FINANCIAL MODELING WITH LEVY´

PROCESSES AND APPLYING LEVY´

SUBORDINATOR TO CURRENT STOCK

DATA

by

GONSALGE ALMEIDA

Submitted in partial fullfillment of the requirements for the degree of

Doctor of Philosophy

Dissertation Advisor: Dr. Wojbor A. Woyczynski

Department of Mathematics, Applied Mathematics and Statistics CASE WESTERN RESERVE UNIVERSITY

January 2020 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

Gonsalge Almeida

candidate for the Doctoral of Philosophy degree

Committee Chair: Dr.Wojbor Woyczynski Professor, Department of the Mathematics, Applied Mathematics and Statis- tics

Committee: Dr.Alethea Barbaro Associate Professor, Department of the Mathematics, Applied Mathematics and Statistics

Committee: Dr.Jenny Brynjarsdottir Associate Professor, Department of the Mathematics, Applied Mathematics and Statistics

Committee: Dr.Peter Ritchken Professor, Weatherhead School of Management

Acceptance date: June 14, 2019

*We also certify that written approval has been obtained for any proprietary material contained therein. CONTENTS

List of Figures iv

List of Tables ix Introduction ...... 1

1 Financial Modeling with L´evyProcesses and Infinitely Divisible Distributions 5 1.1 Introduction ...... 5 1.2 Preliminaries on L´evyprocesses ...... 6 1.3 Characteristic Functions ...... 8 1.4 Cumulant Generating Function ...... 9 1.5 α−Stable Distributions ...... 10 1.6 Tempered Stable Distribution and Process ...... 19 1.6.1 Tempered Stable Diffusion and Super-Diffusion ...... 23 1.7 Numerical Approximation of Stable and Tempered Stable Sample Paths 28 1.8 Monte Carlo Simulation for Tempered α−Stable L´evyprocess . . . . 34

2 Brownian Subordination (Tempered Stable Subordinator) 44

i 2.1 Introduction ...... 44 2.2 Tempered Anomalous Subdiffusion ...... 46 2.3 Subordinators ...... 49 2.4 Time-Changed Brownian Motion ...... 51 2.5 Tempered Stable Subordinator ...... 52 2.6 Numerical Simulation of the TSB Process ...... 54 2.7 Parameter Estimation of the TSB Process ...... 63 2.8 Comparison of TSB Process with Diffusion and Stable Subdiffusive Processes ...... 74 2.9 Error Comparison of Brownian Motion and TSB Processes ...... 75

3 Statistical Analysis of Log-returns of the Stock Data using the Least-Square-Estimation and the method of moments 79 3.1 Introduction ...... 79 3.2 Preliminaries ...... 83 3.2.1 Fourier Transform and Pricing Options ...... 84 3.2.2 Carr-Madan Formulation for Evaluating the European Option Pricing ...... 87 3.2.3 Passing from Characteristic Function to Probability Density Function (PDF) and Cumulative Density Function (CDF) . . 88 3.2.4 Fourier Transforms and Transposition of European Options . . 89 3.3 Stock Return Data ...... 91 3.4 Normal, Subordinated Stable, and Subordinated Tempered Stable Fitting ...... 91 3.4.1 Analyzing Log-returns of the WMK Stock...... 92 3.4.2 Analyzing Log-returns of the APPLE Stock...... 99 3.4.3 Analyzing log-return of the AMAZON stock...... 108

ii 3.5 Statistical Analysis of Log-returns of the Stock Data using the Method of Moments ...... 114 3.5.1 Analyzing Log-return of the WMK Stock using Method of Moments ...... 115

4 The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) with CGMY Jumps 118 4.1 Introduction ...... 118 4.2 The Option Pricing Models ...... 121 4.3 GARCH-CGMY-jumps ...... 128 4.3.1 CGMY Distributions ...... 128 4.3.2 Equivalent Measure Changes for L´evyProcesses ...... 129 4.3.3 Stable Processes ...... 130 4.3.4 CGMY Process as Time Changed Brownian Motion ...... 131 4.3.5 The CGMY-GARCH Option Pricing Model ...... 136

A L´evyProcesses and Infinite Divisibility 140

B 142

iii LIST OF FIGURES

1.1 α−stable densities for α = 0.5, 0.1, 1.5, and 2 with β = 0, σ = 1, and µ =0...... 14 1.2 α−stable densities for β = 0, 0.25, 0.5, 0.75, and 1 with α = 1.25, σ = 0.5, and µ =0...... 15 1.3 Compare cumulative densities for α = 0.5, 0.1, 1.5, and 2 with β = 0, σ = 1, and µ = 0...... 17 √ 1.4 Probability Distributions of Normal (α = 2, β = 0, σ = 1/ 2, µ = 0), Cauchy (α = 1, β = 0, σ = 1, µ = 0), and L´evy(α = 0.5, β = 1, σ = 1, µ =0)...... 18

1.5 The stable process (Sα(σ, β, µ)) using the Monte Carlo method with 1000 simulations. (i) the trajectory, (ii) the random variable, (iii) the

histrogram, (iv) the density of S0.8(1, 0, 0.5, 1)...... 35

1.6 The stable process (Sα(σ, β, µ)) using the Monte Carlo method with 1000 simulations. (i) the trajectory, (ii) the random variable, (iii) the

histrogram, (iv) the density of S0.6(1, 0, −0.5)...... 36

iv 1.7 Trajectory of the stable process (Sα(σ, β, µ)) using the Monte Carlo method with 1000 simulations. The values of the parameters are α = 0.5, 0.9, 1.2, 2, β = −0.8, 0.7, 1, 1, σ = 1, and µ = 0 respectively. . 37

1.8 Trajectory of the Tempered stable process (TSα(λ)) using the Monte Carlo method with 10000 simulations. The values of the parameters are α = 0.8, β = 1, δ = 100, 1000, 10000, 100000 and µ = 0 respectively. 40

1.9 Trajectory of the Tempered stable process (TSα(λ)) using the Monte Carlo method with 10000 simulations. The values of the parameters are α = 0.5, β = 1, δ = 100, 1000, 10000, 100000 and µ = 0 respectively. 41

1.10 Trajectory of the Tempered stable process (TSα(λ)) using the Monte Carlo method with 10000 simulations. The values of the parameters are α = 0.8, β = 1, δ = 100, 1000, 10000, 100000 and µ = 0 respectively. 42

1.11 Densities of the Tempered stable distribution (TSα(λ)) using the Monte Carlo method with 10000 simulations. The values of the pa- rameters are α = 1.2, β = 1, δ = 10, 0.1, 0.001, 0.0001 and µ = 0 respectively...... 43

2.1 The sample trajectories of the strictly increasing Tempered stable process for β = 0.3, 0.5, 0.8 and λ = 10...... 55 2.2 The sample trajectories of the strictly increasing Tempered stable process for β = 0.3, 0.5, 0.8 and λ = 10...... 56 2.3 The sample trajectories of the strictly increasing Tempered stable process for β = 0.3, 0.5, 0.8 and λ = 10...... 57 2.4 The sample trajectories of the strictly increasing Tempered stable process for β = 0.3, 0.5, 0.8 and λ = 10...... 58 2.5 The sample trajectories of the path of Tempered stable subordination driven by brownian motion (TSB) for β = 0.3, 0.5, 0.8 and λ = 10. .. 60

v 2.6 The sample trajectories of the path of Tempered stable subordination driven by brownian motion (TSB) for β = 0.3, 0.5, 0.8 and λ = 10. .. 61 2.7 The sample trajectories of the path of Tempered stable subordination driven by brownian motion (TSB) for β = 0.3, 0.5, 0.8 and λ = 10. .. 62 2.8 The boxplots of the estimators of TSB (0.4, 10, 1). The values are based on the 100 simulations and trajectories of length 1000 each. . . 65 2.9 The boxplots of the estimators of TSB (0.4, 10, 1). The values are based on the 1000 simulations and trajectories of length 1000 each. . 66 2.10 The boxplots of the estimators of TSB (0.4, 10, 1). The values are based on the 10,000 simulations and trajectories of length 1000 each. 67 2.11 The empirical Laplace transform, theoretical Laplace transform for TSB with tempered stable subordinator and theoretical Laplace trans- form for TSB with stable subordinator for the simulated data (β = 0.8, γ = 2 and δ = 10.)...... 75 2.12 Normal fitting for the simulated data of TSB (β = 0.8, γ = 2, δ = 10) using MLE and Laplace transformation...... 76 2.13 MSD of 1000 trajectories of the TSB (β = 0.8, λ = 1, γ = 0.001) process...... 78

3.1 Probability distribution of the empirical log-return compare with the normal distribution for AMAZON, APPLE and S&P500 stocks from 01-01-2000 to 02-12-2019...... 82 3.2 The nomal distributional fitting for Weis Markets log-returns (1500) from 1993 to 2009...... 93 3.3 Q-Q plot for Weis Markets log-returns (1500) from 1993 to 2009. . . . 93 3.4 The empirical Laplace transform of log-returns of WMK stock data compared with the Normal, TSB, and SB using the LS method. . . . 94

vi 3.5 The empirical density (Epdf) compared with the Normal and TSB for log-returns of WMK stock data...... 94 3.6 The empirical density (Epdf) compared with the Stable (the current market model) and TSB for log-returns of WMK stock data...... 95 3.7 The empirical cumulative density (Ecdf) compared with the Normal, and TSB distributions for log-returns of WMK stock data...... 95 3.8 The empirical cumulative density (Ecdf) compared with the Stable (the current market model) and TSB for log-returns of WMK stock data...... 96 3.9 The trajectories of the TSB process for log-returns of WMK stock data using LS method ( first sample )...... 97 3.10 The trajectories of the TSB process for log-returns of WMK stock data using LS method (second sample)...... 97 3.11 The trajectories of the TSB process for log-returns of WMK stock data using LS method (third sample)...... 98 3.12 The examined real data set of APPLE stocks from 01.03.2000 - 02.12.2019. 99 3.13 The examined real data set of log-returns of APPLE stocks from 01.03.2000 - 02.12.2019...... 100 3.14 Q-Q plot for APPLE stocks from 01.03.2000 - 02.12.2019...... 100 3.15 The normal fitting for the log-returns of APPLE stocks from 01.03.2000 - 02.12.2019. The skewness = -4.35 and the kurtosis = 117.89. . . . . 101 3.16 The parameter estimation for the normal fitting using the Laplace transform (Laplace) and the MLE (method of likelihood estimation) from 01.03.2000 - 02.12.2019...... 102

vii 3.17 The empirical Laplace transform (black), theoretical Laplace trans- form for TSB (red), theoretical Laplace transform for SB (blue), and theoretical Laplace transform for Normal (green) for the log-returns of the APPLE stock from 01.03.2000 - 02.12.2019...... 103 3.18 The empirical density (Epdf) compare with the Normal and TSB dis- tributions for log-returns of APPLE stocks from 01.03.2000 - 02.12.2019.103 3.19 The empirical cumulative density (Ecdf) compare with the Normal and TSB distributions for log-returns of APPLE stocks from 01.03.2000 - 02.12.2019...... 104 3.20 The empirical density (Epdf) compare with the Stable (the current market model) and TSB for log-returns of Apple stock data from 01.03.2000 - 02.12.2019...... 105 3.21 The empirical cumulative probability (Ecdf) compare with the Stable (the current market model) and TSB for log-returns of Apple stock data from 01.03.2000 - 02.12.2019...... 105 3.22 The trajectories of the TSB process for log-returns of APPLE stock data using LS method (first sample)...... 106 3.23 The trajectories of the TSB process for log-returns of APPLE stock data using LS method (second sample)...... 106 3.24 The trajectories of the TSB process for log-returns of APPLE stock data using LS method (third sample)...... 107 3.25 The examined real data set of AMAZON stocks from 01.03.2000 - 02.12.2019...... 109 3.26 The examined real data set of AMAZON stocks from 01.03.2000 - 02.12.2019...... 109 3.27 The Q-Q plot for the AMAZON stocks from 01.03.2000 - 02.12.2019. 110

viii 3.28 The normal fitting for the log-returns of the AMAZON stock from 01.03.2000-02.12.2019. The skewness = 0.453 and the kurtosis = 12.142...... 110 3.29 The parameter estimation for the normal fitting using the least square approximation and the Method of likelihood approximation. The considered period is 01.03.2000 - 02.12.2019...... 111 3.30 The empirical Laplace transform (black), theoretical Laplace trans- form for TSB (red), theoretical Laplace transform for SB (blue), and theoretical Laplace transform for Normal (green) for the log-returns of the AMAZON stock from 01.03.2000 - 02.12.2019...... 111 3.31 The trajectories of the TSB process for log-returns of AMAZON stock data using LS method (first sample)...... 112 3.32 The trajectories of the TSB process for log-returns of AMAZON stock data using LS method (second sample)...... 113 3.33 The trajectories of the TSB process for log-returns of AMAZON stock data using LS method (third sample)...... 113 3.34 The empirical Laplace transform compare with TSB distribution us- ing the method of moment estimation...... 117

ix LIST OF TABLES

2.1 Mean, std, and bias of the TSB (0.4, 10, 1) of sample size = 1000 using LS...... 67 2.2 Skewness (ss) and Kurtosis (kk) of TSB distribution for the values of β, γ and δ = 1...... 73 2.3 Skewness (ss) and Kurtosis (kk) of TSB distribution for the values of β, δ and γ = 10...... 73 2.4 Mean, std, and bias for the TSB of 100 simulations using MME for various γ values...... 74 2.5 Mean, std, and bias for the TSB of 100 simulations using the MME for various δ values...... 74

3.1 Parameter estimation for log-returns of WMK stock data...... 96 3.2 Stable distribution parameters extracted from the log-returns of WMK stock data...... 96 3.3 Parameter estimation for log-returns of the APPLE stock. The con- sidered period is 01.03.2000 - 02.12.2019 ...... 104 3.4 Stable distribution parameters extracted from the log-returns of AP- PLE stock data...... 104

x 3.5 Parameter estimation for log-returns of the AMAZON stock from 01.03.2000 - 02.12.2019 ...... 112 3.6 Parameter estimation for log-returns of the WMK data using GMM . 117

xi ACKNOWLEDGEMENTS

I would like to express my gratitude to all those people who have helped me during this project and my course work. This work could not have been possible without the help of many faculty members and administration. I would like to give my sincere gratitude and appreciation to the following people: Professor Wojbor Woyczynski, my advisor and main source of inspiration, for his dedicated support, encouragement, and patience throughout this journey. Professor Peter Ritchken, for assisting in obtaining the necessary materials for the research, his dedication, and support given for this project. Also, I am grateful to Dr. Alethea Barbaro and Dr. Jenny Brynjarsdottir of my thesis committee for their valuable suggestions and taking the time to read my dissertation. I like to acknowledge the contribution of all the faculty members who taught and guided me throughout this journey. Also, I would like to acknowledge Mrs. Kate Camin and Mr. Donyear Thomas for their wonderful help and cheerful attitude throughout my career at Case Western Reserve University. I acknowledge the encouragement given by all of my colleagues in the Department of Mathematics, Applied Mathematics, and Statistics. Finally, I wish to thank my parents, my husband Nalinda, my son Devmal and two daughters, Vinuki and Naduli, for always being there for me, no matter what the circumstances were. Above all, I am grateful for everybody for guiding me throughout the course of this project and for giving me the precious gift of life.

