Black-Litterman Model

LLM-Enhanced Black-Litterman Portfolio Optimization ArXiv ID: 2504.14345 “View on arXiv” Authors: Unknown Abstract The Black-Litterman model addresses the sensitivity issues of tra- ditional mean-variance optimization by incorporating investor views, but systematically generating these views remains a key challenge. This study proposes and validates a systematic frame- work that translates return forecasts and predictive uncertainty from Large Language Models (LLMs) into the core inputs for the Black-Litterman model: investor views and their confidence lev- els. Through a backtest on S&P 500 constituents, we demonstrate that portfolios driven by top-performing LLMs significantly out- perform traditional baselines in both absolute and risk-adjusted terms. Crucially, our analysis reveals that each LLM exhibits a dis- tinct and consistent investment style which is the primary driver of performance. We found that the selection of an LLM is therefore not a search for a single best forecaster, but a strategic choice of an investment style whose success is contingent on its alignment with the prevailing market regime. The source code and data are available at https://github.com/youngandbin/LLM-BLM. ...

Dynamic Factor Allocation Leveraging Regime-Switching Signals ArXiv ID: 2410.14841 “View on arXiv” Authors: Unknown Abstract This article explores dynamic factor allocation by analyzing the cyclical performance of factors through regime analysis. The authors focus on a U.S. equity investment universe comprising seven long-only indices representing the market and six style factors: value, size, momentum, quality, low volatility, and growth. Their approach integrates factor-specific regime inferences of each factor index’s active performance relative to the market into the Black-Litterman model to construct a fully-invested, long-only multi-factor portfolio. First, the authors apply the sparse jump model (SJM) to identify bull and bear market regimes for individual factors, using a feature set based on risk and return measures from historical factor active returns, as well as variables reflecting the broader market environment. The regimes identified by the SJM exhibit enhanced stability and interpretability compared to traditional methods. A hypothetical single-factor long-short strategy is then used to assess these regime inferences and fine-tune hyperparameters, resulting in a positive Sharpe ratio of this strategy across all factors with low correlation among them. These regime inferences are then incorporated into the Black-Litterman framework to dynamically adjust allocations among the seven indices, with an equally weighted (EW) portfolio serving as the benchmark. Empirical results show that the constructed multi-factor portfolio significantly improves the information ratio (IR) relative to the market, raising it from just 0.05 for the EW benchmark to approximately 0.4. When measured relative to the EW benchmark itself, the dynamic allocation achieves an IR of around 0.4 to 0.5. The strategy also enhances absolute portfolio performance across key metrics such as the Sharpe ratio and maximum drawdown. ...

Application of Black-Litterman Bayesian in Statistical Arbitrage ArXiv ID: 2406.06706 “View on arXiv” Authors: Unknown Abstract \begin{“abstract”} In this paper, we integrated the statistical arbitrage strategy, pairs trading, into the Black-Litterman model and constructed efficient mean-variance portfolios. Typically, pairs trading underperforms under volatile or distressed market condition because the selected asset pairs fail to revert to equilibrium within the investment horizon. By enhancing this strategy with the Black-Litterman portfolio optimization, we achieved superior performance compared to the S&P 500 market index under both normal and extreme market conditions. Furthermore, this research presents an innovative idea of incorporating traditional pairs trading strategies into the portfolio optimization framework in a scalable and systematic manner. ...

A Geometric Approach To Asset Allocation With Investor Views ArXiv ID: 2406.01199 “View on arXiv” Authors: Unknown Abstract In this article, a geometric approach to incorporating investor views in portfolio construction is presented. In particular, the proposed approach utilizes the notion of generalized Wasserstein barycenter (GWB) to combine the statistical information about asset returns with investor views to obtain an updated estimate of the asset drifts and covariance, which are then fed into a mean-variance optimizer as inputs. Quantitative comparisons of the proposed geometric approach with the conventional Black-Litterman model (and a closely related variant) are presented. The proposed geometric approach provides investors with more flexibility in specifying their confidence in their views than conventional Black-Litterman model-based approaches. The geometric approach also rewards the investors more for making correct decisions than conventional BL based approaches. We provide empirical and theoretical justifications for our claim. ...

