---
pretty_name: GARCH Densities
license: cc-by-4.0
task_categories: [other]
language: [en]
tags: [garch, finance, density-estimation, time-series]
size_categories: [10M<n<100M]

configs:
  - config_name: default
    default: true
    data_files:
      - split: train
        path: data/train/*.parquet
      - split: test
        path: data/test/*.parquet
      - split: sample
        path: data/sample/*.parquet

---

# GARCH Densities Dataset

GJR-GARCH simulations with Hansen skewed-t innovations, for option pricing and risk modeling. Each row pairs a parameter set Θ with **inverse-CDF quantiles** of terminal returns at a given maturity step.

![Example CDF and derived PDF curves](cdf_and_pdf_comparison.png)  


## Dataset Description

Each example is a tuple *(Θ, ti, x)* where:
- **Θ** = (`alpha`, `gamma`, `beta`, `var0`, `eta`, `lam`)
- **`ti`** = maturity **step index** (integer steps)
- **`x`** = vector of **Q=512** quantiles at probabilities  
  **p = linspace(0.001, 0.999, 512)**

Parameters are sampled by **Sobol** low-discrepancy sequences under the persistence constraint \( \alpha + \gamma + \beta < 0.95 \).

### Features
- `alpha` (float32) — persistence term  
- `gamma` (float32) — effective leverage term (GJR component)  
- `beta`  (float32) — persistence term  
- `var0`  (float32) — initial variance \( \sigma_0^2 \), log-uniform in [1/30, 30]  
- `eta`   (float32) — skewed-t degrees of freedom \( \nu \), log-uniform in [4, 100]  
- `lam`   (float32) — skewed-t skew \( \lambda \) in [-0.98, 0.98]  
- `ti`    (float32) — maturity step index   [1, 1000]
- `x`     (list<float32>[512]) — quantiles of normalized terminal return \( R_T \) at p above

_All numeric columns are float32 for I/O efficiency._

### Parameter Sampling
- (`alpha`, `gamma`, `beta`) with \( \alpha + \gamma + \beta < 0.95 \)  
- \( \nu \in [4, 100] \) (log-uniform), \( \lambda \in [-0.98, 0.98] \) (uniform)  
- \( \sigma_0^2 \in [1/30, 30] \) (log-uniform)

## Dataset Statistics

- **Train:** 110 shards, ~145 MB each  
- **Test:**  29 shards,  ~145 MB each  
- **Row groups:** zstd compression, ~64–128 MiB target (streaming-friendly)

![Parameter Distributions](parameter_distributions.png)  
![Parameter Correlations](parameter_correlations.png)

## Usage

### Recommended loading
```python
from datasets import load_dataset
ds = load_dataset("sitmo/garch_densities", token=False)  # -> DatasetDict(train/test/sample)
train, test = ds["train"], ds["test"]
print(train)
print(train.features)
```

### PyTorch formatting
```python
import torch
from torch.utils.data import DataLoader

param_cols = ["alpha","gamma","beta","var0","eta","lam","ti"]
cols = param_cols + ["x"]
train = train.with_format("torch", columns=cols)

loader = DataLoader(train, batch_size=256, shuffle=True, num_workers=4, pin_memory=torch.cuda.is_available())
batch = next(iter(loader))

params  = torch.stack([batch[c] for c in param_cols], dim=1)  # [B,7]
targets = batch["x"]                                          # [B,512]
print(params.shape, targets.shape)
```

### Inspect a sample
```python
ex = train[0]
len(ex["x"])  # 512
```

## Methodology

1. **Sobol sampling** of Θ in 6D under \( \alpha+\gamma+\beta<0.95 \)  
2. **Monte Carlo**: GJR-GARCH with Hansen skewed-t innovations  
3. **Quantiles**: empirical quantiles of terminal returns on p = linspace(0.001, 0.999, 512)  
4. **Performance**: inverse-CDF accelerations for skewed-t sampling

## Train/Test Split

- **Train:** 10,000,000 paths per case  
- **Test:**  10,000,000 paths per case  
- Independent parameter cases; split preserves Sobol low-discrepancy structure.

## Applications

- Option pricing across strikes (quantile-based)  
- VaR / CVaR risk metrics  
- Learning \( \Theta \mapsto \) return distribution (NNs)  
- Validation/benchmarking of GARCH implementations

## Limitations

- Restricted to GJR-GARCH with Hansen skewed-t  
- Fixed parameter ranges  
- Finite MC precision and finite quantile grid resolution (512 points)

## Provenance

- Synthetic data from our simulator (GJR-GARCH + Hansen skewed-t)  
- Parquet: **zstd**, row groups ~64–128 MiB, shards ~145 MB  

## Citation
```bibtex
@dataset{garch_densities_2025,
  title   = {GARCH Densities Dataset},
  author  = {Thijs van den Berg, Paul Wilmott},
  year    = {2025},
  url     = {https://huggingface.co/datasets/sitmo/garch-densities}
}
```