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Fully Flexible Views: Theory and Practice

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Scenario analysis allows the practitioner to explore the implications on a given portfolio of a set of subjective views on possible market realizations, see e.g. Mina and Xiao (2001). The pathbreaking approach pioneered by Black and Litterman (1990) (BL in the sequel) generalizes scenario analysis, by adding uncertainty on the views and on the reference risk model. Further generalizations have been proposed in recent years. Qian and Gorman (2001) provide a framework to stress-test volatilities and correlations in addition to expectations. Pezier (2007) processes partial views on expectations and covariances based on least discrimination. Meucci (2009) extends the above models to act on risk factors instead of returns, and thus covers highly non-linear derivative markets and views on external factors that influence the p&l only statistically.

In the above techniques, the reference distribution of the risk factors is normal. The COP in Meucci (2006) explores non-normal markets, but correlation stress-testing and non-linear views are not allowed. Furthermore, the COP relies on ad-hoc manipulations.

Here we present the entropy pooling approach (EP in the sequel) which fully generalizes the above and related techniques. The inputs are an arbitrary market model, which we call “prior”, and fully general views or stress-tests on that market. The output is a distribution, which we call “posterior”, that incorporates all the inputs and can be used for risk management and portfolio optimization.

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To obtain the posterior, we interpret the views as statements that distort the prior distribution, in such a way that the least possible amount of spurious structure is imposed. The natural index for the structure of a distribution is its entropy. Therefore we define the posterior distribution as the one that minimizes the entropy relative to the prior. Then by opinion pooling we assign different confidence levels to different views and users.

Among others, the EP handles non-normal markets; views on non-linear combinations of risk factors that impact the p&l directly or only statistically through correlations; views on expectations, but also medians, to handle fat tails; views on volatilities, correlations, tail behaviors, etc.; lax views, such as ranking, on all of the above, thereby generalizing Almgren and Chriss (2006); inputs from multiple users and multiple confidence levels for different views.

Furthermore, in its most general implementation the reference model is represented by Monte Carlo simulations, and the posterior which incorporates all the inputs is represented by the same simulations with new probabilities. Hence the most complex securities can be handled without costly repricing.

In Section 2 we introduce the EP theoretical framework. In Section 3 we present an analytical formula, which generalizes the previous results and provides a benchmark for the numerical implementation. In Section 4 we discuss the numerical routine to implement the EP in full generality. In Section 5 we illustrate a case study: option trading in a non-normal environment with non-linear and ranking views on realized volatility, implied volatility and external macro factors. In Section 6 we conclude, comparing the EP to other related techniques.

To continue reading, click here to download the full article and MATLAB code. 

Attilio Meucci, PhD, CFA, is the chief risk officer at Kepos Capital LP and an adjunct professor at Baruch College. He is also the author of Risk and Asset Allocation and a regular contributor to publications, including Risk Magazine and to GARP Risk Professional Magazine.

Excerpted from A. Meucci, "Fully Flexible Views: Theory and Practice," in Risk 21 (10), p 97-102.


Almgren, R., and N. Chriss, 2006, Optimal portfolios from ordering information, Journal of Risk 9, 1—47.

Black, F., and R. Litterman, 1990, Asset allocation: combining investor views with market equilibrium, Goldman Sachs Fixed Income Research.

Meucci, A., 2006, Beyond Black-Litterman in practice: A five-step recipe to input views on non-normal markets, Risk 19, 114—119.

———, 2009, Enhancing the Black-Litterman and related approaches: Views and stress-test on risk factors, Journal of Asset Management 10, 89—96.

Mina, J., and J.Y. Xiao, 2001, Return to RiskMetrics: The evolution of a standard, RiskMetrics publications.

Qian, E., and S. Gorman, 2001, Conditional distribution in portfolio theory, Financial Analyst Journal 57, 44—51.

Pezier, J., 2007, Global portfolio optimization revisited: A least discrimination alternantive to Black-Litterman, ICMA Centre Discussion Papers in Finance.

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