The Black-Scholes (or Black-Scholes-Merton) Model has become the standard model for the pricing of options and can surely be seen as one of the main reasons for the growth of the derivative market after the model´s introduction in 1973. As a consequence, the inventors of the model, Robert Merton, Myron Scholes, and without doubt also Fischer Black, if he had not died in 1995, were awarded the Nobel prize for economics in 1997.
The model, however, makes some strict assumptions that must hold true for accurate pricing of an option. The most important one is constant volatility, whereas empirical evidence shows that volatility is heteroscedastic. This leads to increased mispricing of options especially in the case of out of the money options as well as to a phenomenon known as volatility smile. As a consequence, researchers introduced various approaches to expand the model by allowing the volatility to be non-constant and to follow a sto-chastic process. It is the objective of this thesis to investigate if the pricing accuracy of the Black-Scholes model can be significantly improved by applying a stochastic volatility model.
Table of Contents
List of Abbreviations
List of Figures and Tables
1 Introduction
1.1 Background
1.2 Problem Definition and Objective
1.3 Course of the Investigation
2 Option Basics
2.1 Classification and Types of Options
2.2 Terminology & Payoff Characteristics
2.3 Put-Call Parity
3 Stochastic Processes
3.1 Markov Property
3.2 Wiener Process and Brownian Motion
3.3 Itô Process and Itô's Formula
3.4 Application of Itô's Lemma to Stock Price Movements
4 The Black-Scholes Model
4.1 Assumptions
4.2 Risk free Evaluation
4.3 Black-Scholes Partial Differential Equation
4.4 Black-Scholes Pricing Formula
4.5 Evaluation of Model Parameters and Assumptions
4.6 Volatility Smile
5 Stochastic Volatility Models
5.1 Overview of Stochastic Volatility Models
5.2 The Heston Model
5.2.1 The Mean-reverting Square Root Diffusion Process
5.2.2 Derivation of the Heston PDE
5.3 Heston Pricing Formula for European Calls
6 Empirical Analysis
6.1 Methodology
6.1.1 Closed Form approximations
6.1.2 Implementation
6.2 Calibration
6.2.1 Model parameters
6.2.2 Calibration methods
6.3 Data
6.4 Results
6.4.1 Calibration results
6.4.2 Pricing Results
7 Model evaluation
7.1 Interpretation of the Results
7.2 Drawbacks of the Model
7.3 Approaches for further Improvements
7.3.1 Time dependent Heston Model
7.3.2 Jump-Diffusion Model
8 Conclusion
Reference List
Appendix
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