STABLE: Shift-Tolerant Allocation via Black-Litterman Using Conditional Diffusion Estimates
Authors: YEJUN SOUN
Hosung Lee
Suyoung Park
U Kang
Venue: ICLR 2026 Poster (2026)
Keywords: portfolio allocation
time-series estimation
generative modeling
Primary area: learning on time series and dynamical systems
TL;DR: We present STABLE, a portfolio allocation method that maximizes expected risk-adjusted return

Abstract: In dynamic financial market characterized by shifting regimes, how can we make effective investment decisions under the changing 1) market regimes and 2) their impact?
Among many research fields in financial AI, portfolio allocation stands out as one of the most practically significant areas.
Consequently, numerous researchers and financial institutions continually seek approaches that improve the risk–reward trade-off and strive to apply them in real-world investment scenarios.
However, achieving robust risk-adjusted performance is extremely challenging, because each asset's return and volatility fluctuate according to the shifting market regime.

In response, modern portfolio theory (MPT) addresses this issue by solving for asset weights that maximize a risk–reward objective, using estimates of the return mean and covariance from historical returns.
Reinforcement learning (RL) frameworks have been introduced to directly decide portfolio allocations by optimizing risk‑adjusted objectives using asset prices and macroeconomic indices.

In this work, we propose STABLE (Shift-Tolerant Allocation with Black-Litterman Using Conditional Diffusion Estimates), which combines a diffusion-based generative model that captures regime shifts with an estimation-based portfolio allocation module that maximizes expected risk-adjusted return. STABLE takes macroeconomic context and asset-specific signals as inputs and generates per-stock return trajectories that reflect the prevailing macro regime while preserving firm-specific dynamics.
This yields regime-aware predictive return distributions at the single-stock level together with a coherent covariance structure, which are then incorporated as investor views within a Black-Litterman allocation module to obtain risk-diversified portfolio weights.
Empirically, STABLE delivers superior portfolio outcomes, achieving up to 122.9% higher Sharpe ratios with reduced drawdowns across major equity markets.
It also attains state‑of‑the‑art time‑series estimation, lowering MSE by up to 15.7% compared with generative baselines.

STABLE: SHIFT-TOLERANT ALLOCATION VIA BLACK-LITTERMAN USING CONDITIONAL DIFFU- SION ESTIMATES
Yejun Soun", Hosung Lee', Suyoung Park U Kang" Scoul National University, Scoul, South Korea DDep'Trade Technologies Inc. ma aathieu775 781 90gmail comm, ukangesnu.ac. acc
ABSTRACT
In dynamic financial market characterized by shifting regimes, how can we make effective investment decisions under the changing 1) market regimes and 2) their impact? Among many research fields in financial AI, portfolio allocation stands out as one of the most practically significant areas. Consequently, numerous re- searchers and financial institutions continually seek approaches that improve the risk-reward trade-off and strive to apply them in real-world investment scenarios. However, achieving robust risk-adjusted performance extremely challenging, because each asset's 's return and volatility fluctuate according to the shifting mar- ket regime. response, modern portfolio theory (MPT) addresses this issue by solving for asset weights that maximize risk-rewart objective, using estimates of return mean and covariance from historical returns. Reinforcement learning (RL) frameworks have been introduced to directly decide portfolio allocations by optimizing risk- adjkstdestedd objectives using asset prices and macroeconomic indices.
In this work, we propose STABLE (Shift-Tolerant Allocation via Using Conditional Diffusion Estimates), which combines diffusion- based gener- ative model that captures regime shifts with an estimation- based portfolio alloca- tion module that maximizes expected risk-adjusted return. STABLE takes macroe- conomic context and asset-specific signals as inputs and generates per-stock return trajectories that reflect the prevailing macro regime while preserving firm-specific dynamics. This yields regime-aware predictive return distributions at the single- stock level together with coherent covariance structure, which are then incor- porated as investor views within a Litteran allocation module to obtain risk-diversified portfolio weights. Empirically, STABLE delivers superior portfolio outcomes, achieving up to 122.9% higher Sharpe ratios with reduced drawdowns across major equity markets. It also attains state-of-the-art time-series estimation, lowering MSE by up to 15.7% compared with generative baselines.
INTRODUCTION
Given historical macroeconomic data and stock data, how can we make effective investment deci- sions under changing market regimes? Because portfolio allocation directly influences investment performance, it is essential not only to aim for high profitability but also to control volatility and
drawdowns for practical deployment (Wang al. 2019 Sun al., 2022 Niu al., 2022 Jeon
et al. 2024
Lee et al.] 2025).
However, to perform robust portfolio optimization under shifting market regimes, we must over- come three key challenges. First, stocks are high-risk assets with substantial exposure to global macro conditions, thus failing to jointly model these macro drivers with firm-specific factors un- dermines predictive accuracy for price dynamics. Second, even when both global factors and local factors, is difficult to know how strongly each factor influences each stock, and the influence varies across assets and over time. Third, individual assets in a portfolio
often correlated Yooo & Kang, 2021 Yoo et al.] 2021 Soun et al., 20022 Kim et al. 2024). so failing to diversify risk when deter- mining weights cause severe drawdowns. There have been existing works relying portfolio

