We propose a unified, fully general methodology to analyze and act on diversification in any environment, including long-short trades in highly correlated markets with complex derivatives. First, we build the diversification distribution, i.e. the distribution of the uncorrelated bets in the...

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...

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

Using the Bayesian posterior distribution of the market parameters we define self-adjusting uncertainty regions for the robust mean-variance problem. Under a normal-inverse-Wishart conjugate assumption for the market, the ensuing robust Bayesian mean-variance optimal portfolios are shrunk by the...

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 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...

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

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

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...

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...