SBBTS: A Unified Schrödinger-Bass Framework for Synthetic Financial Time Series

Alexandre Alouadi; Grégoire Loeper; Célian Marsala; Othmane Mazhar; Huyên Pham

cs.LGApr 8, 2026

SBBTS: A Unified Schrödinger-Bass Framework for Synthetic Financial Time Series

Authors: Alexandre Alouadi, Grégoire Loeper, Célian Marsala, Othmane Mazhar, Huyên Pham

Organizations: BNPP and Ecole Polytechnique. · BNPP and Monash University · BNPP and ENSAE Paris · LPSM, Université Paris Cité and Sorbonne University. · Ecole Polytechnique, CMAP.

Abstract

We study the problem of generating synthetic time series that reproduce both marginal distributions and temporal dynamics, a central challenge in financial machine learning. Existing approaches typically fail to jointly model drift and stochastic volatility, as diffusion-based methods fix the volatility while martingale transport models ignore drift. We introduce the Schrödinger-Bass Bridge for Time Series (SBBTS), a unified framework that extends the Schrödinger-Bass formulation to multi-step time series. The method constructs a diffusion process that jointly calibrates drift and volatility and admits a tractable decomposition into conditional transport problems, enabling efficient learning. Numerical experiments on the Heston model demonstrate that SBBTS accurately recovers stochastic volatility and correlation parameters that prior SchrödingerBridge methods fail to capture. Applied to S&P 500 data, SBBTS-generated synthetic time series consistently improve downstream forecasting performance when used for data augmentation, yielding higher classification accuracy and Sharpe ratio compared to real-data-only training. These results show that SBBTS provides a practical and effective framework for realistic time series generation and data augmentation in financial applications.

Figures & tables

Figure 1: Architecture of the model $s_{\theta}$

\left\{Y_{t_{i}}^{b}=X_{t_{i}}^{b}-\frac{1}{\beta}s_{\theta}^{k}(t_{i},X_{t_{i}}^{b},\Phi_{\theta}^{k}(X_{t_{0}:t_{i}}^{b}))\sim\mathscr{Y}_{t_{i}}^{k}\#\delta_{X_{t_{i}}^{b}}\right\}_{i\leq N-1}

Algorithm 1 SBBTS training algorithm

Panel of Figure 2 — Figure 2: Distribution of estimated Heston parameters using MLE. We show in blue, orange and green the density respectively from the data, SBTS, and SBBTS samples.

Classification metrics
Metric	Zero-Shot	Real + Noise	Real	SBBTS
Accuracy ( $\uparrow$ )	$0.494$	$0.518_{\,(0.008)}$	$0.521_{\,(0.006)}$	$\mathbf{0.532}_{\,(\mathbf{0.005})}$
Log Loss ( $\downarrow$ )	$0.756$	$0.695_{\,(0.004)}$	$0.693_{\,(0.003)}$	$\mathbf{0.691}_{\,(\mathbf{0.002})}$
ROC AUC ( $\uparrow$ )	$0.486$	$0.494_{\,(0.012)}$	$0.497_{\,(0.008)}$	$\mathbf{0.521}_{\,(\mathbf{0.007})}$
Financial metrics
Avg Daily Return (%) ( $\uparrow$ )	$-0.020$	$0.086_{\,(0.035)}$	$0.112_{\,(0.020)}$	$\mathbf{0.143}_{\,(\mathbf{0.015})}$

Table 1: test set performance of TabICL under different training configurations. Results are averaged over 5 seeds; standard deviations are reported in parentheses.

\downarrow

indicates lower is better,

\uparrow

indicates higher is better. Best result per row is bold .

Figure 3: Cumulative return (left) and cumulative excess return (right) on the test period, with S&P 500 index baseline.

Figure 4: Validation ( left ) and test set ( right ) performance as a function of the amount of synthetic data generated by SBBTS . Results are averaged over 5 seeds; error bars indicate one standard deviation.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

$K$	$T$	$\tilde{T}$	$n_{\text{epoch}}$	Batch Size	$lr$	$d_{\text{model}}$	$n_{\text{head}}$	$N^{\pi}$
$5$	$1$	$0.99$	$1000$	$128$	$10^{-3}$	$128$	$16$	$50$

Appendix

Table 2: Parameters used in the numerical experiments.

$\kappa$	$\theta$	$\xi$	$\rho$	$r$
$[0.5,\,4]$	$[0.5,\,1.5]$	$[0.1,\,0.9]$	$[-0.9,\,0.9]$	$[0.01,\,0.1]$

Appendix

Table 3: Parameter ranges used for simulating the Heston process.

Panel of Figure 5 — Figure 5: Autocorrelation functions of returns and squared returns for real and synthetic data across clusters.

Figure 6: Distribution of cluster factors in real and synthetic data.

Figure 7: Correlation matrices of returns for real and synthetic data.

	Real	SBBTS	SBTS
$\text{VaR}_{99\%}$ (%)	$3.60$	$3.57$	$3.49$
$\text{VaR}_{95\%}$ (%)	$2.11$	$2.19$	$2.17$
$\text{ES}_{99\%}$ (%)	$4.65$	$4.44$	$4.36$
$\text{ES}_{95\%}$ (%)	$3.15$	$3.18$	$3.07$
Ann. Ret (%)	$19.02$	$16.31$	$14.68$
Ann. Std (%)	$24.22$	$24.38$	$23.06$

Appendix

Table 4: Comparison of tail-risk metrics (VaR, ES) and annualized performance statistics averaged across instruments.

	Real	SBBTS
Validation	$[-1.73,\ 1.28]$	$[-0.62,\ 2.30]$
Test	$[0.28,\ 3.32]$	$[0.58,\ 3.34]$

Appendix

Table 5: 95% bootstrap confidence intervals for the Sharpe ratios obtained under real-only training and SBBTS -based data augmentation.

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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.

Abstract

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