High-Quality Synthetic Financial Time-Series using a GAN–Diffusion Framework
arXiv:2605.27113v1 [cs.LG] 26 May 2026
GIUSEPPE MASI, Sapienza University of Rome, Italy ANDREA COLETTA∗ , Banca d’Italia, Italy NOVELLA BARTOLINI, Sapienza University of Rome, Italy In recent years, financial institutions and firms have increasingly adopted synthetic data to address data scarcity and to generate counterfactual market scenarios. However, reproducing all the statistical properties of financial time series, commonly known as stylized facts, remains an open challenge for many existing general-purpose architectures. In this paper, we present a quality-aware generative framework that combines two classes of generative methods, demonstrating how their integration addresses existing limitations while enhancing the realism of synthetic data. Specifically, we first introduce CoMeTS-GAN (Correlated Multivariate Time Series GAN), a Conditional Generative Adversarial Network (C-GAN) designed to jointly generate mid-price and volume time-series for correlated stocks. We then show how our GAN architecture can be incorporated into state-of-the-art diffusion models to enhance the quality of generated correlation structures. Specifically, the GAN’s Critic serves as a quality evaluation module that guides the diffusion process, enforcing learned correlation structures in the generated time-series. Our framework offers a lightweight and responsive solution for realistic stock market simulation, explicitly modeling inter-asset correlation structures. We experimentally validate our framework against leading generative architectures, showing that it more effectively captures the stylized facts of stock markets and models inter-asset correlations. CCS Concepts: • Computing methodologies → Machine learning algorithms; • Mathematics of computing → Time series analysis; • Theory of computation → Adversarial learning; • Applied computing → Economics. Additional Key Words and Phrases: Diffusion models, GANs, synthetic data, time-series, financial data ACM Reference Format: Giuseppe Masi, Andrea Coletta, and Novella Bartolini. 2018. High-Quality Synthetic Financial Time-Series using a GAN–Diffusion Framework. ACM J. Data Inform. Quality 37, 4, Article 111 (August 2018), 23 pages. https://doi.org/XXXXXXX.XXXXXXX
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Introduction
The recent advancements in Artificial Intelligence (AI) have had a substantial impact on the financial industry [7, 36], enabling novel algorithms for stock market prediction [40], risk assessment and management tools [29], portfolio hedging solutions [6], or automated credit scoring systems [34]. The effectiveness of AI in the financial domain is — in part — due to the fact that financial firms are early adopters of novel technologies, but it has also been largely driven by the huge amount of historical and high-frequency data. In fact, financial markets are inherently data-driven and require ∗ The opinions expressed in this paper are personal and should not be attributed to Banca d’Italia.
Authors’ Contact Information: Giuseppe Masi, [email protected], Sapienza University of Rome, Italy; Andrea Coletta, [email protected], Banca d’Italia, Italy; Novella Bartolini, [email protected], Sapienza University of Rome, Italy. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. © 2018 Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM 1936-1963/2018/8-ART111 https://doi.org/XXXXXXX.XXXXXXX
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precise, data-driven decision-making. Such an environment seems a natural fit for machine learning models since they could exploit large datasets to extract patterns, model complex dynamics, and improve predictive accuracy. However, the huge amount of data generated within financial systems does not always translate into available datasets for machine learning algorithms, especially for the academic community. This is primarily due to strict regulations — which limit data sharing and usage — as well as the high costs associated with high-frequency data sources. For example, while daily stock price data is generally accessible, more detailed data — such as limit order book data — is often proprietary and not publicly available. Additionally, rare and extreme market events are, by definition, underrepresented in historical data, posing novel challenges for training robust models or conducting thorough backtesting. Synthetic data. Synthetic data offers a promising solution to these challenges by providing data with the same characteristics and structure as real data, while avoiding the associated risks and constraints [22, 39]. Synthetic data tools enable the generation of theoretically unlimited amounts of free data; they support the creation of counterfactual scenarios for testing and what-if analyses; and they allow data sharing without compromising privacy or exposing sensitive information. In recent years, a number of approaches have been designed to generate synthetic data for fundamental financial applications. Some notable examples include market simulations [9], model calibration [13], portfolio construction [28, 38], design of hedging strategies [12], or counterfactual markets scenarios generation [8]. In all these settings, developing data-generation techniques that accurately capture the complexities of global financial markets — thus produce high-quality synthetic data — is essential to the effectiveness of downstream tasks. Financial data have specific statistical properties, often referred to as stylized facts [10], that characterize stock returns, volatility, orders, and all financial data. However, reproducing such properties is not a trivial task. While recent work addresses the specific task of realistic financial time-series generation [45], by conditioning the generative model to preserve the stylized facts, many statistical properties have remained overlooked, and quality-aware frameworks are still missing. In particular, the correlation structure among multiple assets is often neglected, with only a few approaches addressing this issue [30], thereby limiting the realism and applicability of the generated data in multivariate settings. The need for realistic correlated time-series. Correlation dynamics are among the most important statistical properties in the generated time-series. In the financial domain, correlation dynamics play an essential role in managing the risks within a portfolio by ensuring that the individual assets are not overly correlated with one another and impacted by similar market conditions [37]. There are several reasons why stocks may exhibit correlation. Correlation may be sector-specific: for instance, the technology sector is affected by industry-wide regulations and changes in customer demands for a given product. Macroeconomic factors like interest rates or inflation may also impact entire sectors, causing the underlying stocks to react similarly. Furthermore, exogenous news and events can impact stocks belonging to different economic fields. We underline that even stocks that do not have fundamental similarities may show relevant correlation, possibly due to investor sentiment [4, 19]. During periods of market uncertainty or volatility, investors tend to adopt a risk-aware approach, affecting multiple stocks simultaneously. Finally, another source of correlation comes from composite financial instruments such as Exchange-Traded Funds (ETFs). Since these involve a basket of stocks, correlation among otherwise unrelated assets could arise as they align with the funds’ overall performance [24]. To conclude, the existence of the described correlation dynamics cannot be neglected when designing AI tools for the generation of synthetic datasets to simulate financial markets. However, most of the existing approaches attempt to learn stock correlation by solely relying on specific deep ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
High-Quality Synthetic Financial Time-Series using a GAN–Diffusion Framework
CoMeTS-GAN Time-series Dataset
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Di usion Model
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Using the Critic to improve correlation realism
Fig. 1. Overview of the CoMeTS-GAN framework. The system employs a conditional GAN consisting of a Generator (G) and a Critic (C). The Critic not only evaluates realism to properly train the Generator but also can guide a Diffusion Model during sample generation to improve the overall time-series quality.
learning model architectures (e.g., a transformer layer designed to learn feature dependency [46]). Yet, our experiments demonstrate that the resulting time-series do not always preserve the existing correlations.
