stochastic control problem, and pricing variable annuities with guaranteed minimum withdrawal benefits (GMWBs)---an unbounded stochastic control problem. In particular, we use an impulse control formulation for the unbounded stochastic control problem and show that the impulse control formulation is more general than the singular control formulation previously used to price GMWB contracts.

In this thesis, we develop partial differential equation (PDE) based numerical methods to solve certain optimal stochastic control problems in finance. The value of a stochastic control problem is normally identical to the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation or an HJB variational inequality. The HJB equation corresponds to the case when the controls are bounded while the HJB variational inequality corresponds to the unbounded control case. As a result, the solution to the stochastic control problem can be computed by solving the corresponding HJB equation/variational inequality as long as the convergence to the viscosity solution is guaranteed. We develop a unified numerical scheme based on a semi-Lagrangian timestepping for solving both the bounded and unbounded stochastic control problems as well as the discrete cases where the controls are allowed only at discrete times. Our scheme has the following useful properties: it is unconditionally stable; it can be shown rigorously to converge to the viscosity solution; it can easily handle various stochastic models such as jump diffusion and regime-switching models; it avoids Policy type iterations at each mesh node at each timestep which is required by the standard implicit finite difference methods. In this thesis, we demonstrate the properties of our scheme by valuing natural gas storage facilities---a bounded stochastic control problem, and pricing variable annuities with guaranteed minimum withdrawal benefits (GMWBs)---an unbounded stochastic control problem. In particular, we use an impulse control formulation for the unbounded stochastic control problem and show that the impulse control formulation is more general than the singular control formulation previously used to price GMWB contracts.

[pt] O cenário atual da indústria de gás natural brasileira é caracterizado por baixa maturidade e dinamismo de mercado. O comportamento estocástico da demanda por gás, somado volatilidade do preço de mercado do GNL, motiva a utilização de estocagem subterrânea como forma de inserir flexibilidade no suprimento, além de promover proteção contra flutuação no preço. No entanto, a literatura existente carece de uma uma ferramenta analítica mais robusta para apoiar uma análise quantitativa dos benefícios que a atividade UNGS poderia proporcionar à indústria de gás natural. Nesta tese, propomos um modelo de programação dinâmica estocástica para planejamento de longo/médio prazo, a fim de determinar a política ótima de fornecimento juntamente com a possibilidade de armazenamento de gás. Um modelo markoviano caracteriza a demanda termoelétrica, enquanto o preço de GNL é representado por um processo estocástico temporalmente independente. O modelo proposto é eficientemente resolvido usando o algoritmo de programação dinâmica dual estocástica para o estudo de caso brasileiro, considerando dados dos setores de gás e setor elétrico. Para uma escolha exógena, mas significativa, da localização e tamanho do armazenamento subterrâneo, observamos os benefícios operacionais e econômicos da flexibilidade que esta atividade poderia proporcionar. Além disso, comparando os custos de OPEX e CAPEX de investimentos em infraestrutura de armazenamento em campos depletados e cavernas de sal com as economias proporcionadas pelo armazenamento na operação de fornecimento, é possível observar o benefício econômico da atividade de estocagem. A estrutura proposta fornece suporte quantitativo importante para discussões sobre precificação de infraestrutura e modelo de negócios para Armazenamento Subterrâneo de Gás Natural. / [en] The current scenario of the Brazilian natural gas industry is characterized by low maturity and dynamism of the market.The stochastic behavior of Brazilian demand for natural gas, added to its associated market price volatility, motivates the usage of underground storage due to supply flexibility and protection against price fluctuations. However, the existing literature lacks a more robust analytical tool to support a quantitative analysis of the benefits that the UNGS activity could provide to the natural gas industry. In this thesis, we propose a stochastic dynamic programming model for long/medium term planning to determine the supply optimal policy together with the possibility of storing gas. A markovian model characterizes thermoelectric demand while market price is represented by a stagewise independent stochastic process. The proposed model is efficiently solved using the Stochastic Dual Dynamic Programming algorithm for the Brazilian case study considering realistic data for the actual gas network and electric power system. For an exogenous but meaningful choice of underground storage location and size, we observe the operational and economic benefits of the provided storage flexibility. Additionally, comparing the OPEX and CAPEX costs of investments in storage infrastructure in depleted fields and salt caverns with the savings provided by storage in the supply operation, it is possible to observe the economic benefit of storage. The proposed framework provides an important quantitative support for discussion about Underground Natural Gas Storage infrastructure pricing and business models.

This diploma’s thesis analyzes the possibility of accumulation of thermal power plants in the Czech Republic. The thesis is divided into several parts. The first part describes the different types of storage power plants, the historical development of power storage for compressed air and the appropriateness of their location. The second part is devoted to the design of storage power plant for compressed air in South Moravia. In the next chapter, a calculation is made of all equipment storage power plant, including turbo-compressor, combustion chamber, combustion turbines, the volume of storage tanks and two heat exchangers. The last part is the economic analysis of the return on investment of such a project.