Some investment and consumption problems with hidden Markov and regime switching structures are studied in this thesis. The financial market consists of a risky asset with random returns and a riskless asset, which generates random returns with a constant expectation, or simply a constant return rate. At the beginning of each investment period, a consumption decision is made from the total available wealth and then an investment decision is made on the proportion of the remaining wealth to be invested in the risky asset. The riskless asset return rate is assumed to be random with a constant expectation or simply a constant return rate, but the random return rate of the risky asset depends on the market environment, which is characterized by a discrete-time hidden Markov chain. The Bayesian method is used to model the information gathering process on the risky asset in a finite time horizon model. According to the accumulated information, we estimate the transition probabilities of the Markov process, which characterizes the market environment during the next few investment periods, and then predict the expected return from the risky asset. We also investigate the sensitivity of the optimal solution with respect to the investment horizon and other factors such as the expected return rate from the riskless asset and the consumption rate. Therefore the investors investment and consumption decision problem is formulated as an optimal stochastic control problem. The solution of the optimal strategy is characterized for the power utility function. The properties of the optimal strategy and the relevant algorithms are discussed for the case in which the random return rate of the risky asset follows a normal distribution with unknown parameters.

This thesis is concerned with the study of the risk-constrained portfolio selection problem arising from an ordinary investor and the insurer being an investor. We first consider the problem for an insurer who can invest her surplus into financial market. With value at risk VaR) imposed as the dynamic risk constraint, the portfolio selection problem is considered with two objectives: the ruin probability minimization and wealth utility maximization. A closed-form solution is found by solving the associated Hamilton-Jacob-Bellman HJB) equation for the first problem. By using the exponential utility function, we solve the second problem by transforming this stochastic optimal control problem into a deterministic optimal control one and using control parameterization method. Second, we consider the risk-constrained utility maximizing problem with a jump diffusion model and a regime switching model for an ordinary investor. Conditional value at risk CVaR) and maximal value at risk MVaR) are used as the risk constraint in the two models, respectively. The associated HJB equations are treated with numerical techniques.

Volatility estimation plays an important role in the fields of statistics and finance. Many different techniques address the problem of estimating volatilities of financial assets. Autoregressive conditional heteroscedasticity ARCH) models and the related generalized ARCH models are popular models for volatilities. Multivariate approaches to GARCH models, such as Engles Dynamic Conditional Correlation GARCH DCC-GARCH), allow for estimation of multiple financial asset volatilities and covariances. However, the parameters of the DCC-GARCH model are typically estimated with Maximum Likelihood Estimation MLE), which is greatly affected by outliers. Outliers in a DCC-GARCH model affect subsequent estimation of volatilities by the design of the model. These outliers may also affect volatility estimates of other financial assets within the same set of assets due to the correlated nature of the financial asset estimation. This thesis reviews ARCH / GARCH modeling and robust estimation and proposes a robust estimation method for the DCC-GARCH model based on bounded deviance function estimation. This robust method of the DCC-GARCH model better estimates the volatilities of a set of financial assets in the presence of outliers. The thesis presents a study of the consistency of the robust method of the DCC-GARCH model along with simulation results to explore the characteristics of the robust method of the DCC-GARCH model estimation. For a better evaluation of the robust method, the thesis also examines the distribution structure of foreign exchange rate data. The thesis also discusses possible future topics and research in this field of study.