ff
A novel quality-aware framework for correlated time-series generation. In this paper, we present a quality-aware generative framework that consists of a Generative Adversarial Network (GAN) [17] and a Diffusion Model [43]. Specifically, we first introduce CoMeTS-GAN (Correlated Multivariate Time Series GAN) for the synthetic generation of realistic market data (i.e., prices and volumes) that explicitly aims at providing a responsive market environment whilst capturing the inherent correlation among assets. Such approach builds on the GAN architecture [17], which has been widely adopted for financial scenarios [11, 44, 47, 50], as well as in general-purpose time-series generation [21, 25, 41]. Our proposed GAN architecture offers lightweight training and rapid generation, while also leveraging the two-player setup — a Generator and a Critic — to introduce domain-specific constraints. In particular, we incorporate cross-asset correlation coefficients directly into the Critic model, enabling the GAN to learn and reproduce realistic correlation dynamics. Eventually, we show that our Critic can be repurposed as an evaluation module — considering its capability of assessing the realism of correlation dynamics in the generated time series — and we demonstrate how it can be seamlessly integrated into a Diffusion-based Model for time-series generation to guide the sampling process at inference time. We show that such an integrated approach provides a quality-aware generation pipeline that enforces realistic correlation structures. Figure 1 shows the overall proposed framework. Paper Contributions. In detail, the major contributions of this paper are the following: • A Quality-Aware generative framework consisting of a Conditional Generative Adversarial Network (CGAN) and a Diffusion Model for multivariate time-series generation, comprising both price and volume dynamics across multiple correlated assets. • A Conditional GAN architecture that uses a novel cross-correlation distance to quantify the realism of correlation dynamics in generated time-series. A lightweight GAN architecture that enables fast training and generation, allowing for the synthesis of arbitrarily long time-series and responsive market simulation. ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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• A Critic-Guided generation strategy for diffusion models, in which the GAN’s Critic module dynamically steers the denoising process — generation process — of (any) pre-trained diffusion models at inference time, enhancing cross-correlation dynamics in the generated time-series. • A thorough experimental campaign that points out the realism of our approaches, capturing all the known stylized facts and the correlation dynamics. A comparison with existing approaches reveals that our solution outperforms existing work in its ability to capture inter-asset correlation, producing longer time-series, with moderate training effort. • A simulation of counterfactual scenarios, enabling the generation of diverse what-if financial time-series under hypothetical shifts in correlation dynamics. We also evaluate the model’s responsiveness to the behavior of experimental agents. 2
Related Work
In this section, we review existing generative models, and related work, aimed at generating timeseries data, with a particular focus on methods developed for financial and stock price datasets. Time series generation. Early generative models for time-series focused on domains outside finance. C-RNN-GAN [33] utilized LSTM networks in both the generator and discriminator to produce sequences of classical music, while RCGAN [16] adapted a similar framework for medical time-series, introducing a conditioning mechanism distinct from simple auto-regression. These models rely mainly on binary adversarial feedback, which may be insufficient to capture complex temporal dependencies prevalent in financial data. A notable advancement was proposed by Yoon et al. [51] with TimeGAN, a novel framework combining adversarial objectives with stepwise supervised losses. TimeGAN employs four components (embedding function, recovery function, sequence generator, and sequence discriminator), demonstrating superior performance in terms of discriminative and predictive metrics on benchmark datasets. However, despite its merits, TimeGAN generates only fixed-length mini-sequences and is associated with high training costs, both of which are problematic for financial applications that require efficient generation of arbitrarily long sequences. Other recent contributions have further advanced the field. GT-GAN [21] introduced a generalpurpose framework for time-series synthesis based on GANs, designed to handle a diverse range of sequential data by explicitly modeling temporal patterns and dependencies across various domains. COSCI-GAN [41] proposed a novel approach for generating multivariate time-series by coordinating the latent sources underlying different variables, thereby enabling the synthesis of multivariate sequences with common-source dependencies. While these models expand generative capabilities beyond single-variate and fixed-length sequences, they have yet to address the specific challenges of creating realistic, arbitrarily long financial time-series with dynamic interdependencies among multiple assets. Diffusion models for Time-Series generation. Recently, a new class of generative models, namely diffusion models, has been adopted for time-series generation. Diffusion models were originally introduced in [42] and gained widespread popularity through the Denoising Diffusion Probabilistic Models (DDPM) proposed in [20]. They have demonstrated superior performance in sample quality and diversity across modalities [14], and recent work has begun extending their applicability to structured domains such as time-series and dynamical data [27]. Notable examples include DiffTime [8], Diffusion-TS [52], and TSGM [26] for the generation of time-series data. Financial-specific generative models. The aforementioned work mainly focuses on general-purpose generative models, which only partially adapt to the specific requirements of financial time-series generation. In fact, existing models often fail to capture all the stylized facts and statistical properties ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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of financial data. Consequently, recent literature has introduced ad-hoc models specifically tailored to financial data. Takahashi et al.[44] proposed FIN-GAN, adopting classic GANs with a variety of generator/discriminator structures (multi-layer perceptrons, convolutional neural networks, and hybrids) to reproduce stylized facts observed in actual market time-series. Wiese et al.