Aus Sicht der Mathematik spielen Optionen eine wesentliche Rolle seit der bahnbrechenden Arbeit von Black und Scholes im Jahre 1973. Deren Modell basiert jedoch auf der unrealistischen Annahme, das log-returns von Aktienkursen normalverteilt sind. Eberlein und Keller haben 1995 gezeigt, daß solche log-returns hyperbolisch verteilt sind. Die vorliegende Arbeit baut auf dieser Annahme auf und erweitert das Optionsspektrum von Europäischen Optionen auf Asiatische, Amerikanische sowie Multi-Asset-Optionen. Weiters wird das "Standard"-Martingal-Maß, die sogenannte Esscher-Transformation, durch das Entropie-minimierende Maß erweitert. Da jedoch keine exakte Preissetzung solcher Optionen möglich ist, wird auf numerische Simulationen und Approximationen zurückgegriffen. Die verwendeten numerischen Verfahren sind die Monte Carlo-Methode mit verschiedenen Varianzreduktionstechniken und die Quasi-Monte Carlo Methode.
Table of Contents
1 Hyperbolic Lévy processes
1.1 Introduction
1.2 Lévy processes
1.3 Infinitely divisible distributions
1.4 The generalized hyperbolic distribution
1.4.1 The hyperbolic distribution
1.4.2 Asymptotic ML-estimators
1.4.3 Approximating the inverse distribution function
1.4.4 The NIG distribution
1.5 The loghyperbolic distribution
1.6 The hyperbolic Lévy motion
1.6.1 NIG processes
1.6.2 Changing measures
2 Option pricing and Quasi-Monte Carlo methods
2.1 Introduction
2.2 Basic concepts of option pricing
2.2.1 The Black-Scholes model
2.2.2 The generalized hyperbolic model
2.3 Monte Carlo methods and variance reduction
2.3.1 Variance reduction techniques
2.4 Quasi-Monte Carlo methods
3 Asian options
3.1 Introduction
3.2 Options on the general mean
3.3 Approximation of arithmetic average options
4 Asian option pricing in the NIG model
4.1 Introduction
4.2 Stop-loss transforms and upper bounds for arithmetic average options
4.3 Approximations for arithmetic average options
4.4 Arithmetic and geometric average options
4.5 Comparison with the Black-Scholes-model
5 American options
5.1 Introduction
5.2 American option pricing
5.3 Bundling algorithm
5.4 Simulated Tree Algorithm
6 The minimal entropy martingale measure
6.1 Introduction
6.2 Lévy processes and the minimal entropy martingale measure
6.3 The minimal entropy martingale measure for NIG Lévy processes
6.4 Option pricing
7 Options on several assets
7.1 Introduction
7.2 The d-dimensional hyperbolic distribution
7.3 Option pricing
Research Objectives and Key Topics
The primary objective of this thesis is to generalize the hyperbolic option pricing model to include non-standard options, such as Asian and American options, and to enhance pricing accuracy using Monte Carlo and Quasi-Monte Carlo methods, particularly in the context of the Normal Inverse Gaussian (NIG) model.
- Hyperbolic and NIG Lévy processes for modeling stock price dynamics.
- Advanced Monte Carlo and Quasi-Monte Carlo simulation techniques for option pricing.
- Pricing methodologies for exotic options, including Asian and American-style options.
- Comparison of the hyperbolic and NIG models with the classical Black-Scholes framework.
Excerpt from the Book
1.2 Lévy processes
A family {Xt | t ≥ 0} of random variables on R^d with parameter t ∈ [0,∞) defined on a probability space (Ω, F, P) is called a stochastic process. A stochastic process (Yt) is called a modification (or version) of a stochastic process (Xt), if P[Xt = Yt] = 1 for t ∈ [0, ∞).
For any fixed 0 ≤ t1 < · · · < tn, P[Xt1 ∈ B1, . . . , Xtn ∈ Bn] determines a probability measure on B(R^d)^n. The family of the probability measures over all choices of n and t1, . . . , tn is called the system of finite-dimensional distributions of (Xt). Two stochastic processes (Xt) and (Yt) are called identical in law, written as (Xt) =d (Yt), if the systems of their finite-dimensional distributions are identical.
