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We investigate the computational complexity of estimating quantile based risk measures, such as the widespread Value at Risk, via nested Monte Carlo simulations. The estimator is a conditional expectation type estimator where two-stage simulations are required to evaluate the risk measure: an outer simulation is used to generate risk-factor scenarios that govern prices and an inner simulation is used to evaluate the future portfolio value based on each scenario. We propose a new set of non-uniform algorithms to evaluate risk. The algorithms place more importance upon outer scenarios which are more likely to have a direct impact on the estimator and considers the marginal changes in the risk estimator at each additional inner scenario. We demonstrate using experimental settings that our proposed algorithms outperform the uniform algorithm and results in a lower variance and bias with the same initial settings and resources. The results are also robust enough for the multidimensionality of risk factors and the non-linearity of pay-offs.
This is an advanced guide to optimal stopping and control, focusing on advanced Monte Carlo simulation and its application to finance. Written for quantitative finance practitioners and researchers in academia, the book looks at the classical simulation based algorithms before introducing some of the new, cutting edge approaches under development.
In this thesis, we analyze the computational problem of estimating financial risk in nested Monte Carlo simulation. An outer simulation is used to generate financial scenarios, and an inner simulation is used to estimate future portfolio values in each scenario. Mean squared error (MSE) for standard nested simulation converges at the rate $k^{-2/3}$, where $k$ is the computational budget. In the first part of this thesis, we focus on one risk measure, the probability of a large loss, and we propose a new algorithm to estimate this risk. Our algorithm sequentially allocates computational effort in the inner simulation based on marginal changes in the risk estimator in each scenario. Theoretical results are given to show that the risk estimator has an asymptotic MSE of order $k^{-4/5+\epsilon}$, for all positive $\epsilon$, that is faster compared to the conventional uniform inner sampling approach. Numerical results consistent with the theory are presented. In the second part of this thesis, we introduce a regression-based nested Monte Carlo simulation method for risk estimation.
Focusing on what actuaries need in practice, this introductory account provides readers with essential tools for handling complex problems and explains how simulation models can be created, used and re-used (with modifications) in related situations. The book begins by outlining the basic tools of modelling and simulation, including a discussion of the Monte Carlo method and its use. Part II deals with general insurance and Part III with life insurance and financial risk. Algorithms that can be implemented on any programming platform are spread throughout and a program library written in R is included. Numerous figures and experiments with R-code illustrate the text. The author's non-technical approach is ideal for graduate students, the only prerequisites being introductory courses in calculus and linear algebra, probability and statistics. The book will also be of value to actuaries and other analysts in the industry looking to update their skills.
This book describes the new generation of discrete choice methods, focusing on the many advances that are made possible by simulation. Researchers use these statistical methods to examine the choices that consumers, households, firms, and other agents make. Each of the major models is covered: logit, generalized extreme value, or GEV (including nested and cross-nested logits), probit, and mixed logit, plus a variety of specifications that build on these basics. Simulation-assisted estimation procedures are investigated and compared, including maximum stimulated likelihood, method of simulated moments, and method of simulated scores. Procedures for drawing from densities are described, including variance reduction techniques such as anithetics and Halton draws. Recent advances in Bayesian procedures are explored, including the use of the Metropolis-Hastings algorithm and its variant Gibbs sampling. The second edition adds chapters on endogeneity and expectation-maximization (EM) algorithms. No other book incorporates all these fields, which have arisen in the past 25 years. The procedures are applicable in many fields, including energy, transportation, environmental studies, health, labor, and marketing.
Financial Risk Modelling and Portfolio Optimization with R, 2nd Edition Bernhard Pfaff, Invesco Global Asset Allocation, Germany A must have text for risk modelling and portfolio optimization using R. This book introduces the latest techniques advocated for measuring financial market risk and portfolio optimization, and provides a plethora of R code examples that enable the reader to replicate the results featured throughout the book. This edition has been extensively revised to include new topics on risk surfaces and probabilistic utility optimization as well as an extended introduction to R language. Financial Risk Modelling and Portfolio Optimization with R: Demonstrates techniques in modelling financial risks and applying portfolio optimization techniques as well as recent advances in the field. Introduces stylized facts, loss function and risk measures, conditional and unconditional modelling of risk; extreme value theory, generalized hyperbolic distribution, volatility modelling and concepts for capturing dependencies. Explores portfolio risk concepts and optimization with risk constraints. Is accompanied by a supporting website featuring examples and case studies in R. Includes updated list of R packages for enabling the reader to replicate the results in the book. Graduate and postgraduate students in finance, economics, risk management as well as practitioners in finance and portfolio optimization will find this book beneficial. It also serves well as an accompanying text in computer-lab classes and is therefore suitable for self-study.
The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced. The old sections devoted to the smile issue in the LIBOR market model have been enlarged into a new chapter. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered. The fast-growing interest for hybrid products has led to a new chapter. A special focus here is devoted to the pricing of inflation-linked derivatives. The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives -- mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.
The challenges of the current financial environment have revealed the need for a new generation of professionals who combine training in traditional finance disciplines with an understanding of sophisticated quantitative and analytical tools. Risk Management and Simulation shows how simulation modeling and analysis can help you solve risk managemen
Successful investment strategies are specific implementations of general theories. An investment strategy that lacks a theoretical justification is likely to be false. Hence, an asset manager should concentrate her efforts on developing a theory rather than on backtesting potential trading rules. The purpose of this Element is to introduce machine learning (ML) tools that can help asset managers discover economic and financial theories. ML is not a black box, and it does not necessarily overfit. ML tools complement rather than replace the classical statistical methods. Some of ML's strengths include (1) a focus on out-of-sample predictability over variance adjudication; (2) the use of computational methods to avoid relying on (potentially unrealistic) assumptions; (3) the ability to "learn" complex specifications, including nonlinear, hierarchical, and noncontinuous interaction effects in a high-dimensional space; and (4) the ability to disentangle the variable search from the specification search, robust to multicollinearity and other substitution effects.