[50] developed QuantGAN, integrating temporal convolutional networks (TCNs) to effectively model long-range dependencies such as volatility clustering. Li et al. [25] explored transformer-based models for adversarial generation of time series, leveraging self-attention to improve sequence modeling. Nonetheless, none of these methods provides a scalable solution for generating multi-stock series that capture evolving interactions and dependencies among different assets. Broadening the view, some works focused on the generation of specific aspects. Vuletić et al. [48] exploit a GAN architecture trained with an economic-driven loss function for the generator to generate the ETF-excess returns with respect to the underlying stocks to place trades and make profits. Cont et al. [11] proposed an approach based on GANs to generate price traces that retain tail risk features observed in the input dataset, to improve the estimation of loss distributions in dynamic portfolios. However, these approaches do not consider the simultaneous generation of multiple stocks and their correlation dynamics. Masi et al. [30] is the first work that explicitly addresses the generation of correlated stock prices, using a conditional GAN architecture. However, in their framework, volumes and prices are generated independently, which is suboptimal, and the intraday structure of trading activity is largely overlooked. Our quality-aware generation framework builds upon and extends this prior work in several aspects. First, we introduce novel temporal features to better capture intraday stock dynamics (e.g., the characteristic U-shaped volume profile [5]). We extend the GAN framework to jointly generate trading volumes and prices, thereby enhancing realism by capturing the strong dependence between volume dynamics and price volatility. Our analysis of stylized facts is accordingly expanded to encompass empirical regularities in both volume and price series. Finally, we went beyond a single model by also introducing diffusion models: we show that our trained Critic can be integrated into any existing diffusion model to guide the generation of time-series towards more realistic inter-asset correlations. Within this new framework, we also demonstrate the possibility of counterfactual generation, assessing the model’s ability to simulate previously unseen market scenarios - for instance, exploring a hypothetical shift in the correlation between Coca-Cola and Pepsi stock returns from positive to negative. 3
Background
In this section, we briefly review some important pillars that are useful for the reader to approach this paper. 3.1 Conditional GANs Generative Adversarial Networks (GANs) [17] are a subclass of generative models based on game theory. GANs have two neural networks: a generator denoted as 𝐺, and a discriminator denoted as 𝐷. The generator creates synthetic data samples from random noise, while the discriminator distinguishes between real and generated samples. During training, the generator competes against the discriminator to improve its ability to generate synthetic data samples 𝒙 ∼ 𝑝 gen that are realistic, i.e., distributed as the original data distribution 𝑝 real . The generator’s output is 𝒙 = 𝐺 (𝒛; 𝜽 (𝐺 ) ) where 𝒛 ∼ N (0, 𝑰 ) is an input noise to ensure variance at generation time, and 𝜽 (𝐺 ) the parameters of the generator. The discriminator outputs 𝑦 = 𝐷 (𝒙; 𝜽 (𝐷 ) ) ∈ [0, 1] representing the probability of 𝒙 being drawn from 𝑝 real and 𝜽 (𝐷 ) the parameters
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of the discriminator. Both 𝐺 and 𝐷 are trained to optimize their payoff until neither player can unilaterally improve its cost. The discriminator is trained to maximize the probability of assigning the correct label to the data, either real or generated; conversely, the generator is trained to minimize it (by generating realistic samples that the discriminator misclassifies as real). GANs can also be extended to consider conditional input, resulting in what is known as conditional-GANs [31]. In conditional-GANs, or CGANs, both the generator and discriminator networks may take additional conditioning information 𝒄 as input, such as class labels or other contextual features. This allows for generating specific types of data samples based on the provided conditioning information. The generator’s and discriminator’s input vectors are combined with 𝒄. In this work, we use a Wasserstein Generative Adversarial Network (WGAN) [2], which demonstrated better training performance and stability. In a WGAN the discriminator is called critic and outputs a realness score of the input samples, 𝑦 = 𝐷 (𝒙; 𝜽 (𝐷 ) ) ∈ R, even called critic score. Formally, the optimal generator is 𝐺 ∗ : arg min max 𝐺
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E [𝐷 (𝒙 |𝒄)] −
𝒙∼𝑝 data
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[𝐷 (𝐺 (𝒛|𝒄))].
(1)
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Diffusion Model
Diffusion models generate data by learning to invert a Markovian noising process that maps data to a tractable prior, such as a standard Gaussian. Let 𝒙 ∼ 𝑝 real be a real sample. The forward diffusion process progressively corrupts 𝒙 0 into a Î sequence of latent variables (𝒙 1, . . . , 𝒙𝑇 ) using a Markov chain: 𝑞(𝒙 1:𝑇 |𝒙 0 ) = 𝑇𝑡=1 𝑞(𝒙𝑡 |𝒙𝑡 −1 ). Each √︁ transition is typically defined as additive Gaussian noise: 𝑞(𝒙𝑡 |𝒙𝑡 −1 ) = N 𝒙𝑡 ; 1 − 𝛽𝑡 𝒙𝑡 −1, 𝛽𝑡 𝑰 , with a time-dependent noise schedule {𝛽𝑡 }𝑇𝑡=1 . After 𝑇 steps, 𝒙𝑇 becomes approximately standard Gaussian, i.e., 𝑞(𝒙𝑇 ) ≈ N (0, 𝑰 ) for large 𝑇 . The reverse process aims to reconstruct data from noise by learning the reverse conditional Î distributions: 𝑝𝜙 (𝒙 0:𝑇 ) = 𝑝 (𝒙𝑇 ) 𝑇𝑡=1 𝑝𝜙 (𝒙𝑡 −1 |𝒙𝑡 ), where 𝑝 (𝒙𝑇 ) = N (0, 𝑰 ) and each 𝑝𝜙 (𝒙𝑡 −1 |𝒙𝑡 ) is typically parameterized as 𝑝𝜙 (𝒙𝑡 −1 |𝒙𝑡 ) = N 𝒙𝑡 −1 ; 𝝁𝜙 (𝒙𝑡 , 𝑡), 𝚺𝜙 (𝒙𝑡 , 𝑡) , with the mean and covariance learned via a neural network 𝜖𝜙 . In practice, this neural network is trained by maximizing a variational lower bound on the marginal log-likelihood log 𝑝𝜙 (𝒙 0 ), yielding a loss that includes a (reweighted) denoising scorematching term [20]: h i 2 E𝑡,𝒙 0,𝜖 𝜖 − 𝜖𝜙 (𝒙𝑡 , 𝑡) , √ √ Î𝑡 where 𝜖 ∼ N (0, 𝑰 ) and 𝒙𝑡 = 𝛼¯𝑡 𝒙 0 + 1 − 𝛼¯𝑡 𝜖, with 𝛼¯𝑡 = 𝑠=1 (1 − 𝛽𝑠 ). Sampling proceeds by initializing 𝒙𝑇 ∼ N (0, 𝑰 ) and sequentially applying the learned reverse transitions 𝑝𝜙 (𝒙𝑡 −1 |𝒙𝑡 ) for 𝑡 = 𝑇 , . . . , 1, reconstructing data via iterative denoising. 3.3
Stylized Facts
The variation of asset price exhibits several statistical properties that are common across a wide range of markets and timeframes. These properties capture the market behavior over different time periods and are referred to as stylized facts [10, 49]. In this work, we use them to evaluate the quality of samples generated by the model. In particular, we consider the following stylized facts: • Fat-tailed distribution or heavy tails. The distribution of the asset returns displays a power-law or Pareto-like tail, meaning that extreme returns occur more frequently than predicted by a normal distribution. • Aggregational normality. As the time period for computing returns (Δ𝑡) increases, the distribution of returns resembles a normal distribution. ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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• Linear unpredictability or absence of autocorrelation. The linear autocorrelation of the return corr(𝑟𝑡,𝑇 , 𝑟𝑡 +Δ𝑡,𝑇 ) rapidly decays to zero within minutes and becomes insignificant for periods longer than Δ𝑡 = 20 minutes, with 𝑇 the length of the series [49]. • Volatility clustering. Large changes in prices tend to cluster together, that is, volatility shows a positive autocorrelation over several trading days. • Volume/Volatility Correlation. Trading volume and volatility are positively correlated. Specifically, these properties allow us to assess the preservation of the realism of generated traces. 3.4
Correlation Dynamics
