GARCH Model
Instructor: Dr. Ta Quoc Bao
1 Introduction:
GARCH stands for “Generalized AutoRegressive Conditional Heteroskedasticity”and it is a popular approach to model volatility.
We consider, there are four main part at all. We’ll go through what GARCH models are, why we need them, and how to utilize them in Python in Part 1: “GARCH Model Fundamentals.” We’ll look at particular GARCH model assumptions in Part 2, “GARCH Model Configuration,” and learn how to design a workable model for financial time series data. We will study many measures we may use to analyze model performance and pick a better model in Part 3 “Model Performance Evaluation.” We will study some practical applications of GARCH models in financial risk and portfolio management in the last Part, “GARCH in Action,” which includes estimates of value-at-risk, dynamic covariance, and stock Beta.
2 GARCH Model Fundamentals
2.1 What is “volatility”?
Volatility is a statistical measure of asset return dispersion across time used in finance. The standard deviation or variance of price returns is frequently used to calculate it. We’ll use the term “volatility” to refer to both standard deviation and variance in this course. Volatility is a term used to characterize the uncertainty surrounding the price movement of financial assets. It’s a crucial notion that’s used to risk management, portfolio optimization, and other areas. It’s also one of the most active fields of empirical finance and time series analysis study. The riskier a financial asset is, the higher the volatility.
2.2 How to compute volatility
Following three simple procedures, we may calculate volatility as the standard deviation of price returns:
- Step 1: Calculate returns as percentage of price changes \[\text{return}=\frac{P_1-P_0}{P_0}\]
- Step 2: Calculate sample mean return \[\text{mean}=\frac{\sum^{n}_{i=1}\text{return}_i}{n}\]
- Step 3: Calculate the sample standard deviation \[\text{volatility}= \sqrt{\frac{\sum^{n}_{i=1}(\text{return}_i-\text{mean})^2}{n-1}}=\sqrt{\text{varriance}}\]
2.3 Conversion of volatility
If we consider volatility to be the standard deviation of returns, we can calculate monthly volatility by multiplying daily volatility by the square root of 21, the average number of trading days in a month. Daily volatility may be multiplied by the square root of 252, which is the average number of trading days in a year, to get yearly volatility.
2.4 The difficulty of estimating volatility
In time series modeling, it’s a frequent assumption that volatility remains constant throughout time. In financial return data, however, heteroskedasticity, which literally means “various dispersion” in ancient Greek, is commonly found. Volatility tends to rise or fall in a predictable pattern over time.
2.5 Recognize heteroskedasticity
Plotting the data and observing its behavior over time is a simple approach to determine heteroskedasticity in a time series. When compared to homoskedastic data, heteroskedastic data’s volatility does not appear to be steady, but rather shows time-dependent swings.
2.6 What are ARCH and GARCH
The ARCH models came before the GARCH models. ARCH stands for “Auto-Regressive Conditional Heteroskedasticity,” and it was created in 1982 by American economist Robert F. Engle. “Conditional heteroscedasticity” suggests that the data has a time-dependent and unexpected property. Engle was awarded the Nobel Prize in Economics in 2003 for his contributions.
In 1986, Danish economist Tim Bollerslev created GARCH models based on ARCH. “Generalized” is the meaning of the “G” in GARCH. Fun fact: In his Ph.D thesis, Bollerslev wrote about the GARCH models under the supervision of Engle, the author of ARCH models.
2.6.1 Model equations: ARCH
The equation of ARCH model with lag p is defined as follows. In essence it models variance as a weighted average of past residuals up to lag p. Hence ARCH(1) model states: \[\text{ARCH(p)}:\sigma^2_t=\omega+\sum^{p}_{i=1}\alpha_i\epsilon^{2}_{t-i}\] \[\Rightarrow \text{ARCH(1)}:\sigma^2_t=\omega+\alpha\epsilon^{2}_{t-1}\]
2.6.2 Model equations: GARCH
In practice, GARCH models are more commonly used by researchers. The GARCH(p,q) model equation is as follows. Besides p-period lags of residuals, GARCH models add q-period lags of variances for predicting the current variance. Hence a basic GARCH (1,1) model states: \[\text{GARCH(p,q)}:\sigma^2_t=\omega+\sum^{p}_{i=1}\alpha_i\epsilon^{2}_{t-i}+\sum^{q}_{j=1}\beta_j\sigma^{2}_{t-i}\] \[\Rightarrow \text{GARCH(1,1)}:\sigma^2_t=\omega+\alpha \epsilon^{2}_{t-i}+\beta\sigma^{2}_{t-i}\]
2.6.3 Model intuition
Don’t let the calculations intimidate you. GARCH models are simple to comprehend:
- First and foremost, the model is autoregressive. It tries to estimate the volatility at time t based on data available at time t-1.
- Second, it uses a weighted average of historical data to calculate volatility.
2.6.4 GARCH(1,1) parameter constraints
There are two requirements that must be met for a GARCH(1,1) process to be practical. To begin, all of the parameters:\[\omega, \alpha,\beta>0\]. This eliminates the possibility of a negative variance. Second, the sum of alpha and beta should be smaller than one\[\alpha+\beta<1\], ensuring that the model’s predicted variance “means-reverts” to the long-run variance. \[\frac{\omega}{1-\alpha-\beta}\]
2.6.5 GARCH(1,1) parameter dynamics
The rule of thumb for model parameters is that the stronger the alpha, the greater the shocks’ immediate impact. The shocks are represented as residuals, or prediction errors, in this case. If we hold the alpha constant, the bigger the beta, the longer the influence lasts, implying that periods of high or low volatility tend to last.