xii Financial Modeling with LEVY´ Process and Applying LEVY´

Subordinator to Current Stock Data

Abstract

by

GONSALGE ALMEIDA

The normal distribution for financial modeling is frequently encountered, but it is not a good model when the data behavior is skewed and has fat-tailed properties. It is often wrong to employ distributions which have symmetric and rapidly de- creasing tail properties. The properties of the α-stable distribution are important to statisticians for modeling the data for skewness and fat tails. In order to ob- tain a well-defined model for pricing options, the mean, variance, and exponential moments of the return distribution often cannot be considered. For this reason, tempered stable distributions have been proposed for financial modeling. Other modifications for Brownian-type processes are obtained via introduction of a time- changed Brownian model. This extension is related to replacement of the real time in Brownian systems by a non-decreasing L´evyprocess (called subordinator). In this thesis, we analyze a process related to a subordinated Brownian motion which is called a normal tempered subordinator, and also is described as a Brownian mo- tion with drift driven by a tempered stable subordinator. We compare the main statistical analysis of the TSB (Subordinate tempered stable process to Brownian motion) process and diffusive process. In this work, we mention the two techniques of a parameter’s estimation procedures and validate them. In order to show the usefulness of theoretical results, we analyze the system using the real stock data. Dissertation Advisor: Wojbor A. Woyczynski

xiii Introduction

The normal distribution is frequently encountered in financial modeling with assets returns, but it is not a good model for data with skewed and fat-tailed properties, as the distribution is symmetric and has rapidly decreasing tail properties. The prop- erties of the α−stable distribution are important to statisticians for modeling the data for skewness and fat tails. In order to obtain a well-defined model for pricing options, the mean, variance, and exponential moments of the return distribution of- ten cannot be considered. For this reason, tempered stable distributions have been proposed for financial modeling. Other modifications for Brownian-type processes include the introduction of a time-changed Brownian model. This extension is re- lated to the replacement of the real time in Brownian systems by a non-decreasing L´evyprocess (called subordinator). In this thesis, we analyze a process related to subordinated Brownian motion, called the normal tempered subordinator, which also can be described as a Brownian motion with drift driven by a tempered stable subordinator. We compare the primary statistical analysis of the TSB (tempered stable process driven by Brownian motion), SB (stable process driven by Brownian motion), and diffusive processes. In this work, we discuss the two techniques of a parameter’s estimation procedures and validate them. In order to show the useful- ness of theoretical results, we analyze the system using the real stock data. In Chapter 1, we start with the definition of two accessible processes: Brownian motion, and a Poisson process. The beginning of each section explains how the theoretical formation can be used to build the characteristic functions for the pro- cesses and also generate the cumulants from them. Section 1.5 describes the the- ory of α−stable distributions developed in the 1930s by Paul L´evyand Alexander Khinchin. In this section, we develop the theoretical background and framework for applications in finance and physics. Although the normal distribution is encoun-

1 tered in modeling the return distribution of assets, it cannot describe the skewed and fat-tailed properties of the empirical distribution of asset returns. In section 1.6, the properties of the tempered stable distribution are described and the us- ages of it in the financial modeling. In section 1.7, we explain the tempered stable diffusion and super-diffusion. These two sections build the theoretical proof to be used in the simulation algorithm to simulate the α−stable and the tempered stable processes. The simulations are performed based on the Chambers Miller algorithm to the α−stable distribution. We also performed a Monte Carlo simulation for the tempered stable distribution. In Chapter 2, we apply the tempered stable process as a subordinator to data that have a very skewed and fat tail distributional behavior. The subordinated processes have been for fitting the data with many practical applications in stochastic time series. In particular, we have chosen to subordinate the tempered stable process to the Brownian motion with drift (TSB). Also, we will show some statistical analysis for the tempered stable subordinator. This process will be compared with Brownian motion and an α−stable subordinator. Moreover, we analyze a process related to subordinated Brownian motion, which is often called a normal tempered subordina- tor or a Brownian motion with drift driven by a tempered stable subordinator. We compare the primary statistical analysis of the ABM (Arithmetic Brownian Motion) driven by a tempered stable distribution with a stable process and a diffusive pro- cess. Herein, we mention the two techniques of a parameter’s estimation procedure and validate it. In order to show the theoretical result’s usefulness, we apply to the real stock return data. In Chapter 3, we demonstrate and analyse some stock data and show that the data deviation from the normality assumption. We use the empirical characteristic func- tion (fourier transformation) to obtain the three TSB parameters from the stock

2 data and in addition, fit the log returns data to SB and stable distributions to de- termine the closest shape of the data. The Fourier transform approach is a useful tool in pricing options. When the characteristic function of the TSB process is avail- able, the real-time option can be delivered. More generally, the realistic structure of the log-returned of the stock price can be modeled by the TSB process; then, it can adquately capture the excess kurtosis and stochastic volatility. The characteristic function of the TSB process can be evaluated analytically, and the expression for the Fourier transform of the option value can be derived. Since the Fast Fourier Trans- form (FFT) algorithms are computationally more efficient, we can find the option prices, which are the whole spectrum of strikes. The option prices can be obtained by performing Fourier inversion transform. Also, when the characteristic function is readily available for the process, it describes the scenario for calculating the option pricing. We explain how to find the characteristic function for current stock market data using the analysis based on the TSB process. Furthermore, we obtained the data directly from the stock market. We check the models’ validity by analyzing Weis Markets, Inc.(WMK), and the data in the New York Stock Exchanges (NYSE) from January 1993 to Feb 2009. Also, we introduce the parameter estimations using the Least square method (LS). Additionally, we attempt to find an estimation for the Amazon and Apple stocks data from January 01, 2000 to February 12, 2019. End of the this chapter, we directly apply the TSB process to find the method of moment estimators (MME) for log-returns of the Weis Markets (WMK) stock data. The TSB distribution is suited for data that is high kurtosis and heavily tailed. In Chapter 4, we introduce the generalized autoregressive conditional heteroskedas- ticity (GARCH) volatility model with CGMY jumps. Also, we discuss the similar- ities between the CGMY process and Tempered stable (TS) process. Furthermore, we discuss a variant of the tempered stable distributions and apply it to the GARCH

3 option pricing model. This is a continuation of the paper “Jump Starting GARCH: Pricing and Hedging Options with Jumps in Returns and Volatilities” by Jin-Chuan Duan, Peter Ritchken, and Zhiqiang Sun.

4 Chapter 1

Financial Modeling with L´evyProcesses and Infinitely Divisible Distributions

1.1 Introduction

The L´evyprocess describes the innovations of the French mathematician Paul L´evy, who played a significant role in bringing together an understanding and character- ization of processes with stationary independent increments. In the 1940s, L´evy himself referred to them as a sub-class of additive processes. This chapter starts with the definition of the most popular processes such as Brownian motion, and the Poisson process. The beginning of each section explains how the theoretical formation can be used to build the characteristic functions for the processes and also generate the cumulants accordingly. In section 1.5, we describe the theory of α−stable distributions. Furthermore, we also develop the theoretical background necessary for applications in finance. Although the normal distribution has been proposed to model the return distribution of assets, it cannot describe the skewed and fat-tailed properties of the empirical distribution of asset returns. In section 1.6, we describe the properties of the tempered stable distribution and the usages

5 of it in the financial modeling. The tempered stable diffusion and super-diffusion are explained in section 1.7. Finally, the last two sections include the probabilistic proof for an algorithm to simulate the stable and tempered stable processes. The simulations are performed using the Chamber Miller algorithm and Beaumer [1, 24].

1.2 Preliminaries on L´evy processes

Definition 1.2.1 (L´evyprocesses) If process X = {Xt : t ≥ 0} is a stochastic process defined on a probability space (Ω, F, P), then (i) The paths of X are P−almost surely right continuous with left limits.

(ii) P(X0 = 0) = 1.

(iii) For 0 ≤ s ≤ t, Xt − Xs is equal in distribution to Xt−s.

(iv) For 0 ≤ s ≤ t, Xt − Xs is independent of {Xu : u ≤ s}.

Let us now explore the relationship between Levy processes and infinite divisibility.

Definition 1.2.2 (Infinite divisibility) A probability distribution F on R is said to be infinitely divisible if for any integer n ≥ 2, there exist n i.i.d. random variables

Y1,Y2, ..., Yn such that Y1 + Y2 + ... + Yn has distribution F .

Theorem 1.2.3 If X is a L´evyprocess, then the characteristic function (φXt ),

tΨ(u) φXt (u) = e ,

d for each u ∈ R , t ≥ 0, where Ψ(u) := Ψ1(u) is the characteristic exponent of X1, which has an infinitely divisible distribution.

Proof: Let X be a L´evyprocess and for each u ∈ Rd, t ≥ 0, define the characteristic function of Xt :

i u·Xt φt(u) = φXt (u) = E[e ]

6 Since, the increments of the the L´evyprocess are i.i.d., we can express for all s ≥ 0,

φu(t + s) = φXt+s (u) (1.2.1) = φXs (u)φXt+s−Xs (u)

= φXs (u)φXt (u) = φsφt.

the stochastic continuity of t 7→ Xt implies that Xt 7→ Xs in distribution when

s 7→ t. Also, φXs (u) 7→ φXt (u) when s 7→ t so t 7→ φt(u) is a continuous function of t. Together with the above property 1.2.1 implies that t 7→ φt(u) an exponential function. This completes the proof of the theorem 1.2.3 and is according to [2].

Definition 1.2.4 (Brownian motion) A real valued process B = {Bt : t ≥ 0} defined on a probability space (Ω, F, P) is said to be a Brownian motion process if the following conditions are satisfied:

(i) The paths of B are P-almost surely continuous.

(ii) P(B0 = 0) = 1.

(iii) For 0 ≤ s ≤ t, Bt − Bs is equal in distribution to Bt−s.

(iv) For 0 ≤ s ≤ t, Bt − Bs is independent of {Bu : u ≤ s}.

(v) For each t > 0,Bt is equal in distribution to a normal random variable with variance t and mean 0.

The Poisson process will be used for constructing jump processes.

Definition 1.2.5 (Poisson Process) A process valued taking in the set of non- negative integers N = Nt : t ≥ 0, defined on a probability space (Ω, F, P), is said to be a Poisson process with intensity λ > 0 if the following hold:

(i) The paths of N are P-almost surely right continuous with left limits.

(ii) P (N0 = 0) = 1.

7 (iii) For 0 ≤ s ≤ t, Nt − Ns is equal in distribution to Nt−s.

(iv) For 0 ≤ s ≤ t, Nt − Ns is independent of {Nu : u ≤ s}.

(v) For each t > 0,Nt is equal in distribution to a Poisson random variable with parameter t.

Definition 1.2.6 (L´evy-Khinchin formula) If X = {Xt : t ≥ 0} is a L´evypro-

d cess, then φt(u) = exp(−tΨ(u)) for each t ≥ 0, u ∈ R , where

Z 1 iu.y Ψ(u) = −iγ · u + u · Au + [1 − e + iu.y1(||y||<1)(y)]ν(dy), (1.2.2) 2 Rd\{~0} for some γ ∈ Rd, a non-negative definite symmetric d × d matrix A and a Borel d ~ R iu.y measure ν on \{0} for which d [1 − e + iu.y1(||y||<1)(y)]ν(dy) is well R R \{~0} defined.

1.3 Characteristic Functions

The characteristic function plays an essential role in modeling the data using the L´evyprocess. It is the Fourier transform of a random variable, and it describes the distributional behavior. The significant advantage is that it can be derived easily for every distribution whether the density of distribution has a closed form or not.

Definition 1.3.1 The characteristic function of the Rd-valued random variable X is the function

d ΦX : R → R defined by:

Z iu·x ∀u ∈ R, ΦX (u) = E[exp(iu · X)] = e dµX (x). (1.3.1) Rd

The characteristic function of the random variable is always smooth and satisfies the condition ΦX (0) = 1. Extra smoothness of the random variables can be obtained

8 using the moments of the characteristic function. The m − th moments of a random

m variable X on R is defined by Km(X) = E[X ], and also the m−th central moment

Km is defined by the m − th moment of X − E[X]:

m Km(X) = E[(X − E[X]) ]. (1.3.2)

How fast the tails of µX decay at infinity determines the existence of the moments. The tail of the tempered stable distribution decays slower than the normal distri- bution and faster than the tail of a stable distribution. Moments can be obtained in following the way:

m If E[|X| ] < ∞ then ΦX has m continuous derivatives at u = 0 and 1 ∂jΦ (u) ∀j = 1, . . . m, K = E[Xj] = X | . [2] j ıj ∂uj (u=0) The characteristic function of the sum of the random variables Qn {Tn = Y1 + Y2 + ··· + Yn} is ΦTn (u) = j=1 ΦYj (u).

1.4 Cumulant Generating Function

Cumulant generating function is a continuous version of the logarithm of ΦX .

ΨX (0) = 0 and ΦX (u) = exp(ΨX (z)). The cumulants of X are defined by:

1 ∂mΨ (u) C (X) = X | . (1.4.1) m ım ∂um (u=0)

th Cm is called the m cumulant of the X random variable. Also, we can express the

th m cumulant using a polynomial function of the moments Kj, j = 1, . . . n or using the central moments Kj(X). The relation can be expressed as:

9 C1(X) = K1(X) = E(X).

2 C2(X) = K2(X) = K2(X) − K1(X) = V ar(X).

3 C3(X) = K3(X) = K3(X) − 3K2(X)K1(X) + 2K1(X) .

C4(X) = K4(X) − 3K2(X). The skewness (SS) and kurtosis (KK) of the X can be found by:

C3(X) C4(X) SS(X) = 3 ,KK(X) = 2 . (1.4.2) C2(X) 2 C2(X) The invariant properties of the skewness and kurtosis can be expressed by the fol- lowing formula:

∀λ > 0,SS(λX) = SS(X),KK(λX) = KK(X). (1.4.3)

Moreover, we can deduce that, if (Xi)i=1,...,n are independent variables then

n X ΨX1+X2+...,Xn = ΨXj (u) j=1

[2, p. 30-32].

1.5 α−Stable Distributions

The normal distribution is more frequently encountered in financial modeling of assets returns, but it is not applicable when the data display the skewed and fat- tailed properties. It is not consistent with the distribution which has symmetric and rapidly decreasing tail properties. Properties of the α−stable distribution are helpful to statisticians modeling data with skewness and fat tails. This section will discuss in more detail the α−stable distributions [4].

Let Y1,Y2, ..., Yn be independent and identically distributed(i.i.d) random variables.

10 Pn Then, if the summation of the random variables ( i=1 Yi) is following the α−stable distribution then we can express it as follows;

d Y1 + Y2 + ... + Yn = AnX + Bn

1 where, An and Bn are real number such that the constant An = n α which is called the index of stability. The case α = 2 is the normal distribution. The non-normal case is when the 0 < α < 2. Properties of the α−stable process can be seen using the characteristic function Φ(u). Accordingly the definition of the α−stable distribution on Rd, a finite measure λ is defined as in [5]:

Z Z ∞ dr ν(ε) = λ(dε) 1B(rε) α+1 (1.5.1) S 0 r

(0, µ, γ) is infinitely divisible characteristic triplet, and 0 < α < 2. The L´evyprocess with one dimensional L´evymeasure ν(x) is of the form:

A B ν(x) = 1 + 1 (1.5.2) xα+1 x>0 xα+1 x<0 where, the A and B are positive constants.

Definition 1.5.1 A time scale 1 of a α-stable random variable X has the charac- teristic function{Φ(u)} of the following form:

  α α πα −σ |u| {1 − iβsign(u)tan 2 } + iµu, α 6= 1 log Φ(u) = (1.5.3)  2 −σ|u|{1 + iβsign(u) π log|u|} + iµu, α = 1 where,

11    1, t > 0   sign t = 0, t = 0    −1, t < 0

The α−stable distribution is described by four parameters where α is the index of stability of α ∈ (0, 2), β is the skewness parameter of β ∈ [−1, 1], σ is the scale pa- rameter of σ ∈ (0, +∞) and µ is the location parameter of µ ∈ (−∞, +∞). Let a ran- dom variable Y follow the α−stable distribution and be denoted by Y ∼ Lα(σ, β, µ). We can simulate the summations of the independent α−stable random variables as follows:

Assume Yi ∼ Lα(σi, βi, µi))

If Yi and Yj are independent random variables, then Yi + Yj ∼ Lα(σ, β, µ) with β αα + β σα α α 1/α i i j j σ = (σi + σj ) , β = α α and µ = µi + µj [2, P 187]. σi + σj 1/α and also, If Y1 ∼ Lα(σ, β, µ) ,then Yt ∼ Lα(σ(t) , β, µt). This expression leads us to generate a stable random variable with arbitrary drift(µ) and scale parameter(σ) using a Y ∼ Lα(1, β, 0) in the following way:

σY + µ ∼ Lα(σ, β, µ), if α 6= 1 (1.5.4)

2 σY + βσlnσ + µ ∼ L (σ, β, µ), if α = 1 (1.5.5) π α the probability density of the α−stable is not available in closed form except the three special cases such as;

• The Gaussian distribution: (x − µ)2 1 − f(x) = √ e 4σ2 , −∞ < x < +∞ and, also it is denoted by S (σ, 0, µ). 2σ π 2

• The Cauchy distribution:

12 σ f(x) = , −∞ < x < ∞ and, also it is denoted by S (σ, 0, µ). π((x − µ)2 + σ2) 1

• The L´evydistribution: √ σ σ − f(x) = √ e 2(x − µ) , µ < x < ∞ and , also it is denoted by 2π(x − µ)3/2 S1/2(σ, 1, µ).