Interdependency and correlation dynamics need new analytical tools to analyze the set of stocks in relation to each other. We recall the importance of correlation dynamics in portfolio management, the use of correlation as a measure to evaluate portfolio diversification is a common practice. The principal aim is to mitigate the total risk by strategically including assets that exhibit a low correlation with each other. To achieve this, it is imperative to maintain a realistic understanding of the correlation dynamics among stocks. In order to evaluate the effectiveness of the model in capturing these dynamics, we suggest the following quantitative metric, the cross-correlation distance: Cross-correlation distance. This metric measures the distance between the cross-correlations of real and generated stocks. Formally, let 𝜌 (·, ·) be the Pearson correlation coefficient (PCC). For a pair of stocks (𝑆𝑖 , 𝑆 𝑗 ), we can measure: 𝑑 𝜌 (𝑆𝑖 , 𝑆 𝑗 ) = MSE(𝜌 (𝑆𝑖 , 𝑆 𝑗 ), (𝜌 (𝑆b𝑖 , 𝑆b𝑗 )), where 𝑆b𝑖 and 𝑆b𝑗 are the synthetic version of the stocks 𝑆𝑖 and 𝑆 𝑗 and MSE being the mean squared error. We will utilize this metric to scrutinize the model’s performance with respect to stocks that show varying degrees of correlation. For instance, let us consider two prominent companies, CocaCola and PepsiCo, which generally demonstrate a strong positive correlation. This correlation is attributable to their simultaneous presence in the same market, competing for similar consumer demographics, and facing exposure to similar exogenous factors (i.e., suppliers). On the other hand, during times of market turmoil, investors often seek out defensive stocks as opposed to riskier, growth-oriented stocks. This pattern of trading behavior may result in a shift to Coca-Cola, a beverage firm, from a technology company, for instance, Nvidia, leading to a negative correlation between these two stocks. Understanding these correlation patterns is not just academically interesting, it is critical for managing risk and unlocking profit opportunities. 4
System Model
The purpose of our framework is to generate multivariate time-series capturing crucial aspects of the stock markets for multiple stocks simultaneously, which include mid-prices and volumes. A key focus of our approach is to ensure the preservation of the correlation between the considered stocks. The framework’s underlying model is trained to maximize both the similarity and the match in the correlation of the generated time-series to real ones. This architecture can be used to generate time-series also of other domains. Nonetheless, we will mostly stress its strengths in the financial field; more specifically, we are interested in mid-prices and volumes of a chosen set of stock. In the following, we analyze in detail the main components of the framework. Output. The output of our framework is a multivariate time-series of length 𝐹 units of time, two series (price and volume) for each stock in the set S. This output is structured as a matrix ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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𝒙ˆ future ∈ R𝐹 ×2𝑛 , with 𝑛 = |S| (i.e., the number of stocks). To generate longer time-series, an autoregressive approach can be used, that is, iteratively feeding the conditional generator with its own output. More details on the auto-regressive generation are provided in the following paragraphs. Training Data. The dataset used to train the model is generated from historical stock traces in the temporal period [0,𝑇 ]. Let 𝒙𝑠 ∈ R2×𝑇 be the time-series representing the mid-price and the volume of stock 𝑠 ∈ S in the considered temporal period. For the given set of stocks, we stack the individual time-series 𝒙𝑠 along the columns to create the multivariate time-series X S ∈ R𝑇 ×2𝑛 . In order to capture the dependencies between past and future stock movements, we segment the multivariate time-series X S over time to construct the dataset as follows. For each time-step 𝑡, we create a pair consisting of the past 𝑃 time-steps, denoted as 𝒙 past and the future 𝐹 steps denoted as 𝒙 future . By applying this segmentation for 𝑡 ∈ [𝑃,𝑇 − 𝐹 ], we generate the dataset which we denote as: −𝐹 D = {(X S [𝑡 − 𝑃 : 𝑡], X S [𝑡 + 1 : 𝑡 + 1 + 𝐹 ])}𝑇𝑡 =𝑃 (2) The objective of the model is to generate a continuation 𝒙ˆ future of an observed time-series 𝒙 past , by training the model on tuples of the form (𝒙 past, 𝒙 future ) ∈ D. The simultaneous generation of 2𝑛 time-series allows the preservation of the correlation dynamics observed in the training set. 4.1
CoMeTS-GAN Architecture
The underlying model of CoMeTS-GAN is a Conditional Wasserstein Generative Adversarial Network (C-WGAN) [2, 31]. This model is trained to learn the joint probability distribution of the collection of future time-series 𝒙 future in the training dataset D by optimizing the critic score. As for the generator and the critic of the C-WGAN, we used a mixture of vanilla and temporal convolutional neural networks. Figure 2 shows the architecture of our model. The generator 𝐺 is the main component of CoMeTS-GAN. It is responsible for the actual generation of the time-series 𝒙ˆ future , which retains the statistical properties of the real time-series observed by the generator. The architecture used for the generator network is composed of a series of 7 temporal convolutional blocks [3] with increasing dilation size and a final linear layer to adjust the size of the output vector. Each temporal block is followed by a leaky ReLU activation function and a dropout layer. The critic 𝐷 is responsible for assessing the realness of the generated samples. The architecture used for the critic network is made of a series of convolutional layers having increasing filter sizes, followed by a series of linear layers. Each convolutional and linear layer is followed by a leaky ReLU activation and a dropout layer, and spectral normalization is applied on top of each network layer. No activation function is used for the last layer of the network. We denote the output of this sequential network as 𝑜 1 , which is devoted to measuring the realness of the generated time-series. A linear layer is present within the critic (top right of the figure) and is meant to address the correlations of the multivariate time-series. The linear layer takes as input the correlation coefficients computed between all pairs of the mid-prices and volumes of the 𝑛 stocks (i.e., 2𝑛 2 pairs) and it outputs 𝑜 2 . The overall output of the critic (i.e., the critic score) is the sum 𝑜 = 𝑜 1 + 𝛼 · 𝑜 2 , with 𝛼 ∈ [0, 1] being a hyperparameter to weigh the relevance of the correlation. By summing these two scores, we aim to capture the realism and correlation of the 𝑛 stocks. CoMeTS-GAN Training. Upon input pair (𝒙 past, 𝒙 future ) ∈ D, the framework is trained as follows. The input of the generator is the concatenation of the past sequence 𝒙 past and a noise vector 𝒛 ∼ N (0, 𝑰 ) of the same length summed to a vector 𝒕 to embed the time information. The temporal information is computed by first mapping each timestamp to the number of minutes elapsed since
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Linear block N Linear layer
Linear block 1
Convolutional block N Embedding Convolutional block 1 MLP
Linear layer
Temporal convolutional block N
Temporal convolutional block 2
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MLP Temporal convolutional block 1
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Fig. 2. C-WGAN architecture. The ⌢ operator represents the concatenation of two vectors. The or operator iteratively alternates between the real series 𝒙 future and the generated series 𝒙ˆ future .
the start of the trading day. This value is then discretized into uniform intervals, each representing a fixed duration (10 minutes), effectively segmenting the trading session into a sequence of time bins. This variable is then mapped to a high-dimensional embedding using a sinusoidal positional encoding scheme, similar to those used in transformer architectures. In the generator, the time-step 𝑒𝑥 embedding is processed by a 2-layer MLP with a non-linear activation function (𝑓 (𝑥) = 𝑥 · 1+𝑒 𝑥 ) to further enhance its expressiveness. The resulting time embedding is added directly to the noise, effectively conditioning the generative process on the temporal context. This integration allows the model to generate outputs that are sensitive to the progression of time within each sequence, improving its ability to model temporal dynamics. Each temporal block is fed with the output of the previous, concatenated with the noise vector 𝑧 along the dimension of the features. The noise injection technique allows diversity in the model’s output. Finally, the output of the last temporal block is directly fed to the last linear layer. The output of the generator network is the future sequence 𝒙ˆ future .