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Market
Teaperader
AlphaMix
MetaTrader EURO50
STABLE (Proposed)
56,9500
C5/300
KDSPI200
Volatilly (50)
30
25
20
20
20
30
15
10
15
20
Best
10
10
10
25-01
25-03
25-01
25-03
25-01
25-03
25-01
25-03
Date
Date
Date
Date
Figure 1: Annualized portfolio volatility across regions. We measure the annualized portfolio volatility in four real-world stock markets (United States, China, Europe, and South Korea). With estimation-driven Black-Litterman optimization, STABLE achieves effective risk diversification and yields the lowest realized volatility across all regions and periods, delivering robust portfolio man- agement.
optimization to maximize risk-adjusted returns (Pun et al. 2020 Sun et al., 2024 Ma et al., 2022), but they depend too much on past asset returns and lack predictive power for future returns, which causes severe performance drops under regime shifts. Others adopt deep reinforcement learning to derive regime-specifie strategies (Ye et al.) 2020, Hu & Lin 2019) Gao et al.) 2020 Jeon et al,, 2026), but they tend select regimes primarily from macro signals, overfit the prevailing macro state, and fail to capture stock-level idiosyncratic movements.
To overcome the limitations of both classical portfolio theory and deep policy network- mod- els, we propose STABLE (Shift-Tolee Tofft-Toleratt Allocation via Black-Litterman Using Conditional Dif- fusion Estimates) that robustly allocates portfolio weights with maximized expected risk-adjusted return. First, STABLE uses a conditional diffusion mechanism to accurately sample per-stock return paths each step while jointly conditioning on the current macro-regime state and stock-specific features. Second, within the diffusion process, STABLE effectively decomposes per-step noise for each asset into macro impact and micro (fim--specific) impact, yielding strong reconstruction performance. Third, STABLE injects the resulting forecasted views into Black-Littermn alloca- tion module so that the final portfolio weights improve both performance and stability. Fig. shows the cross-region robustness comparison.
Our contributions are summarized as follows:
Method, We present STABLE (Shift-Tolere Allocation via Black-Litterman Using Con- ditional Diffusion Estimates), a portfolio allocation method that maximizes expected risk- adjusted return. STABLE couples conditional diffusion module with Black-Litterman portfolio construction produce regime-awar predictive distributions for portfolio opti- mization. STABLE addresses central shortcomings of conventional mean-vant vvriinne (MVO) approaches and deep RL methods.
Exclusive experiments. We evaluate STABLE on four regional stock markets (United States, China, Europe, and South Korea) with two tasks: portfolio allocation and time-series estimation. STABLE achieves fate-offthheart performance in both tasks. In the portfolio task, STABLE improves the Sharpe ratio by up to 122.9% over the best competitor, demonstrating superior risk-adjusted return. In the estimation task, STABLE improves the MSE metric by up to 15.7% over the best competitor, indicating accurate stock time-series prediction.
Case study. Visual inspection shows that the temporal stock embeddings faithfully re- flect real-world sector relationships and allow stocks with similar characteristics each time step to reference one another. This enables STABLE to condition on intrinsic stock properties, which turn improves time-series estimation accuracy.
Symbols are summarized in Appendix A.I
RELATED WORK
We categorize existing portfolio allocation methods based on how they determine asset weights. Broadly, they can be divided into modern portfolio theory (MPT) methods and deep reinforcement learning approaches.
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Modern portfolio theory. Classical MPT seeks closed-form solutions for weight allocation by solv- ing a meca--vrriance optimization (MVO) problem (Markowitz) 1952b). Representative examples include Black-Littermas and robust portfolio optimization. The Black-Litterman model combines investor views with Markowitz-style portfolio optimization to determine allocation (Black & Litter- man (990), while robust portfolio approaches define an ambiguity set to account for uncertainty in asset returns and then allocate weights based on worst-case scenarios (Goldfarb & Iyengar 2003)) Although these methods strengthen traditional portfolio optimization by plugging in estimates of the mean and covariance of stocks, they are effective only when those estimates are accurate at each rebalancing time. However, because their estimation is typically restricted to historical windows, any post-allocation regime shift that drives the realized distribution depart from the past severely degrades both profitability and stability. Instead, STABLE leverages generative sampling model conditioned on both macro-level and micro-level signals to improve estimation accuracy. The gener- ative sampling process yields per-stock predictive distributions from which the mean and covariance computed, enabling more effective plug-in in to the allocation stage (see Section 3.5)
Deep reinforcement learning. Another line of work applies deep reinforcement learning to port- folio allocation. Under this paradigm, policy network learns output actions (i.e., portfolio weights) to maximize cumulative rewards such as risk-adjusted return metrics. Early approaches rely neural network that directly proposes allocations. Subsequent research incorporates market- regime considerations, aiming to adapt allocations more flexibly. For instance, Alphastock (Wang al. 2019) introduces asset-axis attention to capture correlations among multiple assets, improving weight computation. Similarly, MetaTrader (Niu et al. (2022) proposes a strategy that selects from typical financial domain baselines (c.g.. constant rebalanced portfolio, Markowitz portfolio) under different market regimes. AlphaMix (Sun et al. 2023) employs routing mechanism that switches between multiple neural network models depending on market conditions, thus incorporating regime awareness into the policy. Despite using regime-aware policy networks, these approaches overlook the aspect that the degree to which a given macro state influences returns stock-specific. Conse- quently, their portfolio allocation overfits the macro state and fails to capture stock-level idiosyn- cratic movements. contrast, STABLE employs a learnable guidance scale to decompose per-step noise into macro impact and firm-specific impact, enabling fine-grained per-stock distribution estimation and downstream portfolio allocation (see Sections 3.3 and
3 PROPOSED METHOD
3.1 PROBLEM DEFINITION
Given macro condition m, dR per-stock conditions ct prior mean M E RS and prior covariance Eprior,r E SxS computed from the most recent business days, and investment horizon L,
(e) Rde, prior window length v,
Allocate
E E(w, R
: E arg max Wi
Var(w, RT,
where Rr R denotes the realized adjusted-close returns over the next business days for the S stocks,
Such that the budget constraint holds: 1 [w-- 1.
3.2 OVERVIEW
To solve the portfolio allocation problem defined in Section 3.1 our proposed STABLE executes three stages as summarized in Figure 2 First, the Conditional Diffusion Generator (CDG) gen- erates per-stock return segments conditioned on macro context and firm-specific properties at an individual stock level. Second, Multi-Level Guidance (MLG) constructs the estimated noise by combining the shared impact with the idiosyncratic component, modulated by learnable gate, yielding eegime-aware per-stock distributions. Third, a Black-Literman-bassé Mean-Variance Optimizer (BL-MVO) consumes the iiffusiioninducee moments views and outputs allocation weights that balance diversification and estimation. The main challenges and our ideas are as fol- lows.
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11. Conditional Diffusion Generator
Litterman Mcan Variance Optimizer
Carditional DDIM
multiple (R) astimatioo
Mecro Feesure Ce
(eelre Entcoder W
For andy ce
for vs
R
R
balancing gate
Refine Encodder Wa
Horr E & Ear.r 1t
systematic noise
dicsynoratic noise
+ MVD
2
Xel Mor. R Mair
w:
Conv UNet
Conv UNet
Balancing Gate
10.2%
T34A 84
macco condition
condition
11.8%
12. Multi Level Guidancee
Figure 2: Overview of STABLE. (II) CDG conditions on macro and firm signals to generate per- stock return segments. (I2) MLG decomposes noise into shared (systematic) and idiosyncratic parts via a learnable gate. (I3) BL-MVO fuses diffusion views with rolling prior to produce regime- aware portfolio weights.
Regime shift. How can we accurately estimate future time series markets whose regimes keep changing?
C2 Multi-level factors. How can we separate, at each time step for each stock, the influence of macro-level factors from stock-level factors?
C3 Uncertainty shifts. How can we maintain robust portfolio performance when the certainty of per-step estimates differs over time?
We address the above challenges with the following ideas.
11 Conditional diffusion generator (CDG, Sec. 3.3). We synthesize regime-aware return paths by conditioning diffusion on macro-regime features and stock-specific features. 12 Multi-Level Guidance (MLG, Sec. 3.4). We adapt a guidance modeling framework to financial domain to estimate a shared systematic noise and stock-specific idiosyneratic noise with a learnable gate that adjusts their relative importance over time.
13 Black-Litterman- diffusion-based views with a certainty weighted Black-Litterman update, which yields per-time, per-stock posterior estimates and enables rational, robust allocation.
vean-variance optimizer (BL-MVO, Sec. 3.5). We combine
3.3 CONDITIONAL DIFFUSION GENERATOR (CDG)
Regime-aware sampling, STABLE estimates return segments with a Denoising Diffusion Implicit Model (DDIM)-based conditional diffusion sampler (see A.2] in Appendix for details DDIM). Under the standard random-wall view of log returns, price noise is modeled as Gaussian (Fama) 1995). Diffusion models likewise inject Gaussian perturbations in the forward process and learn to remove them in reverse, the noise assumptions aligned with conditional setting. This alignment makes conditional DDIM a natural mechanism to generate stock-retum paths conditioned on market and firm states.
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Table 1: Macro indices and description, After v-day rolling normalization and log differencing, raw and processed values compose m, E Rdm..
Index
Description
Market index
Region-level equity baseline (overall market condition).
Dollar index
Currency strength affecting equities via FX channels. Bond-marke condition and liquidity measure.
U.S. term spread (10y-3m)
VIX index
Forward- equity volatility (risk expectation). Risk aversion inflation measure.
Gold index
Inputs and refinement. At rebalancing time т, CDG conditions on two inputs that summarize market context and firm identity. The macro feature m,, E IR encodes the global regime at time and is refined by a linear layer W. E Rd xd into a refined macro condition hm,r = m. W. E
The macro features are described in Table 1 The corporate-specife feature el ERo de for stock time r concatenates (i) the temporal stock embedding 3. at T (Das Ghoshal 2010), and (ii) the last normalized adjusted-closs level and daily log returns for A linear layer W. Rd yields a refined stock-level condition h = eT (z)T W. E RRxdd, and the refined full condition becomes
"r.r [hm,r || | E RR122
Given ( CDG produces denoised length-l return segment
p(s)
ER for stock time f0r
Temporal stock embedding. We construct part the stock-specific feature cf²) using temporal stock embedding vector g(*) E Rd at time т. Prior approaches often use static metadata such as sector labels neural embeddings from price series (Dolphin et al. 20022, but these representations are either fixed over time or fail to reflect macroeconomic regimes.
To address this, apply Kalman filtering to estimate time-varying coefficient ((()) with respect to the macro input m,, Given the log return o of stock as the dependent variable and the maero vector my the independent variable,
estimate contemporaneous macro sensitivity vector 3(*) via recursive filtering. This posterior estimate 3f reflects the stock's embedding at time т, incorporating all observations up to that point. It serves as temporal and robust representation that captures regime-aware macro sensitivity of each stock.
Conditioned DDIM synthesis. conditioning and its decomposition are detailed in Section[.4 and the DDIM notation is defined
We first define the base conditioned denoiser. The multi-level
in Section A.1 Let r(a) E 2 be the reverse-chain state for stock s at time and DDIM step
(n = N, N,..., 1,0). The denoiser predicts noise ē with rn, and ((())
i, and DDIM updates
(())
(e)
VI- on
(s) n-1 vān-1 or VI Ōn-1 2,
(17 0)
0.7
an
Training objective. s S, rebalancing times T ET, and DDIM steps n E.W.To mitigate overfitting, we add an &g penalty on the parameters with hyper parameter 3.
STABLE minimizes the diffusion mean-squared error (MSE) across all stocks
an [()) Eg.r,n, e 2 211
B 11012.
Since c~~((0, I) and Eq. minimizes E|| 212 asymptotically we 2 (O,,
3.4 MULTI-LEVEL GUIDANCE (MLG)
Noise decomposition and gate. We use multi-level guidance that decomposes, for each rebalanc- ing time and stock s. the guided noise into a shared (macrocconomic) impact and unshared (firm-speeific) impact with a stock-specific balancing gate. This is motivated by two empirical prop- erties of the financial market. First, macro impact varies over time (Mezei & Sarlin 2014): during
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All labeled terms the equation refer to three well-defined elements: (i) the shared noise term En, (ii) the firm-specific residual oii En, and (iii) the scalar gate 2(4) The two noise terms are evaluations of the convolutional UNet denoiser at DDIM step n with inputs (Ta.r, n, h). We use Ammr to obtain the shared term E,, and h to obtain full-condition term o The gate is produced by linear map 9ø- which adjusts, the stock level, the balance between macro impact crises macro variables dominate and cross- co-movements are pronounced, whereas in calm pe- riods firm-specific signals earry more weight. Second, sensitivities to macro variables differ across stocks (Shiller 1995). By explicitly encoding this multi-level context in the denoising step. the noise estimation becomes more accurate for both regime-driven and idiosyncratic dynamics.
For stock at time at DDIM step n, define
balancing gate
é == co(rlht n, ((t))
E E En.T
n
shared (systematic)
firm- firn-specific (unsystematic)
En,r = uuorrll n,r! n, hm.r),
e(*) n,t = ud(rnh n, (,,))
) = 9m (nn(
E [0, Emax]
(x) and micro dynamics. Here us takes three inputs: the recovered state Th, the step index and the condition vector h E {hmm.r (x)
Training objective and optimization. We train all parameters jointly minimize the diffusion MSE. The optimization drives the gate 27 downward when h indicates macro-level synchroniza- tion, and upward in decoupling regimes to allocate more weight to the firm-specific residual. We replace é with ed ((r1)))
( , and rewrite the diffusion objective in Eq. (1) as
2 ea(r(e) n, (())) + 3 11012
(()) E
{0, n. Wm. Woj.
3.5 BLACK-LITTERMAN BASED MEAN- VARIANCE OPTIMIZER (BL-MVO)
Black-Litterman using generative sampler. STABLE performs portfolio allocation by feeding its stock-wise time-series estimates into the Black-Littermar (BL) algorithm to obtain an updated posterior and then solving vean-variance optimization (MVO), following the standard approach for incorporating views in the Black-Litterman model (Idzorek 2007).
Views from CDG+MLG. For each stock time 7, generate k guided paths and stack them as R(e) Rg whose i-th row constitutes the i-th estimation of the denoised sequence p(() denoted .(s,() as 00r From these samples, we first calculate the view mean vector jkview, E SRS. where the s-th
entry is computed as the sample mean of the generated paths:
(..8)
Mview,
0,
(3)
((s.)) where f( represents the cumulative return derived from the sequence To, Using this mean, we
construct the view covariance matrix Eview,r SxS the unbiased sample covariance of the cumulative-return vectors across the l paths:
Σ Eview
Σ 1=1
ro,r - (view.
(ro0r pliew,
(S..))T E R$ to capture Here, we define the return vector for the i-th path as To.- [=(1.)) ..... 00r the joint estimation errors across all assets. Finally, we fuse these views with the rolling prior (Hppiion, E using prior certainty Φ. = 1-1 Eprior,r and view certainty Ω2 = --1
view.r"
Estimated posterior and allocation. The BL posterior mean is obtained by minimizing the neg- ative log-posterior in Eq. (5) (first-order condition), and the posterior covariance follows from the

