Model update with addition of files requested

submissions/tags/README.md

@@ -1,44 +1,85 @@
-# Team: tags (Edouard Tallent)
-# December 2025
-# First shot
+# tags – IG(3/2, σ₀²) volatility model
 
 ## Model summary
 
-We simulate a continuous-time GBM with **daily stochastic variance** drawn from an
-Inverse-Gamma distribution with shape 3/2:
-
-- Instantaneous variance per year on day t:  
-  $\( V_t \sim \mathrm{InvGamma}(\alpha = 3/2, \beta = \sigma_0^2) \)$
-- Daily time-step: $\( dt = 1/252 \)$
-- Conditional on $\(V_t\)$, the daily log-return is
-  $\[
-  \Delta \log S_t = (\mu - 0.5 V_t) dt + \sqrt{V_t dt} \, \varepsilon_t,\quad
-  \varepsilon_t \sim \mathcal{N}(0,1).
-  \]$
-- Price path:
-  $\[
-  S_{t+1} = S_t \exp(\Delta \log S_t).
-  \]$
-
-This is equivalent to a continuous-time GBM
-$\(
-  dS_t = S_t (\mu dt + \sqrt{V_t} dW_t)
-\)$
-with piecewise-constant volatility on each day.
-
-## Parameters
-
-- Shape of IG: **α = 3/2** (fixed by theory; not tuned).
-- Scale of IG: **σ₀ = 0.259** (one free parameter).
-- Drift: **μ = 0.0** (kept fixed; does not materially affect q-variance).
-- Other settings (S₀, horizon length, RNG seed) are fixed and not tuned for R².
-
-## Result
-
-Using `code/data_loader_csv.py` followed by `code/score_submission.py`,
-we obtain:
-
-- Windows: ~3.09 million
-- Canonical q-variance curve (σ₀ = 0.259, z₀ = 0.021):  
-  **R² ≈ 0.9988**
+This model simulates a single asset price as a GBM with **daily stochastic variance** drawn from an Inverse-Gamma distribution with shape 3/2:
+
+- Instantaneous variance per year on day *t*:
+  \[
+  V_t \sim \mathrm{InvGamma}(\alpha = 3/2,\ \beta = \sigma_0^2)
+  \]
+- Daily time step:
+  \[
+  dt = \frac{1}{252}
+  \]
+- Conditional on \(V_t\), the daily log-return is
+  \[
+  \Delta \log S_t
+  = (\mu - \tfrac{1}{2} V_t)\,dt + \sqrt{V_t\,dt}\,\varepsilon_t,
+  \quad \varepsilon_t \sim \mathcal{N}(0,1).
+  \]
+- The price path is obtained as \(S_{t+1} = S_t \exp(\Delta \log S_t)\).
+
+This is kinda of classical Normal–Inverse-Gamma variance-mixture structure which yields the q-variance relation for a single observation
+\[
+\mathbb{E}[V \mid z] = \sigma_0^2 + \tfrac{1}{2} z^2
+\]
+when \(\alpha = 3/2\) and \(\beta = \sigma_0^2\), and empirically reproduces the canonical q-variance curve with \(R^2 \approx 0.9988\) on the full 100k-day simulation.
+
+- Shape of IG: \(\alpha = 3/2\) (fixed by theory, not tuned).
+- Scale of IG: \(\sigma_0 = 0.259\) (q-variance intercept).
+- Time step: \(dt = 1/252\).
+
+## Drift calibration and log-drift cancellation
+
+In the discrete-time SDE above,
+\[
+\mathbb{E}[\Delta \log S_t]
+= (\mu - \tfrac{1}{2}\mathbb{E}[V_t])\,dt.
+\]
+
+For \(V_t \sim \mathrm{InvGamma}(3/2,\sigma_0^2)\) in the shape–scale parametrisation,
+\[
+\mathbb{E}[V_t] = \frac{\sigma_0^2}{\alpha - 1}
+= 2\sigma_0^2.
+\]
+
+We therefore choose
+\[
+\mu = \sigma_0^2
+\]
+so that
+\[
+\mathbb{E}[\Delta \log S_t] = 0,
+\]
+i.e. the process has **zero expected log-drift** over long horizons.  
+
+**NOTE** In this version of the model, I explicitly account for the 1/252 time step and avoids the systematic long-run decay that would result from taking \(\mu = 0\) with the same volatility structure. The q-variance diagnostics use drift-adjusted returns, so this choice of \(\mu\) affects the **visual behaviour of the price path** but not the underlying q-variance structure.
+
+## Files in this folder
+
+- **`tags_IG32_100Kdays_sim.csv`**  
+  Raw daily price path \(S_t\) from the tags IG(3/2, \(\sigma_0^2\)) model, **100,000 trading days**, single asset, one column `Price` (start \(S_0 = 100\)). This is the “longer simulation” requested for convergence / illustration.
+
+- **`tags_IG32_10Kdays_sim.csv`**  
+  The **first 10,000 days** of `tags_IG32_100Kdays_sim.csv`, one column `Price`. This file is used as input to `data_loader_csv.py` for the 10k-window diagnostics.
+
+- **`dataset.parquet`**  
+  Window-level q-variance dataset generated from the **100,000-day** simulation (`tags_IG32_100Kdays_sim.csv`) using `code/data_loader_csv.py`. Same column structure as the main competition dataset (T, z, sigma, etc.). This is the dataset that yields the full-sample q-variance fit with \(R^2 \approx 0.9988\).
+
+- **`dataset_10K.parquet`**  
+  Same construction as `dataset.parquet`, but using **only the first 10,000 days** of the simulation (`tags_IG32_10Kdays_sim.csv`). This is the input for the q(T) vs T diagnostic.
+
+- **`tags_qT_first10k.png`**  
+  Plot of the **best-fit quadratic coefficient** \(q(T)\) versus horizon \(T\), where for each T we fit
+  \[
+  \sigma^2(z;T) = \sigma_0^2 + q(T)\,(z - z_0)^2
+  \]
+  with \(\sigma_0 = 0.259\) and \(\ z_0 = 0.021\) fixed, using only windows from `dataset_10K.parquet` (first 10k days). The dashed line at 0.5 shows the theoretical q-variance coefficient.
+
+## Interpretation of the 10k-day q(T) plot
+
+The `tags_qT_first10k.png` figure shows that, using only the first 10,000 days of a single simulated path, the estimated quadratic coefficient \(q(T)\) lies in a relatively tight band (roughly 0.28–0.38) and is **approximately stable in T**, i.e. there is no strong systematic horizon dependence. This is the desired scale-invariance feature of the q-variance structure.
+
+The coefficients are systematically below the theoretical value 0.5, reflecting finite-sample effects and the fact that \(\sigma_0\) and \(z_0\) are held fixed while only \(q(T)\) is fitted on a single realisation. When the full 100k-day simulation is used (`dataset.parquet`), the global q-variance fit converges back to the canonical curve with \(R^2 \approx 0.9988\), consistent with the theoretical coefficient of 0.5.