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The input of the critic is the concatenation of the past sequence 𝒙 past with either the real continuation 𝒙 future drawn from the dataset, or a generated one 𝒙ˆ future from the generator. In the discriminator, the temporal information is processed in the same way of the generator, but the resulting time embedding is concatenated as an additional channel to the input sequence along the feature dimension. The result of the concatenation is fed to the convolution and linear blocks to produce a single value in output, representing the standard critic score 𝑜 1 . Additionally, the correlations between the features of the future sequence in the input are computed. All the 2𝑛 2 correlation coefficients are input to the linear layer to produce the score 𝑜 2 . The output of the critic is the sum of the two contributions, as mentioned above. We adopt spectral normalization to enforce the Lipschitz continuity constraint, due to its superiority to other techniques like weight clipping or gradient penalty and for its ease of applicability [32]. 4.2 Diffusion Models: A Critic-Guided generation Sohl-Dickstein et al. [42] and Song et al. [43] show that any pre-trained diffusion model can be conditioned using the gradients of a classifier, steering the denoising (i.e., generation) process towards an arbitrary class label y. This technique, called classifier guidance, in diffusion models typically involves modifying the score function to incorporate the gradient from an external classifier that estimates the likelihood that a sample belongs to a target class. In CoMeTS-GAN, we note that the Critic module is trained to distinguish between real and synthetic stock price sequences; it implicitly learns a representation of realism. Therefore, we propose to utilize our Critic as a surrogate classifier, where its gradient can be leveraged to enhance the generation process of the diffusion model. Mathematically, the diffusion model’s score function, denoted as 𝑠 (𝑥𝑡 , 𝑡), is adjusted by incorporating the gradient of the WGAN critic 𝐷 (𝑥): 𝑠˜ = 𝑠 (𝑥𝑡 , 𝑡) + 𝑤 · ∇𝑥𝑡 𝐷 (𝑥𝑡 ),
(3)
where 𝑤 modulates the guidance strength. This modification encourages the diffusion process to produce samples that align more closely with the distribution learned by the WGAN critic, effectively biasing the sampling trajectory toward regions of higher realism. In the experimental section, we instantiate our Critic-Guided strategy using DiffTime [8], a stateof-the-art Diffusion Model trained on our time-series datasets. We demonstrate how the gradients of the Critic can actually guide the diffusion denoising process at inference time, improving the cross-correlation fidelity of the generated time-series and yielding a quality-aware generation pipeline. More broadly, our approach suggests that any differentiable generative network trained on a domain-specific realism metric can be used to enhance the generation quality of general-purpose diffusion models. 5
Experiments
In this section, we evaluate the performance of our framework against SOTA approaches. We first compare CoMeTS-GAN using several benchmarks and newly curated datasets to ensure a rigorous assessment. Our primary focus is assessing the fidelity of the generated time-series in comparison to real ones. Eventually, we examine the improvement in generation realism achieved by the Critic-Guided strategy, instantiated using DiffTime [8] a recent diffusion models for time-series generation. We ran our Python-based framework on a NVIDIA 2060 GPU.1 1We commit to publicly releasing our source code upon acceptance, enabling other researchers to reproduce our results and
use the framework for their own experiments.
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We test the performance of our framework across data showing varying periodicity, discreteness, level of noise, regularity of time-steps, and correlation across time and features. We also include real stock market time-series. In detail, the considered datasets — including well-known benchmarks from the literature — are the following: • Sines [51]: Multivariate sinusoidal sequences with different frequencies 𝜂 and phases 𝜙. Specifically, the dataset is obtained with 𝑠𝑖 (𝑡) = sin(2𝜋𝜂𝑖 𝑡 + 𝜃 𝑖 ), where 𝑖 ∈ {1, . . . , 5}, 𝜂𝑖 ∼ U [0, 1] and 𝜃 𝑖 ∼ U [−𝜋, 𝜋]. • Multivariate Gaussian Model [51]: Sequences from auto-regressive multivariate Gaussian models, defined as: 𝑔𝑖 (𝑡) = 𝜙𝑖 𝑔𝑖 (𝑡 − 1) + 𝑞 where 𝑞 ∼ N (0, 𝜎𝑖 + (1 − 𝜎𝑖 )), 𝜙𝑖 ∈ [0, 1], and 𝜎𝑖 ∈ [−1, 1]. The coefficients 𝜙𝑖 and 𝜎𝑖 allow to control the correlation across time and features, respectively. • Stock Mid-Prices & Volumes [1]: The source of stock market data is LOBSTER2 , an online limit order book data tool to provide limit order book data for the NASDAQ traded stocks. From limit order book data, it is easy to compute the mid-price with the temporal resolution of choice, that in our case, is minute. We focus on four stocks, namely, Coca-Cola (KO), PepsiCo (PEP), Nvidia (NVDA), and Kansas City Southern (KSU). The time period of choice is from 2018-02-02 to 2018-10-11 for the training set, and from 2018-10-12 to 2018-11-14 for the validation set. Table 1 shows the Pearson correlation coefficient between them. We observe how some are highly positively correlated (KO and PEP, NVDA and KSU), and some are highly negatively correlated (PEP and NVDA, PEP and KSU, KO and KSU, KO and NVDA). To test the scalability of our generative framework, we selected the 30 stocks composing the DJIA index from 2017-02-02 to 2017-10-11 for the training set and from 2017-10-12 to 2017-11-14 for the validation set. For these time-series, we follow the same preprocessing pipeline. For the mid-prices, we compute the log-returns normalized using a 𝑧-score approach. Instead, volumes are scaled in the [−1, 1] with a min-max approach, then followed by tanh activation function, which scales the volumes to the positive domain. KO
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Evaluation Metrics
We measure the performance of our framework in terms of the following aspects: Realism, Diversity, Stylized Facts, Reactivity, Scalability. Realism. We measure the realism of generated time-series by using the Discriminative Score, which offers a quantitative measure of the similarity between real and synthetic (or generated) time-series. It was proposed in [51] and is computed as follows. An LSTM is trained to differentiate between sequences from the original and generated datasets. A standard supervised learning task 2 LOBSTER https://lobsterdata.com/
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is performed and the resulting classification error on the test set (the lower the better) provides a quantitative measure of similarity. Diversity. This aspect refers to the range and variability of patterns and structures present in the generated time-series data. It is crucial because a well-working generative model should produce data that fully explores the natural variations present in real time-series, rather than simply replicating or memorizing the training data (problem known as “mode collapse"). Cross-correlation distance. In order to examine the model’s ability to capture the correlation dynamics that exist between stocks we measure the distance between the cross-correlations of real and generated stocks using the cross-correlation distance defined in 3.4. Formally, at the end of each epoch, for each of the 2𝑛 2 pairs of features (𝑆𝑖 , 𝑆 𝑗 ), we measure the 𝑑 𝜌 (𝑆𝑖 , 𝑆 𝑗 ) with respect to their synthetic counterparts. Stylized Facts. We investigate the properties of the synthetic traces by analyzing the most common stylized facts in the financial domain as discussed in Section 3.3. 5.3
CoMeTS-GAN Results
5.3.1 Realness of Generated Data. We can generate arbitrarily long stock traces thanks to the auto-regressive application of the model to the input time-series. The generation process can be defined as follows: (1) the model is fed with a multivariate of length 𝑃, that is a tensor in R𝑃 ×2𝑛 of original stock data; (2) the output of the model, is a multivariate time-series of length 𝐹 that is a tensor in R𝐹 ×2𝑛 of synthetic stock data; (3) the last 𝑃 time-steps of the synthetic sequence are fed to the model; (4) steps 2 and 3 are repeated until a total number of generative steps are done. By varying the random seed we are able to generate several multivariate time-series. Figure 3 shows the execution of three runs, in addition to the real trace (orange line). As we can observe, the model does not incur self-induction (i.e., it does not generate a trend that remains indefinitely influenced by it). Furthermore, the model respects the diversity property, generating different sequences from the same input if the random noise changes, preserving the correlation dynamics. In fact, as the training of the model proceeds, the average cross-correlation distance between real and synthetic data decreases, as we can see in Figure 4, becoming negligible among all the pairs of stocks.
