Statistics and Actuarial Science
http://hdl.handle.net/10012/9934
2019-08-17T20:39:48ZNumerical Solutions to Stochastic Control Problems: When Monte Carlo Simulation Meets Nonparametric Regression
http://hdl.handle.net/10012/14831
Numerical Solutions to Stochastic Control Problems: When Monte Carlo Simulation Meets Nonparametric Regression
Shen, Zhiyi
The theme of this thesis is to develop theoretically sound as well as numerically efficient Least Squares Monte Carlo (LSMC) methods for solving discrete-time stochastic control problems motivated by insurance and finance problems.
Despite its popularity in solving optimal stopping problems, the application of the LSMC method to stochastic control problems is hampered by several challenges. Firstly, the simulation of the state process is intricate in the absence of the optimal control policy in prior. Secondly, numerical methods only warrant the approximation accuracy of the value function over a bounded domain, which is incompatible with the unbounded set the state variable dwells in. Thirdly, given a considerable number of simulated paths, regression methods are computationally challenging. This thesis responds to the above problems.
Chapter 2 develops a novel LSMC algorithm to solve discrete-time stochastic optimal control problems, referred to as the Backward Simulation and Backward Updating (BSBU) algorithm. The BSBU algorithm has three pillars: a construction of auxiliary stochastic control model, an artificial simulation of the post-action value of state process, and a shape-preserving sieve estimation method which equip the algorithm with a number of merits including obviating forward simulation and control randomization, evading extrapolating the value function, and alleviating computational burden of the tuning parameter selection.
Chapter 3 proposes an alternative LSMC algorithm which directly approximates the optimal value function at each time step instead of the continuation function. This brings the benefits of faster convergence rate and closed-form expressions of the value function compared with the previously developed BSBU algorithm. We also develop a general argument for constructing an auxiliary stochastic control problem which inherits the continuity, monotonicity, and concavity of the original problem. This argument renders the LSMC algorithm circumvent extrapolating the value function in the backward recursion and can well adapt to other numerical methods.
Chapter 4 studies a complicated stochastic control problem: the no-arbitrage pricing of the “Polaris Choice IV" variable annuities issued by the American International Group. The Polaris allows the income base to lock in the high-water-mark of the
investment account over a certain monitoring period which is related to the timing of the policyholder’s first withdrawal. By prudently introducing certain auxiliary state and control variables, we formulate the pricing problem into a Markovian stochastic optimal control framework. With a slight modification on the fee structure, we prove the existence of a bang-bang solution to the stochastic control problem: the policyholder's optimal withdrawal strategy is limited to a few choices. Accordingly, the price of the modified contract can be solved by the BSBU algorithm. Finally, we prove that the price of the modified contract is an upper bound for that of the Polaris with the real fee structure. Numerical experiments show that this bound is fairly tight.
2019-07-30T00:00:00ZOn some topics in Levy insurance risk models
http://hdl.handle.net/10012/14805
On some topics in Levy insurance risk models
Wong, Jeff
Risk management has long been the central focus within actuarial science. There are various risks a typical actuarial company would look into, solvency risk being one of them. This falls under the scope of surplus analysis. Studying of an insurer's ability to maintain an adequate surplus level in order to fulfill its future obligation would be the subject matter, which requires modeling of the underlying surplus process together with de fining appropriate matrices to quantity the risk. Ultimately, it aims at accurately reflecting the solvency status to a line of business, which requires developing realistic models to predict the evolution of the underlying surplus and constructing various ruin quantities depending on the regulations or the risk appetite set internally by the company.
While there have been a vast amount of literature devoted to answering these questions in the past decades, a considerable amount of effort is devoted by different scholars in recent years to construct more accurate models to work with, and to develop a spectrum of risk quantities to serve different purposes. In the meantime, more advanced tools are also developed to assist with the analysis involved. With the same spirit, this thesis aims at making contributions in these areas.
In Chapter 3, a Parisian ruin time is analyzed under a spectrally negative L evy model. A hybrid observation scheme is investigated, which allows a more frequent monitoring when the solvency status to a business is observed to be critical. From a practical perspective, such observation scheme provides an extra degree of realism. From a theoretical perspective,
it uni es analysis to paths having either bounded or unbounded variations, a core obstacle for analysis under the context of spectrally negative L evy model. Laplace transform to the concerned ruin time is obtained. Existing results in the literature are also retrieved to demonstrate consistency by taking appropriate limits.
In Chapter 4, the toolbox of discrete Poissonian observation is further complemented under a spectrally negative L evy context. By extending the classical definition of potential measures, which summarizes the law of ruin time and de cit at ruin under continuous observation, to its discrete counterpart, expressions to the Poissonian potential measures are derived. An interesting dual relation is also discovered between a Poissonian potential measure and the corresponding exit measure. This further strengthens the motivation for studying the Poissonian potential measures. To further demonstrate its usefulness, several problems are formulated and analyzed at the end of this chapter.
In Chapter 5, motivated from regulatory practices, a more conservative risk matrix is constructed by altering the traditional definition to a Parisian ruin time. As a starting point, analysis is performed using a Cram er-Lundberg model, a special case of spectrally negative L evy model. The law of ruin time and its de cit at ruin is obtained. An interesting ordering property is also argued to justify why it is a more conservative risk measure to work with.
