Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

RESUMO No início de 1995 foi adotado no Brasil o Plano Real, tendo como um dos seus tripés de sustentação a busca pelo combate ao processo inflacionário crônico brasileiro que já se estendia por um longo período. Assim, a política monetária passou a ter um papel importante na determinação das variáveis macroeconômicas. Este trabalho busca analisar uma regra de política monetária que capte as variações ocorridas em todo o período do Plano Real, se estendendo até meados de 2005, bem como se deram as relações entre as variáveis econômicas neste período. A especificação proposta consiste na estimação de modelos não-lineares distintos dependendo do estado da economia (em crise ou fora de crise). Utilizamos um modelo com chaveamento Markoviano para a dinâmica da taxa de juros nominal onde a determinação de períodos de crise é feita por uma variável nãoobservada. Além disso, procuramos adotar dois algoritmos distintos de estimação, Expectation-Maximization (EM) e Monte Carlo Markov Chain (MCMC), concluindo que a análise para ambos é bastante próxima, sendo identificados os mesmos períodos entre regimes. Finalmente, motivamos a estimação através de modelos econômicos teóricos cujas dinâmicas são compatíveis com uma regra de fixação de juros não-linear, avaliando os padrões de resposta a impulso condicionados ao estado da economia (regimes de estabilidade e crise econômica). / ABSTRACT In the beginning of 1995, continuing the process of inflation combat, the monetary policy should have been an important role in the determinacy of macroeconomics variables. This work has a target analyzing a monetary rule that reflects the occurred variations in every Real Plan?s period. The specification proposed by the authors consists in an estimation of two independent nonlinear models for different states of the nature (crises or not crises). Here we estimate a model where the dynamic of the nominal interest rate follows a Markov Switching process and the regimes are unobservable variables. In addition, we try adopting two different algorithms to estimation; Expectation-Maximization (EM) and Monte Carlo Markov Chain (MCMC), concluded that the results are very similar. Finally, we motivate the estimations analyzing models where the theoretical dynamics of the economy are compatible with a nonlinear interest rate rule, analyzing the impulse response conditioned to state of economy (regimes of crises or not crises).

The dissertation consists of three papers which apply Bayesian econometric techniques to monitoring macroeconomic and macro-financial developments in the economy. Its aim is to illustrate how Bayesian methods can be employed in standard areas of economic research (estimating systemic risk in the banking sectors, nowcasting GDP growth) and also in a more original area (monitoring developments in sovereign bond markets). In the first essay, we address a task which analytical departments in central banks or commercial banks face very often - nowcasting foreign demand of a small open economy. On the example of the Czech economy, we propose an approach to nowcast foreign GDP growth rates for the Czech economy. For presentation purposes, we focus on three major trading partners: Germany, Slovakia and France. We opt for a simple method which is very general and which has proved successful in the literature: the method based on bridge equation models. A battery of models is evaluated based on a pseudo-real- time forecasting exercise. The results for Germany and France suggest that the models are more successful at backcasting, nowcasting and forecasting than the naive random walk benchmark model. At the same time, the various models considered are more or less successful depending on the forecast horizon....

[pt] E se o risco fiscal não for desprezível? O Banco Central poderia continuar efetivamente trazendo a inflação para a meta ao ignorar o risco de default? Para responder a essas questões, propomos um modelo DSGE de pequena economia aberta com limite fiscal endógeno, onde o governo pode dar calote em seus títulos domésticos, e a autoridade monetária pode responder por isso. Avaliamos a dinâmica sob duas regras de decisão do Banco Central: quando (i) rastreia erroneamente esse risco e (ii) rastreia perfeitamente o risco de inadimplência. O modelo é calibrado com base em dados brasileiros, dado que a recente deterioração orçamentária do país faz dele um caso ideal a ser estudado. Constatamos que inflação alta e moeda desvalorizada coexistem com uma alta taxa de juros quando a autoridade monetária não leva em conta integralmente o risco de inadimplência. Quanto maior a probabilidade de default, maiores as diferenças entre os efeitos dos dois tipos de regras de política que analisamos. Para uma banqueira central restaurar a meta de inflação, ela deve incorporar perfeitamente toda a dinâmica do risco de inadimplência em sua regra de decisão. Além disso, nosso modelo gera um prêmio endógeno entre as taxas de juros dos países devido às diferenças no risco de default soberano. / [en] What if the fiscal risk is not negligible? Could the Central Bank continue effectively bringing inflation to the target when it ignores the default risk? To address those questions, we propose a small open economy DSGE model with an endogenous fiscal limit, where the government can default on its domestic bonds, and monetary authority may account for that. We evaluate dynamics under two different Central Bank decision rules: when (i) it wrongly tracks that risk, and (ii) it perfectly tracks default risk. The model is calibrated based on Brazilian data, as its recent budgetary deterioration makes the country an ideal case to be studied. We find that high inflation and depreciated currency coexist with a high interest rate when the monetary authority does not fully account for the default risk. The higher the default probability, the greater the differences across the effects of the two types of policy rules that we analyzed. For a central banker to restore the inflation target, she must fully track default risk in its decision rule. In addition, our model generates an endogenous premium across countries’ interest rates due to differences in sovereign default risk.