The 2007-2009 financial crisis has shed light on the importance of contagion and systemic risk, and revealed the lack of adequate indicators for measuring and monitoring them. This dissertation addresses these issues and leads to several recommendations for the design of an improved assessment of systemic importance, improved rating methods for structured finance securities, and their use by investors and risk managers. Using a complete data set of all mutual exposures and capital levels of financial institutions in Brazil in 2007 and 2008, we explore in chapter 2 the structure and dynamics of the Brazilian financial system. We show that the Brazilian financial system exhibits a complex network structure characterized by a strong degree of heterogeneity in connectivity and exposure sizes across institutions, which is qualitatively and quantitatively similar to the statistical features observed in other financial systems. We find that the Brazilian financial network is well represented by a directed scale-free network, rather than a small world network. Based on these observations, we propose a stochastic model for the structure of banking networks, representing them as a directed weighted scale free network with power law distributions for in-degree and out-degree of nodes, Pareto distribution for exposures. This model may then be used for simulation studies of contagion and systemic risk in networks. We propose in chapter 3 a quantitative methodology for assessing contagion and systemic risk in a network of interlinked institutions. We introduce the Contagion Index as a metric of the systemic importance of a single institution or a set of institutions, that combines the effects of both common market shocks to portfolios and contagion through counterparty exposures. Using a directed scale-free graph simulation of the financial system, we study the sensitivity of contagion to a change in aggregate network parameters: connectivity, concentration of exposures, heterogeneity in degree distribution and network size. More concentrated and more heterogeneous networks are found to be more resilient to contagion. The impact of connectivity is more controversial: in well-capitalized networks, increasing connectivity improves the resilience to contagion when the initial level of connectivity is high, but increases contagion when the initial level of connectivity is low. In undercapitalized networks, increasing connectivity tends to increase the severity of contagion. We also study the sensitivity of contagion to local measures of connectivity and concentration across counterparties—the counterparty susceptibility and local network frailty—that are found to have a monotonically increasing relationship with the systemic risk of an institution. Requiring a minimum aggregate) capital ratio is shown to reduce the systemic impact of defaults of large institutions； we show that the same effect may be achieved with less capital by imposing such capital requirements only on systemically important institutions and those exposed to them. In chapter 4, we apply this methodology to the study of the Brazilian financial system. Using the Contagion Index, we study the potential for default contagion and systemic risk in the Brazilian system and analyze the contribution of balance sheet size and network structure to systemic risk. Our study reveals that, aside from balance sheet size, the network-based local measures of connectivity and concentration of exposures across counterparties introduced in chapter 3, the counterparty susceptibility and local network frailty, contribute significantly to the systemic importance of an institution in the Brazilian network. Thus, imposing an upper bound on these variables could help reducing contagion. We examine the impact of various capital requirements on the extent of contagion in the Brazilian financial system, and show that targeted capital requirements achieve the same reduction in systemic risk with lower requirements in capital for financial institutions. The methodology we proposed in chapter 3 for estimating contagion and systemic risk requires visibility on the entire network structure. Reconstructing bilateral exposures from balance sheets data is then a question of interest in a financial system where bilateral exposures are not disclosed. We propose in chapter 5 two methods to derive a distribution of bilateral exposures matrices. The first method attempts to recover the balance sheet assets and liabilities “sample by sample”. Each sample of the bilateral exposures matrix is solution of a relative entropy minimization problem subject to the balance sheet constraints. However, a solution to this problem does not always exist when dealing with sparse sample matrices. Thus, we propose a second method that attempts to recover the assets and liabilities “in the mean”. This approach is the analogue of the Weighted Monte Carlo method introduced by Avellaneda et al. 2001). We first simulate independent samples of the bilateral exposures matrix from a relevant prior distribution on the network structure, then we compute posterior probabilities by maximizing the entropy under the constraints that the balance sheet assets and liabilities are recovered in the mean. We discuss the pros and cons of each approach and explain how it could be used to detect systemically important institutions in the financial system. The recent crisis has also raised many questions regarding the meaning of structured finance credit ratings issued by rating agencies and the methodology behind them. Chapter 6 aims at clarifying some misconceptions related to structured finance ratings and how they are commonly interpreted: we discuss the comparability of structured finance ratings with bond ratings, the interaction between the rating procedure and the tranching procedure and its consequences for the stability of structured finance ratings in time. These insights are illustrated in a factor model by simulating rating transitions for CDO tranches using a nested Monte Carlo method. In particular, we show that the downgrade risk of a CDO tranche can be quite different from a bond with same initial rating. Structured finance ratings follow path-dependent dynamics that cannot be adequately described, as usually done, by a matrix of transition probabilities. Therefore, a simple labeling via default probability or expected loss does not discriminate sufficiently their downgrade risk. We propose to supplement ratings with indicators of downgrade risk. To overcome some of the drawbacks of existing rating methods, we suggest a risk-based rating procedure for structured products. Finally, we formulate a series of recommendations regarding the use of credit ratings for CDOs and other structured credit instruments. Keywords: bilateral exposures, collateralized debt obligation, contagion, copula, credit derivatives, credit rating, default clustering, default risk, domino effects, macro-prudential regulation, random graph, relative entropy, scale-free, small-world, systemic risk, structured finance, transition probabilities.

