Submission: Tingjun GARCH Omega fix

code/score_submission.py

@@ -92,7 +92,7 @@ def qvar2(z, s0, s1):    # define q-variance function, parameter is minimal vola
 plt.xlim(-zmax, zmax) 
 plt.ylim(0.0, ymax)
 
-plt.legend(fontsize=12)
+plt.legend(fontsize=12, loc='upper right')
 plt.grid(alpha=0.3)
 plt.tight_layout()
 ##plt.savefig("figure_1_1.png", dpi=300, bbox_inches='tight')
@@ -152,12 +152,14 @@ def qvar2(z, s0, s1):    # define q-variance function, parameter is minimal vola
 ymax2 = 1.2
 ax.axis([-zmax+delz/2, zmax-delz/2, 0, ymax2])   # uses ymax2 for ensemble plot
 
-
+data = df_orig.copy()
 # now plot q-variance for different periods T, check for period-depenedence
 # plot with periods
 TVEC = [5, 10, 20, 40, 80]
 
 plt.figure(figsize=(9,7))
+popt = [0.2586, 0.0214]  # same as optimized fit to data  # for competition score should fit original parabola
+fitted = qvar(binned.z_mid, popt[0], popt[1])  # cols are z_bin, which is a range like (-0.601, -0.55], and qvar
 #plt.plot(binned.z_mid, binned['var'], 'b-', lw=3,label='all T')  
 plt.plot(binned.z_mid, fitted, 'red', lw=3, label=f'σ₀ = {popt[0]:.3f}, zoff = {popt[1]:.3f}, R² = {r2:.3f}')
 
@@ -174,11 +176,13 @@ def qvar2(z, s0, s1):    # define q-variance function, parameter is minimal vola
     r2 = 1 - np.sum((binned["var"] - fitted)**2) / np.sum((binned["var"] - binned["var"].mean())**2)
     print(f"T = {Tcur} σ₀ = {popt[0]:.4f}  zoff = {popt[1]:.4f}  R² = {r2:.4f}")
     colcur = str(Tcur/100)
-    ##plt.plot(binned.z_mid, binned['var'], c=colcur, lw=2,label=f'T = {Tcur/5:.0f}, R² = {r2:.3f}') 
-    plt.plot(binned.z_mid, binned['var'], c=colcur, lw=2,label=f'T = {Tcur/5:.0f}') 
+    plt.plot(binned.z_mid, binned['var'], c=colcur, lw=2,label=f'T = {Tcur/5:.0f}, R² = {r2:.3f}') 
+    #plt.plot(binned.z_mid, binned['var'], c=colcur, lw=2,label=f'T = {Tcur/5:.0f}') 
 
 plt.xlabel('z (scaled log return)', fontsize=12)
 plt.ylabel('Annualised variance', fontsize=12)
+plt.legend(fontsize=10, loc='upper center')
+plt.grid(alpha=0.3)
 plt.title('All stocks T=2, 4, 8, 16 weeks – Q-Variance', fontsize=14)
 
 

submissions/tingjun2/README.md

@@ -3,40 +3,46 @@
 ## Model Overview
 The **GARCH(1,1) Volatility Model** (Generalized Autoregressive Conditional Heteroskedasticity) describes the evolution of a security price $S$ where the variance $V$ is a stochastic process exhibiting **Mean Reversion** and **Volatility Clustering**. 
 
-This implementation maps the discrete-time GARCH process to the theoretical **Q-Variance** relationship. By simulating millions of trading days, we demonstrate that the stochastic variance updates converge to a deterministic parabolic fit: $V(z) = \sigma_0^2 + \frac{(z - z_{off})^2}{2}$.
+This implementation maps the discrete-time GARCH process to the theoretical **Q-Variance** relationship. By simulating millions of trading days, we demonstrate that the stochastic variance updates converge to a deterministic parabolic fit: 
+$$V(z) = \sigma_0^2 + \frac{(z - z_{off})^2}{2}$$
+
+
 
 ## Parameters & Mapping Logic
 The model utilizes four parameters to control the GARCH dynamics and recover the theoretical geometry of the Q-variance parabola:
 
 | Parameter | Value | Influence on Q-Variance Geometry |
 | :--- | :--- | :--- |
-| **Target Vol ($\sigma$)** | **0.095** | **Minimal Volatility ($\sigma_0$):** Sets the vertical baseline (the "floor" of the parabola). |
-| **Annual Return ($\mu$)** | **0.0844** | **Z-Shift ($z_{off}$):** Controls the horizontal asymmetry (displacement from zero). |
-| **Persistence ($\lambda$)** | **0.8** | **Curvature/Steepness:** Determines the quadratic coefficient; lower $\lambda$ makes the parabola **narrower** |
-| **Mean Reversion ($\gamma$)** | **0.01** | **Stability:** The buffer ensures $\lambda + \alpha < 1$, allowing the process to converge to a stationary mean. |
-
+| **Target Vol ($\sigma$)** | **0.425** | **Minimal Volatility ($\sigma_0$):** Sets the vertical baseline (the "floor" of the parabola). |
+| **Annual Return ($\mu$)** | **0.06** | **Z-Shift ($z_{off}$):** Controls the horizontal asymmetry (displacement from zero). |
+| **Persistence ($\beta$)** | **0.8** | **Curvature/Steepness:** Determines the rate of decay; higher $\beta$ maintains "memory" of past volatility. |
+| **Shock Weight ($\alpha$)** | **0.19** | **Sensitivity:** Controls the reaction to market shocks. Fixed such that $\alpha + \beta = 0.99$. |
 
+> **Note:** The "Mean Reversion" strength is implicitly defined as $\gamma = 1 - (\alpha + \beta) = 0.01$.
 
