Assessing Uncertainty in Stock Returns: A Gaussian Mixture Distribution-Based Method

TL;DR

The paper tackles uncertainty in stock returns by modeling the predictive distribution with a Gaussian mixture, enabling capture of skewness and heavy tails beyond traditional single-distribution approaches. It introduces MDNe, a fusion of Crossformer-based time-series processing and stock-code embedding that outputs a nine-component Gaussian mixture for each asset, trained via maximum likelihood. Empirical results on 3226 Chinese A-share stocks (2018–2022) show MDNe outperforms GARCH-family models across CRPS, MSE, and QLIKE, with MDN offering strengths at low-volatility regimes and MDNe delivering stronger performance during high-volatility periods; Diebold-Mariano tests corroborate these advantages. The work further provides a visualization framework through Bag-of-Words stock-code embeddings and t-SNE to reveal clusters of assets with similar risk profiles, aiding portfolio management and risk mitigation, and demonstrates robustness across multiple training runs. This combination of distributional forecasting and interpretable embedding-based visualization advances practical risk modeling in financial markets.

Abstract

This study seeks to advance the understanding and prediction of stock market return uncertainty through the application of advanced deep learning techniques. We introduce a novel deep learning model that utilizes a Gaussian mixture distribution to capture the complex, time-varying nature of asset return distributions in the Chinese stock market. By incorporating the Gaussian mixture distribution, our approach effectively characterizes short-term fluctuations and non-traditional features of stock returns, such as skewness and heavy tails, that are often overlooked by traditional models. Compared to GARCH models and their variants, our method demonstrates superior performance in volatility estimation, particularly during periods of heightened market volatility. It provides more accurate volatility forecasts and offers unique risk insights for different assets, thereby deepening the understanding of return uncertainty. Additionally, we propose a novel use of Code embedding which utilizes a bag-of-words approach to train hidden representations of stock codes and transforms the uncertainty attributes of stocks into high-dimensional vectors. These vectors are subsequently reduced to two dimensions, allowing the observation of similarity among different stocks. This visualization facilitates the identification of asset clusters with similar risk profiles, offering valuable insights for portfolio management and risk mitigation. Since we predict the uncertainty of returns by estimating their latent distribution, it is challenging to evaluate the return distribution when the true distribution is unobservable. However, we can measure it through the CRPS to assess how well the predicted distribution matches the true returns, and through MSE and QLIKE metrics to evaluate the error between the volatility level of the predicted distribution and proxy measures of true volatility.

Figures (7)

• Figure 1: Overall Procedure. During the dataset phase, we collect historical price and volume data for over 3000 A-share stocks and construct a series of indicator data to serve as input for the model. In the model design phase, the Crossformer structure is used to process time-series indicators, and Code Embedding is employed to represent stock tickers, these representations are then fed into fusion Net to predict potential return distributions and analyze asset attributes. In the analysis and evaluation phase, a series of evaluation metrics and visualization methods are used to test the model's performance.
• Figure 2: Mixture of five Gaussian distributions. The x-axis indicates the probability density, indicating how frequently a particular return rate is expected. The y-axis represents the return rate, with values in the range of -0.1 to 0.1 (indicating return rates in the next period from -10% to 10%). Each subfigure demonstrates a different type of distribution formed by a mixture of Gaussian components: (a) Normal distribution: This mixture of Gaussian approximates a normal distribution with a positive mean.(b) Skewed distribution: This distribution demonstrates skewness, formed by the combination of several Gaussian components.(c) Bimodal distribution: This illustrates a bimodal shape, indicating the presence of two distinct peaks. (d) Multimodal distribution: This distribution has multiple peaks, showcasing a more complex multimodal structure.
• Figure 3: Deep learning neural network structure. This figure shows neural network structure. It consists of three main components: (a) Crossformer: This component processes indicators as matrix $X \in \mathbb{R}^{C \times T}$ through three types of layers in Crossformer. (b) Code Embedding: This part handles input stock code by embedding, feed-forward network, add & norm layer, and dimensionality reduction (dense layer). (c) Fusion Net: This module integrates the outputs from Crossformer and Code Embedding, passing through feed-forward network and add & norm layers, producing final outputs $w$, $\mu$, and $\sigma$. The equation $Q = \sum_{i=1}^{n} w_i \cdot \mathcal{N}(\mu_i, \sigma_i)$ represents the probability function corresponding to the output parameters.
• Figure 4: Stock code embedding workflow. This figure shows the process of embedding stock codes into continuous vectors. This process uses the Bag-of-Words approach for stock codes, where each unique stock code is treated as a distinct token, mapped to an integer, and then embedded into a continuous vector space. The embedding vectors are then fed into subsequent layers of the neural network or visualized by dimensionality reduction techniques.
• Figure 5: Comparison of standard deviation for partial stocks. The figure compares the standard deviation for four stock samples, showing the performance of different volatility forecasting models. The green line represents the RRV indicator, serving as a proxy for actual volatility. The blue and yellow lines represent the volatility forecasts from the MDNe and MDN models, respectively. The other lines show forecasts from the GARCH model and its variants. It is observed that the MDN and MDNe models' forecasts are closer to the RRV curve.