# Chaos Dynamics and Stochastic Volatility

**Topics:**Chaos theory, Randomness, Probability theory

**Pages:**24 (5556 words)

**Published:**April 28, 2014

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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