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本文研究美國退休福利保險公司(PBGC)保險價值的計算,延伸 Marcus (1987)模型,提出狀態轉換過程保險價值模型計算,也就是將市場分為兩種情況,正成長率視為正常狀態,負成長率為衰退狀態,利用狀態轉換過程評價 PBGC 契約在經濟困難而終止和介入終止下合理的保險價值。在參數估計方面,本文以 S&P500股價指數和一年期國庫券資料參數估計值及Marcus(1987)和Pennacchi and Lewis(1994)的方式給定參數,以 EM-PSO-Gradient 延伸 EM-Gradient 方法並以最大概似函數值、AIC 準則和 BIC 準則比較估計結果。最後固定其他參數, 探討狀態轉換過程保險價值模型對參數調整後保險價值的影響之敏感度分析。 / In this paper, we evaluate Pension Benefit Guaranty Corporation insurance values through regime switching models, which is the extension of the models of Marcus (1987). That is, we can separate periods of economy with faster growth from those with slower growth when observing long-term trends in economy and calculate the reasonable PBGC insurance values under distress termination and intervention termination by regime switching processes. We set parameters by estimating S&P 500 index and 1-year treasury bills by EM-PSO-Gradient, which is the extensive method of EM-Gradient and refer the methods of setting parameters from Marcus (1987) and Pennacchi and Lewis (1994). After that, we compare the maximum likelihood estimates, AIC and BIC of the estimative results. Finally, we do sensitivity analysis through given the other parameters and look into what would impact on our models of insurance values when adjusting one parameter.

本篇論文以Regime Switching Stochastic Volatility（RSV）作為外匯選擇權市場的波動度模型，採用馬可夫鏈蒙地卡羅法（Markov Chain Monte Carlo）中的GibbS Sampling演算法估計RSV模型的參數，並預測外匯選擇權在RSV模型下的價格。 數值結果方面首先對GibbS Sampling參數估計的結果做討論，再對預測出的選擇權價格與Black and Scholes作比較，最後並提出笑狀波幅與隱含波動度平面的結果。 本研究所得到之結論： 1. RSV模型與MCMC模擬法的組合，具備產生笑狀波幅的能力，提供足夠證據顯示，RSV模型與MCMC演算法所計算出來的選擇權價格，確實反應且捕捉到了市場上選擇權價格所應具備的特色。 2. 本模型能有效解釋期限結構 （Term Stucture of Volatility）、笑狀波幅（Volatility Smile）的現象。 關鍵字：馬可夫鏈蒙地卡羅法、外匯選擇權、貝氏選擇權評價、MCMC、Regime switching Regine change、Gibbs Sampling、currency option、Markov Chain Montec Carlo

[pt] Essa tese discute o papel das expectativas dos agentes a respeito da condução das políticas monetária e fiscal na determinação dos efeitos dessas políticas, na dinâmica da economia e na volatilidades das variáveis macroeconômicas. O primeiro capítulo mostra que considerar as expectativas dos agentes a respeito de possíveis mudanças de regime tem efeitos importantes nas respostas das variáveis macroeconômicas aos choques, mesmo que essa mudança de regime não se materialize ao longo da trajetória observada após o choque. O reconhecimento da possibilidade de mudanças de regime também tem consequências importantes para a volatilidade das variáveis endógenas que são mais altas do que as obtidas no modelo linear e muito dependentes dos parâmetros de política escolhidos pelas autoridades fiscal e monetária em cada regime. O segundo capítulo discute o papel das expectativas a respeito das políticas futuras na determinação da profundidade de uma crise quando a economia atinge o limite inferior de zero para as taxas de juros nominais. Ele mostra que ao analisar o impacto de um estímulo fiscal durante um episódios de taxa de juros zero, deve-se olhar para além dos multiplicadores no curto prazo. Para ter efeitos positivos maiores, as políticas monetária e fiscal devem durar mais do que a crise e precisam ser coordenadas. O terceiro capítulo apresenta uma avaliação dos estímulos fiscais em termos das perdas de bem-estar, tornando claro que essa avaliação deve considerar não apenas o efeitos das políticas sobre a inflação e o produto no curto prazo, mas também o valor presente descontado da inflação e do produto nos períodos futuros. Ele também apresenta uma análise de como se obtém o nível ótimo da taxa de juros nominal uma vez que a economia não está mais em crise se a autoridade monetária pretende usar o canal das expectativas para reduzir a profundidade da crise. / [en] This thesis discusses the role of agents expectations regarding the conduction of monetary and fiscal policies in determining policy outcomes, economic dynamics and the volatilities of macroeconomic variables. The first Chapter shows that accounting for agents’ expectations of a possible regime change has critical effects in the responses of macroeconomic variables to shocks, even if this switch does not materialize itself along the path observed after the shock. Recognizing the possibility of regime switches also have important consequences for the volatilities of endogenous variables, which are higher than those obtained in the linear model and very dependent on the policy parameters chosen by monetary and fiscal authorities in each regime. In the second Chapter, I discuss the role of expectations in determining the depth of a crisis when the economy hits the zero lower bound on nominal interest rates. I show that when analysing the impact of a fiscal stimulus during a zero interest rate episode, there is more than just short-run multipliers. To have larger positive effects on output and inflation, monetary and fiscal policies should last longer than the duration of the shock and be coordinated in their actions. The third Chapter presents a thoughtful evaluation of a fiscal stimulus in terms of the implied welfare losses making clear that it should account not only for the effects of policies on short-run output and inflation, but also for the present discounted value of output and inflation in future periods as well. It also analyses how to obtain the optimal level for the nominal interest rate once the economy gets out of the crisis state, if the monetary authority wants to use the expectations channel to undermine the depth of the crisis.

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.