Why Garman–Klass, Realized Volatility, and GARCH Estimates Differ

# Comparison of results given by volatility estimators: Garman-Klass Vs Garch(1,1)


# Comparison of results given by volatility estimators: Garman-Klass Vs Garch(1,1)












I am pretty new with volatility estimators and I am trying to see if Garman-Klass estimator and Garch(1,1)estimator are closed. So I implemented a python code for the two estimators (an also for the realized volatility estimator as standard deviation of squared returns), but testing it on Tesla data, I have a huge difference between them. That means that my understanding (or my implementation) of these estimators is false. I have unsuccessfully searched for errors.

Below is the python code:

```
import math
import pandas as pd 
from pandas_datareader import data as pdr
import numpy as np
import matplotlib.pyplot as plt
from datetime import date 
import yfinance as yf
from arch import arch_model

observation_period=10# number of months from the last available day
df =  yf.download("TSLA")
df=df.last('%dm'%observation_period)

def annualized_realized_volatility_estimator(df,window_length=observation_period, unit_window_length='m', A=252):
    temp=A*(1/len(df))*np.log(df.last('%d'%observation_period+unit_window_length)).diff().dropna()
    V=temp['Adj Close'].sum() 
    return math.sqrt(V)

def Garman_Klass_estimator_row(H, L, O, C,T=len(df.index), A=252):
    V=A*(1/T)*(1/2)*(math.log(H/L))**2 - (2*math.log(2)-1)*(math.log(C/O))**2
    return V
def Garman_Klass_estimator(df):
    vol=math.sqrt(df.apply(lambda row:Garman_Klass_estimator_row(row['High'], row['Low'], row['Open'], row['Close']),axis=1).sum())
    return vol

def Garch_model(df,training_period,observation_period,horizon=1): #training_period is the number of months for training the Garch
    df_garch=df.last('%dm'%(observation_period+training_period))
    df_garch_train=df_garch.first('%dm'%training_period)
    df_garch_predict=df_garch.last('%dm'%observation_period)
    df_garch_train.loc[:,'Returns']=100 * df_garch_train.loc[:,'Adj Close'].pct_change().dropna()
    garch11 = arch_model(df_garch_train.loc[:,'Returns'].dropna(), p=1, q=1) 
    res = garch11.fit(update_freq=10) 
    forecasts = res.forecast(horizon=horizon)
    return 0.1*forecasts.residual_variance.iloc[-1].values[-1]
```

Calling the three functions with:

```
annualized_realized_volatility_estimator(df,window_length=10, unit_window_length='m', A=252)
Garman_Klass_estimator(df)
Garch_model(df,training_period=6,observation_period,horizon=1)
```

I obtained the following results:

annualized_realized_volatility_estimator=1.643

Garman_Klass_estimator=0.632

Garch_model=2.256

Could someone explain the notable differences between these estimators? Especially between the Garman Klass estimator and the Garch estimator?

Also the code python for the Garch(1,1) is it well-done?

Thanks a lot for your help!