Modelling extremal stock returns in a stable Paretian environment

Inhaltsverzeichnis (Table of Contents)

• INTRODUCTION: THE EMPIRICAL DISTRIBUTION OF STOCK RETURNS
• STOCK RETURNS, VOLATILITY, AND ASSET ALLOCATION
• STYLISED FACTS OF STOCK RETURN DISTRIBUTIONS
• MODELS FOR STOCK RETURN DISTRIBUTIONS
• AN OVERVIEW OF FULL PARAMETRIC RETURN DISTRIBUTION MODELS
• BASIC APPROACHES TO INFERENCE ABOUT EXTREMAL RETURNS
• TECHNICAL BACKGROUND FOR TAIL INFERENCE
• EXTREME VALUE THEORY (EVT)
• THE STABLE PARETIAN MODEL
• ESTIMATION OF THE STABLE PARETIAN INDEX α
• DESIRABLE PROPERTIES OF AN ESTIMATOR AND EVALUATION CRITERIA
• ESTIMATION METHODOLOGIES AND PREREQUISITES
• TAIL ESTIMATORS
• Intuition behind tail estimation
• Where does the tail start?
• Hill estimator (1975)
• Modifications of the Hill estimator
• Generalisations of the Hill estimator
• Pickands estimator (1975)
• Generalisations of the Pickands estimator
• De Haan-Resnick estimator (1980)
• De Haan-Pereira estimator (1999)
• ESTIMATION VIA THE PEAKS OVER THRESHOLD (POT) METHOD
• ML techniques
• Method of probability-weighted moments (PWM)
• Falk-Marohn estimator (1999)
• Choice of the threshold level
• CHARACTERISTIC FUNCTION TECHNIQUES
• Method of Moments estimators (MME)
• Minimum Distance estimators (MDE)
• Regression-type estimators
• MAXIMUM LIKELIHOOD ESTIMATORS (MLE)
• Algorithms for computation of the stable Paretian PDF
• The ML estimation procedure
• QUANTILE-BASED ESTIMATORS
• Quantile estimator by Fama and Roll (1971)
• McCulloch estimator (1986)
• OTHER APPROACHES
• WHICH ESTIMATOR IS THE BEST ONE?
• AN EMPIRICAL COMPARISON OF ESTIMATORS
• SIMULATION STUDY OF TAIL ESTIMATORS
• Performance with Pareto data and small sample properties
• Performance with Fréchet data
• Performance with stable data
• CONSIDERATION OF MODIFIED TAIL ESTIMATORS
• APPLICATION TO STOCK RETURN DATA
• Daily stock returns
• Weekly stock returns

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This dissertation explores the modelling of extreme stock returns in a stable Paretian environment. The primary objective is to examine different estimation methods for the stable Paretian index α, a crucial parameter determining the tail behavior of the distribution. The work aims to provide a comprehensive comparison of these methods, highlighting their strengths and weaknesses, and ultimately guiding researchers in selecting the most appropriate estimator for various applications. Key themes explored include:

• The empirical distribution of stock returns and its stylized facts
• Different models for stock return distributions, including full parametric models
• Tail inference techniques, particularly extreme value theory (EVT) and the stable Paretian model
• A wide range of estimators for the stable Paretian index α
• An empirical comparison of estimators using simulations and real-world stock return data

Zusammenfassung der Kapitel (Chapter Summaries)

The introduction lays the foundation for the study by discussing the importance of stock returns in financial decision-making and examining the stylized facts of stock return distributions, such as fat tails and peakedness. Chapter 2 presents an overview of full parametric models for stock return distributions, followed by a discussion of basic approaches to inference about extreme returns. Chapter 3 provides a technical background for tail inference, introducing extreme value theory (EVT) and the stable Paretian model. Chapter 4 delves into the estimation of the stable Paretian index α, beginning with a detailed discussion of the desirable properties of estimators and evaluation criteria. This chapter then presents a wide range of estimation methodologies, including tail estimators, the peaks over threshold (POT) method, characteristic function techniques, maximum likelihood estimators (MLE), quantile-based estimators, and other approaches. Chapter 5 features an empirical comparison of different estimators using simulations and real-world stock return data. The simulation study examines the performance of tail estimators with different types of data, including Pareto, Fréchet, and stable data. This chapter concludes by applying selected estimators to actual stock return data, focusing on daily and weekly returns.

Schlüsselwörter (Keywords)

This dissertation focuses on the modelling of extreme stock returns, emphasizing the use of stable Paretian distributions and various estimation methods for the tail index α. Key topics include extreme value theory (EVT), stable Paretian distributions, tail estimation, empirical comparison of estimators, simulation studies, stock returns, and financial risk management.