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Table 2: Summary of datasets.
Dataset
Region
#Stocks
Train from
Train cut
Test to
S&P500
United States
55
2013-01
2024-09
2025-03
CS1300
China
55
2013-01
2024-09
2025-03
EUROSTOXX
Europe
37
2013-01
2024-09
2025-03
KOSPI200
South Korea
44
2013-01
2024-09
2025-03
canonical Gaussian form Eq. 7 [A (Ø, 22)) ( Mprior,, Qt (vveww..,)).. Given (JBBL,7; ELL,)) ). the Mlarpe-maximizing weight solves Eq.
EpI.r (Φ, + Q2)
8 and has the closed form
LBL,
" 1 "'p., -RIL
which enforces the budget constraint 1T by normalization.
EXPERIMENT RESULTS
We perform experiments to answer the following research questions.
Q1 Investment performance and robustness. How does STABLE compare with state-of-the-an carfolio-allocation baselines in investment performance and robustness?
Q2 Estimation accuracy. Is STABLE'S DDIM-ccnditioned future time-series estimation supe- rior to competing generative models?
Q3 Stock embedding quality. How effectively does the Kalman-9--bssd stock embedding model each stock's 's latent state?
4.1 EXPERIMENTAL SETTINGS
All experiments run on workstation with RTX 3080 GPUs.
Datasets. To evaluate STABLE fairly across both market-wide and sector-level universes, we build four regional datasets with identical splits. Following the Global Industry Classification Standard (GICS) as of 2025-03-31, we select the top five largest stocks in each of the eleven sectors for S&P500 and CSI300, and the top four per sector for EUROSTOXX and KOSPI200. We exclude stocks whose available histories do not span the entire sample window. Table 2 summarizes the universe and the split period.
Competitors. Q1 aims to demonstrate that STABLE achieves both profitability and robustness in investment performance. We compare against range of classical and learning-basee based portfolio al- location methods. CRP (Kelly) (956) allocates equal weights and rebalances to equality at every decision time. MVO (Markowitz) 1952a) solves the mean-variance program that maximizes pected return for target variance using recent mean and covariance estimates, producing stable allocations under approximately stationary regimes. MOM (Jegadeesh & Titman 1993) assigns larger weights to recent winners based on a lookback momentum score and rebalances on a sched- ule, and suitable when returns tend to continue for the next few months. DeepTrader (Wang al.) 2021) learns to determine portfolio-
long- short balances from macro features using reinforce- ment learning. MetaTrader (Niu al. 2022) classifies the market regime based on macro inputs and switches among CRP. MVO, and MOM accordingly. AlphaMix (Sun et al. (023) routes market state representations to multiple neural allocators and selects the best action based on a soft gating policy.
Q2 aims to demonstrate that STABLE 's conditional generation of future time series is accurate. Ac- cordingly, for portfolio allocation (Section 4.2). evaluate classical allocators and regime-aware RL allocators. For time- series prediction (Section 4.3) we benchmark representative generative forecasters. Diffusion-TS (Yuan & Qiao 2024 is diffusion-basel time-series model that recon- structs past data and forecasts future trajectories using a spectral loss. AEC-GAN (Wang et al.] 2023) augments GAN with adversarial emror correction
autoregressive backbone im-

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Table 3: Portfolio allocation results sector-diversified multi-region datasets. Best performance
per column in bold. RMDD and AVol are expressed percentage units (%)
S&P500 (US)
CS1300 (China)
Method
ASR (1)
RMDD
AVol (L)
ASR (t)
RMDD
AVol (L)
CRP
0.82
8.89
14.44
-0.70
10.96
18.21
MVO
1.18
9.00
21.18
-0.66
13.18
25.93
MOM
0.03
11.87
17.00
-0.47
15.57
28.77
DeepTrader
-0.71
13.40
15.76
13.40
18.76
Meta Trader
1.00
10.88
16.82
-1.09
19.40
24.15
AlphaMix
0.35
9.59
13.92
-0.80
9.59
19.11
STABLE (proposed)
1.85
7.82
13.43
-0.41
8,85
17.17
EUROSTOXX (Europe)
KOSP1200 (South Korea)
Method
ASR (t)
RMDD (L)
AVol (L)
ASR (t)
RMDD (1)
AVol (L)
1.31
5.40
12.96
0.76
8.72
26.55
MVO
0.48
8,35
16.75
0.45
11.84
29.49
MOM
1,42
5.41
12.71
0.33
13.49
23.34
DeepTrader
-2.44
15.99
12.59
0.77
9.62
23.76
MetaTrader
0.50
9.86
14.63
0.57
10.88
22.28
AlphaMix
1.31
5.75
11.77
1.47
9,96
18.76
STABLE (proposed)
2.92
3.84
10.88
1.61
8.34
17.82
prove long-horizon accuracy. KoVAE (Naiman al. 2023) combines variational autoencoder with Koopmann-operatte latent dynamics model linear evolution the latent space.
Evaluation metrics. We report metries for the two experiments. For portfolio allocation experi- ment, we use ASR, RMDD, and AVol. Annualized Sharpe Ratio (ASR) uses daily test returns with a zero risk-free rate and is annualized. Relative Maximum Drawdown (RMDD) is the maximum peak-to-trough loss divided by the peak over the test horizon. Annualized Volatility (AVol) is the standard deviation of daily returns and is annualized. For the time-series prediction task, we report mean squared eттог (MSE) and dynamic time warping (DTW). MSE is computed between the pre- dicted and true return sequences. DTW measures the average temporal alignment cost between the predicted path and the ground-truth trajectory. Across both experiments, to ensure reproducibility.
fixed global seed
Hyperparameters. We grid-search the original segment length E {5, 10, 20), macro and stock encoder width {8, 16, 32). number of reverse DDIM steps (20, 30, 50, 80, 100), and the number of generated paths per stock for BL. views (20, 30, 50). We also investigate the forward noising steps used during training over (100, 200, 400} and the DDIM noise scale n {0. 0.01, 0.1,0.2), The gate cap Amax E {2, 3, 4}, the BL prior window length E {60, 120, 250}, and the l2 regularization weight 3 (0.001, 0.01, 0.1) are tuned per dataset. Baselines follow official implementations or paper-reported settings. We report the selected hyperparameters for each dataset in Appendix A.8 and provide details on computational transparency and reproducibility in Appendix A.9
A.8
4.2 PORTFOLIO MANAGEMENT PERFORMANCE (Q1)
Table reports ASR, RMDD, and AVol across sector-diversified regional universes. STABLE ranked first on all three metrics in every region, delivering higher risk-adjusted returns together with lower drawdowns and lower volatility. We analyze the results for different regimes Appendix[AA4
Classical strategies (CRP, MVO, MOM) are sensitive to regime shifts. In our test window with frequent drawdowns and rebounds, they exhibit larger RMDD and higher AVol, which lowers ASR relative regime-aware approach. MetaTrader, which switches among the classical strategies using macro cues, inherits the same limitations and shows similar patterns.
Among RL baselines, DeepTrader performs the worst. Its macro-only controller struggles during the "new normal" period with persistent yield-curve inversion, where macro signals alone are not sufficient determine long-short balance. AlphaMix is the strongest competitor because it routes
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Table 4: series prediction sector-diversified markets. Lower is better. We report MSE and
DTW aggregated over all stocks and rebalancing times. MSE is shown in x10-1 Normalized DTW shown in x10 Best per column in bold.
S&P500
CS1300
EUROSTOXX
KOSPI200
Method
MSE (4)
DTW (L)
MSE (4)
DTW (L)
MSE (4)
DTW (L)
MSE (4)
DTW (L)
Diffusion-TS
3.90
5.73
5.71
6.78
3.05
5.80
9.41
8.70
AEC-GAN
4.27
6.58
4.57
6.13
3.70
7.40
10.18
9.28
KoVAE
4.58
5.93
5.46
7.28
2.61
5.43
9.83
8.73
STABLE (proposed)
3.51
5.62
3.89
6.09
2.49
4.78
8.15
8.67
Table 5: Top-5 most similar stocks (by Euclidean distance) returned by our dynamic embeddings for
four query snapshots. For TSLA on 2024-12-314 12-31, the global context most strongly reflects the prices of NVDA, AVGO, AAPL, MSFT, and GOOGL.
Query
Top 1
Top 2
Top
Top 4
Top 5
TSLA 2021-06-28
AAPL
AVGO
MA
META
ECL
TSLA @ 2024-12-31
NVDA
AVGO
AAPL
MSFT
GOOGL
BAC @ 2021-06-28
JPM
WELL
WFC
DUK
MCD
BAC @ 2024-12-31
WFC
JPM
ECL.
LIN
APD
among policies by market state, yet it does not model time-varying stock-specifit properties. In contrast, STABLE conditions on temporal Kalman embeddings and adapts to stock-level regime changes, which explains the consistent gains in ASR, RMDD, and AVol.
4.3 TIME-SERIES PREDICTION ACCURACY (Q2)
We assess the accuracy conditional future return sequences predicted by STABLE. At each re- balancing time we generate trajectories per stock. We report error metrics aggregated over all stocks and times: MSE of the mean-of-k forecaster and the average DTW distance across the generated paths. Lower better for both.
Table 4 summarizes mean squared error (MSE) and dynamic time warping (DTW) across all four markets. STABLE achieves the lowest MSE and DTW in every market, indicating the most accurate conditional estimation of future return segments.
Diffusion-TS TS is the strongest baseline but remains behind STABLE. is designed for broad general- ization and does not modulate the importance of conditions at the individual-stoww level. In contrast, STABLE decomposes the noise into systematic noise that captures market-wide regularities and id- iosyncratic noise that captures firm-specific patterns. This dual modeling allows the relative weight of conditions to vary by stock and time, which improves sequence alignment and reduces prediction evaluate the covariance estimation performance derived from these return
error. Furthermore, sequences against non-generative and deep generative models in Appendix (A.5] We also present a stylized facts validation and goodness-of-fit tests the distributions of returns generated by these models in Appendix
4.4 STOCK EMBEDDING QUALITY (Q3)
To assess embedding quality, we perform nearest-nett neighbr analysis on representative U.S. stocks in our universe. For each query date, retrieve the most similar embeddings and verify whether the matches share sector category or display similar price dynamics, as expected when the embedding faithfully encodes stock identity.
Table the five closest neighbours from the dynamic embeddings for TSLA (Tesla) and BAC (Bank of America) at two snapshots (2021 and 2024). In 2021 TSLA is nearest to Big-Tech stocks such as AAPL and AVGO. By late 2024 its closest neighbours shift to Al-focused firms like NVDA and MSFT. This shift illustrates how the embeddings track the market's 's AI boom. BAC stays close to JPM and WFC at both times, confirming a stable financial-sectur relationship that our method captures over time.

