Introducción a Modelos GARCH

Study Notes

Introduction to GARCH Models

GARCH models, or Generalized Autoregressive Conditional Heteroskedasticity models, are statistical models used to describe the time-varying volatility of financial time series data.
They are an extension of ARCH (Autoregressive Conditional Heteroskedasticity) models.
They are widely used in financial econometrics to model volatility clusters and predict future volatility.
The core idea is that the variance of a return or other financial variable isn't constant over time, but rather changes based on past observations.

Key Components of GARCH Models

Conditional Variance: This is the variance of the variable, but it's conditional on past information, including past returns and past conditional variances.
Autoregressive Components (p): These components model the influence of past conditional variances on the current conditional variance.
Moving Average Components (q): These components model the influence of past squared returns on the current conditional variance.
GARCH(p,q): The notation GARCH(p,q) indicates the order of the model—p represents the autoregressive order and q represents the moving average order.

GARCH(1,1) Model Structure

The most common GARCH model is the GARCH(1,1) model.
It models the conditional variance as a function of the previous conditional variance and the previous squared return.
The formula typically includes parameters such as alpha, beta, and omega; all have specific meanings/influences on the model output.

Key Assumptions

The conditional variance should be positive.
The conditional variance should be dependent on past values of the variables (and their squares).
The error terms of the model (residuals) are often assumed to be normally or conditionally Gaussian distributed.

GARCH Model Estimation

Estimators are used to identify the appropriate parameters for a given GARCH model (which are specific to the model type).
Maximum likelihood estimation (MLE) is a common method used to estimate the parameter values in a GARCH model. This method finds the parameters that maximize the likelihood of observing the given data.
Numerical optimization methods are often required to evaluate likelihoods.

Applications of GARCH Models

Risk Management: GARCH models can be used to estimate and forecast volatility, which is crucial for risk management in finance.
Portfolio Optimization: Information about conditional variance can assist in making better decisions on allocating resources to investment portfolios.
Option Pricing: Models of conditional variance can help in identifying optimal pricing for options, based on the expected or modelled risk.
Financial Forecasting: GARCH models help create more accurate predictions of future price movements and risks based on historical variables and behaviours.
Understanding Market Dynamics: Models help clarify and better quantify the complexities of various market interactions.

Limitations of GARCH Models

Model Complexity: Choosing appropriate orders and specifications can present challenges as GARCH models are complex.
Data Dependence: Model accuracy can depend on the quality of the input data.

Extensions of GARCH Models

EGARCH (Exponential GARCH): Allows for asymmetric effects of positive and negative returns on volatility.
TGARCH (Threshold GARCH): Similar to EGARCH, but has a threshold to determine if the effects of returns are different depending on the sign and magnitude of returns.
GJR-GARCH (Glosten-Jagannathan-Runkle GARCH): An extension addressing asymmetric effects.

Data Requirements

A time series of financial data, for example, stock prices or returns of a certain asset.
Historical volatility data; typically includes historical values and patterns of the specific variable being modelled.

Description

Este cuestionario explora los modelos GARCH, que son herramientas estadísticas utilizadas para describir la volatilidad cambiante de las series temporales financieras. Aprenderás sobre componentes clave como la varianza condicional y las partes autorregresivas que ayudan a predecir la volatilidad futura en los datos financieros.