From Single-Point Estimates to Full Probability Distributions.
Most quantitative models split into two worlds: the Alpha World (mean/prediction) and the Risk World (tails/loss). Density forecasting unifies both: the mean of the forecast distribution is your alpha signal, and the tails are your risk limits — one model, one source of truth.
The goal of this project is to build an algorithm that accurately forecasts the probability distribution of
Before introducing Deep Learning, we rigorously established the Parametric Ceiling. We built industry-standard classical models (GARCH, Student-t) and subjected them to a strict 60-Day Independent Block K-S Test across three highly volatile stress-test assets: ARKK (Macro-Regime Shifts), USO (Exogenous Supply Shocks), and BTC-USD (Structural Fat Tails).
Failure Rate by Asset (Percentage of 60-Day Regimes Failed):
| Asset | Naive Gaussian | Student-t (Fat Tails) | GARCH(1,1) (Volatility Clustering) |
|---|---|---|---|
| ARKK | 34.6% | 26.9% | 19.2% |
| USO | 30.8% | 19.2% | 15.4% |
| BTC-USD | 45.0% | 32.5% | 37.5% |
The Classical Flaw: These models are strictly backward-looking. They rely entirely on historical data, adapting to market crashes only after they happen.
To push the parametric equations to their absolute limit, we built a forward-looking hybrid model. By coupling the structural fat tails of the Student-t distribution with the regime-aware, options-implied scaling of the VIX (optimized daily via Nelder-Mead with factor normalization), we drastically reduced the calibration failure rates during crises like the COVID-19 crash.
The Challenge Ahead: The VIX model relies on a rigid, linear equation tied to the US equity market. It cannot capture the idiosyncratic non-linear geometry of individual assets (like Bitcoin). To solve this, we must transition to AI.
We evaluate every model (Classical and AI) through three strict lenses:
- PIT (Probability Integral Transform): Is the model calibrated? (Measured via the Kolmogorov-Smirnov test).
- CRPS (Continuous Ranked Probability Score): How close is the whole distribution to reality?
- Log-Likelihood: How much probability did we assign to what actually happened?
| Phase | Status | Focus |
|---|---|---|
| 01 | ✅ | Problem definition, evaluation framework (CRPS/PIT), and naive baselines. |
| 02 | ✅ | The Parametric Ceiling: GARCH, VIX-Scaled baselines, and optimizer death-loop resolution. |
| 03 | 🔜 | AI Transition: Extracting market geometry using Path Signatures. |
| 04 | 🔜 | Neural SDEs: Training a continuous-time generator to defeat the VIX-Scaled champion. |
This project builds upon foundational research in Rough Path Theory, Deep Learning, and Quantitative Finance:
Path Signatures & Rough Path Theory:
- Lyons, T. (1998). Differential equations driven by rough signals. Revista Matemática Iberoamericana.
- Kidger, P., Bonnier, P., Perez Arribas, I., Salvi, C., & Lyons, T. (2019). Deep Signature Transforms. NeurIPS.
Neural Stochastic Differential Equations:
- Chen, R. T., et al. (2018). Neural Ordinary Differential Equations. NeurIPS.
- Kidger, P., Foster, J., Li, X., & Lyons, T. (2021). Neural SDEs as Infinite-Dimensional GANs. ICML.
Probabilistic Evaluation & Volatility Modeling:
- Gneiting, T., & Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation. JASA.
- Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics.