Submission: equityquant.dev

submissions/equityquant.dev/README.md

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+# EquityQuant.dev Model 
+
+**Team name:** equityquant.dev  
+**Contact:** Yi-Lung (Dragon) Tsai — ytsai@equityquant.dev   
+**Model Description:** please see below
+
+---
+
+## Model Summary
+
+This submission implements a **classical, continuous-time, event-driven price process**.  
+The model is fully specified by its Python implementation and **generates the q-variance relationship endogenously** from the underlying price dynamics, without imposing a quadratic form by construction.
+
+Price dynamics are driven by **stochastic shock activity** rather than by a diffusive volatility process. Volatility is **not modelled directly**; instead, it emerges naturally from the realised aggregation of random shocks.
+
+---
+
+## Simulation Overview
+
+Price paths are generated via the following mechanism:
+
+1. Simulate a latent precision process  
+2. Map precision to stochastic shock intensity  
+3. Aggregate a random number of Gaussian shocks per day  
+
+This construction ensures that:
+
+- Conditional variance increases with realised price movement  
+- The q-variance relationship emerges naturally across time horizons  
+
+---
+
+## Latent Precision Process
+
+A positive latent process $\( \tau_t \)$ governs market activity.  
+It is simulated using a **stationary mean-reverting positive diffusion** (CIR form, used purely for numerical convenience):
+
+$$
+d\tau_t = \kappa(\theta - \tau_t)\,dt + \eta \sqrt{\tau_t}\, dW_t
+$$
+
+
+### Interpretation
+
+- High $\tau$ → calm market regimes with low activity  
+- Low $\tau$ → turbulent regimes with elevated activity  
+
+Mean reversion ensures regime persistence while preventing degeneracy.  
+This latent process does **not** represent volatility.
+
+
+---
+
+## Shock Intensity and Returns
+
+### Shock Intensity
+
+Daily shock intensity is defined as an inverse function of precision:
+
+$$
+\lambda_t = \frac{c}{\tau_t}
+$$
+
+The number of shocks per day is drawn as:
+
+$$
+N_t \sim \text{Poisson}(\lambda_t)
+$$
+
+---
+
+### Return Generation
+
+Conditional on $N_t$, daily log-returns are generated as:
+
+$$
+r_t \mid N_t \sim \mathcal{N}\left(0,\ s_{\text{unit}}^2 \cdot N_t\right)
+$$
+
+This implies:
+
+- Variance is proportional to realised shock activity  
+- Returns are Gaussian *conditional* on $N_t$  
+- Returns are **heavy-tailed unconditionally**, consistent with subordinated stochastic processes  
+
+---
+
+## Volatility Calibration
+
+The per-shock variance $s_{\text{unit}}^2$ is calibrated such that the unconditional long-run daily variance satisfies:
+
+$$
+\mathbb{E}[r_t^2] = \mathbb{E}[N_t] \cdot s_{\text{unit}}^2 = \frac{\sigma_0^2}{252}
+$$
+
+This ensures that:
+
+- $\sigma_0$ represents the **minimum volatility level**  
+- Corresponds to calm market regimes
+
+No additional free parameter is introduced at this stage;  
+the per-shock variance is fully determined by $\( \sigma_0 \)$ and the stationary mean intensity.
+
+---
+
+## Price Process
+
+Log-prices evolve via cumulative daily returns:
+
+$$
+\log P_t = \log P_{t-1} + r_t
+$$
+
+$$
+P_t = \exp(\log P_t)
+$$
+
+This guarantees:
+
+- Strictly positive prices  
+- Correct aggregation of returns across time  
+
+---
+
+## Parameters
+
+### Free Parameters
+
+The model uses **three effective free parameters**:
+
+| Parameter | Value | Description |
+|---------|------|------------|
+| $\sigma_0$ | 0.25 | Long-run annual volatility scale |
+| $\kappa$ | 0.02 | Mean-reversion speed of precision |
+| $c$ | 10.0 | Shock intensity scaling |
+
+These parameters control the scale and persistence of shock activity and were **not tuned** to fit the q-variance parabola.
+
+---
+
+### Fixed Parameters
+
+All remaining parameters are fixed *a priori* for numerical stability and scale normalisation and were **not tuned** to achieve q-variance:
+
+| Parameter | Value | Description |
+|---------|------|--------|
+| `a_shape` | 1.5 | Diffusion discretisation constant |
+| `lam_cap` | 500.0 | Poisson intensity cap (numerical safeguard) |
+| `dt` | 1 / 252 | Trading-day discretisation |
+| `seed` | 3 | Reproducibility only |
+| `s0` | 100.0 | Initial price |
+| `n_days` | 120,000 | Simulation length |
+
+The Poisson cap is set sufficiently high that it is **rarely binding** and does not affect the fitted q-variance curve.
+
+---
+
+## Results
+
+<p align="center">
+  <img src="result1.png" width="650">
+</p>
+
+Simulated price paths reproduce the q-variance relationship with a global
+goodness-of-fit of **R² ≈ 0.996**, exceeding the challenge threshold of **0.995**.
+
+## Dataset Notes
+
+Due to GitHub file size limitations, the submission dataset is provided in three parts:
+`dataset_part1.parquet`, `dataset_part2.parquet`, and `dataset_part3.parquet`.
+
+This format is fully supported by the official `score_submission.py` script, which
+automatically concatenates the split files when `dataset.parquet` is not present.
+No data has been modified or filtered; the three files together form the complete
+submission dataset.
+