In this thesis work, we analyze fit of bivariate BEG and BTLG models to financial asset returns’ episodes of growth and decline. Our data include foreign exchange rates, stock, and stock’s indexes prices, and commoditites. We apply BEG and BTLG models to all data and decide if the models fit reasonably well based on univariate and bivariate fit methods. We also assess “stability” of the returns with respect to their geometric summation. Our results show BEG and BTLG models fitting best the foreign exchange rates.

The traditional Markowitz mean-variance portfolio optimization theory uses volatility as the sole measure of risk. However, volatility is flawed both intuitively and theoretically: being symmetric it does not differentiate between gains and losses； it does not satisfy an expected utility maximization rationale except under unrealistic assumptions and is not a coherent risk measure. The past decade has seen considerable research on better risk measures, with the two tail risk measures Value-at-Risk (VaR) and Expected Tail Loss (ETL) being the main contenders, as well as research on modeling skewness and fat-tails that are prevalent in financial return distributions. There are two main approaches to the latter problem: (a) constructing modified VaR (MVaR) and modified ETL (METL) using Cornish-Fisher asymptotic expansions to provide non- parametric skewness and kurtosis corrections, and (b) fitting a skewed and fat-tailed multivariate parametric distribution to portfolio returns and optimizing the portfolio using ETL based on Monte Carlo simulations from the fitted distribution. It is an open question how MVaR and METL compare with one another and with empirical VaR and ETL, and also how much improvement can be obtained in fitting parametric distributions. In this dissertation, we first show that MVaR and METL are very sensitive to outliers, sometimes rendering complete failure of a portfolio. Then we propose new robust skewness and kurtosis estimates, study their statistical behavior and that of the resulting robust MVaR and METL through the use of influence functions, and show through extensive empirical studies that robust MVaR and METL can effectively curb the failure of the original estimates. We use the same experimental approach to show that the simple empirical ETL optimization yields portfolio performance essentially equivalent to that of the much more complex method of fitting multivariate fat-tailed skewed distributions. Finally we address the following important problem: VaR and ETL based portfolio optimization do not have expected utility maximization rationales. Thus we establish a method of designing coherent spectral risk measures based on non-satiated risk-averse utility functions. We show that the resulting risk measures satisfy second order stochastic dominance and their empirical portfolio performances are slightly improved over ETL.

Many violations of the efficient market hypothesis, such as bubbles, crashes, and “fat tails” in the distribution of returns, are difficult to address using a representative agent framework because in such a setting the departures from equilibrium occur only through some external perturbation. An alternative approach, sometimes referred to as the “complex systems” view, emphasizes the importance of interactions between agents. Even if each individual agent’s optimization problem is known, outcomes of their interactions are probabilistic, implying that markets can evolve “spontaneously” towards an unstable state. Particularly, in a situation where traders may have private information related to the payoff of a financial assets their individual actions may trigger a cascade of similar actions by other traders. While the mechanism of a chain reaction through information revelation can potentially explain a number of stylized facts in finance, such behavior remains notoriously difficult to identify empirically. This is partly because many theoretical underpinnings of herding, such as informational asymmetry, are unobservable and partly because the complex agent-based models of herding do not yield closed-form solutions to be used for direct econometric tests. In addition, such models have been criticized for their lack of economic microfoundations. The following chapters represent a step towards filling both of these gaps. First, I identify evidence of herding behavior by institutional investment managers during the collapse of the recent real estate bubble using an established empirical approach. Then, agent based “stochastic herding” model is introduced and tested with an alternative technique of “detection by distribution”. Subsequently this framework is extended to better understand the mechanisms driving extreme volatility in the dollar-yen foreign exchange market to show that traders’ tendency to herd around information about the possibility of high yield currency crashes can result in self-fulfilling prophecy without a major exogenous shock. The parameter measuring the “thickness” of the tail of the probability distribution of jumps in foreign exchange rates is proportional to the herding intensity by currency speculators. I employ Bayesian econometrics to test the theoretically predicted relationships between this “tail risk” parameter and a number of economic variables related to carry trade activity. The final chapter focuses explicitly on the types of macroeconomic information that traders use to price such extreme events in foreign exchange markets. Since “stochastic herding” provides a plausible data generating mechanism for “rare event,” the empirical units of observation utilized in this work have been carefully selected to match this description. Thus, in looking at domestic stock market we focus on institutional investment managers that liquidate their entire positions, not the incremental adjustments, while the examination of foreign exchange markets abstracts from Gaussian volatility and focuses on rare realized volatility jumps and deep out-of-money options used to price such events.

