In this thesis, we present a stochastic model for stock prices incorporating jump diffusion and shot noise models based on the work of Altmann, Schmidt and Stute ("A Shot Noise Model For Financial Assets") and on its continuation by Schmidt and Stute ("Shot noise processes and the minimal martingale measure"). These papers differ in modeling the decay of the jump effect: Whereas it is deterministic in the first paper, it is stochastic in the last paper. In general, jump effects exist because of overreaction due to news in the press, due to illiquidity or due to incomplete information, i.e. because certain information are available only to few market participants. In financial markets, jump effects fade away as time passes: On the one hand, if the stock price falls, new investors are motivated to buy the stock. On the other hand, a rise of the stock price may lead to profit-taking, i.e. some investors sell the stock in order to lock in gains. Shot noise models are based on Merton's jump diffusion models where the decline of the jump effect after a price jump is neglected. In contrast to jump diffusion models, shot noise models respect the decay of jump effects.
In complete markets, the so-called equivalent martingale measure is used to price European options and for hedging. Since stock price models incorporating jumps describe incomplete markets, the equivalent martingale measure cannot be determined uniquely. Hence, in this thesis, we deduce the so-called equivalent minimal martingale measure, both in discrete and continuous time. In contrast to Merton's jump diffusion models and to the well-known pricing model of Black and Scholes, the presented shot noise models are able to reproduce volatility smile effects which can be observed in financial markets.
Inhaltsverzeichnis (Table of Contents)
- Basics of probability theory
- Probability spaces and random variables
- Probability distributions
- Stochastic processes
- Basic notions
- Martingales
- Local martingales and semimartingales
- Decomposition of stochastic processes
- Doob decomposition
- Doob-Meyer decomposition
- Examples of stochastic processes
- Wiener process / standard Brownian motion
- Homogeneous Poisson process
- Compound Poisson process
- Cox process / doubly stochastic Poisson process
- Stochastic integrals
- Itô integral
- Jump integral
- Itô's formula
- Stochastic differential equations
- Arithmetic Brownian motion
- Geometric Brownian motion
- Cox-Ingersoll-Ross process
- Doléans-Dade exponential / stochastic exponential
- Basics of financial mathematics
- Contingent claims and options
- Discrete market models and continuous-time market models
- Cox-Ross-Rubinstein model
- Price processes in the Cox-Ross-Rubinstein model
- Arbitrage and completeness in the Cox-Ross-Rubinstein model
- Construction of the equivalent martingale measure in the Cox-Ross-Rubinstein model
- Jump diffusion models
- Price processes in jump diffusion models
- Arbitrage and completeness in jump diffusion models
- Construction of equivalent martingale measures in jump diffusion models
- Convergence of the Cox-Ross-Rubinstein model to the Black-Scholes model
- Equivalent minimal martingale measures and locally risk-minimizing strategies
- Shot noise models for stock prices
- Motivation
- Shot noise process
- Shot noise models
- Shot noise models without jumps
- Shot noise models without decay of the jump effect
- Shot noise models with deterministic decay of the jump effect
- Shot noise models with exponential decay of the jump effect
- Shot noise models with stochastic decay of the jump effect
- Equivalent minimal martingale measure in shot noise models with deterministic decay of the jump effect
- Equivalent minimal martingale measure in discrete shot noise models
- Trading strategy in discrete shot noise models
- Equivalent minimal martingale measure in continuous-time shot noise models
- Arbitrage possibilities in continuous-time shot noise models
- Equivalent minimal martingale measure in shot noise models with stochastic decay of the jump effect
- Equivalent minimal martingale measure in continuous-time shot noise models
- Arbitrage possibilities in continuous-time shot noise models
- Conclusion and perspectives
- Conclusion
- Perspectives
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This thesis aims to provide a comprehensive analysis of shot noise models for stock prices. The model is developed based on existing research, incorporating jump diffusion and shot noise models with both deterministic and stochastic decay of the jump effect. The main objective is to construct an equivalent minimal martingale measure, both in discrete and continuous time, and to demonstrate its applicability in financial markets.- Stochastic modeling of stock prices
- Jump diffusion and shot noise models
- Equivalent minimal martingale measure
- Arbitrage and completeness in financial markets
- Volatility smile effects
Zusammenfassung der Kapitel (Chapter Summaries)
- The first chapter introduces the fundamental concepts of probability theory, focusing on stochastic processes, martingales, and the Itô calculus. It lays the groundwork for the mathematical framework used in later chapters.
- The second chapter explores the basics of financial mathematics, presenting discrete and continuous-time market models. Key topics covered include the Cox-Ross-Rubinstein model, jump diffusion models, and the construction of equivalent martingale measures.
- The third chapter delves into shot noise models for stock prices, considering both deterministic and stochastic decay of the jump effect. The construction of the equivalent minimal martingale measure is explained in both discrete and continuous-time models, along with the derivation of the resulting trading strategy for discrete models with deterministic decay.
Schlüsselwörter (Keywords)
This thesis focuses on the application of stochastic modeling, specifically jump diffusion and shot noise models, in the realm of financial markets. It explores the concept of equivalent minimal martingale measures, their construction, and their relevance in pricing derivatives and managing risk. Key terms include stochastic processes, martingales, shot noise models, jump diffusion models, equivalent martingale measure, arbitrage, completeness, and volatility smile effects.- Quote paper
- Daniel Janocha (Author), 2016, Stochastic Modeling of Stock Prices Incorporating Jump Diffusion and Shot Noise Models, Munich, GRIN Verlag, https://www.grin.com/document/335816
