Chaos Dynamics and Stochastic Volatility

Topics: Chaos theory, Randomness, Probability theory Pages: 24 (5556 words) Published: April 28, 2014
STRATHMORE UNIVERSITY
SCHOOL OF FINANCE AND APPLIED ECONOMICS
Year 4, Semester 2
Empirical Finance
Group Assignment

Chaos Dynamics and Stochastic Volatility
Application to Financial Markets

January 29, 2014

Group Members
059831
059882
060139
061074

Oile, Kenneth Ogola
Ndirangu, Evelyn Chaki
Nzesya, Lilian Mwikali
Chege, Eric Theuri

Lecturer
Mr Ferdinand Othieno

Table of Contents
Abstract .............................................................................................................................................................. ii 1

Chaos and Non-Linear Dynamics ..................................................................................................... 1 1.1

Introduction .................................................................................................................................... 1

1.2

The Random Walk Hypothesis ................................................................................................ 1

1.3

Random vs. Unpredictable ........................................................................................................ 2

1.4

Chaos Theory.................................................................................................................................. 2

1.5

Mathematical Abstraction ......................................................................................................... 3

1.5.1 The BDS Statistic....................................................................................................................... 5 1.6
1.7
2

Empirical Examples Demonstrating Random Market Hypothesis Failures ........... 6 Conclusions and Remarks ......................................................................................................... 7

Stochastic Volatility ............................................................................................................................... 8 2.1

Introduction .................................................................................................................................... 8

2.2

Mathematical Abstraction ......................................................................................................... 9

2.3

Stylized Facts about Returns .................................................................................................. 10

2.4

Estimation of the Simple SV Model ...................................................................................... 11

2.4.1 Efficient Method of Moments ............................................................................................ 11 2.4.2 Simulated Method of Moments ......................................................................................... 12 2.4.3 Markov Chain Monte Carlo ................................................................................................. 13 2.5

Conclusion ..................................................................................................................................... 14

Bibliography ................................................................................................................................................... 15

i

Abstract
This paper contains a discussion on two theories that are important in modern empirical finance: chaos theory and stochastic volatility.
Chaos is a deterministic process which looks random. Along with the recognition of sophisticated predictable patterns is a comprehensive mathematical structure to deal with these patterns. The application of this theory to financial markets is a rejection of the Random Walk Hypothesis in stock market prices.

Stochastic volatility (SV) is offered as a more complex alternative to the Generalized Auto-Regressive Conditional Heteroscedasticity (GARCH) framework in modelling timedependent (conditional) return volatility. The main difference is that in SV,...


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