The α−stable distribution is more flexible for modeling nonsymmetric and heavy- tailed data. The normal distribution, which is the particular case when α = 2, is used in many applications in finance [6]. Figure 1 presents the changes in the α−stable distribution, with fixed β and σ. Figure 2 displays changing with the β values. When the α values are lower, then distribution has heavier tail and higher kurtosis.

13 Figure 1.1: α−stable densities for α = 0.5, 0.1, 1.5, and 2 with β = 0, σ = 1, and µ = 0.

The major four properties are useful in financial modeling when use the α−stable distribution [6, pages 61-65]: Property 1: (power tail decay property) The tail of the density function decays like a power function which is slower than exponential decay. Using this property allows us to capture the extreme values of the tail.

P (|X| > x) ∝ A.x−α, x → ∞ (1.5.6) for some constant A.

If X ∼ Sα(α, β, µ) with 0 < α < 2, then

14 Figure 1.2: α−stable densities for β = 0, 0.25, 0.5, 0.75, and 1 with α = 1.25, σ = 0.5, and µ = 0.

1 + β α −α lim P (X > λ) u Aα σ λ (1.5.7) λ→∞ 2

α 1 − β α −α lim λ P (X < −λ) u Aα σ λ (1.5.8) λ→∞ 2 where,

15  1 − α  if α 6= 1 γ(2 − α)cos(πα/2) Aα =  2  if α = 1. π Property 2: E(X)p < ∞ for any 0 < p < α. E(X)p = ∞ for any p ≥ α. Property 3: This is derived from property 2. The first moment is finite only for α > 1 : E(X) = µ, for α > 1, and E(X) = ∞ for 0 < α ≤ 1. This property implies that only the first moment is finite and higher moments are infinite. Property 4: The stability property describes the distributional form of the variable

1 under linear transformations. In particular, α in the Cn = n α . The smaller values of α have to have heavier-tailed distribution. The classical Central Limit Theorem is not valid for the non-Gaussian distributions, since sums of i.i.d. random variables converge to an α−stable random variable instead of the normal random variable. As- sume X1,X2, ..., Xn are i.i.d. random variables and Xi ∝ Sα(σi, βi, µi), i = 1, 2, ..., n where the α is constant. Then, we can obtain the parameters of the distribution of Pn Y = i Xi from the following equations,

Pn α n i βiσi X α 1 X β = , σ = ( σ ) α , µ = µ Pn σα i i i i i i

When the random variable Y = X1 + a, which is α−stable and the parameters are:

β = β1 , σ = σ1 µ = µ1 + a where, a is a constant.

16 The distribution of Y = aX1 follows α−stable law with parameters:

β = (sign a)β1

σ = |a|σ1   aµ for α 6= 1 µ =  2 aµ − 1 − π a(ln a)σ1β1 for α = 1

The distribution of Y = −X1 follows α−stable law with parameters:

β = −β1, σ = σ1, µ = µ1.

Figure 1.3: Compare cumulative densities for α = 0.5, 0.1, 1.5, and 2 with β = 0, σ = 1, and µ = 0.

17 Figure 1.3 describes the changes in the α values, which are decreasing, so the tails decay slower. Also when α = 2 the tails decay faster, since it represents the normal distribution. Figure 1.4 represents the Normal, Cauchy, and L´evydistributional behaviors according to the values of β, σ, and µ.

√ Figure 1.4: Probability Distributions of Normal (α = 2, β = 0, σ = 1/ 2, µ = 0), Cauchy (α = 1, β = 0, σ = 1, µ = 0), and L´evy(α = 0.5, β = 1, σ = 1, µ = 0).

18 1.6 Tempered Stable Distribution and Process

We should be aware of the infinite variance of α−stable distribution before address- ing the proper distribution for financial modeling. In most situations, the infinite variance of the returns will lead to an infinite price for options. Moreover, the skewed and heavily-tailed returns distribution disqualifies the Gaussian distribution as a suitable predictor. The α−stable distributions have desirable properties which were pointed out in section 1.5, but there are not better for modeling option prices. In order to accept modeling options, the mean, variance, and exponential moments of the return distribution have to exit. Because of this reason, various types of tem- pered stable distributions has been proposed to model financial data [8],[124],[?]. The density function of the tempered stable distribution is not given generally by a closed-form formula, but we can use by employing a numerical method. The idea of the tempered stable process was introduced by Rosinski [5]. The tempered stable process is generated by taking a one-dimensional stable process and multiplying the L´evymeasure with a decreasing exponential on each half of the real axis. After this smooth truncating, the small jumps keep their initial α−stable like behavior, whereas the large jumps become much less violent.

Definition 1.6.1 The characteristic function of the tempered stable distribution with the L´evytriplet (σ2, ν, γ) is defined by Le’vy-Khinchin formula without trunca- tion of big jumps;

 Z +∞  iux E(iuXt) = exp t iuγ + (e − 1 − iux)ν(dx) , (1.6.1) −∞

19 where the L´evymeasure

dx ν(dx) = (A e−δ+x1 + A e−δ−x1 ) , (1.6.2) + x>0 − x<0 |x|α+1

Z E[Xt] = γ + xν(dx),, (1.6.3) |x|>1 and,

A+,A−, δ+, δ− > 0, α ∈ (0, 2) and γ ∈ R and E[Xt] = γt. The γ is the drift parameter of Tempered Stable process.

We can find the characteristic function accordingly in Ramacont [2, page 121]. The first, we derive characteristic function for the positive half of the L´evymeasure.

Suppose, α+− 6= 1 and α+− 6= 0;

Z ∞ −δx iux e (e − 1 − iux) 1+α dx 0 x ∞ X (iu)n Z ∞ = xn−1−αe−δx dx n! n=2 0 ∞ X (iu)n = δα−nΓ(n − α) n! n=2  1 iu 2 − α iu (2 − α)(3 − α) iu  = δαΓ(2 − α) ( )2 + ( )3 + ( )4 + ... (1.6.4) 2! δ 3! α 4! α

Applying the power series to the expression in the braces, i.e.

x2 (1 + x)µ = 1 + µx + µ(µ − 1) + .... (1.6.5) 2! and comparing the results of (1.6.4) and (1.6.5), we can obtain that

Z ∞ −δx iux e α iu α iuα (e − 1 − iux) 1+α dx = δ Γ(−α){(1 − ) − 1 + }. (1.6.6) 0 x δ δ

20 Proposition 1.6.2 The characteristic exponent of the tempered stable L´evydistri- bution according to the definition 1.6.1 can be written in the following form,

α Ψ(u) = iγ u + Aδ [a1ζα(−u/δ) + a2ζα(u/δ)], where A, a1, a2 ≥ 0, a1 + a2 = 1, γ is a real number, and satisfying the following conditions,

  α Γ(−α)[(1 − is) − 1], for 0 < α < 1;   ζα(s) = (1 − is)log(1 − is) + isα, for α = 1; (1.6.7)    α Γ(−α)[(1 − is) − 1 + iαs] for 1 < α < 2.

Proof . See appendix. Here, δ is called the tempering parameter (so it is damping big jumps of the stable distribution), and parameter a1, a2 are skewness parameters. When the a1 = a2 = 1 the distribution is called totally skewed. To obtain the symmetric distribution, assume that a1 = a2 = 1/2. Therefore, to obtain the strictly increasing tem- pered α−stable process, we can use TSα(A, 1, δ) and derive the temperd α−stable as TSα(a, p, δ). In general, when defining the tempered α−stable distribution as

TSα(A, a, δ) where α, A, a and δ are parameters, the characteristic exponent is given by the following expression,

s s Ψ(s) = AδαΓ(−α)[a (1 − i )α + a (1 + i )δ − 1], (1.6.8) 1 δ 2 δ and, if p = 1 ,

s Ψ (s) = AδαΓ(−α)[(1 − i )δ − 1] = aΓ(−α)[(δ − is)α − δα], (1.6.9) 1 δ this derived the totally skewed tempered α−stable distribution which we can denote

21 as above, TSα(A, 1, δ).

In the symmetric case a1 = a2 = 1/2, and 0 < α < 2, α 6= 1,

s s Φ(s) = AδαΓ(−α)[1/2(1 − i )α + 1/2(1 + i )δ − 1] (1.6.10) δ δ = AΓ(−α)[(δ2 + s2)α/2cos(α arctan (s/δ)) − δα]. (1.6.11)

in the symmetric case, and α = 1 as denoted by TS1(A, 1/2, δ), The characteristic exponent is,

δ s2 |s| Ψ (s) = A( log(1 + ) − |s| arctan ). (1.6.12) 1 2 δ2 δ

TSα(A, a, δ) for the Xt, with X1 ∼ TSα(1, a1, δ) coinsider with the Y1 ∼ TSα(A, a1, δ). Therefore, the parameter A is called the time scale parameter. Now, we can find the

Laplace transformation for the TSα(A, a = 1, δ), and m = 0 using the characteristic exponent of the form,

α α Ψ(s) = Γ(−α)Aδ (a1(1 − is/δ) − 1) (1.6.13)

This is the one-sided distribution, the Laplace transform of the density given as,

fˆ(s) = exp(Γ(−α)Aδα((1 + s/δ)α − 1). (1.6.14)

The following theorem explains how to derive the Fokker-Planck equation for the probability density function of the tempered stable process{TSα(t)}. We will express it according to [104], as follows:

Theorem 1.6.3 Consider TS(t)(α, C+,C−, δ) is the tempered stable process with the Fourier Transform given in (1.6.7). The probability density function f(., t) of

22 TS(t) satisfies the following equation: here consider C+ = Aa1,C− = Aa2

α,δ,C+,C− ∂tf(x, t) = ∂x f(x), where

∂f(x, t) =(∂/∂t)f(x, t), ∂xf(x, t) = (∂/∂x)f(x, t),

α,δ,C+,C− α α−1 ∂x f(x, t) = −δ (C+ + C−)f(x, t) − αδ (C+ − C−)∂xf(x, t)

−δx α αx α δx α −δx + C+e dx [e f(x, t)] + C−(−1) e dx [e g(x, t)],

α and the fractional operator dx is defined as,

α 1 R +∞ α iux dx z(x) = 2π −∞ (iu) zˆ(u)e du, where zˆ(k) is the Fourier transformation of the function z(x).

Proposition 1.6.4 If g(x) is the probability distribution function of the strictly in- creasing α−stable distribution, 0 < α < 1, then for every fixed δ, there exists one,

−δx and only one, STα distribution with the density f(x) = Cg(x)e . C is the nor- malizing constant.

Proof . See Appendix.

1.6.1 Tempered Stable Diffusion and Super-Diffusion

2 The diffusion equation ∂tP = C ∂xP dictates the transition densities of a Brownian motion B(t), and its solutions spread at the rate t1/2 for all time. The space-

α fractional diffusion equation ∂tP = C ∂x P for 0 < α < 2 gives the transition densities of a totally skewed α stable L´evymotion S(t), and its solutions spread at super-diffusive rate S(at) ∼ a1/αS(t) for any time scale a. This expression can

23 generate large power-law jumps. The random walk with power-law jumps is called

α a L´evyflight[97] .The order of the fractional derivative in the ∂tP = C∂x P equals the power-law index of the jumps. For more general jumps

P (X < −x) = qAx−α (1.6.15) and P (X > x) = (1 − q)Ax−αl (1.6.16) which leads to the following equation,

α α ∂tP = C q∂−xP + (1 − q)C ∂x P (1.6.17) with 0 ≤ q ≤ 1. Fractional diffusion equations are essential in applications to fi- nance. Mantegna and Stanley proposed the idea of the truncated L´evyflight as the modification of the α−stable L´evymotion [20],[98]. Tempering L´evymotion has a different approach. Exponentially tempering has the probability of large jumps, so that large jumps are exceedingly unlikely, and all moments exists. Exponential tem- pering offers technical advantages since the tempered process remains an infinitely divisible L´evy process whose governing equation can and whose transition densities can compute at any scale. Those transition densities solve a tempered fractional diffusion equation, similar to the fractional diffusion equation. The solution of it is similar to the tempered stable process in its early-time behavior like super-diffusive and late-time behavior to diffusive.

2 ∂t P(x, t) = c ∂x P(x, t). (1.6.18)

24 The solution of the fractional equation gives the probability density P(x, t) which is the totally skewed α−stable L´evymotion x = TS(t) for 0 < α < 2, α 6= 1. We can express the Fourier transforms of the density function of f(x), as fˆ(k) = R e−ikxf(x) dx Applying it to the fractional diffusion equation, we can obtain the Fourier transfor- mation for the tempered stable process.

d Pˆ(k, t) = x(ik)αPˆ(k, t), (1.6.19) dt

Pˆ(k, t) = etc(ik)α (1.6.20) where c > 0 for 1 < α < 2, and c < 0 for 0 < α < 1. Also, we can derive the scal- ing property, t1/αP(t1/αx, t) = P(x, 1) from the Fourier transform. Because the L´evyflight has heavy tail with P (TS(t) > t) ∼ Ar−α, hence that any moment < T S(t)n >= R xn P(x, t) dx of order n > α diverges. In financial modeling, we need at least four finite moments in order to capture the tail properties. Because of that significant reason, tempering the tail is needed. Accordingly, the Lemma 2.21 on p.67 in Zolotarev [96] shows that the Fourier transform,

Z e−iuxPx, t) dx = etc(iu)α (1.6.21) when u = k − iδ we can show that

Z e−ikxe−λxP(x, t) dx = etc(δ+ik)α . (1.6.22) where 0 < α < 1. A more straightforward argument using the Laplace transform appears in [5]. Since

25 k = 0 yields the total mass and clearly the density e−tcδα e−δxP(x, t) integrates to one. Now we can obtain the Fourier transform as follows:

etc[(δ+ik)α−δα]. (1.6.23)

d α α The mean of the tempering tail is; i etc[(δ+ik) −δ ]| = −ctαδα−1. then the dk k=0 Laplace transformation gives the following,

ˆ tc[(δ+ik)α−δα−ikαδα−1] Pδ(k, t) = e . (1.6.24)

Invert the Fourier transform to get the mean-zero tempered stable density:

−δx ct(α−1)δα α−1 Pδ(x, t) = e e P(x − ctαδ , t). (1.6.25)

d Since, Pˆ (k, t) = c[(λ + ik)α − δα − ikαδα−1]Pˆ (k, t), Fourier inversion gives us the dt δ δ tempered fractional diffusion equation,

α,δ ∂tPδ(x, t) = c ∂x Pδ(x, t).

α,δ α α α−1 ˆ where ∂x f(x) is the inverse Fourier transofrm of [(δ + ik) − δ − ikαδ ]f(k). Accordingly, by [99] and [24],

α,δ −δx α δxf(x) α α−1 ∂x f(x) = e ∂x [e ] − δ f(x) − αδ ∂xf(x). (1.6.26)

Now we state the following propositions and theorem according to [24].

Proposition 1.6.5 Assume S(t) is a totally skewed α−stable L´evyprocess for 0 <

26 α < 2 with density P(x, t) and Fourier transform

  α exp[ct(ik) ] α 6= 1 E[e−ikS(T )] = Pˆ(k, t) = (1.6.27)  exp[ct ik ln(ik)] α = 1.

Then the tempered α−stable L´evyprocess TS(t) has density with Laplace transform

 e−δx+(α−1)ctδα P(x − c tα δα−1,t) α 6= 1 ˆ  Pδ(x, t) = (1.6.28)  −δx+ctδ e P(x − ct(1 + lnδ), t)) α = 1.

Next, we will express the next theorem according to [24].

Theorem 1.6.6 Let S(t) be an infinitely divisible process with L´evyrepresentation

[γ, σ, ν], where ν is supported on R+. Write the Fourier transform E[e−ikS(t)] = R exp(tΦ(k)) and the probability distribution P (U, t) = U P (dx, t) = P(S(t) ∈ U) for Borel measurable U ⊂ R. Then Φ has an analytic extension to the lower half plane. Given δ > 0, define an infinitely divisible process TS(t) with L´evyrepresentation

−δx [γδ, σ, νδ] where νδ(dx) = e ν(dx). The mean E[TS(t)] exists for all δ > 0, and we can choose γδ ∈ R so that E[TS(t)] = 0. The resulting tempered infinitely divisible L´evyprocess TS(t) has probability distribution with Laplace trasform;

0 ˆ −δx−φ(−iδ)t−iδφ (−iδt) 0 Pδ(dx, t) = e P (dx + iφ (−iδ)t, t). (1.6.29)

Proof . See Appendix.