A stochastic process (Xt) on R^d is stochastically continuous or continuous in probability, if lim s→t P[|Xs − Xt| > ǫ] = 0 for every t ≥ 0 and ǫ > 0. The Brownian motion and Poisson processes belong to the class of Lévy processes, which we will define now:
Definition: A stochastic process (Xt) on R^d is called a Lévy process, if the following conditions are satisfied: (1) For any choice of n ≥ 1 and 0 ≤ t0 < . . . < tn, the random variables Xt0, Xt1 − Xt0, . . . , Xtn − Xtn−1 are independent (independent increments property). (2) X0 = 0 a.s. (3) The distribution of Xs+t − Xs does not depend on s (temporal homogeneity or stationary increments property). (4) The process is stochastically continuous. (5) There is a set Ω0 ∈ F with P(Ω0) = 1 such that, for every ω ∈ Ω0, Xt(ω) is right-continuous for t ≥ 0 and has left limits for t > 0.
Summary of Chapters
Hyperbolic Lévy processes: Provides fundamental definitions of Lévy processes and infinitely divisible distributions, focusing on the generalized hyperbolic and NIG distributions.
Option pricing and Quasi-Monte Carlo methods: Introduces basic concepts of option pricing, the Black-Scholes model, and simulation techniques, including Monte Carlo and Quasi-Monte Carlo methods.
Asian options: Examines pricing methodologies for Asian options on the general mean, employing Monte Carlo methods and variance reduction techniques.
Asian option pricing in the NIG model: Investigates Asian options within the NIG model, deriving upper bounds via stop-loss transforms and providing approximation methods.
American options: Analyzes simulation algorithms for American and Bermudan options across different models using bundling and simulated tree methods.
The minimal entropy martingale measure: Discusses the minimal entropy martingale measure, its connection to the Esscher transform, and its application in the NIG model.
Options on several assets: Extends the modeling framework to multi-asset options, utilizing d-dimensional generalized hyperbolic distributions.
Keywords
Option pricing, Hyperbolic model, Normal Inverse Gaussian model, Lévy processes, Monte Carlo methods, Quasi-Monte Carlo methods, Asian options, American options, Minimal entropy martingale measure, Esscher transform, Variance reduction, Multi-asset options, Financial mathematics.
Frequently Asked Questions
What is the core focus of this dissertation?
The work focuses on generalizing asset price models, specifically using hyperbolic and NIG Lévy processes, to price non-standard options like Asian and American derivatives.
What are the primary financial models discussed?
The dissertation covers the Black-Scholes model as a baseline, focusing on its limitations, and introduces the hyperbolic model and the Normal Inverse Gaussian (NIG) model as more realistic alternatives.
What is the main research objective regarding option pricing?
The primary goal is to overcome the lack of analytical formulas for exotic options by developing robust simulation-based methods, such as Monte Carlo and Quasi-Monte Carlo, and specific approximations for better accuracy.
Which mathematical techniques are employed for pricing?
The author uses Lévy-Itô decompositions, Esscher transforms, the minimal entropy martingale measure, and variance reduction techniques like importance sampling and control variates.
How is the American option pricing addressed?
American option pricing is addressed through backward induction algorithms, specifically the bundling algorithm and the simulated tree algorithm, adapted for hyperbolic and NIG models.
What characterizes the hyperbolic model compared to Black-Scholes?
The hyperbolic model accounts for discontinuous paths and semi-heavy tails, providing a better fit for empirical stock return data, which Black-Scholes typically fails to capture.
How does the author handle multi-asset options?
Multi-asset options are analyzed by introducing d-dimensional generalized hyperbolic distributions and employing the multivariate Esscher transform to price basket or rainbow options.
What role does the minimal entropy martingale measure play?
It is explored as an alternative martingale measure to the Esscher transform, offering potential economic advantages and simplification in calculations for NIG processes.
- Quote paper
- Dr. Martin Predota (Author), 2002, The hyperbolic model: Option pricing using approximation and Quasi-Monte Carlo methods, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/125316