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5 CONCLUSION
We propose STABLE, a regime-adaptive portfolio framework that unifies three modules. First, the Conditional Diffiusion Generator (CDG) uses market regime and stock identity as conditions, and enables accurate per-stock time-series estimation. Second, the Multi-Level Guidance (MLG) esti- mates, for each stock and time, how strongly macro and micro impacts drive the denoising process through a learnable gate. Third, the Black-Litterman-bbrr incorporates sampling certainty into view formation and produces rational and robust allocations. On real-world sector-diversified. multi-region stock market datasets, STABLE outperforms competi- tors portfolio allocation and estimation tasks. For portfolio allocation, Annualized Sharpe Ratio improves by up to 122.9% % relative maximum drawdown decreases by up to 1.56%p, and annual- ized volatility decreases by up to 7.56%. For future time-series estimation, STABLE reduces MSE by up 15.7 % and DTW by up to 13.8 % against the best competitor. Future works include extending our method to exploit more rich features including texts for macro and stock features.
Mewn-Variance Optimizer (BL-MVO)
ACKNOWLEDGMENTS
This work was supported by DeepTrade Technologies Inc. U Kang is the corresponding author.
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A APPENDIX
A.1 SYMBOLS
The symbols used in this paper is summarized in Table
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Table 6: Notation summary of STABLE.
Symbol
Description
s
Number of stocks in the universe.
t
Length of the per-stock return segment reconstructed by the diffusion model.
( E R
Next-period log-retum segment of stock (length E).
Tn,r E R
Reverse-chain state at step n (r0.7 clean; (()) noisy).
hm. E Rdd
Macro representation from my-
h Rdd
Stock-specific representation obtained from e Combined representation, | || h Diffusion schedule (a,, = BR, ag = II o.). Prior mean vector at time r. Prior covariance. View mean vector.
h()) E Rdd
an anion ān
Mprior, E R$
Eprior, RSS8
, R&S
Eview.r E RSXS
View covariance.
MBL. E RS, EBL., E R8x9
Black-Litterman posterior mean and covariance. Parameters of the balancing gate Gr. Parameters of the UNet denoiser llg.
A.2 DENOISING DIFFUSION IMPLICIT MODELS USING GUIDANCE MODELING
Denoising Diffusion Implicit Models (DDIM). DDIM provides a non-Markovian implicit reverse sampler that preserves the training-time marginals while allowing much coarser (fewer-step) dis- cretization of the reverse process than DDPM. This yields faster inference and, under fixed condi- tions, more consistent denoised sequences. These properties are crucial for regime-awar forecasting at each decision time T.
DDIM training objective. Training follows the standard poise-prediction objective used by DDPM. For stock s and time т, let (())) E Rd be the noisy return sequence DDIM reverse step n, and let the condition be either macro vector my E I2d... or per-stock vector (*) E Rdd, A denoiser eg predicts the injected Gaussian noise:
E R.,,,n.c ||e- ea (r!i)) n, cond) [|]] cond E (m,, of*)).
At test period, DDIM replaces the Markovian reverse chain with non-Markovian update, so that the same trained es can be sampled in substantially fewer steps while keeping faithful marginals.
Guidance modeling. Guidance modeling aims to strengthen conditional fidelity by interpolating noise predictions from the same network. We use
Gu (r n, uncond), The scalar = controls the trade-off: larger z pushes sampling toward the conditional mode, while the shared-parameter eu keeps samples realistic. The term (&.- (., acts as a conditional residual that steers the denoising direction toward the conditional mode, while E regularizes the step toward high-likelihon regions of the unconditional data distribution. Despite fewer reverse steps (DDIM), this pundull-puu--regullrizer view improves conditional alignment without sacrificing realism.
c. co(rn, n, cond), غ= ca (c. cw), 2 0.
A.3 BLACK- -IEKK-LITTEMMN-BBEE BASED OPTIMIZATION
(BL) updates bascline prior for asset returns with investor views to form a poste- rior. Given this estimated posterior, BI performs mean-variance-oppina. (MVO) portfolio alloca- tion. The update balances the prior and the views by their certainties: highly certain views tilt the posterior toward the views, while low-certainty views keep it close to the prior.
At time т, let S denote the number of investable stocks in the universe. We use two prior quanti- ties: the prior mean Mprior.r E R and the prior covariance Eprior, E RSXS Under MVO, the market-equilibrimm portfolio weight implied by moments (μ4, Σ) is proportional to Σ E-1u, which maximizes the expected risk-adjusted return. To anchor this reference in a simple way, fix the
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equilibrium benchmark to the equal- -gigh- portfolio Weg E RR with entries 1/S and define
Mprior,r priior, Wegq This construction makes Wen coincide with the MVO direction under the prior and defines certainty via the precision Q,.
Σ prior.7
Let the per-asset view be (Jview,7s Eview,r) with view certainty Q2 E "view.r"
Posterior mean via MAP. Our goal is the posterior mean returns PBL, that balances the prior Mptior,7 and the view ,view.r according to their certainties. Under Gaussian prior and view, the max- imum posteriori (MAP) estimator is obtained by minimizing the negative log-posterior (constants omitted):
((p) = ( - Mprior,7) )) T 2 ) + 74 - (view,)) T (1 (view,7).
(5)
Setting the gradient zero yields the closed-form solution
MBL. = (#, + Ω2) ( Uprioo,, + P.
(6)
Posterior covariance via canonical form.
Expanding 5 yields
((n) = 20 H / n7 + const,
H, D, + S2, = Pr + Se Hview...
Hence
p(p prior, view) x exp( - J()) = exp( 1/2²² H 1TTT Tm);;
which is the canonical form of multivariate Gaussian. Therefore,
MBL -H-1 (4, + + 02)
MVO with the BL posterior. With the Black Littermmnn posterior moments (UBL,T, EBL,T);
the goal is to maximize the Sharpe ratio
W- MBL.T
" E arg max m.
(8)
w EBL
Let 2BI. w, and b EBI. BB Then
n. MBLL
a
V / EBL,
||a|
By the Scchwar inequality, ||ll $ [|b]]. with equality iff is colinear with b. Thus the maximizing direction satisfies a = Ab for some À > 0, i.e.
W, À En RB..
Finally, enforce the budget constraint 1 Wr = 1 to fix the scale:
EBL,
- 1 E "BB,
A.4 PORTFOLIO MANAGEMENT PERFORMANCES OVER MULTIPLE MARKET REGIMES
We evaluate whether STABLE sustains superior risk-adjusted portfolio performance across hetero- geneous market regimes and regions. Predictive performance for financial time series often varies with the dataset and the sample window. To verify consistency. measure results across disjoint periods and markets.
present experimental results for two contrast- ing market regimes, excluding the recent period. The first regime, the COVID-19 crisis (2019-09-0) to 2020-03-31). is global stock market crisis characterized by enormous crash and extreme volatility. The second regime is the Zero Interest Rate Policy (ZIRP) period (2020-04-01 to 2022-03-31). Following the crisis, global quantitative easing policies kept bond yields and credit spreads near zero, fueling a global stock market rally.
Building on the recent- recent-period results in Section 4.2
Results. STABLE attains the best ASR, RMDD, and AVol almost all markets across the two contrasting regimes (Tables 7 and In the COVID-19 window (Table 77). STABLE shows strong crisis-resilience. For example, it improves ASR to 1.61 in the US (from 0.07 for MOM) and reduces RMDD significantly. An exception is South Korea (KOSPI200), where STABLE'S ASR (-1.30) is
14

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Table 7: Portfolio performance during COVID-19 (2019-09-01 to 2020-03-31). Best performance per column in bold. RMDD and AVol are in percentage units (%).
S&P500 (US)
CS1300 (China)
Method
ASR (1)
RMDD
AVol (L)
ASR (t)
RMDD ())
AVol (L)
CRP
-0.69
35.63
51.59
-0.17
16.28
26.09
MVO
-0.62
32.34
49.47
-0.05
14.91
25.36
MOM
0.07
26.64
46.58
-1.00
9.64
24.72
DeepTrader
-0.32
25.53
38.09
11.71
19.15
MetaTrader
0.05
37.05
34.65
1.09
10.70
21.66
AlphaMix
-0.65
38.69
38.79
-0.35
9.70
19.40
STABLE (proposed)
1.61
23.77
34.10
1.18
9.20
18,50
EUROSTOXX (Europe)
KOSP1200 (South Korea)
Method
ASR (t)
RMDD (L)
AVol (L)
ASR (t)
RMDD (
AVol (L)
-1.74
39.97
40.92
-1.28
37.18
38.25
MVO
-1.11
37.02
39.04
-1.29
36.74
36.68
MOM
-3.70
50.16
45.81
-2.64
46.71
41.91
DeepTrader
-136
39.71
32.50
-1.64
36.87
40.61
MetaTrader
-0.84
37.03
37.15
-1.59
53.00
40.37
AlphaMix
-0.92
42.49
34.20
-1.38
43.30
37.80
STABLE (proposed)
-0.70
28.17
30.30
-1.30
37.65
31.19
similar to CRP (-1.28). This is an expected outcome, as our BL-MVO framework converges to the equgh-weight vector Weq (which CRP represents) when predicted covariances become extremely high, prioritizing stability. Notably, STABLE still achieves the lowest AVol (31.19) in this market, confirming its stable performance.
In the ZIRP window ()a STABLE demonstrates strong performance in the market rally, achiev- ing the highest ASR all regions. In China (CS1300), while its AVol (18.40) slightly higher than MVO's (17.63), it secures the best ASR (1.67), indicating superior risk-adjusted returns. These outcomes are consistent with the regime-shiff-toleran design STABLE. The diffusion sampler generates regime-aware paths and the MLG mechanism separates macro impact from firm-specific effects at the stock and time level. As noted for the KOSPI200 case, within BL-MVO, as the view precision Ω2 Vieww, decreases (e.g., in high uncertainty), the posterior places greater weight the prior and the allocation approaches the equal-weight vector Weq, which explains the observed stability during extreme episodes.
A.5 INDEPENDENT TEST OF COVARIANCE ESTIMATION PERFORMANCE
We isolate the covariance estimation to assess whether the diffusion-derivee covariance of STABLE delivers meaningful gains relative to simple prediction models and deep generative forecasters. The goal is to test the covariance view in isolation because risk-adjusted portfolio allocation is driven by risk control, which depends on accurate covariance estimation. We conduct this experiment over multiple regimes including the periods examined in Section 4.2 and Section A.4
In this experiment STABLE CDG with MLG generates per-stock return paths and forms the return view pliew.7. To decouple the contributions of mean and covariance estimation, fix fview, from STABLE for all methods and replace only the covariance view Eview, Specifically, for each alternative generative model sample l paths per asset and compute Eview,r via Eq. (4) using those paths; for the simple prediction models we directly use their covariance forecast. BI-MVV then solves for the portfolio weights given the shared μview.r and the method-specificc Eview,r- All settings including datasets and data split follow those of Section 4.2 and Section [A.4] In these settings STABLE as the covariance estimator shows consistent improvements compared with all alternatives. We report ASR, RMDD, and AVol on the four regional universes.
Simple generative forecasters. We compare STABLE against simple prediction models that include regression model, deep learning model, and tree-based model, each producing
15

