Dynamic Black-Litterman
Anas Abdelhakmi, Andrew E. B. Lim
Operations Research · 2026 · Volume 74 · Issue 1
Abstract
The Black-Litterman model is a framework for incorporating forward-looking expert views in a portfolio optimization problem. Existing work focuses almost exclusively on single-period problems with the forecast horizon matching that of the investor. We consider a generalization where the investor trades dynamically and views can be over horizons that differ from the investor. By exploiting the underlying graphical structure relating the asset prices and views, we derive the conditional distribution of asset returns when the price process is geometric Brownian motion and show that it can be written in terms of a multidimensional Brownian bridge. The components of the Brownian bridge are dependent one-dimensional Brownian bridges with hitting times that are determined by the statistics of the price process and views. The new price process is an affine factor model with the conditional log-price process playing the role of a vector of factors. We derive an explicit expression for the optimal dynamic investment policy and analyze the hedging demand for changes in the new covariate. More generally, the paper shows that Bayesian graphical models are a natural framework for incorporating complex information structures in the Black-Litterman model. The connection between Brownian motion conditional on noisy observations of its terminal value and multidimensional Brownian bridge is novel and of independent interest. Funding: A. Abdelhakmi received partial financial support from Fondation Ibn Rochd Pour Les Sciences et L’Innovation, under its 2021 SEEDS Initiative. A. E. B. Lim received financial support from the Ministry of Education, Singapore, under its 2021 Academic Research Fund Tier 2 grant call [Award MOE-T2EP20121-0014]. Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results are available at https://doi.org/10.1287/opre.2024.1010 .
Keywords
Financial Engineering, Black-Litterman model, forward-looking views, stochastic programming, multidimensional Brownian bridge, portfolio allocation, hedging strategies, optimal control
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