Chapter 1: Introduction to Volatility Modeling

ARCH and GARCH models are essential tools for capturing non-constant conditional variance in time series data. Unlike ARIMA models, which assume a steady variance, Autoregressive Conditional Heteroskedasticity (ARCH) specifically addresses the fluctuations in variance (volatility) over time. ARCH(1) serves as the simplest form of GARCH, while the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model enhances ARCH by incorporating a moving average component along with the autoregressive part.

In financial models such as Black-Scholes, volatility (standard deviation) is a critical parameter. When the variance of a time series shows a consistent upward trend, leading to variations that are not uniform, it is termed heteroskedasticity. If this variance change is dependent on previous observations, it can be effectively modeled using an autoregressive approach like ARCH.

However, it is crucial to apply ARCH models only to series that do not exhibit trends or seasonal patterns, where no significant serial correlation is observed. In situations where such characteristics exist, it is advisable to utilize ARIMA models (or Seasonal ARIMA) prior to considering ARCH models as a viable solution.

# Setting up the environment for time series analysis

!pip install arch

import math

from time import time

import pandas as pd

import numpy as np

import matplotlib.pyplot as plt

from datetime import datetime, timedelta

from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

from sklearn.model_selection import train_test_split

from sklearn.metrics import mean_squared_error, mean_absolute_error

from arch import arch_model

import warnings

plt.style.use('fivethirtyeight')

warnings.filterwarnings('ignore')

%matplotlib inline

This code snippet is crucial for establishing the environment necessary for conducting time series analysis and constructing an ARCH model. Below is an overview of its functionality:

• Library Installation: Install the ARCH package, which provides vital tools for modeling conditional variance.
• Importing Libraries: Necessary libraries are imported for mathematical operations, data manipulation, visualization, and statistical analysis.
• Plotting Style: The plotting style is set for consistent visualization.
• Warnings Management: Warnings are suppressed to enhance output clarity.

Chapter 2: Analyzing Bitcoin Prices

Visualizing the stock price over time is an important step in understanding market movements.

# Read and plot Bitcoin price data

df = pd.read_csv('/content/drive/MyDrive/Yoctobe Fintech/Datasets/btcusd30mn.csv', parse_dates=True)

df.rename(columns={'datetime': 'date'}, inplace=True)

df.index = pd.to_datetime(df['date'])

df['close'].plot(figsize=(16,8))

This snippet employs the pandas library to load Bitcoin price data from a CSV file into a DataFrame, renames the 'datetime' column to 'date', and sets it as the index. It subsequently visualizes the closing prices over time.

From Basics to Mastery: ARCH & GARCH Models in Financial Analysis - YouTube

This video provides foundational knowledge about ARCH and GARCH models, emphasizing their applications in financial analysis.

Next, we resample the data, calculate returns, and visualize the results:

ddf = df.resample("D").mean()

returns = 100 * ddf['close'].pct_change().dropna()

ax = returns.plot(figsize=(16,8))

ax.set_xlim(returns.index.min(), returns.index.max())

This code resamples the DataFrame to compute daily averages and calculates daily percentage changes, facilitating the analysis of volatility trends over time.

Chapter 3: Fitting GARCH Models

The standard approach results in a model with a constant mean and GARCH(1,1) conditional variance.

# Fit a GARCH model to returns

am = arch_model(returns)

res = am.fit(update_freq=5)

print(res.summary())

This process establishes a GARCH model instance and fits it to the returns data, allowing for a deeper understanding of volatility dynamics.

GARCH Model: Time Series Talk - YouTube

In this video, the speaker dives into the GARCH model, explaining its significance in time series analysis.

The GJR-GARCH model enhances the GARCH framework by allowing for asymmetries in volatility:

# Fit GJR-GARCH model to financial returns

am = arch_model(returns, p=1, o=1, q=1)

res = am.fit(update_freq=5, disp="off")

print(res.summary())

This fitting process captures the complexities in financial data, leading to more accurate risk assessments.

Chapter 4: Value-at-Risk (VaR) Forecasting

# Generate forecasts and quantile values

forecasts = res.forecast(start="2020-1-1", reindex=False)

cond_mean = forecasts.mean["2020":]

cond_var = forecasts.variance["2020":]

q = am.distribution.ppf([0.01, 0.05], res.params[-2:])

print(q)

In this section, we compute the Value at Risk (VaR) using conditional means and variances, enabling better risk management decisions.

This comprehensive guide covers the essential techniques for utilizing ARCH and GARCH models in financial analysis, providing readers with the knowledge to implement these models effectively for risk forecasting and volatility assessment.