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Table 8: Portfolio performance during the ZIRP window (2020-04-01 to 2022-03-31).
Best
performance per column in bold. RMDD and AVol are in percentage units (%)
S&P500 (US)
CS1300 (China)
Method
ASR (1)
RMDD
AVol (L)
ASR (t)
RMDD
AVol (
CRP
2.16
9.46
14.71
1.15
14.37
18.24
MVO
2.50
7.84
13.94
1.09
15.63
17.63
MOM
-0.19
30.17
23.03
-0.01
18.19
23.20
DeepTrader
-1.93
58.13
19.44
16.57
18.96
MetaTrader
1.41
14.38
21.44
1.36
19.76
21.77
AlphaMix
1.60
12.10
16.70
1.27
16.40
19.30
STABLE (proposed)
2.58
7.14
13.50
1.67
14.01
18.40
EUROSTOXX (Europe)
KOSPI200 (South Korea)
Method
ASR (t)
RMDD (L)
AVol (L)
ASR (t)
RMDD (
AVol (L)
1.09
19.93
18.97
1.71
14.90
17.11
MVO
1.18
19.64
18.59
1.75
14.40
17.94
MOM
0.46
25.10
19.20
0.25
19.40
23.00
DeepTrader
-1.05
46.23
22.03
-0.94
44.71
21.50
MetaTrader
1.04
20.03
22.76
1.36
19.76
21.77
AlphaMix
1.04
20.67
20.66
1.50
18.90
19.50
STABLE (proposed)
1.83
16.17
17.12
1.87
14.03
16.14
a full covariance forecast over time. For the neural and tree models we follow established volatility- forecasting settings for LSTM (Bucei) 2020) and LightGBM (Zhang) 2022). Both pre- dict the lower-trian swerrttranggllrr Cholesky factor L, of the covariance, which reduces the prediction target and improves computational efficiency. The simple prediction models are as follows.
DCC-GARCH(I,) (Engle 2002). multivariate GARCH that captures time-varying covari- ances through univariate volatility updates coupled with dynamic correlation recursion. The notation (1, 1) denotes one lag of the innovation and one lag of the conditional variance. It is widely used regression baseline that reflects evolving correlations.
LSTM-Cholesky (Nelson al., 2017). A vanilla LSTM uses the macro sequence in Table over the last y business days and predicts the entries of L,, The diagonal is constrained positive via a softplus map. The loss is the Gaussian negative likelikel Ly.
3. LightGBM-Cholesk (Ke et al. 2017). We train one madient-bbossedd tree regressor per entry Lij.r- Inputs include the most recent values from the previous rebalance Lıj.r Li Lij.r-e together with macro features aggregated over the last business days from Table
Deep generative forecasters. We also include diffusion- and autoencoding models (Diffusion-TS (Yuan & Qiao 2024), AEC-GAN (Wang et al.) (2023), ) and KoVAE (Naiman et al. 2023)) as covariance forecasters by sampling paths per asset and taking the sample covari- each T. STABLE forms Eview,r directly from its conditional diffusion paths with multi-level guidance.
deep generative
and summarize portfolio outcomes under the fixed-return setting across multiple regimes. In general, using STABLE for the covariance view achieves the best performance across all metrics in almost every regime and region. However, in South Korea during the COVID- 19 period, where STABLE previously showed lower performance than CRP, replacing the covariance with a simple regression model DCC- (CC-GARCH((11) yields the highest performance.
Results. Tables
During the COVID-19 crisis (Table 99. replacing STABLE's covariance with other deep generative models leads to severe performance degradation, with the exception KoVAE South Korea. As shown in the stylized facts comparison in Section A.6 these models tend to estimate Kurtosis and Skewness close to normal distribution. Consequently. they fail to account for the extreme tail risks inherent in market crashes, resulting in significantly worsened stability metries. For instance, in the S&P500 market, STABLE improves ASR by 92.7% (from 0.84 to 1.61) compared to the best alternative (Diffusion-TSS), while reducing annualized volatility by 1.49 4.49%p. Similarly, in the
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Table 9: Portfolio results with fixed return view from STABLE (CDG+MLG) during the COVID-19 crisis. We fix the return view jview,r to the per-stock estimates from STABLE CDG+MLG and vary only the covariance view 2view.r using each forecaster. BL-MVO then uses (Jvviow.r r, Eview,7) to produce the allocation. Performance of STABLE corresponds to the results in Table Replacing the covariance view of STABLE generally degrades performance, except for South Korea where (CC-GARCH(1,I) outperforms the others. Best per column in bold.
S&P500 (US)
CS1300 (China)
Covariance forecaster
ASR (t)
RMDD (4)
AVol (})
ASR (t)
RMDD (1)
AVol (L)
DCC-GARCH(1.)
-0.48
39.28
36.62
0.33
12.48
18.39
LSTM LSTM-Cholese Cholesky
-0.35
45.78
35,09
0.53
11.79
19,80
LightGBM-Cholese
-0.15
26.35
37.66
0.46
15.60
19.11
Diffusion-TS
0.84
26.51
35.59
0.63
12.17
21.26
AEC-GAN
-0.73
33.86
37.78
1.02
13.29
20.58
KoVAE
-1.04
41.36
39.57
0.87
14.51
21.15
STABLE (proposed)
1.61
23.77
34.10
1.18
9.20
18.50
EUROSTOXX (Europe)
KOSPI200 (South Korea)
Covariance forecaster
ASR (t)
RMDD (4)
AVol (L)
ASR (t)
RMDD (4)
AVol (L)
DCC-GARCH(L.)
-1.38
37.18
32.86
0.01
27.60
25.54
-2.00
34.31
35.37
-1.60
32.11
33.37
LightGBM-Cholesky
40.69
34.41
-0.75
47.56
40.22
Diffusion-TS
-0.93
36.32
35.26
-1.88
42.55
34.18
AEC-GAN
-1.00
48.82
36.75
-1.49
41.02
30.93
KoVAE
-2.61
46.57
39.48
-1.26
39.22
41.78
STABLE (proposed)
-0.70
28.17
30.30
-1.30
37.65
31.19
EUROSTOXX market, STABLE reduces RMDD by 6.14%p (from 34.31% to 17.17%) compared to the strongest baseline (LSTM), demonstrating superior stability during crises.
In the ZIRP period (Table I0 STABLE'S covariance estimation yields the superior investment per- formance across regions. This dominance aligns with the findings Section A.6 where the distribution of returns estimated by STABLE exhibits stylized facts highly similar to those of real- ized market data, enabling effective risk diversification. Notably, in the S&P500 market, STABLE improves ASR by 25.1% (from 2.06 to 2.58) and reduces RMDD by 2.49% compared to the best competitor (AEC-GAN). In the EUROSTOXX market, STABLE achieves 29.5 higher ASR (1.83 1.41) than the LSTM baseline, confirming that accurate covariance estimation via STABLE translates directly into enhanced risk-adjusted returns.
For the recent period (Table I1 the performance degradation from replacing STABLE's covariance the lowest in the EUROSTOXX market. This result is consistent with the findings in Table 4 where the deep generative models exhibit their strongest average MSE and DTW scores. Even here, STABLE improves the ASR by 9.8% (from 2.66 to 2.92) and reduces RMDD by 14.1 % (from 4.47% 3,84%) compared to the strongest competitor (KoVAE). Conversely, in regions where generative models have higher MSE and DTW scores, such as CSI300 and KOSPI200 (Table 4 replacing STABLE's covariance leads severe performance degradation. This result aligns with the time- series estimation performance reported in Table 4 and underscores the effectiveness of STABLE'S multi-level guidance. By separating systematic impacts from idiosyncratic characteristics, STABLE maintains robust covariance estimation and risk control even in markets where simpler generative models fail.
A.6 STYLIZED FACTS ANALYSIS AND GOODNNESS-OF-PEE Tests
We conduct a comprehensive analysis to validate the distributional properties of the return segments generated by STABLE and the deep generative models from Section 4.3 We conduct this exper- iment over multiple regimes including the periods examined in Section 4.2 and A.4 We perform both stylized facts validation and goodness-of of--ft tests on the generated segments against the realized return segments. The objective is to verify that the generated segments, particularly from STABLE, capture not only simple time-series similarity but also the key dynamic properties and underlying stochastic structure of realized return.. For stylized facts validation, conduct statistical tests to
17

