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.
Contents
1. Basics of probability theory
1.1. Probability spaces and random variables
1.2. Probability distributions
1.3. Stochastic processes
1.3.1. Basic notions
1.3.2. Martingales
1.3.3. Local martingales and semimartingales
1.4. Decomposition of stochastic processes
1.4.1. Doob decomposition
1.4.2. Doob-Meyer decomposition
1.5. Examples of stochastic processes
1.5.1. Wiener process / standard Brownian motion
1.5.2. Homogeneous Poisson process
1.5.3. Compound Poisson process
1.5.4. Cox process / doubly stochastic Poisson process
1.6. Stochastic integrals
1.6.1. Itô integral
1.6.2. Jump integral
1.7. Itô’s formula
1.8. Stochastic differential equations
1.8.1. Arithmetic Brownian motion
1.8.2. Geometric Brownian motion
1.8.3. Cox-Ingersoll-Ross process
1.8.4. Doléans-Dade exponential / stochastic exponential
2. Basics of financial mathematics
2.1. Contingent claims and options
2.2. Discrete market models and continuous-time market models
2.3. Cox-Ross-Rubinstein model
2.3.1. Price processes in the Cox-Ross-Rubinstein model
2.3.2. Arbitrage and completeness in the Cox-Ross-Rubinstein model
2.3.3. Construction of the equivalent martingale measure in the Cox-Ross-Rubinstein model
2.4. Jump diffusion models
2.4.1. Price processes in jump diffusion models
2.4.2. Arbitrage and completeness in jump diffusion models
2.4.3. Construction of equivalent martingale measures in jump diffusion models
2.5. Convergence of the Cox-Ross-Rubinstein model to the Black-Scholes model
2.6. Equivalent minimal martingale measures and locally risk-minimizing strategies
3. Shot noise models for stock prices
3.1. Motivation
3.2. Shot noise process
3.3. Shot noise models
3.3.1. Shot noise models without jumps
3.3.2. Shot noise models without decay of the jump effect
3.3.3. Shot noise models with deterministic decay of the jump effect
3.3.4. Shot noise models with exponential decay of the jump effect
3.3.5. Shot noise models with stochastic decay of the jump effect
3.4. Equivalent minimal martingale measure in shot noise models with deterministic decay of the jump effect
3.4.1. Equivalent minimal martingale measure in discrete shot noise models
3.4.2. Trading strategy in discrete shot noise models
3.4.3. Equivalent minimal martingale measure in continuous-time shot noise models
3.4.4. Arbitrage possibilities in continuous-time shot noise models
3.5. Equivalent minimal martingale measure in shot noise models with stochastic decay of the jump effect
3.5.1. Equivalent minimal martingale measure in continuous-time shot noise models
3.5.2. Arbitrage possibilities in continuous-time shot noise models
4. Conclusion and perspectives
4.1. Conclusion
4.2. Perspectives
Objectives & Topics
This thesis aims to develop a stochastic model for stock prices that integrates jump diffusion and shot noise processes. The primary research focus lies in constructing an equivalent minimal martingale measure to effectively price and hedge European options in incomplete markets, overcoming the limitations of standard models like Black-Scholes.
- Stochastic modeling of asset prices with jump-decay dynamics.
- Construction of equivalent minimal martingale measures in discrete and continuous time.
- Evaluation of arbitrage possibilities and derivation of optimal trading strategies.
- Comparison of shot noise models with existing jump diffusion and Black-Scholes frameworks.
Book Excerpt
1.8.1 Arithmetic Brownian motion
First, we introduce the arithmetic Brownian motion. For this purpose, let μ ∈ ℝ and σ ∈ ℝ⁺₀. In (1.35) we set μt(Xt) = μ = const. and σt(Xt) = σ = const. Hence, we consider the SDE dXt = μdt + σ dWt, t ∈ [0, T]. (1.38)
According to Theorem 1.8.2, the SDE (1.38) with X₀ ∈ L²(ℙ) independent of the natural filtration generated by W = {Wt}t∈[0,T] has a unique strong solution since μt and σt are constant (set L := 0 and C := |μ|+|σ|). We use the jumpless version of Itô’s formula (cf. Corollary 1.7.4) to solve this SDE: Setting f(t,Wt) := Xt and comparing the coefficients of dt and dWt in (1.38) and (1.33) yields a system of two ordinary differential equations possessing a unique solution. This solution can be rewritten to Xt = X₀ + μt + σWt, t ∈ [0, T]. (1.39)
Summary of Chapters
1. Basics of probability theory: Provides the foundational stochastic concepts, including filtration, martingales, and stochastic differential equations, necessary for modeling market dynamics.
2. Basics of financial mathematics: Covers fundamental market models like Cox-Ross-Rubinstein and jump diffusion, focusing on pricing contingent claims and establishing the requirements for arbitrage-free, complete markets.
3. Shot noise models for stock prices: Introduces shot noise processes to generalize jump diffusion models, specifically addressing jump decay and deriving equivalent minimal martingale measures and optimal trading strategies.
4. Conclusion and perspectives: Summarizes the thesis's results regarding the construction of the equivalent minimal martingale measure and discusses future research directions, such as stochastic decay parameters.
Key Words
Stochastic processes, Jump diffusion, Shot noise models, Martingale measure, Arbitrage-free, Asset pricing, Financial mathematics, Brownian motion, Hedging, European options, Stochastic differential equations, Incomplete markets, Risk minimization, Volatility, Stochastic decay.
Frequently Asked Questions
What is the core focus of this thesis?
The work primarily deals with the stochastic modeling of stock prices using shot noise processes to better reflect market behavior, particularly the decay of jump effects, and to price options in incomplete markets.
Which financial models are used as a basis?
The thesis builds upon the Cox-Ross-Rubinstein binomial model for discrete settings and classical jump diffusion and Black-Scholes models for continuous-time frameworks.
What is the primary goal regarding option pricing?
The objective is to derive the equivalent minimal martingale measure, which provides a consistent way to price contingent claims in markets where standard arbitrage pricing is not uniquely determined.
Which mathematical tools are employed?
The research relies on advanced stochastic calculus, including Itô’s formula, Girsanov’s theorem, Doob-Meyer decomposition, and the analysis of semimartingales.
What does the main part of the thesis cover?
It details the construction of shot noise processes, evaluates their properties under deterministic and stochastic jump decay, and solves optimization problems for locally risk-minimizing trading strategies.
How can this work be characterized by keywords?
Key characterizations include stochastic modeling, jump-decay dynamics, martingale measures, and locally risk-minimizing hedging strategies.
How do shot noise models improve upon jump diffusion?
Unlike jump diffusion models, where jump effects remain constant, shot noise models allow jump impacts to diminish over time, providing a more realistic representation of market recovery after shocks.
What is the significance of the "dampening function" introduced?
The dampening function quantitatively regulates how quickly the impact of a market shock (jump) fades, which is essential for capturing volatility smile effects observed in real-world trading.
- Arbeit zitieren
- Daniel Janocha (Autor:in), 2016, Stochastic Modeling of Stock Prices Incorporating Jump Diffusion and Shot Noise Models, München, GRIN Verlag, https://www.grin.com/document/335816