27 1.7 Numerical Approximation of Stable and Tem-

pered Stable Sample Paths

We used the Chambers-Mallows-Stuck algorithm to simulate the skewed random variables [1]. Before explicitly starting the algorithm, it is worth examining the proof that shows the equations, which will set up the above algorithm. When we consider the essential properties of the α−stable distributions, they are identically distributed random variables and follow the limiting laws of normalized sums of independents. Gaussian distribution is a particular case, with α = 2, and the properties are easy to understand. However, after the failure of the Black-Scholes model in 2008, non-Gaussian stable has been widely used in fields as diverse as economics [1]. With this situation, the α−stable random variable became a fast generator to apply to the financial data. Using the inverse transformation method on the simulation of the stable distribution failed because it does not have the inverse F −1 for the cumulative densities of the α−stable distribution except special cases α = 1/2, 1, 2 called L´evy, Cauchy and Gaussian. Consistent with the proof of in [96], we will consider the definition below.

Definition 1.7.1 (see,[1]) A random variable X is α-stable iff its characteristic function is given by

  α α π −σ |u| exp{−iβsign(u) 2 K(α)} + iµu, α 6= 1 logΦ(u) = (1.7.1)  π 2 −σ|u|{ 2 + iβsign(u) π log|u|} + iµu, α = 1 where

28   α, α < 1, K(α) = α − 1 + sign(1 − α) = (1.7.2)  α − 2, α > 1.

The parameters are σ2, β2, σ, and β as defined in 1.5.1. The new skewness parameter

β2 for α 6= 1 is πK(α) πα tanβ
= βtan , (1.7.3) 2 2 2 and the new scale parameter

πα σ = σ1 + β2tan2
1/(2α). (1.7.4) 2 2

2 where α = 1, β2 = β and σ2 = π σ.

Proposition 1.7.2 (see,[1]) Let

(α) = sign(1 − α), (1.7.5)

π K(α) γ = − β , (1.7.6) 0 2 2 α 1 C(α, β )− = 1 − (1 + β K(α)/α)(1 + (α)) (1.7.7) 2 4 2 sin α(γ − γ )α/(1−α) cos(γ − α(γ − γ )) U (γ, γ ) = 0 0 , (1.7.8) α 0 cosγ cos γ and

π 2 + β2γ 1 π
U1(γ, β2) = exp ( + β2γ) tan γ (1.7.9) cos γ β2 2

Then the cumulative density function F (x, α, β2) of a standard stable random vari- able can be written as a function of the characteristic function in the representation

29 of (1.7.1),

π Z 2 (α) α/(α−1) F (x, α, β2) = C(α, β2) + exp[−x Uα(γ, γ0)]dγ, (1.7.10) π γ0 when α 6= 1 and x > 0

π 1 Z 2 −x/β2 F (x, 1, β2) = exp[−e U1(γ, β2)]dγ, (1.7.11) π π − 2

when α = 1 and β2 > 0

F (−x, α, β2) + F (x, α, −β2) = 1 (1.7.12)

for any real x and any admissible parameter α, β2 (or β). Since the Gaussian, Cauchy and L´evydistributions have closed forms of the density function, the simulations for specific cases (α = 1/2, 1 and 2) can be done using the

−1 inverse F . Kanter gave a direct method for simulating Sα(1, 1, 0) when α < 1 [23]. Chambers [1] introduced the formulas for the α-stable simulation. We will follow the explicit proofs of these formulas according to Weron [1].

Lemma 1.7.3 Let γ0 and Uα(γ, γ0) be defined as in Proposition (1.7.2). For α 6= 1

π and γ0 < γ < 2 , iff for x > 0, X is a Sα(1, β2, 0) random variable (accordingly derive in (1.7.1))

 π  1 Z 2 P (0 < X < x), α < 1 exp[−xα/(α−1)U (γ, γ )]dγ = (1.7.13) π α 0 γ0  P (X ≥ x), α > 1.

30 Proof . Case of 0 < α < 1. From (2.11) we have

π 1 − β 1 Z 2 F (x, α, β ) = P (X ≤ x) = 2 + exp[−xα/(α−1)U (γ, γ )]dγ 2 2 π α 0 γ0 (1.7.14) 1 − β = 2 + P (0 < X ≤ x), 2

Theorem 1.7.4 Consider γ0 as in Proposition (1.7.2). Let γ be uniformly dis-

π π tributed on (− 2 , 2 ) and W be an independent exponential random variable with mean 1. Then for α 6= 1

sin α(γ − γ )cos(γ − α(γ − γ ))(1−α)/α X = 0 0 . (1.7.15) (cos γ)1/α W for α = 1

π  W cos γ  X = ( + β2γ)tan γ − β2 log π (1.7.16) 2 2 + β2γ is S1(1, β2, 0)

Proof .

When γ > γ0 then the quantity of the right hand side of (1.7.15) can be expressed as,

a(γ)(1−α)/α , (1.7.17) W where

sin α(γ − γ )α/(1−α)cos(γ − α(γ − γ )) a(γ) = 0 0 . (1.7.18) cos γ cos γ

Case (i) for 0 < α < 1.

31 1−α Equation (1.7.15) implies that X > 0 iff γ > γ0. Since α > 0 ,

P (0 < X ≤ x) = P (0 < X ≤ x, γ > γ0)

(1−α)/α = P (0 < (a(γ)/W ) ≤ x, γ > γ0)

α = P (W ≥ x /(α − 1) a(γ), γ > γ0)

α/(α−1) = Eγ exp[−x a(γ)] 1{γ>γ0} π 1 Z 2 = exp[−xα/(α−1) a(γ)] dγ. π γ0

From Lemma (1.7.3) and (1.7.12), we conclude that X ∼ Sα(1, β2, 0). Case (ii) for 1 < α ≤ 2.

α−1 Since α > 0, for x > 0

P (X ≥ x) = P (X ≥ x, γ > γ0)

(1−α)/α = P ((a(γ)/W ) ≥ x, γ > γ0)

(1−α)/α = P ((W/a(γ)) ≥ x, γ > γ0)

α/(α−1) = P (W ≥ x a(γ), γ > γ0)

α/(α−1) = Eγ exp[−x a(γ)] 1{γ>γ0} π 1 Z 2 = exp[−xα/(α−1) a(γ)] dγ. π γ0

From Lemma (1.7.3) and (1.7.12), we conclude that X ∼ Sα(1, β2, 0). Case (iii) for α = 1.

π For β2 = 0, the quantity of the right hand of (1.7.16) simplifies to 2 tanγ whose behavior as Cauchy distribution (in representation (1.7.1)). When β2 6= 0, it can expressed as following:

a (γ) β log 1 , (1.7.19) 2 W

32 where

π 2 + β2γ  1 π  a1(γ) = exp ( + β2)tanγ . (1.7.20) cos γ β2 2

For , β2 > 0

P (X ≤ x) = P (β2 log(a1(γ)/W ) ≤ x)

−x/β2 = P (W ≥ e a1(γ))

−x/β2 = Eγ exp[−e a1(γ)] π 1 Z 2 −x/β2 = exp[−e a1(γ)] dγ. π γ0

From (1.7.10) and (1.7.11), we conclude that for all β2, X ∼ Sα(1, β2, 0). Now we can construct the simulation method for a skewed random variable X ∼

Sα(1, β, 0), according to the definition of the α− stable distribution in (1.10). For α ∈ (0, 2] and β ∈ [−1, 1] :

ALGORITHM (Simulate the α−stable random variables)

−π π • Generate a uniformly distributed (V) random variable on ( 2 , 2 ) and an in- dependent exponentially distributed random variable (W) with mean 1;

• Compute

sin(α(V + B )) cos(V − α(V + B ))(1−α)/α X = S × α,β × α,β (1.7.21) α,β (cos (V )1/α) W

for α 6= 1. where

arctan(β tan πα ) 1/(2α) B = 2 , α,β α

33 h παi1/(2α) S = 1 + β2 tan2 ; α,β 2

• Compute

2 h π W cos V i X = ( + β V ) tan V − β log π . (1.7.22) π 2 2 + βV for α = 1.

Bα,β is the parameter change from β2 to β, γ0 in (1.7.15), and Sα,β is representing the parameter change from σ2 to σ (see 1.7.4). To generate α−stable variables for any parameter values of α, σ, β and µ, we can use the following expressions,

  σX + µ, α 6= 1 Y = (1.7.23)  π σX + 2 β σ log σ + µ, α = 1

Then,

Y ∼ Sα(σ, β, µ) if X ∼ Sα(1, β, 0).

1.8 Monte Carlo Simulation for Tempered α−Stable

L´evy process

More theoretical and experimental studies show that the tempered α−stable L´evy process (TS(t)) is also infinitely divisible. Therefore, we can approximate a random walk at given ∆t.

34 Figure 1.5: The stable process (Sα(σ, β, µ)) using the Monte Carlo method with 1000 simulations. (i) the trajectory, (ii) the random variable, (iii) the histrogram, (iv) the density of S0.8(1, 0, 0.5, 1).

Proposition 1.8.1 Consider X is an α−stable variable with the density f(x), and

e−δxf(x) define the tempered α−stable density fδ(x) = R ∞ −δu . Choose Y exponentially 0 e f(u) −1 distributed with mean δ , independent of X. Then generate (Xi,Yi) independent and identically distributed with (X,Y ), and let N = min{n : Xn ≤ Yn}. Then XN has the exponentially tempered density function fδ(x).

This proof is provided by Fama [24]:

Z ∞ −δy P(XN ≤ x, XN ≤ Yn) = P(X ≤ x, X ≤ y)δ e dy 0 Z x −δy −δx = P(X ≤ y)δe dy + P(X ≤ x)e (1.8.1) 0

35 Figure 1.6: The stable process (Sα(σ, β, µ)) using the Monte Carlo method with 1000 simulations. (i) the trajectory, (ii) the random variable, (iii) the histrogram, (iv) the density of S0.6(1, 0, −0.5).

therefore, the density of XN is

 Z x  d −1 d −δy −δx P(XN ≤ x) = w P(X ≤ y)δe dy + P(X ≤ x)e dx dx 0  Z x  (1.8.2) −1 −δy −δx −δx = w P(X ≤ x)δe + f(x)e − P(X ≤ x)e 0

R ∞ −δy which simplifies to fδ(x), since w = 0 f(y)e dy via integration by parts.

The following algorithm simulates the {TS(t)}t≥0 process according to [24].

Let X be the exponentially tempered probability density fδ(x) as defined in 1.8.1. Here S(t) is stable with index α < 1 and its density has Fourier transform p(k, t) = etc(ikt)α . Simulate for TS(t) when α < 1 :

36 Figure 1.7: Trajectory of the stable process (Sα(σ, β, µ)) using the Monte Carlo method with 1000 simulations. The values of the parameters are α = 0.5, 0.9, 1.2, 2, β = −0.8, 0.7, 1, 1, σ = 1, and µ = 0 respectively.

I. Generate exponential random variable E with mean δ−1. II. Generate totally skewed α−stable random variable S(α < 1), using the following formula,[25]

π π (1−α)/α 1 sin α(γ + 2 )cos(γ − α(γ + 2 )) S(t) = (|c|t) α . (cos γ)1/α W

Here , γ is uniformly distributed on [−π/2, π/2], and W has exponential distribution

37 with mean 1; Then, tempered stable random variable (TS(t)) can be simulated by the rejection method: III. If E > S replace TS(t) = S(t) + ctαλα−1, otherwise go to step I [24]. Pn To simulate the entire path, set TS(t) = i=1[TS(k∆t) − TS((k − 1)∆t)], where t = n∆t. When α > 1, stable density p(x,t) with Fourier transform Pˆ(k, t) = etc(ik)α is positive for all −∞ < x < ∞ and all t > 0; i.e. the support is all R. The exponentially

−tcδα −δx tempered density Pδ(x, t) = e e P(x, t) is still integrable because P(x, t) → 0 faster than any exponential function as x → 0. To simulate these random variables, we will use the procedure for α < 1 and use the following theorem as in [24].

Theorem 1.8.2 Let X be a random variable with density f(x) such that integral I =

R ∞ −δ −δx −∞ e xf(x)dx exists. Define the α−stable tempered density fδ(x) = e f(x)/I. Assume Y be exponentially distributed with mean δ−1, independent of X. Generate

(Xi,Yi) independent and identically distributed with (X,Y ) and for a ≥ 0 let Na =

1 min{n : Xn ≤ Yn − a}. Then the density of XNa converges in L to fδ as a → ∞, and hence d(XNa ,X) → 0 in the Kolmogorov-Smirnov distance.

Proof . See Appendix.

Here is the calculation of the Kolmogorov-Smirnov distance between XNa and X,

Z x Z x d(XNa ,X) = supx∈R|Fδ(x) − Fa(x)| = sup| fδ(y)dy − fa(y)dy| −∞ −∞ Z x ≤ sup |fδ − fa(y)|dy −∞ Z ∞ 3ga = |fδ(y) − fa(y)|dy ≤ → 0. −∞ I − ga + ha

38 Simulate for TS(t) when α > 1 : choose a > 0 so that P (S(t) < −a) is small. draw S(t) using

π  π π (1−α)/α 1 sin α(γ − 2 ) cos(γ − α(γ − 2 + α )) S(t) = (|c|t) α . (cos γ)1/α W

and E which is exponential with mean δ−1. If S > E − a, reject and draw again; otherwise , set TS(t) = S(t) + ctαδα−1, Simulate for TS(t) when α = 1 : Using the same way as the above, compute the TS(t) = S(t) + ct(1 + ln δ). where,  π  π π 2 W cosγ S(t) = (ct)(ln( )ct) + ( + γ) − ln π . 2 2 2 + γ

39 Figure 1.8: Trajectory of the Tempered stable process (TSα(λ)) using the Monte Carlo method with 10000 simulations. The values of the parameters are α = 0.8, β = 1, δ = 100, 1000, 10000, 100000 and µ = 0 respectively.

40 Figure 1.9: Trajectory of the Tempered stable process (TSα(λ)) using the Monte Carlo method with 10000 simulations. The values of the parameters are α = 0.5, β = 1, δ = 100, 1000, 10000, 100000 and µ = 0 respectively.

41 Figure 1.10: Trajectory of the Tempered stable process (TSα(λ)) using the Monte Carlo method with 10000 simulations. The values of the parameters are α = 0.8, β = 1, δ = 100, 1000, 10000, 100000 and µ = 0 respectively.

42 Figure 1.11: Densities of the Tempered stable distribution (TSα(λ)) using the Monte Carlo method with 10000 simulations. The values of the parameters are α = 1.2, β = 1, δ = 10, 0.1, 0.001, 0.0001 and µ = 0 respectively.

43 Chapter 2

Brownian Subordination (Tempered Sta- ble Subordinator)

2.1 Introduction

The Brownian motion was considered in many situations and has found many appli- cations [59, 60, 61, 62]. However, the normality assumption for the log-returns is not reasonable in the number of examined phenomena. Therefore, various modifications of the Brownian motion have been proposed for financial modeling [83, 84, 85]. The replacement of Gaussian distributions by another distribution is among the simplest modifications. The early studies of Mendelbrot and Fama made preliminary sup- port available for a L´evy α−stable model [86, 64]. These distributions appeared to capture the leptokurtic property observed in log- returns. However, the model is not perfect for modeling the log-returns. Moreover recent studies have shown that all α−stable distributions, including the normal distribution, may not provide the best fitting of log-returns. Studies of Hsu, Miller and Wichern, Blattberg, and Gonedes and Haggerman precisely reveal that with the holding period, the characteristic ex- ponent increases [83, 84, 85]. This is a condition that is violating the single stable

44 law assumption. Also, in the non-degenerate case α < 2, any finite interval almost surely contains an infinite number of discontinuities of α−stable process. This prop- erty implies that the Itˆo calculus cannot be initiated to create hedge positions and valuation relationships. Mantegna and Stanley estimated the probability density function for the increments (V) of the Standard & Poor 500 Index from January, 1984 to December, 1989 [20]. They discovered that although the variance is finite, the empirical density can be approximated by the α−stable density in the interval of |V | ≤ 6σ, also the α−stable decay is like a stretched exponential outside of the interval of |V | ≤ 6σ. The L´evy α−stable (or, more generally, fractal) processes are a tool that was well established in financial model building [18, 21, 20, 12, 14, 46, 17, 98, 19]. According to Ritchken and Woyczynski [9], the stochastic deferential equation below represents the model price changes:

dS(t) = γ dt + σ dL (t), t ∈ (0,T ],S(0) = S , S(t) α 0

where {Lα(t): t ≥ 0, 0 < α ≤ 2} is the representation of the symmetric α−stable L´evymotion. In other words, this is a stochastic procedure that possesses increments that are both independent and stationary, with Lα(0) = 0, and E exp(isLα(t)) = exp (−t|s|α). Furthermore, additional details of α−stable random variables and pro- cedures, as well as stochastic integrals, and the numerical and statistical simulation methods of theoretical properties are precisely explained by Kwapien and Woyczyn- ski, Samorodnitsky and Taqqu, and Janicki and Weron [87, 94, 63]. The processes based on stable distribution were popular in modeling the financial data [89, 90, 88]. The problem with non-Gaussian option pricing is that the market is incomplete, i.e., there may be more than possible pricing formula [81]. This is

45 an indication of the likelihood of more than one possible pricing formula existence. Likewise, this is undesirable, and a number of selection principles, like minimization of entropy, have been utilized to overcome the problem. One possible modification for the Brownian-type processes is that subordinate real- time is represented by a strictly increasing continuous L´evyprocess. The new process is called the subordinator. Bochner [91] introduced the subordination idea and il- lustrated it in his book [92]. There is also a detailed exploration of the subordinated processes theory in [156] and its analysis in [104]. Increasing L´evyprocesses are also called subordinators because they can be used as time changes for other L´evypro- cesses. These processes are an essential component for building L´evy-basedmodels in finance.We have specifically chosen to subordinate the tempered stable process to a real-time process. In this chapter, we analyze the process related to a subordinated Brownian motion, which is called normal tempered subordinator. This is also known as the Brownian motion with drift driven by the tempered stable subordinator (TSB). We compare the primary statistical analysis of the TSB process with the Brownian process, and the α−stable subordinated processes (SB). Also, we discuss the two techniques of parameter’s estimation procedure and validate them. In order to show the usefulness of theoretical results, we analyze the system using the real stock returns data.

