GARCH Volatility Estimation - The Quant Science

The GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model is a statistical model used to forecast the volatility of a financial asset. This model takes into account the fluctuations in volatility over time, recognizing that volatility can vary in a heteroskedastic (i.e., non-constant variance) manner and can be influenced by past events.

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The general formula of the GARCH model is:

σ²(t) = ω + α * ε²(t-1) + β * σ²(t-1)

where:

σ²(t) is the conditional variance at time t (i.e., squared volatility)
ω is the constant term (intercept) representing the baseline level of volatility
α is the coefficient representing the impact of the squared lagged error term on the conditional variance
ε²(t-1) is the squared lagged error term at the previous time period
β is the coefficient representing the impact of the lagged conditional variance on the current conditional variance

In the context of financial forecasting, the GARCH model is used to estimate the future volatility of the asset.

HOW TO USE
This quantitative indicator is capable of estimating the probable future movements of volatility. When the GARCH increases in value, it means that the volatility of the asset will likely increase as well, and vice versa. The indicator displays the relationship of the GARCH (bright red) with the trend of historical volatility (dark red).

USER INTERFACE

Alpha: select the starting value of Alpha (default value is 0.10).
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Beta:select the starting value of Beta (default value is 0.80).
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Lenght: select the period for calculating values within the model such as EMA (Exponential Moving Average) and Historical Volatility (default set to 20).
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Forecasting: select the forecasting period, the number of bars you want to visualize data ahead (default set to 30).
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Design: customize the indicator with your preferred color and choose from different types of charts, managing the design settings.
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Catatan Rilis

Pine Script 6 update
Introduction to Maximum Likelihood Estimation
Handling Two Statistical Distributions (Gaussian vs. Student’s t)
Advanced Backcasting and EMA on Residuals
Multi-Step Forecast (Forecast Horizon) and Volatility Regime
Statistics Table
Chart Optimization

Description Update | 2026

💎 Introduction to Maximum Likelihood Estimation (MLE)
❌ In the old version alpha and beta parameters were either fixed or estimated heuristically using a rudimentary loop based on fixed tolerance thresholds.
✅ In the new version model now uses a grid search for Maximum Likelihood Estimation. It dynamically optimizes the values of alpha sensitivity and beta persistence by analyzing a historical time window, mathematically determining the parameters that best explain the actual market data.

💎 Handling Two Statistical Distributions (Gaussian vs. Student’s t)
❌ In the old version variance calculation assumed a purely normal distribution.
✅ In the new version can choose between the Gaussian (normal) distribution and the Student’s t-distribution.

Introduction to Student’s t-distribution
The Student’s t-distribution is a fundamental statistical concept that, in the world of finance and trading, serves to solve a major problem: the reality of the markets is much more volatile than classical mathematics would have us believe.

The Problem with the “Normal Bell Curve” (Gaussian)
Normally, when making statistical predictions traders use a classic bell curve. This curve assumes that extreme events are extremely rare, almost impossible. In financial markets, sudden crashes, flash crashes, or wild rallies occur much more frequently than the normal bell curve predicts.

The Student’s t-Distribution Solution: “Fat Tails”
The Student’s t-distribution is a bell curve that is slightly more “flattened” in the center and has higher and wider sides (the tails). This means the model accounts extreme and unexpected events with a more realistic probability. It recognizes that a market movement of 5% or 10% in a single day, while rare, is part of the nature of financial markets and should not be considered an impossible mathematical error.

The Role of “Degrees of Freedom”
It's the “slider” that adjusts the shape of this distribution:
The lower the number, the higher the tails of the curve rise, indicating an extremely volatile, unpredictable market prone to sharp swings.
The higher the number, the more the Student’s t-distribution changes and resembles the normal Gaussian bell curve (a calm and orderly market).

💎 Advanced Backcasting and EMA on Residuals
❌ In the old version simple difference from an EMA of the price was used to calculate the error.
✅ In the new version calculation is much more rigorous:

Log returns are calculated
Residuals are isolated based on an exponential moving average of returns
A backcasting system with exponential decay is implemented to estimate the initial variance, avoiding distortions at the beginning of the time series

💎 Multi-Step Forecast (Forecast Horizon) and Volatility Regime
❌ In the old version forecast was managed primarily through a visual offset parameter (shifting the bars forward).
✅ In the new version true theoretical h-step forecast is calculated based on long-term persistence, providing a mathematical estimate of future volatility.

💎 Statistics Table
❌ In the old version there was no on-screen text interface; users had to infer the values from the graphs.
✅ In the new version dynamic monitoring table has been added in the lower-right corner of the chart, displaying the following in real time:

The optimal Sensitivity and Persistence found by the model
The Degrees of Freedom (if using Student’s t-test)
The Total Persistence (alpha + beta) with a visual alert if it is close to 1
The Backcast Variance and Half-Life (half-life of the volatility shock expressed in bars)

💎 Chart Optimization
The Current Volatility vs. Expected Volatility overlay allows you to immediately see whether the market expects an increase (thin line above the thick line) or a decrease (thin line below the thick line) in volatility in the short term, acting as a leading indicator of market risk sentiment.

The colors are determined by a conditional logic based on the forecast:
🔴 Bright red indicates an imminent phase of rising volatility, meaning an expected increase in risk and price fluctuations.
🟢 Bright green indicates an impending phase of falling volatility, meaning a forecast of greater stability and calm in the markets.

Without looking at any numbers, the chart’s color scheme immediately conveys the direction of risk:
🔴 Red --> Increased Risk
🟢 Green --> Reduced Risk

Semantic Fill
To make the contrast even more evident, the space between the current line and the projected line is dynamically filled:
🔴 Semi-transparent red fill --> the market is expected to expand, the area between the two lines turns red, visually emphasizing the “pressure” of rising risk.

🟢 Semi-transparent green fill --> the market is in an expected contraction phase, the area turns green, providing calming visual feedback.

The filled area amplifies the color signal, making it impossible to miss the shift in market conditions, even when viewing the chart from a distance or for just a few seconds.

This chart optimization eliminates the need to interpret complex trends. At a single glance, the user can tell whether the GARCH model is signaling an acceleration or a slowdown in market activity, enabling a much faster tactical response to changing risk conditions.

Skrip open-source

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thequantscience

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Pernyataan Penyangkalan

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Skrip open-source

thequantscience

🔥 Unlock the power of Symbiosis, the first Python XGBoost model natively built for Pine Script. Grab the last available spots on Patreon: patreon.com/c/xgboost_symbiosis