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Monte Carlo simulation with applications to finance
\"Preface This book can serve as the text for a one-semester course on Monte Carlo simulation. The intended audience is advanced undergraduate students or students on master's programs who wish to learn the basics of this exciting topic and its applications to finance. The book is largely self-contained. The only prerequisite is some experience with probability and statistics. Prior knowledge on option pricing is helpful but not essential. As in any study of Monte Carlo simulation, coding is an integral part and cannot be ignored. The book contains a large number of MATLAB coding exercises. They are designed in a progressive manner so that no prior experience with MATLAB is required. Much of the mathematics in the book is informal. For example, randomvariables are simply defined to be functions on the sample space, even though they should be measurable with respect to appropriate algebras; exchanging the order of integrations is carried out liberally, even though it should be justified by the Tonelli-Fubini Theorem. The motivation for doing so is to avoid the technical measure theoretic jargon, which is of little concern in practice and does not help much to further the understanding of the topic. The book is an extension of the lecture notes that I have developed for an undergraduate course on Monte Carlo simulation at Brown University. I would like to thank the students who have taken the course, as well as the Division of Applied Mathematics at Brown, for their support. Hui Wang Providence, Rhode Island January, 2012\"-- Provided by publisher.
Book
Riemann manifold Langevin and Hamiltonian Monte Carlo methods
The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis-Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from http://www.ucl.ac.uk/statistics/research/rmhmc allows replication of all the results reported.
Journal Article
Particle Markov chain Monte Carlo methods
Markov chain Monte Carlo and sequential Monte Carlo methods have emerged as the two main tools to sample from high dimensional probability distributions. Although asymptotic convergence of Markov chain Monte Carlo algorithms is ensured under weak assumptions, the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and/or if highly correlated variables are updated independently. We show here how it is possible to build efficient high dimensional proposal distributions by using sequential Monte Carlo methods. This allows us not only to improve over standard Markov chain Monte Carlo schemes but also to make Bayesian inference feasible for a large class of statistical models where this was not previously so. We demonstrate these algorithms on a non-linear state space model and a Lévy-driven stochastic volatility model.
Journal Article
Sequential Monte Carlo samplers
We propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space, each distribution being known up to a normalizing constant. These probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential Monte Carlo methods. This methodology allows us to derive simple algorithms to make parallel Markov chain Monte Carlo algorithms interact to perform global optimization and sequential Bayesian estimation and to compute ratios of normalizing constants. We illustrate these algorithms for various integration tasks arising in the context of Bayesian inference.
Journal Article
Hamiltonian Monte Carlo sampling in Bayesian empirical likelihood computation
We consider Bayesian empirical likelihood estimation and develop an efficient Hamiltonian Monte Carlo method for sampling from the posterior distribution of the parameters of interest. The method proposed uses hitherto unknown properties of the gradient of the underlying log-empirical-likelihood function. We use results from convex analysis to show that these properties hold under minimal assumptions on the parameter space, prior density and the functions used in the estimating equations determining the empirical likelihood. Our method employs a finite number of estimating equations and observations but produces valid semiparametric inference for a large class of statistical models including mixed effects models, generalized linear models and hierarchical Bayes models. We overcome major challenges posed by complex, non-convex boundaries of the support routinely observed for empirical likelihood which prevent efficient implementation of traditional Markov chain Monte Carlo methods like randomwalk Metropolis—Hastings sampling etc. with or without parallel tempering. A simulation study confirms that our method converges quickly and draws samples from the posterior support efficiently. We further illustrate its utility through an analysis of a discrete data set in small area estimation.
Journal Article
Intraclass correlation – A discussion and demonstration of basic features
A re-analysis of intraclass correlation (ICC) theory is presented together with Monte Carlo simulations of ICC probability distributions. A partly revised and simplified theory of the single-score ICC is obtained, together with an alternative and simple recipe for its use in reliability studies. Our main, practical conclusion is that in the analysis of a reliability study it is neither necessary nor convenient to start from an initial choice of a specified statistical model. Rather, one may impartially use all three single-score ICC formulas. A near equality of the three ICC values indicates the absence of bias (systematic error), in which case the classical (one-way random) ICC may be used. A consistency ICC larger than absolute agreement ICC indicates the presence of non-negligible bias; if so, classical ICC is invalid and misleading. An F-test may be used to confirm whether biases are present. From the resulting model (without or with bias) variances and confidence intervals may then be calculated. In presence of bias, both absolute agreement ICC and consistency ICC should be reported, since they give different and complementary information about the reliability of the method. A clinical example with data from the literature is given.