To ensure that the thesis flows smoothly, Chapter 1 and 2 are devoted to the background reading. Literature reviews and existing tools necessary for subsequent derivations are provided at the beginning of each chapters to ensure self-containment. A summary and concluding remarks can be found in Chapter 6.
2019-07-16T00:00:00ZA Statistical Response to Challenges in Vast Portfolio Selection
http://hdl.handle.net/10012/14792
A Statistical Response to Challenges in Vast Portfolio Selection
Guo, Danqiao
The thesis is written in response to emerging issues brought about by an increasing number of assets allocated in a portfolio and seeks answers to puzzling empirical findings in the portfolio management area. Over the years, researchers and practitioners working in the portfolio optimization area have been concerned with estimation errors in the first two moments of asset returns. The thesis comprises several related chapters on our statistical inquiry into this subject. Chapter 1 of the thesis contains an introduction to what will be reported in the remaining chapters.
A few well-known covariance matrix estimation methods in the literature involve adjustment of sample eigenvalues. Chapter 2 of the thesis examines the effects of sample eigenvalue adjustment on the out-of-sample performance of a portfolio constructed from the sample covariance matrix. We identify a few sample eigenvalue adjustment patterns that lead to a definite improvement in the out-of-sample portfolio Sharpe ratio when the true covariance matrix admits a high-dimensional factor model.
Chapter 3 shows that even when the covariance matrix is poorly estimated, it is still possible to obtain a robust maximum Sharpe ratio (MSR) portfolio by exploiting the uneven distribution of estimation errors across principal components. This is accomplished by approximating the vector of expected future asset returns using a few relatively accurate sample principal components. We discuss two approximation methods. The first method leads to a subtle connection to existing approaches in the literature, while the second one named the ``spectral selection method" is novel and able to address main shortcomings of existing methods in the literature.
A few academic studies report an unsatisfactory performance of the optimized portfolios relative to that of the 1/N portfolio. Chapter 4 of the thesis reports an in-depth investigation into the reasons behind the reported superior performance of the 1/N portfolio. It is supported by both theoretical and empirical evidence that the success of the 1/N portfolio is by no means due to the failure of the portfolio optimization theory. Instead, a major reason behind the superiority of the 1/N portfolio is its adjacency to the mean-variance optimal portfolio.
Chapter 5 examines the performance of randomized 1/N stock portfolios over time. During the last four decades these portfolios outperformed the market. The construction of these portfolios implies that their constituent stocks are in general older than those in the market as a whole. We show that the differential performance can be explained by the relation between stock returns and firm age. We document a significant relation between age and returns in the US stock market. Since 1977 stock returns have been an increasing function of age apart from the oldest ages. For this period the age effect completely dominates the size effect.
2019-07-04T00:00:00ZRisk Management with Non-Convex and Non-Monotone Preferences
http://hdl.handle.net/10012/14775
Risk Management with Non-Convex and Non-Monotone Preferences
Wei, Yunran
This thesis studies two types of problems, the theory of risk functionals and the risk sharing problem. We put a special focus on a class of non-monotone law-invariant risk functionals, called the signed Choquet integrals.
The contribution can be seen as three portions.
The first portion of this thesis contains various results on signed Choquet integrals. A functional characterization via comonotonic additivity is established, along with some theoretical properties including six equivalent conditions for a signed Choquet integral to be convex. We proceed to address two practical issues currently popular in risk management, namely, robustness (continuity) issues and risk aggregation with dependence uncertainty, for signed Choquet integrals. Our results generalize in several directions those in the literature of risk functionals. From the results obtained in this chapter, we see that many existing elegant mathematical results in the theory of risk measures hold for the general class of signed Choquet integrals; thus they do not rely on the assumption of monotonicity.
In the second portion, we analyze the “convex level sets” (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability, and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the context of multi-dimensional risk functionals. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one-dimensional signed Choquet integrals, and that of all two-dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that a comonotonic-additive coherent risk measure is co-elicitable with a Value-at- Risk if and only if it is a convex combination of the mean and the corresponding Expected Shortfall. The new findings generalize several results in the recent literature and partially answer an open question on the characterization of multi-dimensional elicitability.
In the third portion, we study a risk sharing problem. Unlike classic risk sharing problems based on expected utilities or convex risk measures, quantile-based risk sharing problems exhibit two special features. First, quantile-based risk measures (such as the Value-at-Risk) are often not convex, and second, they ignore some part of the distribution of the risk. These features create technical challenges in establishing a full characterization of optimal allocations, a question left unanswered in the literature. In this paper, we address the issues on the existence and the characterization of (Pareto-)optimal allocations in quantile-based risk sharing problems. It turns out that negative dependence, mutual exclusivity in particular, plays an important role in the optimal allocations, in contrast to positive dependence appearing in classic risk sharing problems. As a by-product of our main finding, we obtain some results on the optimization of the Value-at-Risk and the Expected Shortfall.
2019-06-24T00:00:00Z