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The fact that the model is able to capture the correlation dynamics is also evident in Figure 5, where we plot the prices of the stocks, and the average cross-correlation distance is 0.04. Figure 6 shows the corresponding volumes. ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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We measure the stylized facts on the auto-regressive generation of stock mid-prices and volumes for 24 trading days, given that a trading day lasts for 6.5 hours, we generate 9360 minutes in total. Notice that most of existing approaches are unable to generate such a long stock time-series data. Results are averaged over 10 runs where we vary the seed and the randomization of the generation. ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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Figure 7a and 7b show the histogram of the intraday log-returns with periods of Δ𝑡 = 1 and Δ𝑡 = 15 minutes, respectively. Results show a fat-tailed distribution as expected for the first one, and as we increase the period Δ𝑡, the distribution of returns better fits a normal distribution. The described property is often referred to as aggregational normality. As seen in the figures, both real and generated data closely follow these known properties for all the stocks. In Figure 8, we plot the autocorrelation of the minutely intraday log-returns of the stocks with different time lags of 1, 10, 20, and 30 minutes. As we can observe, at increasing lag, the returns tend to cluster around a null correlation value. The stylized fact related to the absence of autocorrelation is thus present in both real and generated traces. Figure 9 shows the autocorrelation coefficients of the volatility at varying day lag. The volatility is computed as the standard deviation of the returns. We observe that the correlation is much higher when the lag is short and decreases as the lag increases, confirming that the volatility tends to cluster in time. The plot also confirms that all the considered stocks show this behavior, both in the real and generated traces. 5.3.2 Comparison with State-of-the-Art. We now compare CoMeTS-GAN with several state-of-the-art approaches, namely TTS-GAN [25], COSCI-GAN [41], GT-GAN [21], TimeGAN [51], RCGAN [16] and C-RNN-GAN [33]. For purely auto-regressive approaches, we compare against RNNs trained with teacher-forcing (T-Forcing) [18] as well as professor-forcing (P-Forcing) [23]. For additional comparison, we consider the performance of WaveNet [35] as well as its GAN counterpart WaveGAN [15]. We first focus on standard general-domain benchmarks from the literature, and the we focus on financial data. Synthetic data realism on literature benchmarks. We first show the results obtained on the Gaussian model dataset and on the Sines dataset in Table 2 in terms of discriminative score (i.e., realism). Numerical results show that our model performances are somehow comparable with respect to more complex state-of-the-art approaches. In fact, our model is specifically designed for financial data while the literature benchmarks do not focus on financial datasets. This experiment highlights both the potential performance and the versatility of our model, beyond the specific domain for which it was originally developed. It is also important to note that our model is structurally different from more advanced models such as TimeGAN, COSCI-GAN, and GT-GAN. In such work, the original time-series is factorized in short sub-sequences, and the model outputs a synthetic set of such sub-sequences at inference time, but there is no natural and intuitive way to recombine them to obtain a longer sequence, ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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retaining temporal continuation. Instead, our model is able to generate a single time-series with up to 9360 observations (24 trading days). Furthermore, the training times of such complex models are generally higher — typically in the range of tens of hours (e.g., TimeGAN required 39 hours of training, compared to just 4 hours and 20 minutes for our model). Finally, Table 2 presents also an ablation study to assess the contribution of the cross-correlation term in the critic score. In particular, rows one and two of the table report the discriminative score ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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for 𝛼 = 0 and 𝛼 = 1, respectively. The results indicate that incorporating the cross-correlation term improves model performance, especially in settings with high inter-feature correlation (i.e., high values of 𝜙), thereby demonstrating the effectiveness of the cross-correlation coefficients. Correlation dynamics on financial synthetic data. As previously mentioned, our model has comparable performance wrt more recent approaches, though it does not outperform them in terms of realism. However, our model is specifically designed for financial data, and to improved correlation dynamics of synthetic data. Therefore, we now evaluate the most promising approaches in Table 2, i.e. COSCI-GAN [41] and GT-GAN [21], with respect to our financial dataset, to highlight the advantages of CoMeTS-GAN over the aforementioned general-purpose approaches. Figure 10 depicts the pair-wise correlations of the stocks’ price over windows of one trading day (390 time-steps) showing that CoMeTS-GAN is the best model in respecting the correlation values of the real data, as also reported in Table 3 in terms of Wasserstein distance between the histograms. CoMeTS-GAN KO - PEP KO - NVDA KO - KSU PEP - NVDA PEP - KSU NVDA - KSU
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Table 3. Wasserstein distance of the distributions in Figure 10.