Journal Article
A Molecular Dynamics Simulation for Thermal Activation Process in Covalent Bond Dissociation of a Crosslinked Thermosetting Polymer
A novel algorithm for covalent bond dissociation is developed to accurately predict fracture behavior of thermosetting polymers via molecular dynamics simulation. This algorithm is based on the Monte Carlo method that considers the difference in local strain and bond-dissociation energies to reproduce a thermally activated process in a covalent bond dissociation. This study demonstrates the effectiveness of this algorithm in predicting the stress–strain relationship of fully crosslinked thermosetting polymers under uniaxial tensile conditions. Our results indicate that the bond-dissociation energy plays an important role in reproducing the brittle fracture behavior of a thermosetting polymer by affecting the number of covalent bonds that are dissociated simultaneously.
Journal Article
Handbook of Markov Chain Monte Carlo
Handbook of Markov Chain Monte Carlo brings together the major advances that have occurred in recent years while incorporating enough introductory material for new users of MCMC. Along with thorough coverage of the theoretical foundations and algorithmic and computational methodology, this comprehensive handbook includes substantial realistic case studies from a variety of disciplines. These case studies demonstrate the application of MCMC methods and serve as a series of templates for the construction, implementation, and choice of MCMC methodology.
eBook
Publication bias examined in meta-analyses from psychology and medicine: A meta-meta-analysis
Publication bias is a substantial problem for the credibility of research in general and of meta-analyses in particular, as it yields overestimated effects and may suggest the existence of non-existing effects. Although there is consensus that publication bias exists, how strongly it affects different scientific literatures is currently less well-known. We examined evidence of publication bias in a large-scale data set of primary studies that were included in 83 meta-analyses published in Psychological Bulletin (representing meta-analyses from psychology) and 499 systematic reviews from the Cochrane Database of Systematic Reviews (CDSR; representing meta-analyses from medicine). Publication bias was assessed on all homogeneous subsets (3.8% of all subsets of meta-analyses published in Psychological Bulletin) of primary studies included in meta-analyses, because publication bias methods do not have good statistical properties if the true effect size is heterogeneous. Publication bias tests did not reveal evidence for bias in the homogeneous subsets. Overestimation was minimal but statistically significant, providing evidence of publication bias that appeared to be similar in both fields. However, a Monte-Carlo simulation study revealed that the creation of homogeneous subsets resulted in challenging conditions for publication bias methods since the number of effect sizes in a subset was rather small (median number of effect sizes equaled 6). Our findings are in line with, in its most extreme case, publication bias ranging from no bias until only 5% statistically nonsignificant effect sizes being published. These and other findings, in combination with the small percentages of statistically significant primary effect sizes (28.9% and 18.9% for subsets published in Psychological Bulletin and CDSR), led to the conclusion that evidence for publication bias in the studied homogeneous subsets is weak, but suggestive of mild publication bias in both psychology and medicine.
Journal Article
Statistically Controlling for Confounding Constructs Is Harder than You Think
Social scientists often seek to demonstrate that a construct has incremental validity over and above other related constructs. However, these claims are typically supported by measurement-level models that fail to consider the effects of measurement (un)reliability. We use intuitive examples, Monte Carlo simulations, and a novel analytical framework to demonstrate that common strategies for establishing incremental construct validity using multiple regression analysis exhibit extremely high Type I error rates under parameter regimes common in many psychological domains. Counterintuitively, we find that error rates are highest--in some cases approaching 100%--when sample sizes are large and reliability is moderate. Our findings suggest that a potentially large proportion of incremental validity claims made in the literature are spurious. We present a web application (http://jakewestfall.org/ivy/) that readers can use to explore the statistical properties of these and other incremental validity arguments. We conclude by reviewing SEM-based statistical approaches that appropriately control the Type I error rate when attempting to establish incremental validity.
Journal Article