Stochastic Modeling of Stock Prices Incorporating Jump Diffusion and Shot Noise Models

Abstract

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, [2]) and on its continuation by SCHMIDT and STUTE ( Shot noise processes and the minimal martingale measure, [23]). These papers differ in modeling the decay of the jump effect: Whereas it is deterministic in [2], it is stochastic in [23]. 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 [15] 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 [15] and to the well-known pricing model of BLACK and SCHOLES [5], the presented shot noise models are able to reproduce volatility smile effects which can be observed in financial markets.

This thesis is structured as follows: The first chapter deals with the basics of probability theory. In particular, we present the stochastic processes which are incorporated in jump diffusion and shot noise models.

In the second chapter, the basics of financial mathematics are presented, especially the Cox-Ross-Rubinstein model as a discrete market model and jump diffusion models as continuous-time market models.

The third chapter deals with shot noise models for stock prices with deterministic decay of the jump effect – presented in ALTMANN, SCHMIDT and STUTE [2] – and with stochastic decay of the jump effect – presented in SCHMIDT and STUTE [23]. The equivalent minimal martingale measure is constructed in discrete and continuous-time models. Moreover, the resulting trading strategy is derived in discrete models with deterministic decay of the jump effect.

In the fourth chapter, a conclusion and perspectives are given.

The thesis is concluded by an appendix where we present a financial interpretation of the mathematical tools used in shot noise models.

The front page shows one sample path of a stock price process according to a shot noise model with exponential decay of the jump effect. The model parameters are chosen as follows: drift μ = 0, volatilityσ = 31.62%, intensity λ = 2, dampening parameter a = 5.

Contents

1. Basics of probability theory.. 1

1.1. Probability spaces and random variables.. 1

1.2. Probability distributions.. 3

1.3. Stochastic processes.. 5

1.3.1. Basic notions.. 5

1.3.2. Martingales.. 6

1.3.3. Local martingales and semimartingales.. 7

1.4. Decomposition of stochastic processes.. 9

1.4.1. Doob decomposition.. 9

1.4.2. Doob-Meyer decomposition.. 10

1.5. Examples of stochastic processes.. 12

1.5.1. Wiener process / standard Brownian motion.. 12

1.5.2. Homogeneous Poisson process.. 14

1.5.3. Compound Poisson process.. 17

1.5.4. Cox process / doubly stochastic Poisson process.. 19

1.6. Stochastic integrals.. 19

1.6.1. Itô integral.. 19

1.6.2. Jump integral.. 20

1.7. Itô’s formula.. 21

1.8. Stochastic differential equations.. 22

1.8.1. Arithmetic Brownian motion.. 23

1.8.2. Geometric Brownian motion.. 24

1.8.3. Cox-Ingersoll-Ross process.. 26

1.8.4. Doléans-Dade exponential / stochastic exponential.. 26

2. Basics of financial mathematics.. 28

2.1. Contingent claims and options.. 28

2.2. Discrete market models and continuous-time market models.. 29

2.3. Cox-Ross-Rubinstein model.. 30

2.3.1. Price processes in the Cox-Ross-Rubinstein model.. 30

2.3.2. Arbitrage and completeness in the Cox-Ross-Rubinstein model.. 32

2.3.3. Construction of the equivalent martingale measure in the Cox-Ross-Rubinstein model.. 34

2.4. Jump diffusion models.. 36

2.4.1. Price processes in jump diffusion models.. 37

2.4.2. Arbitrage and completeness in jump diffusion models.. 41

2.4.3. Construction of equivalent martingale measures in jump diffusion models.. 43

2.5. Convergence of the Cox-Ross-Rubinstein model to the Black-Scholes model.. 46

2.6. Equivalent minimal martingale measures and locally risk-minimizing strategies.. 48

3. Shot noise models for stock prices.. 51

3.1. Motivation.. 51

3.2. Shot noise process.. 51

3.3. Shot noise models.. 52

3.3.1. Shot noise models without jumps.. 53

3.3.2. Shot noise models without decay of the jump effect.. 53

3.3.3. Shot noise models with deterministic decay of the jump effect.. 53

3.3.4. Shot noise models with exponential decay of the jump effect.. 54

3.3.5. Shot noise models with stochastic decay of the jump effect.. 55

3.4. Equivalent minimal martingale measure in shot noise models with deterministic decay of the jump effect.. 57

3.4.1. Equivalent minimal martingale measure in discrete shot noise models.. 58

3.4.2. Trading strategy in discrete shot noise models.. 63

3.4.3. Equivalent minimal martingale measure in continuous-time shot noise models.. 67

3.4.4. Arbitrage possibilities in continuous-time shot noise models.. 73

3.5. Equivalent minimal martingale measure in shot noise models with stochastic decay of the jump effect.. 75

3.5.1. Equivalent minimal martingale measure in continuous-time shot noise models.. 76

3.5.2. Arbitrage possibilities in continuous-time shot noise models.. 78

4. Conclusion and perspectives.. 81

4.1. Conclusion.. 81

4.2. Perspectives.. 82

A. Appendix.. I

A.1. Financial interpretation of the mathematical tools.. I

A.2. Technical lemmata.. III

A.3. Semimartingale decomposition of the shot noise process and the stock price process in shot-noise models.. VI

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