The topic motivating this dissertation is functionally generated portfolios and their capacity to deliver relative arbitrage, an aspect of stochastic portfolio theory SPT). The aim is to relax some of the common assumptions of SPT and explore the performance of functionally generated portfolios in this more general setting, with an eye towards arbitrage. In particular, the assumption of a constant number of companies in the market model is relaxed, as well as the assumption that all changes in capitalizations are passed on as returns to investors through the stochastic integral. On the way to these goals, the notions of a piecewise semimartingale taking values in &cup；infinityn＝1Rn and piecewise stochastic integration are developed as useful mathematical tools. Many properties of the stochastic integral with respect to an Rn -valued semimartingale are shown to extend to this setting. For example, the following fundamental theorems of asset pricing carry over: “No free lunch with vanishing risk” is equivalent to the existence of an equivalent sigma-martingale measure and “No arbitrage of the first kind” is equivalent to the existence of an equivalent local martingale deflator for the integrator. An important idea of SPT is the notion of diversity of equity markets, meaning that relative capital does not become arbitrarily concentrated in a single company. In [38] Robert Fernholz observed an inconsistency between the normative assumption of existence of an equivalent local martingale measure ELMM) for the price process and the empirical reality of diversity in equity markets. Here an alternative model is examined, in which diversity is maintained not through smaller returns of the largest company, but via a type of antitrust regulation that is compatible with ELMMs. The regulatory procedure breaks up companies that become too large, while assuming that total capital in the market is conserved. In this case and in several other examples where the price process is a piecewise Ito process, straightforward functionally generated relative arbitrage is found to be less readily available than in n-dimensional Ito process models.

The huge amount of tick-by-tick data provides rich and timely information regarding fluctuations of traded assets and their co-movements. Nevertheless, the existence of market microstructure noise interferes with estimation, especially when the sampling frequency increases to beyond every five minutes. The first chapter introduces the Quasi-Maximum Likelihood Estimator of volatility as a solution to this problem. In theory, the proposed approach is consistent, rate-efficient, and shares the model-free feature with non-parametric alternatives. In practice, it is also convenient and has better small sample performance without any tuning parameters. When measuring covariance and correlation, the fact that the two assets may not trade or otherwise be observed at exactly the same time, known as observation asynchronycity, is another issue that may distort the estimates. The second chapter, jointly written with Yacine Ait-Sahalia and Jianqing Fan, extends the Quasi-Maximum Likelihood Estimator to explore asynchronous and noisy data, with the help of generalized synchronization scheme. A fundamental problem in option pricing is to find explicit pricing formulae or efficient pricing algorithms. However, closed-form pricing formulae are very sparse, whereas numerical or simulation-based methods are computationally expensive and deliver no insight into the structure of the option price. The third chapter fills in the gap between closed-form solutions and numerical methods through expansions of option prices. This approach works with general dynamics without any requirement on affine dynamics or explicit characteristic functions, and quantitatively characterizes the relative importance of model parameters as the option approaches expiration. With closed-form expansions, we translate model features into option prices, such as stochastic interest rate, mean-reverting drift, and self-exciting or skewed jumps. Numerical examples illustrate the accuracy of this approach.

Statisticians often encounter data in the form of a combination of discrete and continuous outcomes. A special case is zero-inflated longitudinal data where the response variable has a large portion of zeros. These data exhibit correlation because observations are obtained on the same subjects over time. In this dissertation, we propose a two-part mixed distribution model to model zero-inflated longitudinal data. The first part of the model is a logistic regression model that models the probability of nonzero response； the other part is a linear model that models the mean response given that the outcomes are not zeros. Random effects with AR1) covariance structure are introduced into both parts of the model to allow serial correlation and subject specific effect. Estimating the two-part model is challenging because of high dimensional integration necessary to obtain the maximum likelihood estimates. We propose a Monte Carlo EM algorithm for estimating the maximum likelihood estimates of parameters. Through simulation study, we demonstrate the good performance of the MCEM method in parameter and standard error estimation. To illustrate, we apply the two-part model with correlated random effects and the model with autoregressive random effects to executive compensation data to investigate potential determinants of CEO stock option grants.