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Table 10: Portfolio results with fixed return view from STABLE (CDG+MLG) during the ZIRP period. Performance of STABLE corresponds to the results Table STABLE consistently achieves the best performance across all regions. Best per column in bold.
S&P500 (US)
CS1300 (China)
Covariance forecaster
ASR (1)
RMDD (4)
AVol (U)
ASR (1)
RMDD (1)
AVol (L)
(CC-GARCH(L.)
1.39
20.81
24.27
1.47
16.96
27.15
LSTM-Cholesky
1.77
10.19
19.30
1.30
14.64
18.80
LightBMM-Cholekyy Cholesky
1.36
18.81
21.44
1.11
20.70
24.92
Diffusion-TS
0.82
12.69
22.38
1.16
21.34
24.26
LEC-GAN
2.06
9.63
18.94
15.68
21.53
KoVAE
1.42
11,47
20.43
1.35
14.26
21.81
STABLE (proposed)
2.58
7.14
13.50
1.67
14.01
18,40
EUROSTOXX (Europe)
KOSPI200 (South Korea)
Covariance forecaster
ASR (1)
RMDD (4)
AVol (U)
ASR (1)
RMDD (1)
AVol (L)
DCC-GARCH(1.))
1.11
19.17
21.71
-0.62
31.78
19.14
LSTM-Cholesky
1.41
18.02
20.65
1.57
21.31
20.21
LightGBM Cholesky
0.95
25.92
24.48
0.50
29.03
27.70
Diffusion-TS
0.87
22.61
27.94
1.27
16.94
23.74
AEC-GAN
0.78
21.48
25.56
1.78
14.25
18.90
KoVAE
0.66
23.54
26.69
1.66
15.64
21.98
STABLE (proposed)
1.83
16.17
17.12
1.87
14.03
16.14
Table II: Portfolio results with a fixed return view from STABLE (CDG+MLG) during the
recent period. Performance of STABLE corresponds to the results in Table
STABLE consistently
achieves the best performance across all regions. Best per column in bold. RMDD and AVol are in %.
S&P500 (US)
CS1300 (China)
Covariance forecaster
ASR (t)
RMDD (4)
AVol (L)
ASR (1)
RMDD (1)
AVol (L)
DCC- (CC-GARCH(L.))
0.81
10.18
16.10
-0.95
14.12
19.91
LSTM-Cholesky
10.22
16.94
-1.23
13.94
20.12
LightGBM-Chen Cholesky
0.95
13.63
20.94
-1.22
14.20
21.70
Diffusion-TS
0.53
12.68
25.85
-1.01
13.58
20.57
AEC-GAN
0.80
12.70
19.90
-0.74
16.14
21.41
KoVAE
0.90
12.36
16.13
-0.68
16.60
22.52
STABLE (proposed)
1.85
7.82
13.43
-0.41
8.85
17.17
EUROSTOXX (Europe)
KOSP1200 (South Korea)
Covariance forecaster
ASR (t)
RMDD (1)
AVol (L)
ASR (t)
RMDD (1)
AVol (L)
DCC (CC-GARCH(L.)
2.53
5.31
11.78
0.79
11.09
18.34
LSTM-Cholesky
2.54
5.36
14.14
1.13
12.46
19.69
-igheeyM-Choleskyy
1.52
7.12
14,66
1.18
9.60
23.10
Diffusion-TS
5.21
14.55
21.93
24.98
AEC-GAN
2.41
6.11
14.94
-0,82
13.09
21.13
KoVAE
2.66
4.47
14.59
0.66
21.70
26.29
STABLE (proposed)
2.92
3.84
10.88
1.61
8.34
17.82
determine if the realized kurtosis, skewness, and autocorrelation of volatility exhibit significant dif- ferences from the generated values. For goodness-of-fi tests, perform the Kolmogorov-Smin -mmirno (KS) (Smirnoy] 1948) and Anderson-Darling (AD) (Anderson Darling 1952) tests.
Setup. We apply stratified bootstrapping procedure for all tests. Let S be the number of stocks and T' the total number of rebalancing timesteps. Let = S x T be the total number of realized segments. Let D N.N be the set of N realized return segments of length and let VD IR. Ne be its flattened ID vector. Let î represent the set of generated returns, structured as (N E) array, where for each of the N conditions T), we have l candidate segments of length We perform M bootstrap iterations. In each iteration i, we construct a sample set ( Dample Nx by randomly selecting exactly segment (out of k) for each of the N conditions. This sample is then flattened
18

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into a ID vector sample IRAV, This procedure is applied STABLE, Diffusion-TS, AEC-GAN, and KoV.
Stylized facts validation. We analyze three key stylized facts: kurtosis, skewness, and volatility clustering. The results are visualized in Figures 13 14 and [5] respectively. The validation for each statistic is performed as follows:
Kurtosis: We test whether the single kurtosis value calculated from the flattened realized data VD RNE falls within the 95% confidence interval of the distribution formed by the M kurtosis values, where each kurtosis is calculated from a corresponding flattened V bootstrap sample sample E R.C.
Skewness: We test whether the single skewness value calculated from the flattened real- ized data VD falls within the 95% confidence interval of the distribution formed by the M skewness values calculated from each v:(() sample'
Volatility clustering: We measure the lag-1 autocorrelation of absolute returns (ACF(I)) for each f-length segment. We test whether the mean ACF(1) calculated across all N segments in the realized data set D E R falls within the 95% confidence interval of the distribution formed by the M mean ACF(1) values, where each mean acf, is calculated
from the N segments within a bootstrap sample set (
sample E R NXe
Goodness-of-ft tests. We use the KS and AD tests to assess the similarity between the generated and realized distributions from two perspectives: overall shape (KS) and tail behavior (AD). The
results are visualized in Figures 16 and
Kolmogorov-Smirnat (KS) Test: For each of the M bootstrap samples, we perform a two- sample KS test between the flattened realized data VD E R and the flattened bootstrap v-( sample sample E IRNE, We compute the KS statistic based on the maximum discrepancy between the two empirical CDFs (Smirnov] 1948). Instead of relying on asymptotic dis- tributions, we calculate the p-value using a permutation test with B = 200 iterations to ensure robustness against finite sample sizes and non-normal distributions. We then plot the ECDF of these M p-values. If the generated samples are statistically indistinguishable from the realized data, this ECDF plot should be close to the diagonal line (i.e., a uniform distribution).
Anderson-Darling (AD) Test: We conduct similar procedure using the two-sample AD test. The test statistic is computed as a squared, weighted difference between the ECDFs, giving higher weight tail deviations (Anderson & Darling 1952). Consistent with the KS test, the significance levels are determined via a permutation test. We again plot the ECDF of the M resulting p-values check for uniformity.
STABLE
Officion-TS
ABC GAN
KoVAE
Rosioed Volue
S6P500
CS1300
EUROSTOXX
KOSP1200
0.6
0.30
Real-20.
Real-5.60
0.5
Real-22:00
0.5
12.5
0.6
0.4
0.35
Densky
0.4
8.30
0.2
0.05
0.2
0.1
0.0
Kurfosis Valuee
Kurtowis Velue
Kurtosis Value
Kurtesis Value
Figure 3: Stylized Fact Validation (COVID-19): Kurtosis. Distribution of kurtosis in the COVID- 19 regime. Note the extreme realized kurtosis in the S&P500 (20.46). STABLE generally provides robust estimates, though the extreme tails in non-CS1300 markets challenge all models.
Results. Overall, STABLE exhibits distribution characteristics the most similar to realized returns across all regimes and stock markets, with the exception of KOSPI200 during the COVID-19 regime. KoVAE is confirmed to show high similarity for skewness in the COVID-19 regime. How- ever, KoVAE consistently overestimates the volatility clustering (ACF(1)) of stock returns across all
19

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STABLE
MlsiinnTS
AEC GAN
NOMAE
Roo Foed VOhU0
SSP500
EUROSTOKX
KOSP1200
0.7
Aeale-1.52
Reale 6-80
Reali
2.30
Reall 6,70
0.5
0.6
0.5
0.4
044
0.3
0.2
0.0
0.0
0.0
0.0
Figure 4: Stylized Fact Validation (COVID-19): Skewness. Distribution of skewness in the COVID-19 regime. VoVAE captures the negative skewness in KOSPI200 well. reflecting the market crash dynamics.
STAGLE Dif.alen-T5
ACC-GAN
SAP500
CS1300
EUROSTOIX
KOSP1200
RamI-0.039
F0s1-000022
Flell-2.000
Saati-- 0.007
0.00 ACT(2 [Retums]
oãe ACF110 [Returns]
ACF11) [Retumss]
ACF:11 (Reburs)
Figure 5: Stylized Fact Validation (COVID-19): Volatility Clustering (ACF(1)). Realized values
near zero. STABLE remains consistent, while Ko VoVAE tends overestimate volatility clustering.
0.6
0.4
9..0
0.2
0.6
0.6
0.8
1.0
0.2
0.4
06
0.8
3.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
p-value
p-vatue
pivalue
p-vaue
Figure 6: Goodness-of-Fit (COVID-19): KS Test. ECDF of p-values. STABLE demonstrates superior fit across most markets, with KoVAE showing strength in KOSPI200.
STABLE
Diffusion:T5 C5:300
ABC GAN
Maai Unifforms Line
1.0
$6/5000
EUROSTOOS TOXX
KOSP1200
0.8
0.2
0.60
p-oelue
p-ealae
prvelue
pevalue
Figure 7: Goodness-of-Fit (COVID-19): AD Test. ECDF of p-values for the tail-sensitive AD test. The rankings remain consistent with KS tests.
20

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STABLE
D flusion-T
ABC GAN
KOSAC
Realloed VSAAS
SAP500
CS1900
EUROSTOXX
KOSP1200
Real-9.00
Real-d..5 1.18
0.30
Read-S.
0.5
1.0
0.25
0.20
0.6
0.4
0.1
0.10
0.2
0.2
0.05
0.0
0.00
Kurtosis
Kurtosis
Kurtosis Value
Kurfosss Value
Figure 8: Stylized Fact Validation (ZIRP): Kurtosis. Distribution of kurtosis in the ZIRP regime. STABLE successfully captures the kurtosis within confidence intervals in most markets.
STARLE
D OifisionSSS
AFC-GAN
KIVIE
Reed Virue
SSP500
CS1300
EUROSTOKX
KOSPI200
Realn0.09
0.0
Realn0.25
Realno.27
Realno.92
0.6
0.6
0.6
0.6
Densly
0.4
0.4
0.4
0.2
0.2
0.0
0.0
0.0
Value
Figure 9: Stylized Fact Validation (ZIRP): Skewness. Positive skewness is observed due to the market rally. STABLE and others (except KoVAE) capture this feature well.
STAGLE
Dffailen-75
ALC-GAN
Taallord Varue
Feale-0.031:
100
Reeel-8.00 0 04
Realm-0.02
Reate-0.02
Denity
.0.50 .0.55
0.50 8.75 1.00 (Reburn]
40.50 .0.5
0.00 0.75 0.50
1.00
4.50 -0.75 0.00 n25 0.50
40.5
0.25 0.50 0.75
[Returss]
[Returs]]
ACF(3 [Retume]
Figure 10: Stylized Fact Validation (ZIRP): Volatility Clustering (ACF(1)). STABLE provides accurate estimates near zero, whereas competitors show significant deviations.
STABLE
Diffusion-TS
ABC
Uniiform
0.6
0.5
0.4
0.2
2.4
0.6
0.8
1.0
0.0
0.2
0.4
06
0.8
3.0
0.0
0.2
0.4
0.6
0.8
3.0
0.0
0.2
0.4
0.5
0.8
1.0
p-oalum
produe
p-oal
prveuue
Figure 11: Goodness-of-Fit (ZIRP): KS Test. STABLE shows the best fit, aligning closely with the uniform distribution line across all markets.
21