Figure 11 shows the distribution of the correlation between the average traded volume and the volatility over two trading days. According to the stylized fact, trading volume and volatility are positively correlated, and it is clear that CoMeTS-GAN’s data are the best in reproducing this expected behavior. 5.3.3 Reactivity. An important what-if question in finance concerns the market impact of metaorders and the subsequent propagation of price shocks [5]. In other words, how would the market react if a given amount of money were traded? In this section, we demonstrate how our conditional framework can be leveraged to simulate interactive market dynamics, and address such questions. In particular, we conducted an experiment in which a perturbation was manually introduced to the Gaussian Dataset Temporal Correlations (fixing 𝜎 = 0.8) Model
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0.211 ± 0.007 0.187 ± 0.003 0.494 ± 0.004 0.167 ± 0.007 0.364 ± 0.082 0.175 ± 0.006 0.177 ± 0.012 0.391 ± 0.006 0.500 ± 0.000 0.498 ± 0.002 0.337 ± 0.005 0.336 ± 0.011
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0.137 ± 0.012 0.102 ± 0.006 0.437 ± 0.025 0.070 ± 0.018 0.172 ± 0.120 0.105 ± 0.005 0.133 ± 0.019 0.220 ± 0.016 0.499 ± 0.001 0.396 ± 0.018 0.229 ± 0.013 0.230 ± 0.023
0.187 ± 0.006 0.180 ± 0.009 0.305 ± 0.019 0.091 ± 0.020 0.085 ± 0.091 0.181 ± 0.006 0.186 ± 0.012 0.198 ± 0.011 0.499 ± 0.001 0.460 ± 0.003 0.217 ± 0.010 0.192 ± 0.012
0.206 ± 0.018 0.164 ± 0.011 0.311 ± 0.037 0.075 ± 0.030 0.147 ± 0.110 0.152 ± 0.011 0.190 ± 0.012 0.202 ± 0.010 0.499 ± 0.001 0.408 ± 0.016 0.226 ± 0.011 0.205 ± 0.015
0.137 ± 0.012 0.102 ± 0.006 0.437 ± 0.024 0.070 ± 0.018 0.172 ± 0.120 0.105 ± 0.005 0.133 ± 0.019 0.220 ± 0.016 0.499 ± 0.001 0.396 ± 0.018 0.229 ± 0.013 0.230 ± 0.023
−− 0.013 ± 0.007 0.092 ± 0.045 0.042 ± 0.019 0.012 ± 0.014 0.011 ± 0.008 0.022 ± 0.008 0.229 ± 0.040 0.495 ± 0.001 0.430 ± 0.027 0.158 ± 0.011 0.277 ± 0.013
Table 2. Discriminative score for the Gaussian and Sines datasets. First and second best are highlighted in bold and underline respectively.
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time-series of KO, to observe how the model reacts in generating the other stocks. In particular, we focus on the reaction of PEP, which is highly positively correlated with KO. Specifically, let 𝒙ˆ𝑡 :𝑡 +𝐹 be the sub-sequence generated by the model starting at time 𝑡. A perturbation is introduced by substituting what the model actually generates with 𝒙ˆ𝑡 :𝑡 +𝐹 + 𝛼 · 𝜎 ( 𝒙ˆ𝑡 :𝑡 +𝐹 ), where 𝜎 (·) being the standard deviation function. Figure 12 shows the perturbation introduced on KO in the red window. It is interesting to observe how the model “propagates" the shock and adjusts the generation of the non-perturbed stock in order to preserve the correlation following the perturbation event. Figure 13 shows the correlation of KO and PEP, varying the intensity of the perturbation. The orange line is the correlation between real KO and real PEP, which is highly positive, close to 1. The c and PEP, d which is still positive green line is the correlation between their synthetic versions KO d and high. Let KO𝑝 be the synthetic version of KO subject to the perturbation. The red line is the d𝑝 and PEP. d Let PEP 𝑟 be the synthetic version of PEP generated by the model correlation between KO d𝑝 and in response to the perturbation observed in KO. The purple line is the correlation between KO ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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Fig. 12. Price evolution of KO (top) and PEP (bottom) under two scenarios: with and without a perturbation applied to KO (𝜎 = 0.4, highlighted period).
𝑟 . The experiment was repeated for several seeds, generating the standard error evidenced by PEP the bands in the figure. As we can see, the model punctually intervenes in adjusting the generation of PEP to preserve the correlation with KO after the perturbation. 1.00 0.75
Correlation coefficient
0.50 0.25 0.00 0.25 0.50 1.00 0.1
(KOp, PEP) (KOp, PEPr)
(KO, PEP) (KO, PEP)
0.75 0.2
0.3
0.4
0.5
0.6
0.7
Fig. 13. Correlation of the KO and PEP stocks varying the intensity of the perturbation. The experiment demonstrates that a shock to KO’s price induces a correlated response in PEP, illustrating CoMeTS-GAN ’s capacity to simulate realistic cross-asset market reactions, preserving inter-stock dependencies following external interventions.
5.3.4 Scalability. Considering the model’s architecture in Figure 2, the size of the input and output of the generator and the input of the discriminator linearly depend on 𝑛. On the other hand, the intermediate hidden layers of the networks remain constant with respect to 𝑛. The linear layer in the critic takes as input a vector of size 2𝑛 2 . To test the model’s scalability, we run the auto-regressive application on all 30 stocks included in the DJIA index (Figure 14), obtaining realistic correlation dynamics. Indeed, the synthetic traces still show the expected correlation properties, even when considered in the simultaneous generation of numerous time-series. Furthermore, all the stylized facts that characterize the returns are preserved.
ACM J. Data Inform. Quality, Vol. 37, No. 4, Article 111. Publication date: August 2018.
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Observed
Prices
Real
AAPL
AMGN
AXP
CRM
CSCO
GS
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Synthetic BA
CAT
CVX
DIS
GE
HD
HON
IBM
INTC
JNJ
JPM
KO
MCD
MMM
MRK
MSFT
NKE
PG
TRV
UNH
V
VZ
WBA
WMT
Fig. 14. Concurrent price generation of the 30 components of the DJIA, demonstrating the framework’s scalability.
5.4
A Critic-as-a-Guide framework
While WGANs leverage a critic to estimate the Wasserstein distance and enforce realistic data distributions, diffusion models utilize iterative noise conditioning to generate high-fidelity samples. In this section, we explore the theoretical and practical implications of repurposing a pre-trained WGAN critic as a guidance mechanism for classifier-guided diffusion models, with a focus on stock price generation. To assess efficacy, we propose benchmarking both guided and unguided diffusion models by examining the correlation dynamics of the generated time-series. Specifically, we employ a diffusion model based on DiffTime [8] for the unconditional generation windows of 150 time-steps across four selected stocks: KO, PEP, NVDA, and KSU. For each generated window, we compute the pairwise correlations between the stocks, resulting in the distributions visualized in Figure 15a. It illustrates the correlation distributions derived from the unguided diffusion model (𝑤 = 0 in Eq. 3), while Figure 15b shows those obtained with guided generation (𝑤 = 250), highlighting a much closer alignment with the real data distribution and demonstrating a substantial improvement due to guidance. To quantify the improvement, we report the Wasserstein distance in Table 4. To further highlight the value of re-purposing the WGAN critic to guide the generation, Figure 15c presents the generation of a counter-factual scenario. In particular, the guidance is applied in the opposite direction, i.e., by flipping the classifier gradients (𝑤 = −250). By reversing the guidance signal, the generated samples are explicitly discouraged from aligning with the real data distribution, resulting in correlation patterns that diverge markedly from the empirical ones and from those generated by both the guided and unguided models. This experiment underscores the importance of effective guidance not only for enhancing sample quality but also for generating meaningful counterfactuals under adversarial guidance.
