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

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:

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..

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:

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:

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 |

University of Applied Sciences:

- Online Application
- International Students
- Bachelor Program Applied Mathematics (in german)
- Master Program Applied Mathematics (in german)