Dynamic Black-Litterman ArXiv ID: 2404.18822 “View on arXiv” Authors: Unknown 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 multi-dimensional 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 multi-dimensional Brownian bridge is novel and of independent interest. ...

Combining Transformer based Deep Reinforcement Learning with Black-Litterman Model for Portfolio Optimization ArXiv ID: 2402.16609 “View on arXiv” Authors: Unknown Abstract As a model-free algorithm, deep reinforcement learning (DRL) agent learns and makes decisions by interacting with the environment in an unsupervised way. In recent years, DRL algorithms have been widely applied by scholars for portfolio optimization in consecutive trading periods, since the DRL agent can dynamically adapt to market changes and does not rely on the specification of the joint dynamics across the assets. However, typical DRL agents for portfolio optimization cannot learn a policy that is aware of the dynamic correlation between portfolio asset returns. Since the dynamic correlations among portfolio assets are crucial in optimizing the portfolio, the lack of such knowledge makes it difficult for the DRL agent to maximize the return per unit of risk, especially when the target market permits short selling (i.e., the US stock market). In this research, we propose a hybrid portfolio optimization model combining the DRL agent and the Black-Litterman (BL) model to enable the DRL agent to learn the dynamic correlation between the portfolio asset returns and implement an efficacious long/short strategy based on the correlation. Essentially, the DRL agent is trained to learn the policy to apply the BL model to determine the target portfolio weights. To test our DRL agent, we construct the portfolio based on all the Dow Jones Industrial Average constitute stocks. Empirical results of the experiments conducted on real-world United States stock market data demonstrate that our DRL agent significantly outperforms various comparison portfolio choice strategies and alternative DRL frameworks by at least 42% in terms of accumulated return. In terms of the return per unit of risk, our DRL agent significantly outperforms various comparative portfolio choice strategies and alternative strategies based on other machine learning frameworks. ...

Portfolio Construction using Black-Litterman Model and Factors ArXiv ID: 2311.04475 “View on arXiv” Authors: Unknown Abstract This paper presents a portfolio construction process, including mainly two parts, Factors Selection and Weight Allocations. For the factors selection part, We have chosen 20 factors by considering three aspects, the global market, different assets class, and stock idiosyncratic characteristics. Each factor is proxied by a corresponding ETF. Then, we would apply several weight allocation methods to those factors, including two fixed weight allocation methods, three optimisation methods, and a Black-Litterman model. In addition, we would also fit a Deep Learning model for generating views periodically and incorporating views with the prior to achieve dynamically updated weights by using the Black-Litterman model. In the end, the robustness checking shows how weights change with respect to time evolving and variance increasing. Results using shrinkage variance are provided to alleviate the impacts of representativeness of historical data, but there sadly has little impact. Overall, the model by using the Deep Learning plus Black-Litterman model results outperform the portfolio by other weight allocation schemes, even though further improvement and robustness checking should be performed. ...

Black-Litterman Asset Allocation under Hidden Truncation Distribution ArXiv ID: 2310.12333 “View on arXiv” Authors: Unknown Abstract In this paper, we study the Black-Litterman (BL) asset allocation model (Black and Litterman, 1990) under the hidden truncation skew-normal distribution (Arnold and Beaver, 2000). In particular, when returns are assumed to follow this skew normal distribution, we show that the posterior returns, after incorporating views, are also skew normal. By using Simaan three moments risk model (Simaan, 1993), we could then obtain the optimal portfolio. Empirical data show that the optimal portfolio obtained this way has less risk compared to an optimal portfolio of the classical BL model and that they become more negatively skewed as the expected returns of portfolios increase, which suggests that the investors trade a negative skewness for a higher expected return. We also observe a negative relation between portfolio volatility and portfolio skewness. This observation suggests that investors may be making a trade-off, opting for lower volatility in exchange for higher skewness, or vice versa. This trade-off indicates that stocks with significant price declines tend to exhibit increased volatility. ...