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Real
Density
Density
Density
4
KO - PEP
2 0
1
0
1
DiffTime
4 KO - NVDA
4
2
2
0
1
0
0
1
DiffTime + Critic
KO - KSU
1
0
1
DiffTime + Critic (CF)
4 PEP - NVDA 4 PEP - KSU
4 NVDA - KSU
2
2
0
2 1
0
1
0
1
0
1
(a) Synthetic.
0
1
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4 NVDA - KSU
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1
0
1
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1
0
1
(b) Synthetic with guidance.
4 KO - PEP
0
1
2
0
1
KO - KSU
1
0
1
0
1
0
1
4 PEP - NVDA 4 PEP - KSU
4 NVDA - KSU
2
2
0
2 1
0
1
0
1
0
1
0
1
0
1
(c) Synthetic with guidance for counterfactual generation. Fig. 15. Distributions of price correlations between asset pairs for real data and synthetic series generated by three diffusion model variants: DiffTime, DiffTime + Critic, and DiffTime + Critic (CF). Adding the Critic as guidance (middle row) results in synthetic correlations that more closely match the real market structure, while counterfactual guidance (bottom row) produces distinctly different correlation patterns, illustrating the model’s flexibility for simulating both realistic and counterfactual scenarios.
DiffTime + Critic (𝑤 = 250) DiffTime (𝑤 = 0) KO - PEP KO - NVDA KO - KSU PEP - NVDA PEP - KSU NVDA - KSU
0.07 0.09 0.15 0.04 0.17 0.04
0.26 0.21 0.20 0.23 0.19 0.30
Table 4. Wasserstein distance of the distributions in Figure 15.
6
Discussion and Conclusion
In this work, we discuss novel solutions for generating high-quality interdependent markets using deep learning models. Our quality-aware framework introduces three key contributions: First, we create a new critic score for a C-WGAN, specifically suitable for time-series generation, based on the degree of realism of the cross-correlation between the features; Then we introduce a new architecture based on a C-WGAN, to capture aspects of interdependence among financial markets and to generate time-series with similar characteristics to real ones; Finally, we discuss how our
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Critic can be repurposed to enhance the performance of any existing diffusion model framework. We thoroughly analyze the model’s performance both on benchmark datasets and new stock market traces measuring the performance in terms of Discriminative Score, Diversity, Stylized Facts, Reactivity, and Scalability. The results are promising: in all respects, the proposed architecture yields good results with respect to state-of-the-art approaches such as TimeGAN, solving its inherent issues related to training time and temporal continuation of long time-series. We believe this work demonstrates that a small GAN architecture can be effectively employed in specific domains, to preserve particular statistical properties of synthetic data. Most importantly, GANs’ critics can be reused within more sophisticated diffusion models without the need for retraining. We release our code to foster future research in this direction. Code and Data availability Data are extracted from LOBSTER3 , an online limit order-book data provider, which is publicly available for the research community with an annual fee. We commit to publicly releasing our source code upon acceptance. References [1] [n. d.]. LOBSTER: Limit Order Book System - The Efficient Reconstructor. https://lobsterdata.com/. [2] Martin Arjovsky, Soumith Chintala, and Léon Bottou. 2017. Wasserstein Generative Adversarial Networks. In Proceedings of the 34th International Conference on Machine Learning. PMLR, 214–223. [3] Shaojie Bai, J Zico Kolter, and Vladlen Koltun. 2018. An empirical evaluation of generic convolutional and recurrent networks for sequence modeling. arXiv preprint arXiv:1803.01271 (2018). [4] Malcolm Baker and Jeffrey Wurgler. 2007. Investor sentiment in the stock market. Journal of economic perspectives 21, 2 (2007), 129–151. [5] Jean-Philippe Bouchaud, Julius Bonart, Jonathan Donier, and Martin Gould. 2018. Trades, quotes and prices: financial markets under the microscope. Cambridge University Press. [6] Hans Buehler, Lukas Gonon, Josef Teichmann, and Ben Wood. 2019. Deep hedging. Quantitative Finance 19, 8 (2019), 1271–1291. [7] Longbing Cao. 2022. Ai in finance: challenges, techniques, and opportunities. ACM Computing Surveys (CSUR) 55, 3 (2022), 1–38. [8] Andrea Coletta, Sriram Gopalakrishnan, Daniel Borrajo, and Svitlana Vyetrenko. 2023. On the constrained time-series generation problem. Advances in Neural Information Processing Systems 36 (2023), 61048–61059. [9] Andrea Coletta, Matteo Prata, Michele Conti, Emanuele Mercanti, Novella Bartolini, Aymeric Moulin, Svitlana Vyetrenko, and Tucker Balch. 2021. Towards Realistic Market Simulations: A Generative Adversarial Networks Approach. In Proceedings of the Second ACM International Conference on AI in Finance (ICAIF ’21). 1–9. [10] Rama Cont. 2001. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative finance 1, 2 (2001), 223. [11] Rama Cont, Mihai Cucuringu, Renyuan Xu, and Chao Zhang. 2025. Tail-gan: Learning to simulate tail risk scenarios. Management Science (2025). [12] Rama Cont and Milena Vuletić. 2025. Data-driven hedging with generative models. Available at SSRN 5282525 (2025). [13] Christa Cuchiero, Wahid Khosrawi, and Josef Teichmann. 2020. A Generative Adversarial Network Approach to Calibration of Local Stochastic Volatility Models. Risks 8, 4 (Dec. 2020), 101. doi:10.3390/risks8040101 [14] Prafulla Dhariwal and Alexander Nichol. 2021. Diffusion models beat gans on image synthesis. Advances in neural information processing systems 34 (2021), 8780–8794. [15] Chris Donahue, Julian McAuley, and Miller Puckette. 2018. Adversarial audio synthesis. arXiv preprint arXiv:1802.04208 (2018). [16] Cristóbal Esteban, Stephanie L. Hyland, and Gunnar Rätsch. 2017. Real-Valued (Medical) Time Series Generation with Recurrent Conditional GANs. arXiv:1706.02633 [cs, stat] (Dec. 2017). arXiv:1706.02633 [cs, stat] [17] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. 2014. Generative Adversarial Nets. In Advances in Neural Information Processing Systems, Vol. 27. Curran Associates, Inc. [18] Alex Graves. 2013. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850 (2013). 3 https://lobsterdata.com/
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Received 20 February 2007; revised 12 March 2009; accepted 5 June 2009
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