Attilio Meucci

Simulations with Exact Means and Covariances

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.

Toda Equations, bi-Hamiltonian Systems, and Compatible Lie Algebroids

We present the bi-Hamiltonian structure of Toda3, a dynamical system studied by Kupershmidt as a restriction of the discrete KP hierarchy. We derive this structure by a suitable reduction of the set of maps from Zd to GL(3,R), in the framework of Lie algebroids.

Neither 'Normal' nor 'Lognormal': Modeling Interest Rates Across All Regimes

We introduce a simple approach to managing portfolio interest rate risk that is consistent and performs well across different interest rate regimes, including when interest rates are low or even negative. Inspired by Black (1995), this approach uses a novel inverse-call transformation methodology to convert interest rates into shadow rates. We show that this methodology is more appropriate than the standard normal and lognormal models for forecasting and managing the distribution of the profits and losses of portfolios affected by the term structure of interest rates, producing more reliable forecasts and thus risk estimates for purposes of both internal and regulatory risk management.

Fully Integrated Liquidity and Market Risk Model

We introduce a new framework to integrate liquidity risk, funding risk and market risk, which goes beyond the simple bid-ask spread overlay to a VaR number. In our approach, we overlay a whole distribution of liquidity uncertainty to each future market-risk scenario. Then we allow for the liquidity uncertainty to vary scenario by scenario, depending on what liquidation policy or funding policy is implemented in that scenario.The result is one easy-to-interpret and easy-to-implement formula for the total liquidity-plus-market-risk PL compute total risk and decompose it into a novel liquidity-plus-market risk formula; and define a liquidity score as a monetary measure of portfolio liquidity.Our approach relies on three pillars: first, the literature on optimal execution, to model liquidity risk as a function of the actual trading involved; second, an analytical conditional convolution, to blend market risk and liquidity/funding risk; third the Fully Flexible Probabilities framework, to model and stress-test market risk even in highly non-normal portfolios with complex derivatives. Our approach can be implemented efficiently with portfolios of thousand of securities. The code for the case study is available for download

Fully Flexible Extreme Views

We extend the Fully Flexible Views generalization of the Black-Litterman approach to effectively handle extreme views on the tails of a distribution. First, we provide a recursive algorithm to process views on the conditional value at risk, which cannot be handled directly by the original implementation of Fully Flexible Views. Second, we represent both the prior and the posterior distribution on a grid, instead of by means of Monte Carlo scenarios: this way it becomes possible to cover parsimoniously even the far tails of the underlying distribution. Documented code is available for download.

Parametric Stress-Testing in Non-Normal Markets via Copula-Marginal Entropy Pooling

A novel approach for stress-testing (portfolios of) financial assets is presented. The technique extends the parametric Entropy Pooling approach to skewed and thick-tailed markets. The technique rests on a copula-marginal decomposition for the entropy together with several approximation schemes which renders the numerical computations feasible for real-life problems. An illustration with a portfolio of European options is presented.

Portfolio Construction and Systematic Trading with Factor Entropy Pooling

The Entropy Pooling approach in Meucci (2008) is a versatile, general framework to process market views in portfolio construction and generalized stress-tests in risk management. Here we present an efficient algorithm to implement Entropy Pooling with fully general views in multivariate normal markets. Then we discuss two applications. First, we use normal Entropy Pooling to estimate a market distribution consistent with the CAPM equilibrium, which improves on the “implied returns” a-la-Black and Litterman (1990) and can be used as the starting point for portfolio construction. Second, we use normal Entropy Pooling to process ranking signals for alpha-generation.

Estimating the distribution of the market invariants

In this chapter we discuss how to estimate the distribution of the market invariants from empirical observations. In Section 4.1 we define the concept of estimator, which is simply a function of current information that yields a number, the estimate. Such a general definition includes estimators that perform poorly, i.e. functions that yield an estimate which has little in common with the real distribution of the market invariants. Therefore we discuss optimality criteria to evaluate an estimator. After defining estimators and how to evaluate them, we need to actually construct estimators for the market invariants. Nevertheless, constructing es-timators by maximizing the above optimality criteria is not possible. First of all, the search of the best estimator among all possible functions of current information is not feasible. Secondly the optimality criteria rarely yield a uni-vocal answer. In other words, an estimator might perform better than another one in given circumstances, and worse in different circumstances. Therefore, we construct estimators from general intuitive principles, making sure later that their performance is acceptable, and possibly improving them with marginal corrections. In this spirit, we proceed as follows. In Section 4.2 we introduce nonparametric estimators. These estimators are based on the law of large numbers. Therefore, they perform well when the number of empirical observations in the time series of the market invariants is large, see Figure 4.1. When this is the case, nonparametric estimators are very flexible, in that they yield sensible estimates no matter the underlying true distribution of the market invariants. In particular we discuss the sample quantile, the sample mean, the sample covariance and the ordinary least square estimate of the regression factor loadings in an explicit factor model, stressing the geometrical properties of these estimators. We conclude with an overview of kernel estimators. When the number of observations is not very large, nonparametric esti-mators are no longer suitable. Therefore we take a parametric approach, by assuming that the true distribution of the market invariants belongs to a restricted class of potential distributions. In Section 4.3 we discuss maximum

Multi-Period Optimal Asset Allocation for a Multi-Currency Hedged Portfolio

An asset allocation strategy is presented to support a fund manager who wants to outperform a constant weights, constant hedging benchmark. This strategy is a continuous time, multi-period extension of the classical one-period mean-variance optimization framework.