The Valuation of Multivariate Options
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Diploma Thesis from the year 2004 in the subject Business economics - Investment and Finance, grade: 1,0, Friedrich-Alexander University Erlangen-Nuremberg (Wirtschafts- und Sozialwissenschaftliche Fakultät), course: Statistik und Ökonometrie, language: English, abstract: Inhaltsangabe:Abstract:During the last decades, capital markets have transformed rapidly. Derivative securities or more simply derivatives like swaps, futures, and options supplemented the trading of stocks and bonds. These financial products can frequently be seen in the media: Due to derivatives, Procter & Gamble lost $15...
Diploma Thesis from the year 2004 in the subject Business economics - Investment and Finance, grade: 1,0, Friedrich-Alexander University Erlangen-Nuremberg (Wirtschafts- und Sozialwissenschaftliche Fakultät), course: Statistik und Ökonometrie, language: English, abstract: Inhaltsangabe:Abstract:
During the last decades, capital markets have transformed rapidly. Derivative securities or more simply derivatives like swaps, futures, and options supplemented the trading of stocks and bonds. These financial products can frequently be seen in the media: Due to derivatives, Procter & Gamble lost $150 million in 1994, Barings Bank lost $1.3 billion in 1995 and Long-Term Capital Management (LTCM) lost $3.5 billion in 1998.
Though these figures seem daunting, derivatives can be useful financial instruments. Applications include risk management, speculation, reduced transaction costs, and regulatory arbitrage.
Theory and practice of option valuation were revolutionized in 1973, when Fischer Black and Myron Scholes published their celebrated Black Scholes formula in the landmark paper The pricing of options and corporate liabilities . Afterwards, a vast amount of papers on option valuation was published which employ all kinds of stochastic processes. Thereby, the special features of financial return data are tried to be taken into account.
Advancing option valuation theory to options with multiple underlyings, lead to the problem that the dependence structure of the underlying securities needs to be considered. Though linear correlation is a widely used dependence measure, it may be inappropriate for multivariate return data. Throughout the last years, dependence modelling through copulas has become common.
Copulas are multivariate distributions on the d-dimensional unit-hyper-square which couples d marginal distributions to a joint distribution. Copulas can be used to construct dependence measures like the rank correlation coefficients of Spearman or Kendall. They are also a useful tool in the context of option valuation.
The prices of multivariate options depend on the distributional assumptions of stock price changes and the dependence structure. This thesis exhibits the features of multivariate financial return data. Evidence of (multi-)non-normality is presented. A general overview on multivariate option valuation theory is given. A nonparametric model and two Esscher models are introduced in detail. Then, the multivariate normal and the multivariate normal inverse Gaussian distribution are assumed as return distributions for an empirical study. The study exhibits the influence of the dependence structure and the properties of the assumed return distribution on option prices.
Inhaltsverzeichnis:Table of Contents:
List of FiguresVI
List of TablesVII
Frequently Used NotationsVIII
1.Introduction1
2.Derivatives and Options in Particular3
2.1Standard Options3
2.2Exotic Options5
2.3Multivariate Options6
3.Characteristics of Financial Returns8
3.1Stylized Facts of Univariate Return Distributions8
3.2Digression: Dependence & Copulas12
3.2.1Shortcomings of Linear Correlation14
3.2.2Copulas and Rank Based Dependence Concepts16
3.2.3Further Remarks on Copulas20
3.3Stylized Facts of Multivariate Return Distributions24
4.Option Valuation Approaches30
4.1Binomial Option Pricing30
4.2The Black-Scholes Formula34
4.3Risk-Neutral Valuation36
5.Multivariate Option Pricing Models39
5.1Literature Review39
5.1.1Margrabe: The Value of an Option to Exchange one Asset for Another39
5.1.2Johnson: Options on the Maximum or the Minimum of Several Assets41
5.1.3Boyle: A Lattice Framework for Option Pricing With two State Variables42
5.1.4Carmona and Durrleman: Generalizing Black-Scholes to Multivariate Contingent Claims4...
During the last decades, capital markets have transformed rapidly. Derivative securities or more simply derivatives like swaps, futures, and options supplemented the trading of stocks and bonds. These financial products can frequently be seen in the media: Due to derivatives, Procter & Gamble lost $150 million in 1994, Barings Bank lost $1.3 billion in 1995 and Long-Term Capital Management (LTCM) lost $3.5 billion in 1998.
Though these figures seem daunting, derivatives can be useful financial instruments. Applications include risk management, speculation, reduced transaction costs, and regulatory arbitrage.
Theory and practice of option valuation were revolutionized in 1973, when Fischer Black and Myron Scholes published their celebrated Black Scholes formula in the landmark paper The pricing of options and corporate liabilities . Afterwards, a vast amount of papers on option valuation was published which employ all kinds of stochastic processes. Thereby, the special features of financial return data are tried to be taken into account.
Advancing option valuation theory to options with multiple underlyings, lead to the problem that the dependence structure of the underlying securities needs to be considered. Though linear correlation is a widely used dependence measure, it may be inappropriate for multivariate return data. Throughout the last years, dependence modelling through copulas has become common.
Copulas are multivariate distributions on the d-dimensional unit-hyper-square which couples d marginal distributions to a joint distribution. Copulas can be used to construct dependence measures like the rank correlation coefficients of Spearman or Kendall. They are also a useful tool in the context of option valuation.
The prices of multivariate options depend on the distributional assumptions of stock price changes and the dependence structure. This thesis exhibits the features of multivariate financial return data. Evidence of (multi-)non-normality is presented. A general overview on multivariate option valuation theory is given. A nonparametric model and two Esscher models are introduced in detail. Then, the multivariate normal and the multivariate normal inverse Gaussian distribution are assumed as return distributions for an empirical study. The study exhibits the influence of the dependence structure and the properties of the assumed return distribution on option prices.
Inhaltsverzeichnis:Table of Contents:
List of FiguresVI
List of TablesVII
Frequently Used NotationsVIII
1.Introduction1
2.Derivatives and Options in Particular3
2.1Standard Options3
2.2Exotic Options5
2.3Multivariate Options6
3.Characteristics of Financial Returns8
3.1Stylized Facts of Univariate Return Distributions8
3.2Digression: Dependence & Copulas12
3.2.1Shortcomings of Linear Correlation14
3.2.2Copulas and Rank Based Dependence Concepts16
3.2.3Further Remarks on Copulas20
3.3Stylized Facts of Multivariate Return Distributions24
4.Option Valuation Approaches30
4.1Binomial Option Pricing30
4.2The Black-Scholes Formula34
4.3Risk-Neutral Valuation36
5.Multivariate Option Pricing Models39
5.1Literature Review39
5.1.1Margrabe: The Value of an Option to Exchange one Asset for Another39
5.1.2Johnson: Options on the Maximum or the Minimum of Several Assets41
5.1.3Boyle: A Lattice Framework for Option Pricing With two State Variables42
5.1.4Carmona and Durrleman: Generalizing Black-Scholes to Multivariate Contingent Claims4...
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