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Abstract: We walk the reader through the Black-Litterman approach, providing all the proofs. We show how minor modifications of the original model greatly improve its range of applications. We discuss full generalizations of this and related models. MATLAB code is available through MATLAB Central
estimation risk, shrinkage estimation, decision theory
Abstract: The copula-opinion pooling (COP) approach extends in principle the Black-Litterman methodology to non-normally distributed markets and views. However, the implementations of the COP framework presented so far rely on restrictive quasi-normal assumptions. Here we present a general recipe to implement the COP approach in practice under all possible market and views specifications.
opinion pooling, views, copula, skewness, fat tails, Bayesian prior, posterior, Monte Carlo, quantitative portfolio management, asset allocation, CVaR, expected shortfall, Student t copula, non-parametric estimation
Abstract: We propose a unified methodology to input non-linear views from any number of users in fully general non-normal markets, and perform, among others, stress-testing, scenario analysis, and ranking allocation. We walk the reader through the theory and we detail an extremely efficient algorithm to easily implement this methodology under fully general assumptions. As it turns out, no repricing is ever necessary, hence the methodology can be readily applied to books with complex derivatives. We also present an analytical solution, useful for benchmarking, which per se generalizes notable previous results. Code illustrating this methodology in practice is available through author's homepage.
Black-Litterman, stress-test, scenario analysis, entropy, opinion pooling, Bayesian theory, Kullback-Leibler, Monte Carlo simulations, importance sampling, fat-tails, median, regime shift, normal mixtures, multi-manager, skill, ranking, ordering information, option trading, macro views
Abstract: The Black-Litterman and related approaches modify the return distribution of a normally distributed market according to views or stress-test scenarios. We discuss how to broaden the range of applications of these approaches significantly by letting them act on the risk factors underlying the market, instead of the returns of the securities.
scenario analysis, option trading, views on macro factors, non mean-variance optimization
Abstract: We introduce the multivariate Ornstein-Uhlenbeck and discuss how it generalizes a vast class of continuous-time and discrete-time multivariate processes. Relying on the simple geometrical interpretation of the dynamics of the Ornstein-Uhlenbeck process we introduce cointegration and its relationship to statistical arbitrage. We illustrate an application to swap contract strategies. Fully documented code illustrating the theory and the applications is available at MATLAB Central.
alpha, z-score, signal, half-life, vector-autoregression (VAR), moving average (MA), VARMA, stationary, unit-root, mean-reversion, Levy processes
Abstract: We extend the Black-Litterman methodology to generic non-normal market distributions and non-normal views. We draw on the copula and opinion pooling literature to express views directly on the market realizations, instead of the market parameters as in the Black-Litterman case. We compare the two approaches and we show an application to a thick-tailed, skewed and highly dependent market, where the views are expressed as uncertainty ranges.
opinion pooling, copula, views, fat tails, Bayesian prior, posterior, Monte Carlo, quantitative portfolio management, asset allocation, skew t distribution, CVaR, expected shortfall
Abstract: Exercises and case studies for a rigorous approach to risk- and portfolio-management. This booklet stems from the assignments for a course on advanced risk and portoflio management taught several times by the author at the Master's in the Mathematics of Finance of the Courant Institute - NYU. This booklet complements the textbook A. Meucci 'Risk and Asset Allocation,' Springer, 2005, to which the exercises refer in multiple locations. All the solution code is available at MATLAB Central File Exchange under the same author/title.
multivariate statistics, invariance quest, estimation theory, factor modeling, dimension reduction, pricing, VaR, CVaR, robust optimization, estimation risk, copula, cointegration, shrinkage, robustness, Bayesian, Black-Litterman
Abstract: We review the main processes used to model financial variables. We emphasize the parallel between discrete-time processes, mainly used by econometricians for risk- and portfolio-management, and their continuous-time counterparts, mainly used by mathematicians to price derivatives. We highlight the relationship of such processes with the building blocks of stochastic dynamics and statistical inference, namely the invariants. Figures and practical examples support intuition. Fully documented code illustrating these processes in practice is available at MATLAB Central File Exchange.
invariants, random walk, Levy processes, autocorrelation, ARMA, Ornstein-Uhlenbeck, long memory, fractional integration, fractional Brownian motion, volatility clustering, GARCH, stochastic volatility, subordination, real measure, risk-neutral measure, fat tails
Abstract: Using the Bayesian posterior distribution of the market parameters we determine self-adjusting uncertainty regions, which take the investor's prior into account, for the robust mean-variance problem. Under the standard normal-inverse-Wishart conjugate assumption for the market, the ensuing robust Bayesian mean-variance efficient frontier simplifies to a parsimonious set. This set is parametrized by the exposure to overall risk, which includes market risk, estimation risk for the expected values and estimation risk for the covariances.
Bayesian estimation, robust optimization, asset allocation, estimation risk
Abstract: We propose a unified, fully general methodology to analyze and act on diversification in any environment, including long-short trades in highly correlated markets. First, we build the diversification distribution, i.e. the distribution of the uncorrelated bets in the portfolio that are consistent with the portfolio constraints. Next, we summarize the wealth of information provided by the diversification distribution into one single diversification index, the effective number of bets, based on the entropy of the diversification distribution. Then, we introduce the mean-diversification efficient frontier, a diversification approach to portfolio optimization. Finally, we describe how to perform mean-diversification optimization in practice in the presence of transaction and market impact costs, by only trading a few optimally chosen securities. Fully documented code illustrating our approach can be downloaded from MATLAB Central File Exchange
entropy, mean-diversification frontier, transaction costs, market impact, selection heuristics, systematic risk, idiosyncratic risk, principal component analysis, principal portfolios, r-square, risk contributions, random matrix theory
Abstract: We describe a simple recursive routine to estimate by maximum likelihood the correlation matrix and the degrees of freedom of the t-copula, when structure needs to be imposed on the eigenvalues for dimensionality issues.
isotropy, shrinkage, structured correlation, estimation-maximization, maximum likelihood, radial generator
Abstract: We present a simple method to generate scenarios from multivariate elliptical distributions where the sample mean and covariances match the respective population moments. This methodology easily applies to large numbers of scenarios and large-dimensional distributions. We show an application to the risk management of a book of options.
matrix Riccati equation, antithetic variables, affine equivariance, affine transformations, copula-marginal factorization, correlation stress-testing
Abstract: We draw on regression analysis to decompose volatility, VaR and expected shortfall into arbitrary combinations or aggregations of risk factors and we present a simple recipe to implement this approach in practice
risk attribution, marginal contributions, Euler identity, risk factorization, volatility, tracking error, expected shortfall, positive homogeneous risk measures, principal component analysis, regression analysis
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