2.2 Tempered Anomalous Subdiffusion

√ The fractional subdiffusion is defined by the formula X (t) = γ B(Θ(t)), t ≥ 0, where B is the Brownian motion process and random time {Θ(t) = min{θ : T (θ ≥ t)}} has the inverse β−stable p.d.f., 0 < β < 1, for all times t. The behavior of

2 2 β hX (t)i = Γ(β+1) γ t confirms that the β-stable random time, T (θ), has the infinite

46 mean and infinite second moment. From the physical perspective, it is desirable to have a model that avoids the infinite-moment difficulty while conserving the subdiffusive behavior, at least for small times. For that reason, we will consider the tempered anomalous diffusion for modeling. The tempered anomalous subdiffusion process can be expressed as:

√ X (t) = γ B(Θ(t)), (2.2.1) where Θ(t) is the inverse tempered β-stable random time, with the tempered β- stable random time T (θ). It is defined by the following Laplace transform;

  ˆ −sT (θ) θΦ(s) β β fβ,tmp(s; δ) = he i = e = exp δ − (s + δ) , 0 < β < 1, (2.2.2) where

Z ∞ β Φ(s) = (e−sz − 1)φ(z) dz, φ(z) = z−β−1 e−δzχ(z) , (2.2.3) 0 Γ(1 − β) for some δ > 0. The expression of the L´evyintensity function φ(z) ensures its β- stable behavior for small jump sizes z. The tempering of the tail by multiplying the L´evymeasure by the exponential term gives the existence of all moments of T (θ). Therefore, tempered stable processes occupy an intermediate place between pure stable processes and the classical Brownian motion diffusion processes [102]. We can derive the mean square of the process X (t) using the cumulative distribution function of Fβ,tmp(t; θ) of the random time T (θ):

Z ∞ 2 2 2 (σ )(t) = h X (t)i = hhW (Θ(t)iBiΘ = h2Θ(t)i = 2γ Fβ,tmp(t; θ) dθ . (2.2.4) 0

47 The Laplace image of h X 2(t)i in variable t is

Z ∞ 2γ Z ∞ 2γ (σ2)(s) = e−sth X 2(t)i dt = (fˆ )θ(s) dθ = − , (2.2.5) d β,tmp ˆ 0 s 0 s ln fβ,tmp(s)

ˆ ˆ where fβ,tmp(s) = fβ,tmp(s; 1) is the mother Laplace transform of the tempered β- stable random time T (θ). Substituting into 2.2.5, the Laplace image of fβ(s; δ) from 2.2.2, we obtain 2γ (dσ2)(s) = . (2.2.6) s[(s + δ)β − δβ]

For a dimensionless variable q = s/δ the above expression takes the form

2γ 1 s (dσ2)(s) = kˆ(q; β) , where kˆ(q; β) = , q = . (2.2.7) δ1+β q[(q + 1)β − 1] δ

So the mean square

2γ h X 2(t)i = k(τ; β) , τ = δ t , (2.2.8) δβ where k(τ; β) is the inverse Laplace transform of kˆ(p; β). The asymptotic of the function k(τ; β) is then determined by the asymptotic of kˆ(q; β) which is

1 1 kˆ(q; β) ∼ (q → ∞); kˆ(q; β) ∼ (q → 0) . qβ+1 β q2

The corresponding asymptotics of k(τ; β) is as follows:

τ β τ k(τ; β) ∼ (τ → 0) ; k(τ; β) ∼ (τ → ∞) . Γ(1 + β) β

Thus, for small τ, the mean square h X 2(t)i grows according to the subdiffusive law τ β while, for large τ, the subdiffusive law is replaced by the classical Fickian law of

48 the Gaussian linear diffusion.

2.3 Subordinators

A subordinator is a one-dimensional L´evyprocess which can be defined as a random model of time evolution. Let S = (S(t), t ≥ 0) is a subordinator then S(t) ≥ 0 for each t > 0 a.s. S(t1) ≤ S(t2) whenever t1 ≤ t2.

Theorem 2.3.1 If S is a subordinator then its L´evymeasure takes the form

Z ν(s) = ibs + (eisy − 1)λ(dy), (2.3.1) (0,∞) where b ≥ 0, and L´evymeasure λ satisfies the additional requirements of λ(−∞, 0) = R 0 and (0,∞)(y ∧ 1)λ(dy) < ∞. The pair (b, λ) is the characteristics of the subordinator (S) mapping from Rd → C.

Since S(t) is a positive random variable for all t, we will describe the Laplace trans- form rather than Fourier transform. Before obtaining the Laplace transformation for the developments, we will provide some essential definitions and proofs according to [2].

Theorem 2.3.2 (Subordination of a L´evyprocess) Fix a probability space (Ω, F, P).

b Let (Xt)t≥0 be a L´evyprocess on R with characteristic exponent Ψ(u) and triplet

(A, ν, γ), and let (St)t≥0 be a subordinator with Laplace exponent l(u)and triplet

(0, ρ, b). Then the process (Yt)t≥0 defined for each ω ∈ Ω by Y (t, ω) = X(S(t, ω), ω) is a L´evyprocess. Its characteristic function is

E[eiuYt = etl(Ψ(u))] (2.3.2)

49 where

E[euSt ] = etl(u). i.e., the characteristic exponents of Y is obtained by composition of the Laplace exponent of S with the characteristic exponent of X. The triplet (AY , νY , γY ) of Y is given by

AY = bA,

Z ∞ Y X d ν (B) = bν(B) + ps (B)ρ(ds), ∀B ∈ B(R ), 0 Z ∞ Z Y X γ = bγ + ρ(ds) xps (dx), 0 |x≤1

X where pt is the probability distribution of Xt and (Yt)t≥0 is said to be subordinate to the process (Xt)t≥0.

Proof . We will use the proof as in [2]. We first prove that Y is a L´evyprocess.

S S S Let Ft is the filtration of (St)t≥0 with F ≡ F∞. Using the L´evy-Khinchin formula for X and the independent increments property of X and S, for every sequence of times t0 < t1 < ... < tn,

n n h Y iu (X(S )−X(S ))i h Y iu (X(S )−X(S )) Si E e i ti ti−1 = E{E e i ti ti−1 |F } (2.3.3) i=1 i=1 n h Y iu (X(S )−X(S )) Si = E{ Ee i ti ti−1 |F } (2.3.4) i=1 n Y (S −S )Ψ(u ) = E {e ti ti−1 i } (2.3.5) i=1 n Y (S −S )Ψ(u ) = E[e ti ti−1 i ] (2.3.6) i=1 n Y iu (X(S )−X(S )) = E[e i ti ti−1 ]. (2.3.7) i=1

50 The above proof confirmed that Y has independent increments. Next show that Y is continuous in probability. For every ε > 0 and δ > 0,

P {|X(Ss)−X(St) > ε} ≤ P {|X(Ss)−X(St)| > ε |Ss −St| < δ}+P {|Ss −St| ≥ δ}.

Hence, X is uniformly continuous in probability, since the above first term can be made arbitrarily small simultaneously for all values of s and t by changing δ. As for the second term, its limit as s → t is always zero, because S is continuous in probability.

Therefore, P {|X(Ss) − X(St)| > ε} → 0 as s → t. Also, we can obtain 2.3.2 by the conditioning on F s :

h i EeiuX(St) = E{E eiuX(St)|F S = E{eStΨ(u)} = etl(Ψ(u)).

The detailed proof by using the characteristic function is in [101, P 168].

2.4 Time-Changed Brownian Motion

Consider the pure jump process Θ = (Tt≥0) as positive and nondecreasing, that is,

Tt ≥ 0 for t > 0, and Tt1 ≤ Tt2 for t1 ≤ t2. Then the process Θ can be used as the subordinator or instrinsic time process. The Poisson, Gamma, Inverse Gaussian process, and the β-stable (0 < β < 1) process can be considered as subordinators for the Brownian motion. We assume an additional condition, E[Tt] = t, which means the expected intrinsic time is the same as real time.

51 2.5 Tempered Stable Subordinator

We use the tempered stable subordinator {TS(t), t ≥ 0}, which is a strictly increas- ing L´evyprocess with tempered stable increments. Consider a subordinator of the inverse time of the process {Θ(t) = min{θ : T (θ ≥ t)}} independent from the standard Brownian motion B(t)t≥0. Then, substituting t = TS(t) in the Brownian motion with volatility σ and drift b, we have a new process

XTSB = (XTSB(t))t≥0 with

XTSB(t) = γT S(t) + σB(TS(t)). (2.5.1) which is called the time-changed Brownian motion. Rosi´nskiwas first introduced the Laplace transforms for the tempered stable distribution, and Saichev and Woy- czy´nskiexamined the particular case of β (0 < β < 1) [5, 102]. According to [103], we can express the Laplace transformation of tempered stable subordinator, i.e.

< e−sT S(t) > = et(δβ −(δ+s)β ), δ > 0, 0 < β < 1. (2.5.2)

When δ = 0, then {TS(t)} becomes totally skewed β-stable L`evy process. The probability density function of tempered stable subordinator can be expressed in the following form:

−δx+δβ t fβ,tmp(x) = e fβ,stab(x), (2.5.3)

where fβ,stab(t)(.) is the pdf of the totally skewed β−stable L´evymotion {fβ,stab, t ≥ 0}. Related proofs and details are available in reference [101] and [102]. Also, we

52 can approximate the tail of the β stable density [105] as follows:

−δx+δαt β −(β+1) fβ,tmp(x) ∼ 2βcβe t x , x → ∞ (2.5.4)

where cβ is a constant. Moreover the right tail of the cumulative distribution can be approximated by:

−δx+δβ β −(β+1) 1 − Fβ,temp(x) ∼ e t x , x → ∞ (2.5.5)

where Fβ,temp(x) is the distribution function of TS(t). The process {XTSB(t)} is known as normal tempered stable subordination (TSB). For simplicity, we consider σ = 1. We can find the Laplace transformation of TSB process according to [101]:

1 < e−s XTSB (t) > = exptδα − δ + γs − s2)β
. (2.5.6) 2

Z ∞ fβ,tmp(x) = fX(z)(x)fβ,,tmp(z)dz (2.5.7) 0

Z ∞ 1 −(x−µz)2/2z−δz+ fβ,temp(t)(x) = √ e fβ,,stab(z)dz, (2.5.8) 0 2πz

The first and second moments of the prosess {XTSB(t)},

Z ∞ β−1 < XTSB(t) > = xfβ,tmp(x)dx = γtβδ . (2.5.9) 0

cov(t, s) = < XTSB(t), XTSB(s) > = − < XTSB(t) >< XTSB(s) > (2.5.10)

= min(s, t)(βδβ−1 + γ2β(1 − β)δβ−2).

53 The most useful tool for comparison between diffusion and anomalous diffusion models is the mean squared displacement (MSD).

Z ∞ 2 2 2 2 β−1 2 β−1 2 β−2 < XTSB(t) > = x fβ,tmp(x)dx = t (γ βδ ) + t(βδ + γ β(1 − β)δ ). 0 (2.5.11)

2.6 Numerical Simulation of the TSB Process

Here, the estimation methods will be investigated theoretically and computationaly. The simulation process for the TSB is based on Monte Carlo simulation. The class of tempered β−stable process time was developed in [5]. This process is conve- nient for modeling financial data since it has the finite moments of all orders. The property of the infinite divisibility can be approximated by a given mesh ∆t. The general method to simulate any exponentially tempered stable random variables is explained in [24]. The simulation of the tempered stable subordinator is based on the procedure ex- plained in [103]. ALGOTRITHM(Generating the TS process on a fixed time grid) Generating tempered stable random variable (TS) with Laplace transform of

ˆ β β fβ,temp(s) = (s + δ) − δ . i. Simulate the exponential random variable (E) with mean λ−1. ii. Compute totally skewed β-stable random variable (S) as follows:

(1−β) π π 1 sin(β(U + ))cos(U − β(U + )) β β 2 2 S = ∆t 1 . cos(U) β E iii. If E > S, then TS = S, otherwise assign to the step i. The sample trajectories of the strictly increasing tempered stable process are shown

54 in Figures 2.1 - 2.4.

Figure 2.1: The sample trajectories of the strictly increasing Tempered stable process for β = 0.3, 0.5, 0.8 and λ = 10.

55 Figure 2.2: The sample trajectories of the strictly increasing Tempered stable process for β = 0.3, 0.5, 0.8 and λ = 10.

56 Figure 2.3: The sample trajectories of the strictly increasing Tempered stable process for β = 0.3, 0.5, 0.8 and λ = 10.

57 Figure 2.4: The sample trajectories of the strictly increasing Tempered stable process for β = 0.3, 0.5, 0.8 and λ = 10.

58 The Brownian motion driven by tempered stable subordinator {XTSB(t), t ≥ 0} is XTSB(t) = B(TS(t)), where TS(t) is strictly incrasing L´evyprocess with β−tempered stable increments. Simulation is based on the algorithms proposed in [2]. ALGORITHM (Generating the subordinated Brownian motion (TSB) on a fixed time grid)

Simulation of (XTSB(t1), XTSB(t2), ...XTSB(tn)) for n fixed times t1, t2, ..., tn where

XTSB(t) = B(TS(t)) is Brownian motion with volatility σ = 1 and drift γ, time changed with subordinator (TSt).

i. Simulate increments of the subordinator: ∆TSi = TSti − TSti−1 where S0 = 0. ii. Simulate n standard Gaussian random variables N1,N2, .., Nn. √ Pi iii. Define ∆Xi = Ni ∆TSi + γ∆TSi and set Xi = k=1 ∆Xi. Simulated trajectories of the TSB(t) according to the above algorithm are shown in Figures 2.5 - 2.7.

59 Figure 2.5: The sample trajectories of the path of Tempered stable subordination driven by brownian motion (TSB) for β = 0.3, 0.5, 0.8 and λ = 10.

60 Figure 2.6: The sample trajectories of the path of Tempered stable subordination driven by brownian motion (TSB) for β = 0.3, 0.5, 0.8 and λ = 10.

61 Figure 2.7: The sample trajectories of the path of Tempered stable subordination driven by brownian motion (TSB) for β = 0.3, 0.5, 0.8 and λ = 10.

62 2.7 Parameter Estimation of the TSB Process

Here, we discuss two parameter estimation methods, the least square method (LS) and the generalized method of moments (GMM). Every parameter estimation method is based on the empirical characteristic functions. Not all of the L´evy processes can be described using the closed-form of the density function. However, the closed-form of the characteristics function is available for every L´evyprocesses. The empirical characteristics function (ECF) method is the most efficient way to find the param- eter estimates of the TSB process. We will analyse the performance of the methods using error analysis. It is important to find parameters estimation from historical data when modeling the financial data. Therefore, intense research is necessary to establish the most accurate estimation methods for financial modeling. Most of the financial data are either skewed or high kurtosis. For that reason, the α−stable distributional ap- proach is heavily used in financial modeling. Zolotarev developed an integral representation of stable laws, and the results were used to develop the parameter estimation techniques [110]. The maximum likelihood (ML) approach is widely used in financial modeling because of the asymptotic effi- ciency. To utilize the ML method for financial data, the densities of models should be in closed-form. However, any density function can be expressed as its Fourier transformation. The Fourier transformation is always bounded, but likelihood func- tion might not be in closed-form. Because of these reasons, we are interested in pursuing the empirical characteristic function (ECF) method for finding parameters for the TSB process. A detailed description of the method can be found in [115]. In this chapter, we will investigate the method for densities which do not have a closed-form expression. A useful software package for estimating the stable distribu-

63 tion is provided in [116]. We will develop the theory and computational estimation methods to the TSB using the method of least square (LS) and the method of the moment (MME) [117].