 ## Simulation Methodology
-The simulation generates a synthetic price history using independent paths of **2,500 trading days** each. To eliminate "local path luck" and ensure statistical smoothing, we utilize **2,000 samples** to create a total dataset of **5,000,000 trading days**.
+The simulation generates a synthetic price history using independent paths of **2,500 trading days** each. To eliminate "local path luck" and ensure statistical smoothing, we utilized **2,000 samples** to create a total dataset of **5,000,000 trading days**.
 
 ### Discrete-Time Updates
 
 - **Variance Update (GARCH Logic):**
+The variance at step $i$ is driven by the weighted average of the long-run variance (Variance Targeting), the previous variance, and the most recent market percentage-return shock ($\epsilon_{i-1}$):
+
+$$V_i = \omega + \alpha \cdot \epsilon_{i-1}^2 + \beta \cdot V_{i-1}$$
+
+Where:
+- $\epsilon_{i-1} = \frac{S_{i-1} - S_{i-2}}{S_{i-2}}$
+- $\omega = \frac{\sigma^{2}}{252} \cdot 0.01$ (Targeting a long-run annual volatility $\sigma$)
+- $\alpha = 0.99 - \beta$
 
-$$
-V_i = \omega + \lambda V_{i-1} + (1 - \lambda - \gamma) \cdot \left( \frac{S_{i-1} - S_{i-2}}{S_{i-2}} \right)^2
-$$
 
-Where $\omega = \frac{\sigma^{2}}{252} \cdot (1 - \lambda)$.
 
 - **Price Update:**
+The price follows a Geometric Brownian Motion (GBM) step, adjusted for the current stochastic variance $V_i$:
 
-$$
-S_i = S_{i-1} \exp\left( \frac{\mu}{252} - \frac{V_i}{2} + \sqrt{V_i} \epsilon \right)
-$$
+$$S_i = S_{i-1} \exp\left( \frac{\mu}{252} - \frac{V_i}{2} + \sqrt{V_i} Z \right)$$
 
-where $\epsilon \sim \mathcal{N}(0,1)$.
+where $Z \sim \mathcal{N}(0,1)$.
 
 ---
 
@@ -47,12 +53,14 @@ The model was optimized using the 5M-day horizon to achieve a high-fidelity fit
 ### Optimized Model Performance
 | Total Days | $\sigma_0$ (Target) | $z_{off}$ (Target) | $R^2$ | Notes |
 | :--- | :--- | :--- | :--- | :--- |
-| **5,000,000** | 0.2586 | 0.0214 | **0.9967** | High-fidelity structural convergence. |
-| **100,000** | 0.2586 | 0.0214 | **0.9701** | Lower $R^2$ due to idiosyncratic noise. |
+| **5,000,000** | 0.2586 | 0.0214 | **0.997** | High-fidelity structural convergence. |
+| **100,000** | 0.2586 | 0.0214 | **0.966** | Lower $R^2$ due to idiosyncratic noise. |
 
 ### Comparative Visualizations
 The transition from 100k days to 5M days shows the "clearing" of statistical noise. In the large sample, the stochastic variance updates converge almost perfectly to the theoretical parabola.
 
+
+
 #### **100,000 Day Sample (Noise-Dominant)**
 <div style="display: flex; justify-content: space-around;">
   <img src="Figure_1_100k.png" width="32%">
@@ -78,6 +86,20 @@ By plotting $R^2$ as a function of total simulated days, we identified a clear t
 ![Convergence Analysis](convergence_analysis.png)
 Figure: $R^2$ score vs. Total Days. The 0.995 threshold is maintained after the 1M-day mark.
 
+---
+## Acknowledgments & Version History
+
+> **Version 2.0 Update**
+> An earlier version of this model utilized an incorrect definition of the intercept term ($\omega$), which led to a scaling mismatch in the long-run variance. Special thanks to **@Orrell** for pointing out this discrepancy.
+
+In this current version, I have:
+* **Fixed the $\omega$ definition:** Implemented proper **Variance Targeting** logic, where $\omega = V_{target} \cdot \gamma$.
+* **Standardized Nomenclature:** Renamed parameters to follow standard GARCH(1,1) conventions ($\alpha$ for the shock component and $\beta$ for the persistence component).
+* **Re-optimized Parameters:** Recalibrated the parameter set specifically for the 5M-day horizon to ensure maximum fit.
+
+Despite these structural corrections, the results remain consistent with previous findings, further confirming the original hypothesis of **Q-variance emergence** in GARCH-governed systems.
+
+---
 
 ## Project Structure
 - [price_generator2.ipynb](price_generator2.ipynb): Vectorized simulation and optimization logic.