Diffolio: A Diffusion‑Model Framework for Risk‑Aware Portfolio Optimization

Background

The paper addresses the problem of generating diverse, high‑quality portfolios that respect a given user risk preference and historical market data. Deterministic deep‑learning methods lack flexibility for varying risk tolerances, while stochastic methods often rely on multi‑stage pipelines that are complex and misaligned with the true optimization objective.

Problem Definition

Given historical price data X, market index G, and a target risk level γ, the goal is to find a sequence of portfolio weights w_τ for each trading step τ that maximizes cumulative return R while satisfying risk constraints ρ (risk must lie within the range defined by adjacent risk levels) and weight constraints (weights sum to 1).

Method

3.1 Diffusion‑Model‑Based Portfolio Optimization

3.1.1 Pseudo‑Optimal Portfolio Synthesis – For each step τ and N assets, a return vector is defined and transformed by a normalization function f to obtain a pseudo‑optimal portfolio that emphasizes assets with higher returns while assigning smaller weights to under‑performing assets.

3.1.2 Conditional Diffusion Model – The model conditions on pseudo‑optimal portfolios, historical asset prices, and index data. The forward process adds Gaussian noise with a fixed schedule β_t. The reverse process is parameterized to reconstruct the original data.

3.1.3 Market Dynamics Encoder – A time‑convolution network T_θ with self‑attention extracts queries Q, keys K, and values V from historical data, computes an affinity matrix S, applies softmax, and produces market‑dynamic embeddings via residual connections and layer normalization.

3.1.4 Auxiliary Loss Optimization – The diffusion objective is reformulated as predicting the un‑noised portfolio from a noisy one. An auxiliary loss encourages accurate market‑dynamic representations.

3.2 Risk‑Aware Denoising Network

A risk‑aware denoising network F_γ learns distinct denoising trajectories for each target risk level γ. It redefines pseudo‑optimal portfolios per risk level using a risk‑specific normalization function f_γ that selects the top k_γ assets by return.

3.3 Risk‑Guided Denoising Diffusion

A risk‑guidance term adjusts the mean of the reverse diffusion step based on a proxy risk metric ρ', steering the denoising path toward the desired risk level and reducing sampling bias.

3.4 Training and Sampling Portfolio Weights

3.4.1 Training – Pseudo‑optimal portfolios conditioned on sampled risk levels are used as denoising targets. The model minimizes a weighted sum of mean‑squared error and the auxiliary loss.

3.4.2 Sampling – Starting from pure noise, the trained denoising network iteratively applies the risk‑guided reverse steps. After the final step, a risk‑specific normalization f_γ yields the risk‑aware portfolio weights.

Experiments

4.1 Experimental Setup

Six real‑world market datasets (US, KR, CN, JP, UK equities and cryptocurrency) are split 7:1:2 for training, validation, and testing. Evaluation metrics include Annualized Return (ARR), Annualized Sharpe Ratio (ASR), Maximum Drawdown (MDD), Annualized Volatility (AVol), Calmar Ratio (CR), and Sortino Ratio (SoR).

4.1.2 Baselines

Diffolio is compared against classic methods (ARIMA, CSM, BWSL, Buy‑Hold), recent two‑stage generative models (FactorVAE, D‑Va, TMDM), and modern deterministic deep‑learning / reinforcement‑learning approaches (MetaTrader, EarnMore, MILLION).

4.2 Quantitative Evaluation

Across most datasets, Diffolio achieves statistically significant improvements in profitability and risk‑adjusted performance over baselines. While deterministic neural baselines sometimes yield higher raw profits, Diffolio provides a better balance between return and risk, especially at multiple risk levels. Performance degrades on the cryptocurrency dataset due to extreme volatility.

4.3 Ablation Study

Variants removing risk guidance (DF‑nRG) or using perfect risk knowledge (DF‑γ*) demonstrate that conditioning on discrete risk levels is crucial. Dynamic risk selection benefits stable markets (e.g., UK) but may favor low‑risk models in highly volatile crypto markets.

4.4 Effectiveness of Risk Guidance

Ranking correlation between generated portfolio risk and target risk (Spearman) improves consistently when risk guidance is enabled, confirming finer risk control during denoising.

4.5 Denoising Path Visualization

Visualization of 50 sampled trajectories shows that early diffusion steps produce random portfolios, while later steps increasingly differentiate and converge toward the intended risk level.

4.6 Risk‑Driven Weight Allocation

Analysis of asset weight distributions across risk levels reveals that Diffolio allocates higher weights to more volatile assets as the target risk increases, confirming adaptive risk‑aware behavior.

Conclusion

Diffolio demonstrates that a diffusion‑model‑centric, end‑to‑end framework can directly learn portfolio distributions aligned with user‑defined risk preferences, achieving superior return‑risk trade‑offs compared to both deterministic and multi‑stage stochastic baselines.

All images above illustrate the mathematical definitions, model architectures, and experimental results described in the paper.