Thesis
Here on this webpage you can find my diploma thesis (in German) with the title “Optionspreisbewertung mit stochastischer Volatilität und Sprungprozessen – Eine Untersuchung am Deutschen Aktienindex” supervised by Prof. Dr. Daniel Rösch.
Abstract
The pricing of financial instruments, including options, require the proper modeling of the return distribution of financial assets. Since Black and Scholes (1973) and Merton (1973) introduced the pricing and hedging theory for the options markets, their model based on the assumption of normally distributed returns has been a very popular model for option pricing. Despite this fact, numerous studies have shown that the assumptions underlying the Black–Scholes market model are not consistent with the empirical stylized facts. Alternative models for describing return distributions have been proposed since the 1960’s, with strong support for the family of Lévy processes. Another stylized phenomena in the empirical observation of financial markets is the time-varying volatility of asset returns. This thesis deals with models designed to address these empirical facts which are commonly referred to as heavy tails, leptokurtosis, volatility smile and stochastic volatility.
The models under consideration include the classical Black–Scholes model as benchmark model, the Merton (1976) jump-diffusion model, the pure jump Finite Moment Log Stable process (FMLS) proposed by Carr and Wu (2003) and the Heston (1993) stochastic volatility model. In addition to these specifications, the jump-diffusion and the FMLS model are extended with seperate stochastic volatility processes.
The central part of the thesis consists of a simulation study with respect to the Deutscher Aktienindex (DAX). The model parameters are estimated with cross-sectional DAX index options data over a period of 30 business days. Two calibration procedures are applied, one naïve approach by minimizing the error between market and model values, and the second relies on a regularization method by adding a penalty term to the objective function. The resulting paramater estimates are reported and finally the out-of-the-sample forecasting performance for those models are investigated.
Keywords
option pricing, characteristic functions, Fourier transform, Lévy processes, stochastic volatility, calibration, regularization.
Miscellanea
This work has been nominated for the “Hochschulpreis David–Kopf 2008” and participated at the “Karriere-Preis der DZ Bank Gruppe” in 2009.
