The main objective of this thesis is to develop novel Monte Carlo techniques with emphasis on various applications in finance and economics, particularly in the fields of risk management and asset returns modeling. New stochastic algorithms are developed for rare-event probability estimation, combinatorial optimization, parameter estimation and model selection. The contributions of this thesis are fourfold. Firstly, we study an NP-hard combinatorial optimization problem, the Winner Determination Problem (WDP) in combinatorial auctions, where buyers can bid on bundles of items rather than bidding on them sequentially. We present two randomized algorithms, namely, the cross-entropy (CE) method and the ADAptive Mulitilevel splitting (ADAM) algorithm, to solve two versions of the WDP. Although an efficient deterministic algorithm has been developed for one version of the WDP, it is not applicable for the other version considered. In addition, the proposed algorithms are straightforward and easy to program, and do not require specialized software. Secondly, two major applications of conditional Monte Carlo for estimating rare-event probabilities are presented: a complex bridge network reliability model and several generalizations of the widely popular normal copula model used in managing portfolio credit risk. We show how certain efficient conditional Monte Carlo estimators developed for simple settings can be extended to handle complex models involving hundreds or thousands of random variables. In particular, by utilizing an asymptotic description on how the rare event occurs, we derive algorithms that are not only easy to implement, but also compare favorably to existing estimators. Thirdly, we make a contribution at the methodological front by proposing an improvement of the standard CE method for estimation. The improved method is relevant, as recent research has shown that in some high-dimensional settings the likelihood ratio degeneracy problem becomes severe and the importance sampling estimator obtained from the CE algorithm becomes unreliable. In contrast, the performance of the improved variant does not deteriorate as the dimension of the problem increases. Its utility is demonstrated via a high-dimensional estimation problem in risk management, namely, a recently proposed t-copula model for credit risk. We show that even in this high-dimensional model that involves hundreds of random variables, the proposed method performs remarkably well, and compares favorably to existing importance sampling estimators. Furthermore, the improved CE algorithm is then applied to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. We present two empirical examples to demonstrate the proposed approach. The first example involves women's labor market participation and we compare three different binary response models in order to find the one best fits the data. The second example utilizes two vector autoregressive (VAR) models to analyze the interdependence and structural stability of four U.S. macroeconomic time series: GDP growth, unemployment rate, interest rate, and inflation. Lastly, we contribute to the growing literature of asset returns modeling by proposing several novel models that explicitly take into account various recent findings in the empirical finance literature. Specifically, two classes of stylized facts are particularly important. The first set is concerned with the marginal distributions of asset returns. One prominent feature of asset returns is that the tails of their distributions are heavier than those of the normal---large returns (in absolute value) occur much more frequently than one might expect from a normally distributed random variable. Another robust empirical feature of asset returns is skewness, where the tails of the distributions are not symmetric---losses are observed more frequently than large gains. The second set of stylized facts is concerned with the dependence structure among asset returns. Recent empirical studies have cast doubts on the adequacy of the linear dependence structure implied by the multivariate normal specification. For example, data from various asset markets, including equities, currencies and commodities markets, indicate the presence of extreme co-movement in asset returns, and this observation is again incompatible with the usual assumption that asset returns are jointly normally distributed. In light of the aforementioned empirical findings, we consider various novel models that generalize the usual normal specification. We develop efficient Markov chain Monte Carlo (MCMC) algorithms to estimate the proposed models. Moreover, since the number of plausible models is large, we perform a formal Bayesian model comparison to determine the model that best fits the data. In this way, we can directly compare the two approaches of modeling asset returns: copula models and the joint modeling of returns.

Consider distinct latin squares, L and M, of order n. Then the pair (T1, T2) where T1 = L \M and T2 = M \ L is called a latin bitrade. Furthermore T1 and T2 are referred to as latin trades, in which T2 is said to be a disjoint mate of T1 (and vice versa). Drápal (1991) showed that, under certain conditions, a partition of an equilateral triangle of side length n, where n is some integer, into smaller, integer length sided equilateral triangles gives rise to a latin trade within the latin square based on the addition table for the integers modulo n. A partial latin square P of order n is said to be completable if there exists a latin square L of order n such that P ⊆ L. If there is only one such possible latin square, L, of order n then P is said to be uniquely completable and P is called a defining set of L. Furthermore, if C is a uniquely completable partial latin square such that no proper subset of C is uniquely completable, C is said to be a critical set or a minimal defining set. These concepts, namely latin trades and defining sets in latin squares, are intimately connected by the following observation. If L is a latin square and D ⊆ L is a defining set, then D intersects all latin bitrades for which one mate is contained in L. In Part I of this thesis Dr´apal’s result is applied to investigate the structure of certain defining sets in latin squares. The results that are obtained are interesting in themselves; furthermore, the geometric approach to the problem yields additional appealing results. These geometric results are discussed in Chapters 3, 4, 5 and 6. They pertain to partitioning regions (polygons in R2 that satisfy certain obvious conditions) into equilateral, integer length sided, triangles, such that no point, in the region, is a corner of more than three distinct triangles. In Chapter 2 one of the main two theorems on defining sets is established, as is a method for using the above geometric results to prove the nonexistence of certain types of defining sets. In Part II of this thesis, intersection problems, for latin squares and Steiner triple systems, are considered. The seminal works, for problems of these types, are due to Lindner and Rosa (1975) and Fu (1980). A natural progression, from the established literature, for intersection problems between elements in a pair of latin squares or Steiner triple systems is to problems in which the intersection is composed of a number of disjoint configurations (isomorphic copies of some specified partial triple system). In this thesis solutions to two intersection problems for disjoint configurations are detailed. An m-flower, (F,F), is a partial triple system/configuration, such that: F = {{x, yi, zi} | {yi, zi} ∩ {yj , zj} = ∅, for 0 ≤ i, j ≤ m − 1, i 6= j}; and F = UX∈FX. The first such problem considered in this thesis asks for necessary and sufficient conditions for integers k and m ≥ 2 such that a pair of latin squares of order n exists that intersect precisely in k disjoint m-flowers. The necessary terminology, constructions, lemmas and proof for this result are contained in Chapters 7, 8 and 9. The second such problem considered in this thesis asks for necessary and sufficient conditions for integers k such that a pair of Steiner triple systems of order u exists that intersect precisely in k disjoint 2-flowers. This result relies on the solution to the latin square problem and an additional result from Chapter 9. The further constructions and lemmas used to prove this result are detailed in Chapter 10.