Empirical Laplace function methods

(i) Least square method (LS):

Suppose a set of observations: ∆Xi = Xi+1 − Xi for i = 1, 2, ..., n − 1 follow the TSB distribution. Then we can approximate the empircal Laplace transformation function of these i.i.d. observations (∆Xi) by the Monte Carlo approach based on the law of large numbers, i.e.

n−1 1 X fˆ (s) = E[e−s∆Xi ] ≈ e−s∆Xi (2.7.1) β,temp n − 1 i=1

By using the LS method, we can find the estimators β,ˆ δ,ˆ γˆ which satisfy the fol- lowing formula:

 1 2 (β,ˆ δ,ˆ γˆ) = min fˆ (s) − exp{δβ − (δ + γs − s2)β} (2.7.2) β,δ,γ β,tmp 2

Simulations of the trajectories are based on the above two agrorithms of the TSB process. The LS estimates are computed for each simulated sample. Results on the biases (estimate-actual) and std (standard deviation) of the estimates are used to validate the LS estimation method. Simulations are based on n = 100, 1000, 10000 trajectories of length 100 each for various values of β, γ, and δ. We used the β = 0.4, γ = 10, and δ = 1 as the parameter values. For a length of 1000 trajectory , the LS estimates of parameters β, γ, δ are computed and the proces is repeated for 100, 1000 and 10,000 times. Average of bias and std of the LS estimates are computed.

64 The results are displayed in Table 2.1. Also, to verify the accuracy of the parameter estimation method, we used the boxplots of estimates for TSB (X (t)). The box plots of least squares estimates are presented in Figures 2.8 - 2.10.

Figure 2.8: The boxplots of the estimators of TSB (0.4, 10, 1). The values are based on the 100 simulations and trajectories of length 1000 each.

65 Figure 2.9: The boxplots of the estimators of TSB (0.4, 10, 1). The values are based on the 1000 simulations and trajectories of length 1000 each.

66 Figure 2.10: The boxplots of the estimators of TSB (0.4, 10, 1). The values are based on the 10,000 simulations and trajectories of length 1000 each.

According to the Table 2.1, the theoretical values are very close to the estimated values because of the less biassness of the parameters estimation method. Also, Figures 2.8 - 2.10 show that the theoretical values are between appropriate quantiles which confirmed the estimation method works properly.

Table 2.1: Mean, std, and bias of the TSB (0.4, 10, 1) of sample size = 1000 using LS. Simulation (iter) 100 1000 10000 mean std bias mean std bias mean std bias β = 0.4 0.403 0.007 -0.003 0.403 0.007 -0.0030 0.403 0.0071 -0.0031 γ = 10 9.876 0.269 0.124 9.884 0.268 0.116 9.870 0.286 0.129 δ = 1 1.029 0.047 -0.029 1.019 0.039 -0.019 1.018 0.042 -0.018

67 The empirical Laplace trasform for simulated data of TSB (0.8, 2, 10) is shown in Figure 2.11. In order to show that TSB process (X (t)) is better than Brownian motion with drift process (Normal) and stable subordinator driven by Brownian motion (SB) processes (TSB (β, γ, 0)), we compare the empirical Laplace transform among the theoretical Laplace trasforms of Normal, TSB and TS processes. (ii) Method of moments estimator (MME): In this subsection, the closed-form expression of expectation and formula for com- puting the higher order non-central moments are derived. The method of finding the moments for TSB distribution is introduced by Rosinski [5]. Later they have been studied by Terdik and Woyczynski [118]. The L´evymeasure for infinitely divisible tempered stable distribution can be written in the form [5]:

R R ∞ 1 −t d L(B) = d 1B(tx) α+1 e dtR(dx),B ∈ ( ), R 0 t R where R is a unique measure on Rd such that, ({0}) = 0 and R (||x||2||x||)R(dx) < ∞. The characteristic function of a tem- R Rd pered stable random variable X with parameter β, γ, and R, is also defined by the following formula: R If 0 < α < 1 and ||x||≤1 ||x||R(dx) then,

Z Φ(s) = exp{ Ψβ(< s, x >)R(dx) + i < s, γ >}, Rd

β where Ψβ(s) = Γ(−β)[(1−is) −1]. We can express moments and cumulant of L´evy process as described in [2, 5].

Proposition 2.7.1 Let (Xt)t ≥ 0 be a L´evyprocess on R with characteristic triplet

n (A, ν, γ). The absolute moment of Xt,E[|Xt| ] is finite for some t or, equivalently,

R n for every t > 0, if and only if |x|≥1 |x| ν(dx) < ∞. In this case moments of Xt can compute from its characteristic function by differentiation. In particular, the form

68 of cumulants of Xt is especially as follows:

Z E[Xt] = t(γ + xν(dx)), |x|≥1

Z +∞ 2 cum2(Xt) = V ar(Xt) = t(A + x ν(dx)), ∞ Z +∞ n cumn(Xt) = t x ν(dx) for n ≥ 3. −∞

We will describe the cumulant generating function for the TSB process using the proposition of 2.7.1. Let us consider the measure ν for the tempered stable process as:

e−δx ν(dx) = c 1 , xβ+1 x≥0

where 0 < β < 1, c > 0 and δ > 0, and consider the pure jump process T = (Tt)t≥0 is defined by ν and γ , where

Z 1 γ = xν(dx). 0

Then the characteristic function for ΦTSt ;

Z e−δx Φ (s) = E[eis∗TSt ] = exp(tc (eisx − 1) dx). (2.7.3) TSt xβ+1

Solving the above integral of the equation 2.7.3, we can express the characteristic function for the tempered stable process in the following way:

is∗TSt β β ΦTSt (s) = E[e ] = exp(tcΓ(−β)((δ − is) − δ )). (2.7.4)

69 And also, 1 ∂ E[TS ] = log Φ (s)| = tcΓ(1 − β)δβ−1. (2.7.5) t i ∂s TSt s=0

Considering the mean time which is equal to the actual time, E[TSt] = t, we can find:

1 c = . (Γ(1 − β)δβ−1)

Lets denote the tempered stable process as TS(β, (Γ(1 − β)δβ−1)−1, δ). The char- acteristic function of Xt can be obtained according to the equation 2.3.2 such that

E[iuXt] = exp{tl(φ(u))}, where, l(u) is the Laplace exponent of the TSt subor- dinator, and φ(u) is the characteristic exponent for the Brownian motion with a drift. Now we use the characteristic function of the TS process to find the moment generating function of the TSt:

i (TSt)∗s ΦTSt (−is) = E[e ] (2.7.6)

= exp(tcΓ(−β)((δ − i(−is))β − δβ))

= exp(tcΓ(−β)((δ − s))β − δβ)).

Substituting s for the characteristic exponent for the Brownian motion with drift φ(s) = iγs − 0.5σ2s2 and considering σ = 1, we obtain the characteristic function of the TSB:

i T SBt∗s ΦTSBt (s) = Ee (2.7.7)

= exp(tcΓ(−β)((δ − iγs + 0.5s2)β − δβ)) Γ(−β) = exp(td ((δ − iγs + 0.5s2)β − δβ)) Γ(1 − β)δβ−1 −t = exp( ((δ − iγs + 0.5s2)β − δβ)). βδβ−1

70 Since we have the charateristic function for the TSB process, then we can obtain the moment generating function as follows:

MX (s) = ΦTSBt (−is) (2.7.8) −t = exp( ((δ − iγ(−is) + 0.5(−is)2)β − δβ)) β δβ−1 −t = exp( ((δ − γs − 0.5s2)β − δβ)). β δβ−1

Cumulant generating function can be expressed as:

dm C (X) = ln(M (0)), k dsk X

d m (X) = E[Xk] = M (0), k dsk X where, mk(X) is the k-th moment of the TSB distribution for t = 1. We use Math- ematica to find the cumulants and moments of TSB distribution as follows:

C1 = γt (2.7.9) (β − 1)(γ2) C = t[1 − ] 2 δ 3 2 C3 = [(−1 + β)(−3δγ + (−2 + β)γ )]t/δ

β Substituting the value c = Γ(1−β) to equation 2.7.5, the analytical results of the

71 parameters are as follows:

−1+β C1 = βδ γt (2.7.10)

−2+β 2 C2 = βδ (δ − (−1 + β)γ )t

−3+β 2 C3 = (−1 + β)βδ γ(−3δ + (−2 + β)γ )t

−4+a 2 2 2 4 C4 = −(−1 + β)βδ (3δ − 6(−2 + β)δγ + (6 − 5β + β )γ )t −(((−1 + β)δ−β(3δ2 − 6(−2 + β)δγ2 + (6 − 5β + β2)γ4)) Kurtosis = (β(δ − (−1 + β)γ2)2)) (2.7.11)

Skewness = ((−1 + β)βδ−3+β(−3δγ + (−2 + β)γ3))/(βδ−2+β)(δ − (−1 + β)γ2))3/2

C2/C1 = 1/γ + γ(1 − β)/δ

Then we can express the parameter δ using only the β and γ parameters:

C 1 δ = γ(1 − β)/[ 2 − ] (2.7.12) C1 γ

Since, 0 < β < 1, we can conclude that

γ δ < (2.7.13) C2 − 1 C1 γ

These results make much easier to find parameter using the method of moments estimator.

72 Table 2.2: Skewness (ss) and Kurtosis (kk) of TSB distribution for the values of β, γ and δ = 1. γ = 0.01 γ = 0.1 γ = 1 γ = 10 γ = 100 β ss kk ss kk ss kk ss kk ss kk 0.1 -0.307 13.746 0.464 10.472 3.618 22.789 4.846 27.776 5.134 34.605 0.5 0.505 3.497 0.400 2.266 1.344 3.825 2.466 7.520 2.495 9.131 0.8 0.103 0.324 0.212 0.601 0.634 1.516 2.586 16.411 2.951 17.010

Table 2.3: Skewness (ss) and Kurtosis (kk) of TSB distribution for the values of β, δ and γ = 10. δ = 0.1 δ = 1 δ = 10 δ = 100 δ = 1000 β ss kk ss kk ss kk ss kk ss kk 0.1 0.928 5.190 2.128 7.935 4.508 31.884 7.435 78.767 9.046 107.375 0.5 0.138 0.609 1.559 4.728 3.192 12.507 6.214 55.131 9.452 104.893 0.8 -0.056 0.199 1.195 5.043 7.724 103.291 14.962 280.690 13.617 229.884

Table 2.2 and Table 2.3 give the skewness (ss) and kurtosis (kk) for some pa- rameter combinations of (β, γ, δ). These values confirmed that we can model the financial data using the TSB process when there are high skewness and kurtosis . Because of the complexity of the analytical results of the parameters, we solve the system using the GMM function in the R-package. A method for numerical cal- culations of stable densities and distribution is introduced in [119, 116]. GMM is one of the method that was suggested by Carrasco and Florens for the continuum of moment conditions [120]. They used the fact that the characteristic function,

E(eixiτ ), where i is the imaginary number and τ ∈ R has a closed-form expression. The GMM package does not yet include continuum of moment conditions; however we can choose a certain grid τ1, ..., τq over a given interval. Parameter estimation using GMM is based on the following condition:

ixiτl E[e − φ(θ; τl)] = 0 for l = 1, 2, 3, ...q. where, φ(θ; τ1) is the characteristic function of the TSB distribution. We imple-

73 mented R-codes for TSB process according to [140]. To see the accuracy of the MME, we simulate 100 samples of n = 100, 1000, 10000 trajectories for TSB (0.4,10,1), TSB (0.4, 100, 1), TSB (0.4, 10, 10), and TSB (0.4, 10, 1) by setting with set.seed (134) in R-programming. We calculate the mean,the standard deviation, and the bias for the estimators which are presented in Table 2.4, and Table 2.5.

Table 2.4: Mean, std, and bias for the TSB of 100 simulations using MME for various γ values. Sample size 100 1000 10000 mean std bias mean std bias mean std bias β = 0.4 0.382 0.035 0.965 0.402 0.013 0.925 0.404 0.007 -0.004 γ = 10 9.480 1.630 0.124 9.535 0.881 0.114 9.610 0.547 0.391 δ = 1 0.965 0.521 0.035 0.925 0.465 0.075 0.944 0.066 0.056 β = 0.4 0.621 0.162 -0.221 0.598 0.193 -0.198 0.600 0.190 -0.200 γ = 100 58.765 193.17 41.235 102.4 267.9 -2.359 94.940 254.300 5.061 δ = 1 0.520 0.426 0.480 0.5012 0.422 0.498 0.5147 0.423 0.485

Table 2.5: Mean, std, and bias for the TSB of 100 simulations using the MME for various δ values. Sample size 100 1000 10000 mean std bias mean std bias mean std bias β = 0.4 0.506 0.174 -0.106 0.509 0.175 -0.1085 0.503 0.170 -0.103 γ = 10 9.701 1.167 0.298 9.622 1.238 0.377 9.614 1.239 0.386 δ = 10 1.493 1.899 8.507 1.398 1.751 8.602 1.4059 1.755 8.594 β = 0.4 0.376 0.0443 0.0236 0.467 0.102 -0.0669 0.465 0.101 -0.0651 γ = 10 9.467 1.616 0.533 7.447 2.828 2.5528 7.449 2.809 2.551 δ = 1 0.956 0.144 0.0436 1.831 1.495 -0.831 1.821 1.489 -0.8207

2.8 Comparison of TSB Process with Diffusion

and Stable Subdiffusive Processes

We use the simulation of the TSB (0.8, 2, 10) process and compare the empirical Laplace transform with theoretical Laplace transforms of the TSB, Gaussian and

74 SB processes. Figure 2.11 shows a comparison of the TSB, Gaussian, and SB. All the plots coincided perfectly except the plot of theorical laplace transform of the Gaussian process.

Figure 2.11: The empirical Laplace transform, theoretical Laplace transform for TSB with tempered stable subordinator and theoretical Laplace transform for TSB with stable subordinator for the simulated data (β = 0.8, γ = 2 and δ = 10.)

2.9 Error Comparison of Brownian Motion and

TSB Processes

We use the characteristic function to compare the accuracy of the processes. Also, in the real data analysis, the most popular statistic is the mean square displacement (MSD) to compare the processes. The MSD for a stochastic process calculates from

75 Figure 2.12: Normal fitting for the simulated data of TSB (β = 0.8, γ = 2, δ = 10) using MLE and Laplace transformation. the following formula:

2 2 MSD(t) =< ∆r(t) >= [r(t + t1) − r(t)] ,

where r(t) is the position of the stochastic time series at time t and t1 is the time lag between two positions taken by the stochastic time series. That can be calcu- lated as the displacement ∆r(t) = r(t + t1) − r(t). We calculate the MSD for the process using the 1000 trajectories by only performing the average of the MSD at time t1. The ensemble average MSD is defined for the process as a second moment. The second moment of the process can be calculated using the following integration:

Z 2 2 E[XT (t)] = x fβ,tempT (x)dx

76 where, fβ,tempT is a pdf of the TSB process. Also, we can calculate the ensemble average MSD for real stochastic time series data only when more than one trajectory is available. However, MSD can be calculated from a single trajectory by only taking

β a time average. Without loss of generality, we consider the c = Γ(1−β) to calculate theoretical MSD value. To calculate the second moment of the TSB process, we can choose the characteristic function of the TSB process as follows:

2 2 V ar(XT (t)) = C2(XT (t)) = E(XT (t)) − [E(XT (t))]

where, C2 is the second cumulant of the process of TSB. Therefore, MSD of the TSB process is,

2 2 2 MSD(XT (t)) = E(XT (t)) = V ar(XT (t)) + [E(XT (t))]

2 = C2(XT (t)) − (C1(XT (t)))

= βδ−2+β(δ − (−1 + β)γ2)t + (βδ−1+βγt)2

When γ → 0, the ensemble averaged MSD scales as t. To present this characteristic, we simulated 1000 trajectories of the TSB processes for β = 0.8, δ = 1, and γ = 0.001. We can see clearly from Figure 2.13, assemble MSD of the TSB process with α = 0.8, δ = 1, and γ = 0.001 is fitted closely with the second-order polynomial. In chapter 3, we will apply the TSB to real stock data analysis. Also, will compare the TSB process with diffusion and stable subdiffusive processes.