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STABLE
DiTTustonTS
Uniffom
0.0
0.0
0.4
0.2
244
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
3,0
0.0
0.2
0.4
0.5
0.0
1.0
p-oolae
prosuee
prosl
proelue
Figure 12: Goodness-of-Fit (ZIRP): AD Test. Similar to KS results, STABLE outperforms com- petitors.
ABC
Rozilaed
3.4
Realized-
Rcoliood=4.08
Real/pol-6,
Rcbiized=6.99 99
0.5
0.35
32
8.30
0.55
0.00
10
15
-2.5
0.0
5.0 Burtos's Value
2.5
7.5
10.0
-2.5
3.0
5.0 Kurtosis Value
2.5
7.5
15.0
10
Kurtosis Value
Murtosis Value
Figure 13: Stylized Fact Validation (Recent): Kurtosis. Distribution of kurtosis in the recent regime. The black line is the realized value. All models systematically underestimate the S&P500 kurtosis (10.83). STABLE (blue) provides the closest estimate in CSI300 and KOSPI200, while KoVAE (orange) is closest in EUROSTOXX.
STAGLE
flalee-75
ALC GAN
KIVAE
Roalloed Warua
1.2
Realizod=-00 2007
Mealicede 0.31
1,2
1.0
1.0
3.0
0.6
0.8
0.8
0.5
0.6
0.6
0.4
0.7
0.0
0.0
0.0
0.0
Figure 14: Stylized Fact Validation (Recent): Skewness. Distribution of skewness in the recent regime. STABLE (blue) provides the closest estimate in EUROSTOXX and KOSPI200. Diffusion- TS (green) is closest in S&P500, and KoVAE (orange) is closest in CSI300.
Defusien755
ACC-DAN
Bealized Value
SGP500
CS1300
EUROSTOOXK
KOSP1200
Realicedm-d 034
asalizedm-O.com
Realizedeo..0z
Realiced
0.2
6.0
0.6
0.2
0.0
0.0
0.2
ACF:11 Retuma)
ACF133 of [Retums]
ACF(3) [Poturns]
ACF(11 of [Rotums]
Figure 15: Stylized Fact Validation (Recent): Volatility Clustering (ACF(1)). Distribution of the mean ACF(I) in the recent regime. Realized values are near zero. STABLE (blue) provides the closest estimate in all markets. KoVAE (orange) consistently fails, predicting extreme outliers in three markets.
22

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2.0
(Cumulaties
0.4
0.2
98.0
Ub
0.2
Ub
0.4
0.8
0.0
0.A
08
p-value
p-value
prealue
proalce
Figure 16: Goodness-of-Fit (Recent): KS Test. ECDF of p-values. The STABLE curve (blue)
is closest to the ideal uniform line (black dash) in all markets, aligning with its top ASR rank in Table
STABLE
DifiisionsT
AEC GAN
Uniiformes
6.8
0.6
(Cumuletie
8..0
0.2
0.6
3.0
0.2
0.6
0.8
0.2
0.4
0.8 1.0
0.0
0.4
1.0
Figure 17: Goodness-of-Fit (Recent): AD Test. ECDF of p-values for the tail-sensitive AD test. STABLE again provides the best fit in all four markets.
regimes. Consequently, its prediction accuracy for realized returns ment performance similar to STABLE in Table Notably, excluding the CSI300 market, the regimes we select exhibit very high kurtosis (specifically, 20.46 for the S&P500 in the COVID-19 regime). This indicates that the market regimes during our experimental periods contain high volatility due to crashes, surges, and political issues, whereas the CSI300 market relatively limited tail values likely due to price fluctuation regulations. Additionally, volatility clustering based on ACF(1) is confirmed to be nearly zero across all regimes.
high, resulting invest-
The stylized facts validation results
follows.
COVID-19 (Figures 3 STABLE demonstrates the most homogeneous stylized facts across most kurtosis
markets. The r in the S&P500 is extremely high (20.46) (Figure which challenges all models. However, STABLE (7.74) provides distribution closer to the realized value compared to KoVAE (4.30). In KOSPI200, consistent with the results in Table 9 Ko is confirmed gen- erate the return distribution the most similar to reality. In particular, Ko VA''sss generated skewness (-1.18) (Figure(4) closely matches the realized value (-0.80), accurately capturing the negative skew- ness during the crash period, whereas STABLE (-0.13) underestimates this asymmetry. Regarding volatility clustering (Figure 5). the realized values are near zero across all markets (e.g., S&P500 0.04), but Ko VAE predicts extreme outliers (e.g., S&P500 0.63, CSI300 0.86), showing significant deviation from reality.
ZIRP (Figures 8 10h: STABLE shows the best stylized facts across all markets. The realized kurtosis the S&P500 is 9.00 (Figure 8. and STABLE (8.21) successfully captures this within its confidence interval, whereas Diffusion-TS (0.36) significantly underestimates it. A unique characteristic of this regime the positive skewness observed due to the market rally (e.g., KOSPI200 0.92, CSI300 0.25). Notably, all models except KoVAE include this feature within their confidence intervals. For instance, in KOSPI200 (Figure 9). STABLE (0.27) and Diffusion-TS (0.40) show positive skewness, while Ko VAE predicts excessive value (1.59). In terms of ACF(1) (Figure VoV again pre- dicts unrealistic values (e.g.. CSI300 0.94), whereas STABLE (0. provides a much closer estimate to the realized value (-0.04).
Recent (Figures 13 15
The realized kurtosis is high, confirming fat tails. For the S&P500, the outlier that all models systematically underestimate, although STABLE's
realized value (10.83) distribution is centered the closest (7.27) to this value (Figure 13p. The realized skewness market- dependent. STABLE provides the closest estimate EUROSTOXX and KOSPI200 (Figure 14 For
23

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Bast Compatitor
Best Compatiter
3.0
Bast Compettorr
1.6
Bast Competttrr
1.8
1.6
1.4
0.50
1.2
1.5
1.3
0055
1.0
1.0
1.2
STABLE .FFE
.MLG BLNVO
STABLE
KFF
.LGG -BLVVO
STABLE
KFEE
MMG -LM
STABLE KFF
.MGG -LL-VC
Figure 18: Ablation study of STABLE on ASR. Across all regions the BL-MVO module contributes the most risk adjusted return.
volatility clustering, STABLE consistently centers its distribution very close to the realized value, whereas Ko VAE predicts extreme outliers in three markets (Figure 15
In terms of the goodness-of fit, of-fit, STABLE consistently exhibits the most superior goodness-of- fit in both the KS and AD tests across all evaluated market regimes. In the COVID-19 regime (Figures 6 markets, with the only exception being the KOSPI200 market where KoVAE demonstrates strength. This robust performance is consistently observed through the ZIRP regime (Figures I1 [2) and the Recent regime (Figures [T). where the ECDF curves of the p-values for STABLE are unambiguously the closest to the ideal uniform distribution line across all datasets. Overall, the AD test results remain highly consistent with the test results, confirming that STABLE outperforms the competitors in accurately modeling the underlying distributions across diverse market conditions.
7. STABLE shows the best fit in
A.7 ABLATION STUDY
STABLE is organized into a feature-engineering module and three core modules. The in- puts are enriched by the Kalman-filtered stock embedding (KFE), and the core pipeline com- prises the Conditional Diffusion Generator (CDG), the Multi-Level Guidance (MLG), and the Black-Litterman
Mean-Variance Optimizer (BL-MVO). To quantify each component"s 's con- tribution, remove modify one module time while keeping all other settings identical to those of Section 4.2, We do not include variant that removes CDG because allocation requires diffusion-derived views. Accordingly, the ablation variants as follows.
STABLE without KFE. The temporal stock embedding 3/ is set to zero, that the corporate- specific feature CT contains only the last normalized adjusted-close level and daily log returns.
STABLE without MIG. The noise decomposition and the learnable gate are disabled by fix- ing the balancing gate to a constant 2/ = 0.5 for all stocks and times. Sampling reduces to (s)
conditional DDIM with the full condition h f.r without adaptive separation of systematic and idiosyneratic effects.
3. STABLE without BL-MVO. The BL update is removed and plug-in MVO uses the diffusion view moments (/ ? directly. This is equivalent to skipping posterior blending.
Results. Figure 18 summarizes ASR across the four regional universes. The full STABLE at- tains the highest ASR in every market. However, removing a module can sometimes results in performance below the best competitor. For instance, in KOSPI200, the variants without KFE or BL-MVO show a lower ASR than AlphaMix (see Table 3. In CSI300, all three ablation variants fall below the ASR of MOM. The degree which each module contributes varies by dataset. For example, the KFE's influence larger than MLG's in the S&P500, but their importance is reversed in CSI300. Despite this, BL-MVO consistently plays the most critical role; removing it causes the largest drop in ASR in all markets. This is natural, as the BL-MVO module determines the final portfolio weights by deciding how much trust the estimates from the other modules.
24

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SGP500
KOSP1200
1.6
1,75
ASR
-0.5
1.25
1.2
-0.4
3.0
1.6
ASR
1.50
1.4
2.0
0.6
0.001 0.1
02
0.5
0.001 0.1
022
0.5
0.001 0.1
0.2
0.
0.001 0.1
0.2
0.5
Figure 19: ASR sensitivity the most influential hy hyperparameters. Top row varies the number k of guided paths. ASR saturates when k 2 50. Bottom row varies the fz weight 3. Values above 0.001 induce excessive regularization and reduce ASR.
Table 12: Selected hyperparameters by dataset. Dataset-level grid-search selections for all hy- perparameters defined in Section 4.1 Except for the DDIM noise scale v which controls sampling stochasticity, all four datasets share the identical optimal values for the remaining hyperparameters.
Hyperparameter
S&P500
CS1300
EUROSTOXX
KOSPI200
Segment length
20
20
20
Macro & stock emhedding dim
16
16
16
16
DDIM reverse steps
30
30
30
30
Number & paths
50
50
50
forward diffusion steps
200
200
200
200
DDIM noise ?)
0.00
0.00
0.00
0.01
Balancing gate cap Zmax
2
2
2
2
BL prior window length L2 weight
120
120
120
120
0.001
0.001
0.001
0.001
A.8 HYPERPARAMETER ANALYSIS
We measure how STABLE'S hyperparameters affect the risk-adjusted return metric ASR. This sen- sitivity analysis assesses how responsive the method to hyperparameter choices and informs generalization of the selected configuration in practice. In particular, we sweep the number k of guided paths per stock and the l regularization weight / because these two hyperparameters the most strongly affect ASR, while fixing all remaining settings to the dataset-wise choices in Table I2
The ASR sensitivity to k and / is visualized in Figure 19 For k, = 50 is optimal across regions and the ASR curve saturates beyond this point. For the best value is 3 0.001 and larger values suppress performance due to over-regularization.
A.9 COMPUTATIONAL TRANSPARENCY AND REPRODUCIBILITY
For computational transparency and reproducible results, we report training and inference latencies in Table [3 and the detailed hardware and software environment in Table [4] We document the compute budget and environment to support reproducibility and fair comparison. We report wall- clock time, estimated GPU hours, and per-rebalance as well as per-asset inference time by dataset, and disclose the hardware and software stacks used for runs.
25

