diff --git a/submissions/tags/README.md b/submissions/tags/README.md index 6724944..ff5d2f6 100644 --- a/submissions/tags/README.md +++ b/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. diff --git a/submissions/tags/dataset_10K.parquet b/submissions/tags/dataset_10K.parquet new file mode 100644 index 0000000..1db837c Binary files /dev/null and b/submissions/tags/dataset_10K.parquet differ diff --git a/submissions/tags/tags_IG32_10Kdays_sim.csv b/submissions/tags/tags_IG32_10Kdays_sim.csv new file mode 100644 index 0000000..66ebd92 --- /dev/null +++ b/submissions/tags/tags_IG32_10Kdays_sim.csv @@ -0,0 +1,10001 @@ +Price +100.49707432148736 +100.43490517519508 +100.07341861061796 +100.22905102413976 +98.87197578166985 +94.30646119442048 +89.37350055653197 +91.37317960253569 +95.0849061008644 +90.14035847462834 +91.39550230884426 +91.3805239739374 +90.64418436731472 +91.6583004417369 +95.86435075495906 +97.32967775758831 +99.16521645766557 +98.69826631528188 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