77 Figure 2.13: MSD of 1000 trajectories of the TSB (β = 0.8, λ = 1, γ = 0.001) process.

78 Chapter 3

Statistical Analysis of Log-returns of the Stock Data using the Least-Square-Estimation and the method of moments

3.1 Introduction

Why do we need financial modeling with subordination? Major failure in financial markets started with the bankruptcy of Lehman Brothers in September 2008. This made financial risk managers, portfolio managers, and reg- ulators think more critically about the extreme events occurring. These significant failures in the financial markets have led them to increase their efforts to improve existing statistical models to capture the dynamics of financial variables. The in- ability of existing financial models has been identified by some researchers who serve as significant contributors to solving the latest global financial crisis. At that time, risk managers, portfolio managers, and regulators did not think of the intricacies of real-world financial markets. More precisely, the underlying assumption regarding asset returns and prices failed to capture real-world movement of these quantities.

79 The underlying assumption in most financial models is that log-returns of the prices are normally distributed. This model is known as the Gaussian model. The empir- ical evidence of the log-returns has led to the following three facts: (1) the heavy tails (2) the skewness (3) the volatility clustering The extreme values occur in the tails of the distributions. The normal distribution would not predict extreme values. It can be predicted only by modest changes in prices and returns. However, there will be periods of prominant crashes and booms. Figure 3.1 displays that the Gaussian distribution does not follow distributions of empirical log returns. In 1960s, Mandelbrot first explained the fundamentals of the assumption that price or return distribution is not normally distributed [106, 108]. He did not accept that the log-returns are normally distributed as based on evidence of studies about the commodity returns and interest rates. Mandlebrot speculated that a non-normal stable distribution could describe financial returns. Furthermore, he believed that log-return distribute as non-normal distribution. Later those con- cepts were further examined by Fama [48, 49]. Black and Scholes (1973) and Merton (1974) introduced pricing and hedging theory for the options market using a stock price model which was developed based on the exponential Brownian motion. In 1997, Merton and Scholes were awarded the Nobel Prize in Economic Science for the Black and Scholes model. This model significantly impacts the way market participants price and hedge options. Which can be written as follows:

−rt C = S × N(d) − Xe × N(d). where,

2 C0( S, t, r, σ ) are the value of a European option at time t equal to zero, the current

80 value of the underlying asset, the life to expiration of the option, the risk free interest rate, variance (value) of the underlying assets respectively.

σ ln(S/X) + (r +  t) d = √ σ t

√ d = d − σ t and, N(d) is the cumulative probabilty of the standardized normal distribution. Also the other forms of the Balck and Scholes model are as follows:

S(t) = S0 + σB(t).

S(t) = S0 exp (γt + σB(t)) . or, the local form dS(t)  σ2  = σdB(t) + γ + dt. S(t) 2

In spite of the valuability of option theory as created by Black, Scholes, and Merton, on Black Monday, October 19, 1987, the Black-Scholes formula broke down. The reason for the failure of the model is that the major assumption broke down during volatile periods. It is assumed that returns are following the normality and that volatility of the return is constant over the options’ life. In recent time, researchers who work in the financial market, focus on option pric- ing models whose underlying asset price processes are following the Levy processes [2],[131]. In this chapter, we demonstrate the numerical results of fitting the log-returns using the normal, stable, and the TSB distributional approaches. The Fourier transforma- tion approach is a useful tool in pricing options. When the characteristic function is

81 Figure 3.1: Probability distribution of the empirical log-return compare with the normal distribution for AMAZON, APPLE and S&P500 stocks from 01-01-2000 to 02-12-2019. available to the TSB process, the real-time option can be delivered. More generally, the log-returns of the stock price can be modeled by the TSB process; then, it is allowed for capturing the excess kurtosis and stochastic volatility. The characteristic function of the TSB process can be evaluated analytically. Then, the expression for the Fourier transform of the option value can be derived. Through computationally faster implementation of Fourier transform, and the FFT (fast Fourier transform) algorithms, we can find the option prices, which are the whole spectrum of strikes. It can be obtained by performing a Fourier inversion transform. Throughout this chap- ter, the scenario for calculating the option pricing when the characteristic function is readily available for the TSB process is described.

82 3.2 Preliminaries

Accordingly in [130], ”If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks”. An option is security giving the right to buy or sell an asset, subject to certain conditions, within a specified period of time. An ”American option” is one that can be exercised at any time up to the date the option expires. A ”European option” is one that can be exercised only on a specified future date. The price that is paid for the asset when the option is exercised is called the ”exercise price” or ”sticking price”. The last day on which the option may be exercised is called the ”expiration date” or maturity date. In general, the Black-Scholes-Mertons Geometric Brownian process can not be used when the financial instrument has some jumps. Stein and Heston direct us to find an analytic representation of the European option price function in the Fourier domain [132, 133]. Duffie came up with transform methods for European pricing options under the affine jump-diffusion processes [134]. The flexibility of finding the characteristic function (Fourier transform of the density function) may be easier to compute with the Levy process than the density function itself. Nowadays, it may be easier to compute the integral of their Fourier transform using a library of the R-package. Also, the L´evy-Khinchine formula clearly described the way to find the characteristic function for a Levy process.

The TSB process (Xt) can be described by its characteristic function Φ(s).

83 3.2.1 Fourier Transform and Pricing Options

The Fourier transform of f(x) is defined by,

Z ∞ F(u) = eiuy f(y) dy. −∞ where, f(x) is satisfies the following condition of,

Z ∞ |f(x)|dx < ∞. −∞

Also, the function f(x) can be derived by using the inverse Fourier trasform as fol- lows:

1 Z ∞ f(x) = e−iuxF(u) du. 2π −∞

We can check the validity of the above Fourier inversion formula using the Dirac function δ(y − x):

1 Z ∞ δ(y − x) = eiu(y−x) du. 2π −∞

Using the above Dirac function , we can define the f(x) as:

Z ∞ f(x) = f(y)δ(y − x) dy ∞

Then,

Z ∞ 1 Z ∞ 1 Z ∞  1 Z ∞  f(x) = f(y) eiu(y−x) du dy = eiux f(y)eiuydy du. −∞ 2π −∞ 2π −∞ 2π −∞ 1 Z ∞ = e−iuxF(u) du. 2π −∞

84 We can express the relation between the Fourier transformation and the character- istic function of Xt as follows;

Z ∞ F(u, t) = eiuxp(x, t) dx = E[eiuXt ] −∞

where, p(x, t) is the density function of the stochastic process Xt. The characteristic function of a L´evyprocess Xt can be derived by the representation of the L´evy Khinchine formula. Such as:

isXt ΦX (s, t) = E[e ] 2 Z  σ 2 isx  = exp γ ist − ts + t (e − 1 − isx1|x|≤1)ν(dx) 2 R\{0}

= exp(tψX (s, t)). where,

Z 2 2 min(1, x )ν(dx) < ∞, γ ∈ R, σ ≥ 0. R

Here, ψX (s) is called the characteristic exponent of Xt. We can derive all the moments of the characteristic function which generates the moments to a com- plex domain. Any L´evyprocess can be fully characterized by its characteristic function ΦX (s). When analytic form of the characteristic function is readily avail- able for the L´evyprocess, then we can derive the formal analytic representation of European option price using cumulative distribution functions [135].

Under the risk neutral measure Q, we can express the underlying asset price process as in the following form,

St = S0 exp(Xt), t > 0

85 where, Xt is a L´evyprocess. Let denote by FvT the Fourier transform of the ter- minal payoff function VT (x), where x = log ST . The European option value can be calculated as follows [136]:

−rT Z iµ+∞ −rT e h −izx i V (St, t) = e EQ[VT (x)] = EQ e FVT (z) dz 2π iµ−∞ −rT Z iµ+∞ e −izx = e ΦXT (−z)FVT (z) dz 2π iµ−∞

where µ = lm z and φXT (z) is the characteristic function of the XT . The call option C(S,T,K), can be derived in the following way [136]:

−rT Z iµ+∞ −izk −K e e ΦXT (−z) C(S,T,K) = 2 dz 2π iµ−∞ z − iz −rT Z iµ+∞ Z iµ+∞ −K e h −izk i −izk i i = e ΦXT (−z) dz − e ΦXT (−z) dz 2π iµ−∞ z iµ−∞ z h1 1 Z ∞ eiu log kΦ (u) i = S + Re XT du 2 π 0 iu h1 1 Z ∞ eiu log kΦ (u) i − Ke−rT + Re XT du 2 π 0 iu where, S is the current stock price, T is the maturity of call option with terminal S payoff (S − K)+ and k = log . T K The above integrand function and its singularity will appear when u = 0, which createss a discontinuity of the payoff function at ST = K. As a solution to this problem, Carr and Madan produce the tempering of the high frequency terms by multiplying the payoff with exponential decay function [137].

86 3.2.2 Carr-Madan Formulation for Evaluating the European

Option Pricing

The analytical expression of the characteristic function of the underlying asset price process is the most popular way to find the call option value since the desirable L´evyprocess doesn’t have the close-form of the density function. Carr and Madan compute the corresponding Fourier inversion to recover the call price using the FFT. Assume, k = log K.

The Fourier transformation of the call option price C(K) does not exist because the C(k) is not square-integrable. Consider the call price c(k) = eαkC(k), for α > 0.

Accordingly, in [137], we can write the Fourier transform of c(k) as ψT (u).

Z ∞ iuk ψT (u) = e c(k)dk −∞ Z ∞ Z s −rT s+αk (1+α)k iuk = e pT (s) [e − e ]e dk ds −∞ −∞ e−rT Φ (u − (α + 1)i) = T α2 + α − u2 + i(2α + 1)u

and, the call option C(k) can be calculated by,

−αk Z ∞ e −iuk C(k) = e ΨT (u) du, 2π −∞

−αk Z ∞ e −iuk = e ΨT (u)du. π 0 where pT (s) is the density of the underlying asset price process, s = log ST , ψT (u), is the Fourier transformation of c(k), and the φT (u) is the characteristic function of

87 the pT (s). This integral can be calculated using the FFT.

3.2.3 Passing from Characteristic Function to Probability

Density Function (PDF) and Cumulative Density Func-

tion (CDF)

Consider the characteristic function as Φ(s), then we can find the CDF as;

Z ∞ iux −iux 1 1 e ΦX (−u) − e ΦX (u) FX (x) = + du. 2 2π 0 iu

Proof . Before we prove the above formula for CDF, we will consider the following preliminary results: Since, eiux = cos ux + i sin ux. then,

1 Z +∞ eiuτ 1 Z +∞ sinτu du = = sgn(τ). π −∞ iτ π −∞ u

For fixed x,

Z +∞ Z x Z ∞ sgn(y − x)dF (y) = − dF (y) + dF (y) = 1 − 2F (x). −∞ −∞ x

Then,

Z ∞ eiuxΦ (−u) − e−iuxΦ (u) Z ∞ Z +∞ eiuxe−iuz − e−iuxeiuz X X du = dF (z) du 0 iu 0 −∞ u

Z ∞ Z +∞ 2sin u(x − z) Z +∞ = dudF (z) = πsgn(x − z)dF (z) = π(2F (x) − 1) 0 −∞ u −∞

88 Therefore,

1 1 Z ∞ eiuxΦ (−u) − e−iuxΦ (u) F (x) = + X X du. 2 2π 0 iu

The formula for PDF:

Z ∞ Z ∞ 0 1 iux −iux 1 −iux fX (x) = F (x) = (e Φ(−u) + e Φ(u))du = e Φ(u)du. 2π 0 π 0

3.2.4 Fourier Transforms and Transposition of European

Options

The option forward price c(k) can be derived from the following formula;

rt Q st k c(k) = e c(K, t)/F0 = E0 [(e − e )1St≥k],

where, st = ln Ft/F0 and k = ln K/F0. The analytical form of the Fourier transform of the call option in terms of the characteristic function of the log return st is always available. The result below is the analog of how we can derive the European call option value using the characteristic function of the log-returns.

0 Let us consider the option trasform as χc(u), then;

Z +∞ Z +∞ 0 iuk iuk ∞ iuk χc(u) ≡ e dc(k) = e c(k)|−∞ − c(k)iue dk. −∞ −∞

Z +∞ 0 −iu∞ iuk χc(u) = −e − c(k)iue dk −∞

89 Z +∞ Z +∞ −iu∞ st k iuk = −e − iu [ (e − e )ist≥k dF (s)]e dk −∞ −∞ Z +∞ Z +∞ −iu∞ st k = −e − iu [ (e − e )ist≥kdk] dF (s) −∞ −∞

Z +∞ Z st = −e−iu∞ − iu [ (eiut+st − e(iu+1)k)dk]dF (s) −∞ −∞ Z +∞ h iuk (iu+1)k i −iu∞ st e e st = −e − iu e − |−∞ dF (s). −∞ iu iu + 1

(iu+1)k We need to check the boundary values again. limk→∞ e = 0 given the real component e−∞. The other boundary value is non-convergent este−iu∞, which we pull out and take the expectation to have

Z +∞ est e−iu∞ iu dF (s) = e−iu∞, −∞ iu which cancels out the other nonconvergent term.

Z +∞ (iu+1)st (iu+1)st 0 −iu∞ he e i χc(u) = −e − iu − dF (s). −∞ iu + 1 iu + 1

Z ∞ e(iu+1)st Φ(u − i) = − dF (s) = − ∞ iu + 1 iu + 1

Let take the option transform:

Z ∞ 0 iuk Φ(u − i) χc(u) = e dc(k) = − −∞ iu + 1

Then, the inversion formula is analogous to the inversion of a CDF:

1 1 Z ∞ eiuxχ0 (−u) − e−iuxχ0 (u) c(x) = + c c du. 2 2π 0 iu

90 3.3 Stock Return Data

According to the section 3.2, the option value can be calculated if we can derive the characteristic function or Fourier transform of the log-returns (ln ST /ST −1) an- alytically. The following sections describe the details of finding the characteristic function for current stock market data. The analysis is based on the TSB process which is explained precisely in chapter 2. We test the model directly from the stock data and validate it using the Weis Mar- kets, Inc. (WMK) data in the New York Stock Exchange (NYSE), from January 1993 to Feb 2009. We are interested in using the WMK data for modeling, since it is one of the major data sources for quandmod package in R. Also, we find the parameters using the LS method. Furthermore, we analyse the system using the Amazon and Apple stocks data from 01.01.2000 to 02.12.2019. Data was obtained from the Yahoo finance through quantmode package in R.

3.4 Normal, Subordinated Stable, and Subordi-

nated Tempered Stable Fitting

First, we will use the Q-Q plot and the normality fitting to see the lack of fitting of normal distributional approaches. Second, we assumed the log-returns of the stocks are behaved as TSB distribution, and obtained the parameter estimations using LS. Also, we find the parameter estimation for the Normal, SB and STABLE processes. We validate the processes by comparing them with their Laplace transforms.

91 3.4.1 Analyzing Log-returns of the WMK Stock.

Data were obtained from the Gmm package through R. We can observe from Figure 3.2 and Figure 3.3 the log-returns of WMK data are not only high peaked but they also have left and right skinny tails with extreme outliers. Table 3.6 provides the parameter estimation values for the TSB process. The comparison of the empirical Laplace transforms of the normal, TSB, and SB is shown in Figure 3.4. We could not obtain properly the least square error for the SB fitting using the nls function in R package. The SB model minimizes the least square error until 4.7 which confirmed that the flexibility of using the TSB and normal processes are greater than with the SB process. Figure 3.5 displays the comparison between the empirical and theoretical densities of the TSB process for log-returns of WMK stock data. Also, the comparison of the cumulative densities in Figure 3.7 confirmed that the tempered stable subordination process is better than the Brownian motion. According to Figure 3.5 and Figure 3.7, we can predict more accurately the option values and hedging using the TSB subordination than with the Brownian motion with drift. The reason is that the option price is based on the cumulative densities of the distribution. In Figure 3.6 and Figure 3.8 we provide the densities and cumulative densities of empirical log-returns for the TSB and the STABLE (current market model) processes. Also, the parameter estimation using the α−stable process are shown in Table 3.2.

92 Figure 3.2: The nomal distributional fitting for Weis Markets log-returns (1500) from 1993 to 2009.

Figure 3.3: Q-Q plot for Weis Markets log-returns (1500) from 1993 to 2009.