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Table 13: The compute budget by each dataset for one full training run with 200 epochs using four GPUs (4xRTX 3080). Estimated GPU hours assume full utilization. Latency (rebalance) represents the end-to-end per-rebalance inference time on the full universe (55 assets for US/China, 37 for Europe, and 44 for South Korea). Latency (asset) denotes the corresponding per-asset time.
Dataset
Wall time
Est. GPU hours
Latency (rebalance)
Lateney (asset)
S&P500 (US)
2h 0lm 13s
8.08
31.21
568 ms
CSI300 (China)
Ih 59m 42s
7.98
29.64
539
EUROSTOXX (Europe)
21m 05s
5.41
21.37
578 ms
KOSPI200 (South Korea)
1h 35m 28s
6.36
25.84
587 ms
Table 14: Hardware and software environment. Specification
Item
GPUs
4x NVIDIA RTX 3080 GiB each)
CPU
2x Intel Xcon Silver 4214 @ 2.20 GHz (12 cores/socket, 24 cores, 48 threads total)
RAM
503 GiB
Storage
33 TiB root volume (PERC H730P RAID)
OS
Ubuntu 20.04.5 LTS, Linux kernel 5.4.0-216-
Driver and CUDA
NVIDIA driver $35. 183.08..0,, CUDA runtime 12.2; CUDA Toolkit 11.6 (nvec 11.6.124)
Deep learning framework
PyTorch 1.13.1 (cul16, cuDNN 8.3.2): torchaudio 0.13.1
torchvision 0.14.1;
26


Reviews:
[{"review_id": "O3dQsRcRgx", "reviewer": "Reviewer_zHmG", "summary": "This paper presents STABLE, a Shift-Tolerant Allocation framework that integrates a conditional diffusion generator with Black–Litterman (BL) portfolio optimization to achieve regime-aware portfolio allocation.", "questions": "How sensitive are results to the number of generated paths (k) for BL view estimation?\n\nHave you compared STABLE’s performance during crisis windows (e.g., COVID-19) vs. stable regimes?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The problem formulation and derivations (Eqs. 1–6) correctly extend diffusion sampling to a conditional macro-financial setting.\n\nThe BL posterior update and Sharpe-maximizing allocation are mathematically consistent with canonical forms.\n\nExperiments use proper multi-region datasets, fixed seeds, clear metrics, and ablations verifying each module’s contribution.", "weaknesses": "Ablation depth: while individual module ablations exist, a joint ablation of CDG + MLG + BL contributions would clarify interdependencies.\n\nCompute transparency: GPU hours and sampling latency per rebalance step are not reported.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Iv8Z66heLB", "reviewer": "Reviewer_b8bB", "summary": "The paper proposes STABLE, a portfolio allocation framework that combines a conditional diffusion model with the Black–Litterman optimizer. The diffusion model is used to predict future return paths for each stock, conditioning on both macroeconomic indicators and firm-specific features. It introduces a multi-level guidance mechanism to separate shared market noise from idiosyncratic noise, making the predictions regime-aware. These diffusion-based forecasts are then treated as “views” in a Black–Litterman update to produce robust and stable portfolio weights. Across multiple markets, STABLE achieves higher Sharpe ratios and lower risk metrics than classical and deep RL baselines.", "questions": "My concerns are summarized above. My question is that`, can you provide experiments isolating the covariance estimation—e.g., keeping returns fixed and comparing portfolio volatility using covariance from different models—to show that the diffusion-based covariance is truly better?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "As far as I am concerned, the paper’s most interesting idea is using a diffusion process to get a better estimate of how assets move together (their covariance). This is very important because good covariance estimates are key to controlling risk and building better portfolios.\n\nThe method has been tested in several major markets and consistently beats older methods in both return and stability.\n\nThe model’s internal patterns also make economic sense, showing realistic links between companies and sectors.", "weaknesses": "Indeed, the paper tackles two key challenges in portfolio construction: (1) return prediction and (2) covariance estimation, both crucial for building effective, risk-controlled portfolios. The proposed diffusion-based framework is designed to improve both — by generating realistic future return paths and by learning dynamic relationships among assets.\n\nThe authors do provide some prediction performance analysis. In Section 4.3, they compare STABLE’s return forecasting accuracy (using MSE and DTW) against other generative models such as Diffusion-TS, AEC-GAN, and KoVAE, showing that STABLE achieves the lowest prediction errors across multiple markets. However, these comparisons are limited to deep generative baselines. There is no evidence that diffusion modeling improves predictive power over simpler and widely used models like regression, deep learning, or tree-based approaches.\n\nFor covariance estimation, which directly drives risk control, the paper also lacks clear evidence. It does not explicitly test whether the diffusion-derived covariance matrices are superior. Ideally, this would involve fixing the same return forecasts and showing that the covariance estimates from the proposed method lead to better risk-adjusted portfolios.\n\nIn short, while the framework is novel and conceptually appealing, the benefits of using the diffusion process are not rigorously demonstrated for either return prediction or covariance estimation. Moreover, the comparison setup is not entirely fair, since the proposed model uses richer macro- and firm-level inputs than the RL baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fPvFm6CkJI", "reviewer": "Reviewer_FycL", "summary": "This paper introduces a novel portfolio allocation framework named STABLE (Shift-Tolerant Allocation via Black-Litterman Using Conditional Diffusion Estimates), designed to tackle the dynamic allocation challenges in financial markets characterized by \"regime shifts.\"\nThe method innovatively integrates deep generative models (Diffusion) with classical financial theory (Black-Litterman). The framework primarily consists of three stages:\n\n1.\tConditional Diffusion Generator (CDG): A Conditional Diffusion Model (DDIM) generates future stock return paths based on macroeconomic conditions and stock-specific signals. The stock-specific signal includes a \"time-varying stock embedding\"  estimated via Kalman filtering to capture the dynamic sensitivity of the stock to macro factors.\n\n2.\tMulti-Level Guidance (MLG): An improved classifier-free guidance technique is used during the diffusion model's denoising process. This technique decomposes noise into shared \"systematic noise\" (macro-driven) and \"idiosyncratic noise\" (firm-specific), and dynamically balances their influence using a learnable gate.\n\n3.\tBlack-Litterman Mean-Variance Optimizer (BL-MVO): The K return paths generated by CDG+MLG are used to form a predictive \"investor view,\" including a view mean and covariance. This \"view\" is merged with a historical \"prior\" using the Black-Litterman framework to obtain a posterior mean and covariance. Finally, these posterior estimates are fed into a standard Mean-Variance Optimizer (MVO) to compute the final portfolio weights.\n\nThe authors conducted experiments across four regional stock markets (US, China, Europe, and Korea). Results show that STABLE achieves SOTA performance on both portfolio allocation and time-series forecasting tasks. For instance, it shows up to a 122.9% improvement in Sharpe Ratio on the portfolio task and up to a 15.7% reduction in MSE on the time-series estimation task.", "questions": "See weakness above", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1.\tHigh Novelty (Connecting Generative Models and Classic Finance): The main contribution of this paper is building a sophisticated bridge between state-of-the-art deep learning technology (Conditional Diffusion Models) and established classic financial theory (Black-Litterman). Utilizing the predictive paths generated by the diffusion model as the \"investor view\" within the BL framework is a very novel and insightful idea.\n\n2.\tElegant Architectural Design (Macro/Micro Decoupling): STABLE is not a simple application of diffusion models. The design of the Multi-Level Guidance (MLG) module is a major highlight. By decomposing noise into systematic (macro) and idiosyncratic (micro) components and using a learnable gate, the model can more finely capture the varied reactions of individual stocks under different market conditions. This directly addresses the shortcomings of previous models that often overfit macro signals and neglect stock-specific dynamics.\n\n3.\tPowerful Feature Engineering (Dynamic Embeddings): The authors innovate by using a Kalman filter to estimate the time-varying stock embedding instead of static industry labels or simple price embeddings. This allows the model to capture the dynamic sensitivity of stocks to macro factors.\n4.\tRigorous Experimental Validation: The experimental setup is comprehensive:\n\nMulti-Task: Evaluation on both portfolio allocation (Q1) and time-series forecasting (Q2) tasks, proving the model's effectiveness from different angles.\n\nMulti-Market: Tested across four diverse regional markets (US, China, Europe, Korea), demonstrating robustness and cross-market applicability.\n\nSOTA Results: The experimental results are impressive, with STABLE outperforming all baseline methods across almost all key metrics (ASR, RMDD, AVol, MSE, DTW) in all markets.", "weaknesses": "1.\tLack of Ablation Study: STABLE is a very complex, multi-stage pipeline with Kalman Filter, CDG, MLG and BL-MVO. Although the final results are good, I cannot determine which component contributes the most to the performance gains. Without ablations, the actual value of each innovative element of the model is hard to assess.\n\n2.\tShort Testing Period: According to Table 2, the training set ends in 2024-09, and the test set ends in 2025-03. This means the testing period is only 6 months. For a model claiming to address \"regime shifts,\" a 6-month window is far too short to evaluate its robustness across different macroeconomic cycles (e.g., a full recession, a high-inflation period, a zero-rate period). The model's excellent short-term performance might simply be due to its parameters fitting the market dynamics of these specific six months.\n\n3.\tHyperparameter Tuning and Reproducibility: The model introduces a large number of hyperparameters. The authors state these parameters were tuned \"per dataset.\" This raises concerns about potential overfitting and presents certain challenges to its generalization ability.\n\n4.\tLack of Stylized Facts Validation: Since STABLE uses a diffusion model, it is expected to capture the non-Gaussian properties of real financial returns. However, the paper lacks evidence that its generated distributions successfully replicate essential financial stylized facts such as high kurtosis (heavy tails), negative skewness, and volatility clustering. This omission undermines the validation of the diffusion model's intrinsic advantage over simpler generative baselines.\n\n5.\tMissing Distribution Goodness-of-Fit Tests: Given the unpredictability of financial time series, I would prefer that the distribution of the generated results be similar to the actual distribution, rather than having a sufficiently small error and a sufficiently close trajectory. But the paper fails to perform rigorous statistical goodness-of-fit tests (e.g., Kolmogorov-Smirnov, Anderson-Darling tests) on the distribution of the K generated return paths, leaving the similarity between the generated and true return distributions unquantified.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)