Time Series Models: ARCH, GARCH and Stochastic Volatility

                                     - with R-Implementations -


Chapter 1: Brief Introduction into Using R
Chapter 2: Typical Properties of Financial Time Series
Chapter 3: Time Series Models with Return-Driven Volatilities: A Common Sense Approach
Chapter 4: The ARCH Time Series Model
Chapter 5: Maximum Likelihood Estimation of the ARCH(1)-Model
Chapter 6: Maximum Likelihood Estimation of the ARCH(d)-Model
Chapter 7: Exponentially Weighted Volatilities and Introduction to GARCH
Chapter 8: Maximum Likelihood Estimation of the GARCH(1,1)-Model
Chapter 9: Stochastic Volatility Models
Chapter 10: The Likelihood Function for Stochastic Volatility Models
Chapter 11: Simulated Likelihood Inference for Stochastic Volatility Models





Key Figures and Illustrations:


Chapter 2: Typical Properties of Financial Time Series

Daily closing prices of S&P500 (adjusted close) on logarithmic scale, 1950 - 2015:




Returns of S&P500:




Realized d-day historical volatilities of S&P500:
(measured in daily units, not yearly)




Comparison of normalized returns to standard normal distribution:






Chapter 3: Time Series Models with Return-Driven Volatilities:
A Common Sense Approach


Simulated Black-Scholes path with 1% daily volatility:
(corresponding to about 16% yearly volatility)




Simulated path for the "naive ARCH model":
apparently, the model needs to be adjusted..






Chapter 4: The ARCH Time Series Model

Simulated ARCH(5)-path: pronounced volatility clustering




Simulated ARCH(15)-path: returns cluster into large and small size
regions:




Simulated ARCH(250)-path: returns more uniformly distributed:






Chapter 5: Maximum Likelihood Estimation of the ARCH(1)-Model

Contour plot of the log-likelihood function logL(bsvol,w0) for S&P500-Data,
ARCH(1)-model parameter bsvol on x-axis, w0 on y-axis:




Contour plot of the log-likelihood function logL(bsvol,w0) for DAX-Data,
ARCH(1)-model parameter bsvol on x-axis, w0 on y-axis:




Contour plot of the log-likelihood function logL(bsvol,w0) for GE-Data,
ARCH(1)-model parameter bsvol on x-axis, w0 on y-axis:





Data Sets (from Yahoo Finance):
SPX.txt: Daily adjusted close of S&P500, 1950 - 2015
GE.txt: Daily adjusted close of General Electric Stock, 1962 - 2015
DAX.txt: Daily adjusted close of German DAX30, 2005 - 2014




Hochschule RheinMain Wiesbaden,
University of Applied Sciences:

Bachelor and Master Program in Applied Mathematics:



Option Pricing
Time Series Models
















Last Update: Aug 11th, 2017
contact@option-pricing.de