Zhoufan Zhu

Formulaic alpha discovery is a core challenge in quantitative trading, as identifying alphas that work well together remains difficult. Recent reinforcement learning (RL) methods formulate this task as a Markov decision process (MDP), but two important issues remain unresolved. First, as the alpha pool evolves, the reward function changes accordingly, making the MDP inherently non-stationary. Second, most existing methods optimize a single objective, typically predictive power, while ignoring other important properties of a high-quality alpha pool. Motivated by these challenges, we propose AlphaPareto, an RL method for formulaic alpha discovery. To address non-stationarity, AlphaPareto augments the state to include both the alpha under construction and the current alpha pool, and applies a large language model (LLM) to encode the pool. This design allows the agent to adapt to the evolving search environment. To overcome the limitation of single-objective reward design, AlphaPareto replaces the scalar reward with a multi-objective vector-valued reward that simultaneously captures predictive power, temporal stability, perturbation robustness, and diversity, and optimizes these objectives through a Pareto-regularized learning procedure. Empirical applications to real-world datasets show that our AlphaPareto method outperforms its competitors.

Large-scale portfolio selection requires accurate forecasts of high-dimensional dynamic conditional covariance matrices (CCMs). This task is challenging because the number of covariance elements grows quickly with the number of assets, while the available time-series observations are limited and noisy. In this paper, we propose a novel graph-based quantiled (GRAND) CCM estimator in the spirit of "divide and conquer". By extracting information from the extensive asset features, the GRAND method leverages a new graph-based deep learning network, called the self-grouping autoencoder (SGA), to learn the conditional quantiles of returns. Using those learned conditional quantiles, the GRAND method further learns (co)variances of CCM element-wisely via the quantiled conditional moments (QCM) method, and finally, it makes the resulting assembled CCM estimator positive semidefinite through a projection. Due to the use of SGA network and QCM method in an element-wise way, the GRAND method is blessing dimensionality in learning CCM of a large-scale portfolio, while keeping the computation manageable and maintaining causal semantics. From an extensive empirical analysis of the Chinese stock market, we show that the GRAND method can produce much better tangency portfolios than its competitors, and provide dynamic causal graphs to study variance and quantile connectedness of all stocks for interpreting the systemic risk and forecasting business cycle.

For researchers and practitioners in finance, finding synergistic formulaic alphas is very important but challenging. In this paper, we reconsider the discovery of synergistic formulaic alphas from the viewpoint of sequential decision-making, and conceptualize the entire alpha discovery process as a non-stationary and reward-sparse Markov decision process. To overcome the challenges of non-stationarity and reward-sparsity, we propose the AlphaQCM method, a novel distributional reinforcement learning method designed to search for synergistic formulaic alphas efficiently. The AlphaQCM method first learns the Q function and quantiles via a Q network and a quantile network, respectively. Then, the AlphaQCM method applies the quantiled conditional moment method to learn unbiased variance from the potentially biased quantiles. Guided by the learned Q function and variance, the AlphaQCM method navigates the non-stationarity and reward-sparsity to explore the vast search space of formulaic alphas with high efficacy. Empirical applications to real-world datasets demonstrate that our AlphaQCM method significantly outperforms its competitors, particularly when dealing with large datasets comprising numerous stocks.

This paper proposes a new graph-based conditional moments (GRACE) method to do portfolio selection based on thousands of stocks or even more. The GRACE method first learns the conditional quantiles and mean of stock returns via a factor-augmented temporal graph convolutional network, which is guided by the set of stock-to-stock relations as well as the set of factor-to-stock relations. Next, the GRACE method learns the conditional variance, skewness, and kurtosis of stock returns from the learned conditional quantiles via the quantiled conditional moment method. Finally, the GRACE method uses the learned conditional mean, variance, skewness, and kurtosis to construct several performance measures, which are criteria to sort the stocks to proceed the portfolio selection in the well-known 10-decile framework. An application to NASDAQ and NYSE stock markets shows that the GRACE method performs much better than its competitors, particularly when the performance measures are comprised of conditional variance, skewness, and kurtosis.

We propose a new asset pricing model that is applicable to the big panel of return data. The main idea of this model is to learn the conditional distribution of the return, which is approximated by a step distribution function constructed from conditional quantiles of the return. To study conditional quantiles of the return, we propose a new conditional quantile variational autoencoder (CQVAE) network. The CQVAE network specifies a factor structure for conditional quantiles with latent factors learned from a VAE network and nonlinear factor loadings learned from a “multi-head” network. Under the CQVAE network, we allow the observed covariates such as asset characteristics to guide the structure of latent factors and factor loadings. Furthermore, we provide a two-step estimation procedure for the CQVAE network. Using the learned conditional distribution of return from the CQVAE network, we propose our asset pricing model from the mean of this distribution, and additionally, we use both the mean and variance of this distribution to select portfolios. Finally, we apply our CQVAE asset pricing model to analyze a large 60-year US equity return dataset. Compared with the benchmark conditional autoencoder model, the CQVAE model not only delivers much larger values of out-of-sample total and predictive R-sqaures, but also earns at least 30.9% higher values of Sharpe ratios for both long-short and long-only portfolios.