93 Figure 3.4: The empirical Laplace transform of log-returns of WMK stock data compared with the Normal, TSB, and SB using the LS method.

Figure 3.5: The empirical density (Epdf) compared with the Normal and TSB for log-returns of WMK stock data.

94 Figure 3.6: The empirical density (Epdf) compared with the Stable (the current market model) and TSB for log-returns of WMK stock data.

Figure 3.7: The empirical cumulative density (Ecdf) compared with the Normal, and TSB distributions for log-returns of WMK stock data.

95 Figure 3.8: The empirical cumulative density (Ecdf) compared with the Stable (the current market model) and TSB for log-returns of WMK stock data.

Table 3.1: Parameter estimation for log-returns of WMK stock data. βˆ γˆ δˆ µˆ σˆ RSS TSB 0.831922 0.0372221 0.475918 - - 0.001176158 SB 1.00000 0.500000 - - - 9.569116 B - - - 0.101501 1.09172 0.002928001

Table 3.2: Stable distribution parameters extracted from the log-returns of WMK stock data. αˆ µˆ βˆ σˆ 1.7766 -0.01697 0.2489 0.5803

96 Figure 3.9: The trajectories of the TSB process for log-returns of WMK stock data using LS method ( first sample ).

Figure 3.10: The trajectories of the TSB process for log-returns of WMK stock data using LS method (second sample).

97 Figure 3.11: The trajectories of the TSB process for log-returns of WMK stock data using LS method (third sample).

98 3.4.2 Analyzing Log-returns of the APPLE Stock.

This data set includes the APPLE stock from 01-01-2000 to 02-12-2019. The goal is to find the characteristic function of the log-returns {ln ST /ST −1} numerically. Figure 3.12 and Figure 3.13 display the chart series of the APPLE stock (AAPL) and log-returns of the APPLE stock data respectively. Our goal is to show the ability of the better hedging option using the TSB rather than diffusion and TB processes. Therefore, we consider the financial crash in 2008. To determine the nature of the details in normality, we run the Q-Q plot, shown in Figure 3.14. Also, the kurtosis of the log-returns of the APPLE stock data is 117.89 and the skewness is -4.35 are the major properties to look for the non-normal distributional fittings. We notice that the log-returns data exhibit high skewness which is reflected in the fitting of the distribution. Therefore, we can conclude that the log-returns of APPLE data do not follow the normality assumption.

Figure 3.12: The examined real data set of APPLE stocks from 01.03.2000 - 02.12.2019.

99 Figure 3.13: The examined real data set of log-returns of APPLE stocks from 01.03.2000 - 02.12.2019.

Figure 3.14: Q-Q plot for APPLE stocks from 01.03.2000 - 02.12.2019.

100 Figure 3.15: The normal fitting for the log-returns of APPLE stocks from 01.03.2000 - 02.12.2019. The skewness = -4.35 and the kurtosis = 117.89.

Figure 3.16 illustrates the comparison of the estimation using the MLE (method of likelihood estimation) and LS (least square estimation) for the normal distribu- tional approaches. We compare the two ways of parameter estimation for the Laplace transformation using the LS and the MLE. Figure 3.16 shows a slight overfitting of the MLE than LS. Table 3.3 provides parameters estimations for APPLE stock using the TSB, SB and Normal (diffusion with drift) processes. Figure 3.17 represents the comparison of the numerical values of the Laplace transformation using these three methods. By looking at Figure 3.17, we can guarantee that the TSB and Normal estimation is the best estimation by their model, but SB can not find estimation for the Laplace transformation. However, the minimization proceeded up to the residual sum of square (RSS) error of approximately 8.8. We used the CharFun package in R to estimate the PDF and CDF using the inverse Fourier transforma- tion. Statistically, Figure 3.17 confirmed the best estimation we can find using the

101 different model approximations. Figure 3.18 illustrates the estimated probability densities for the log-returns of the APPLE stock using the Normal and TSB distri- butions. Also, Figure 3.19 displays the comparison of cumulative densities of the log-returns. These graphs suggest that the TSB distribution is the best fit for the log-returns of the APPLE stock. Furthermore, Figure 3.20 and Figure 3.21 provide the densities and cumulative densities of empirical log-returns for the TSB and the STABLE (current market model) processes. Also, the parameter estimation using the α−stable process are shown in Table 3.4. Figures 3.22 - Figure 3.24 show the simulated trajectories using the estimated parameters of the TSB distribution. We can see clearly the trajectories perform some jumps.

Figure 3.16: The parameter estimation for the normal fitting using the Laplace transform (Laplace) and the MLE (method of likelihood estimation) from 01.03.2000 - 02.12.2019.

102 Figure 3.17: The empirical Laplace transform (black), theoretical Laplace transform for TSB (red), theoretical Laplace transform for SB (blue), and theoretical Laplace transform for Normal (green) for the log-returns of the APPLE stock from 01.03.2000 - 02.12.2019.

Figure 3.18: The empirical density (Epdf) compare with the Normal and TSB dis- tributions for log-returns of APPLE stocks from 01.03.2000 - 02.12.2019.

103 Figure 3.19: The empirical cumulative density (Ecdf) compare with the Normal and TSB distributions for log-returns of APPLE stocks from 01.03.2000 - 02.12.2019.

Table 3.3: Parameter estimation for log-returns of the APPLE stock. The considered period is 01.03.2000 - 02.12.2019 βˆ γˆ δˆ µˆ σˆ RSS TSB 0.1562598 1.0293586 545.9552007 - - 2.657751e-11 SB 1.00000 0.500000 - - - 8.823782 B - - - 0.000788618 0.0276827 2.571696e-11

Table 3.4: Stable distribution parameters extracted from the log-returns of APPLE stock data. αˆ µˆ βˆ σˆ 1.5858 0.00094 -0.0258 0.0125

104 Figure 3.20: The empirical density (Epdf) compare with the Stable (the current mar- ket model) and TSB for log-returns of Apple stock data from 01.03.2000 - 02.12.2019.

Figure 3.21: The empirical cumulative probability (Ecdf) compare with the Sta- ble (the current market model) and TSB for log-returns of Apple stock data from 01.03.2000 - 02.12.2019.

105 Figure 3.22: The trajectories of the TSB process for log-returns of APPLE stock data using LS method (first sample).

Figure 3.23: The trajectories of the TSB process for log-returns of APPLE stock data using LS method (second sample).

106 Figure 3.24: The trajectories of the TSB process for log-returns of APPLE stock data using LS method (third sample).

107 3.4.3 Analyzing log-return of the AMAZON stock.

In this subsection we also analyze the AMAZON stock data by assuming TSB dis- tribution for log-returns of AMAZON stock data from 01-03-2000 to 02-12-2019. The characteristic function method was employed and we obtained the parameter estimation using the LS method. The results are shown in Table 3.5. The Amazon stock data also have skewness 0.4529 and kurtosis is 12.1415 ( Figure 3.1). The log-returns of Amazon stock data exhibit a tail. We can determine the nature of the tail from the Q-Q plot as shown in Figure 3.27. Clearly, the Q-Q plot does not follow a straight line. Figure 3.28 confirmed that the AMAZON stock data failed to follow the normality. Figure 3.30 shows the empirical and theoritical Laplace transforms. Subordination with stable (SB) does not fit the parameter estimation using the LS method. The estimation relies on the minimization of the residual sum of square 0.73. The TSB and Normal approaches decreased until 4.248e−14 and 1.69996e−13 respectively. Therefore, it is obvious that the TSB and Normal models performed the best. The simulated trajectories of the TSB process for the log-returns of the Amazon stock are shown in Figure 3.31 - Figure 3.33.

108 Figure 3.25: The examined real data set of AMAZON stocks from 01.03.2000 - 02.12.2019.

Figure 3.26: The examined real data set of AMAZON stocks from 01.03.2000 - 02.12.2019.

109 Figure 3.27: The Q-Q plot for the AMAZON stocks from 01.03.2000 - 02.12.2019.

Figure 3.28: The normal fitting for the log-returns of the AMAZON stock from 01.03.2000-02.12.2019. The skewness = 0.453 and the kurtosis = 12.142.

110 Figure 3.29: The parameter estimation for the normal fitting using the least square approximation and the Method of likelihood approximation. The considered period is 01.03.2000 - 02.12.2019.

Figure 3.30: The empirical Laplace transform (black), theoretical Laplace trans- form for TSB (red), theoretical Laplace transform for SB (blue), and theoretical Laplace transform for Normal (green) for the log-returns of the AMAZON stock from 01.03.2000 - 02.12.2019.

111 Table 3.5: Parameter estimation for log-returns of the AMAZON stock from 01.03.2000 - 02.12.2019 βˆ γˆ δˆ µˆ σˆ RSS TSB 0.0400000 0.5552079 42.6264349 - - 4.4285e-14 SB 1.00000 0.500000 - - - 0.7398943 B - - - 0.0006044 0.0329927 1.69965e-13

Figure 3.31: The trajectories of the TSB process for log-returns of AMAZON stock data using LS method (first sample).

112 Figure 3.32: The trajectories of the TSB process for log-returns of AMAZON stock data using LS method (second sample).

Figure 3.33: The trajectories of the TSB process for log-returns of AMAZON stock data using LS method (third sample).

113 3.5 Statistical Analysis of Log-returns of the Stock

Data using the Method of Moments

All the necessary theories and proofs are explained in Section 2.7. We directly apply the TSB process to find the method of Moment estimators to log-returns of the stock data. The following calculations are based on the explanation in [145]. Let us again consider the characteristic function of the TSB process as follows:

1 Φ (s) = exp {δβ − (δ + γs − s2)β} X 2

where ΦX (s) is the characteristic function of the TSB distribution. We consider the ˆ log-returns as TSB behavior. In general, the moment conditions E(f(θ, xi)) = 0 is a q × 1 vector of nonlinear functions of θ,ˆ where θˆ ∈ R. Our goal is to estimate the parameter for the TSB when the density does not have closed-form expression. The direct inference is explained in [147]. According to

ˆ ixiτl [148], we can use f(θ, xi) = e − Φ(θ; τl), where {τ1, ..., τq} is a certain grid. Let assume,

¯ 1 Pi=1 f(θ) ≡ n n f(θ, xi) = 0 and Φ(θ, τl) is the charracteristic function of TSB. Then, 0 minimizing the quadratic function f(¯θ) W f¯(θ), we can find the parameters of the TSB model in the following way; W is a positive definite and symmetric q × q matrix of weights. The algorithm for the Generalized Method of Moment (GMM) can be described as follows [146]:

ˆ ¯ 0 ˆ ∗ −1 ˆ θ = arg minθf (θ) Ω(θ ) f(θ)

114 is the optimal matrix and also, W produces the efficient estimators derived as:

√ ∗ ˆ −1 W = {limn→∞ V ar( nf(θ0)) ≡ Ω(θ0)} .

The general form of the optimal matrix Ω which is the optimum result can be estimated by an heteroskedasticity and auto-correlation consistent matrix (HAC), using the formulas,

1 X 0 ωˆ (θ∗) = f(θ∗, x )f(θ∗, x ) s n i i+s i

n−1 ˆ X ∗ Ω = kh(s)ˆωs(θ ), s=−(n−1)

∗ ¯ 0 ˆ ∗ −1 ˆ (i) Compute θ = arg minθf (θ) Ω(θ ) f(θ) (ii) Compute HAC matrix Ω(ˆ θ∗)

ˆ ¯ 0 ˆ ∗ −1 ¯ (iii) Compute the 2SGMM θ = arg minθf(θ) [Ω(θ )] f(θ). We will find the parameters estimation using the GMM library in the R- package.

3.5.1 Analyzing Log-return of the WMK Stock using Method

of Moments

Weis Markets, Inc.(WMK) log-returns data from January 1993 to February 2009 were obtained from the GMM package . The results of GMM for the log-returns are presented below. The P-value of the out-put, 0.16353, is greater than 0.05 which confirmed that the null hypothesis of the moment condition is satisfied. Therefore, the null hypothesis is not rejected. For nonlinear models, the J-test may indicate whether either it has reached the global minimum or it did not. Clearly, we can justfy the fitting of the nonlinear model using the J-test. Also, since the convergence code is equal to zero, this confirms that it has reached the global minimization.

115 Call: gmm(g = g2, x = x3, t0 = t0, optfct = "nlminb", lower = c(0.1, 0.01, 0.1), upper = c(2, 1, 1))

Method: twoStep

Kernel: Quadratic Spectral(with bw = 0.80658 )

Coefficients: Estimate Std. Error t value Pr(>|t|) beta 6.9849e-01 3.2065e-02 2.1783e+01 3.3416e-105 gamma 4.9456e-02 2.7884e-02 1.7736e+00 7.6128e-02 delta 6.7746e-01 1.3853e-01 4.8904e+00 1.0066e-06

J-Test: degrees of freedom is 17 J-test P-value Test E(g)=0: 22.57705 0.16353

Initial values of the coefficients beta gamma delta 0.6873439 0.0100000 0.4511383

############# Information related to the numerical optimization

116 Convergence code = 0 Function eval. = 13 Gradian eval. = 51 Message: relative convergence (4)

Table 3.6: Parameter estimation for log-returns of the WMK data using GMM βˆ γˆ δˆ TSB 0.698 0.0494 0.677

Figure 3.34: The empirical Laplace transform compare with TSB distribution using the method of moment estimation.

117 Chapter 4

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) with CGMY Jumps

4.1 Introduction

Financial markets respond nervously to political disorders and economic crises. In such periods, prices of financial assets tend to fluctuate a lot. From a statistical point of view, it means that the conditional variance for the given past is not constant over time and the process Yt,

V ar(Yt|Yt1 ,Yt2 , ...) is conditionally Heteroskedastic. The GARCH volatility models are a major tool in the analysis of time series data. These models are used to analyze and forecast volatility. This chapter describes the foundation used to build the environment of GARCH model and illustrates its usefulness in examining portfolio risk and also discusses the extension of the model

118 briefly. Using the evidence of the empirical data, the volatility,

p σt = V ar(Yt|Yt1 ,Yt2 , ...) changes over time. It is very important to many macroeconomic and financial appli- cations to understand the volatility of the time series of the financial data, such as option pricing, asset pricing, etc. The financial Market attempts to make financial decisions on the basis of the observed price asset data St in discrete time. Prices St are considered to be non stationary. Usually, prices transformed into the log returns

Yt = log St − log St−1. are supposed to be stationary. The GARCH model incorporates a moving aver- age component together with the autoregressive component. The empirical analysis of daily stock returns shows that volatility behavior changes like a stochastic time series. According to the empirical evidence of time-varying volatility, we need to estimate the volatility of future returns. In this situation, The GARCH model plays an important role. Since Black and Scholes introduced the pricing and hedging the- ory for the option market, their model has been the most popular model for option pricing [100]. Although, the model which assumes heteroskedasticity and the log normality fails to explain the observed skewness, heavy tails and volatility cluster- ing of the stock returns. Mandelbrot is the pioneer who introduce the non-normal L´evyprocess [20, 98]. Hurst, Platen, and Rachev, used a model based on stable processes to price options [162]. Because the stable distributions have infinite moments of the second or higher order, the tempered stable (TS) process has been introduced by Koponen [72]. In 2002, the ”CGMY” process was introduced by Carr et. al. [139]. He used the

119 generalized Fourier transform for the distribution of the stock price under the as- sumption of the Markov property which helped to obtain a closed-form solution of the European option price. For example, the Markov property is often rejected by the empirical evidence as in the case of volatility clustering. The GARCH models have been developed to price options under the assumption of volatility clustering. GARCH models of Duan, Heston and Nandi [164] are remarkable works on the non- Markovian structure of asset returns but disregarded the conditional leptokurtosis and skewness of asset returns [67]. Duan and Ritchken [151] introduced the classical GARCH model by adding jumps to the innovation processes. Moreover, Menn and Rachev [80] introduced an enhanced GARCH model with innovations which follow the smoothly truncated distribution. Tempered stable distribution (TS) has a finite variance and at the same time allows the conditional leptokurtosis and skewness. L´evyprocesses are increasingly being used to model the asset returns. The conse- quence is that pricing distributions display high levels of excess kurtosis. Also, the asset returns are modeled using the representations of the L´evyprocesses as time changed Brownian motions. The gamma process is subordinated to the Brownian motion is called variance gamma process while the inverse Gaussian process sub- ordinated to the Brownian motion called normal inverse Gaussian process. Some other L´evy processes such as the CGMY process were introduced by Carr, Geman, Madan and Yor [79], and Boyarchenko and Levendorskii [163]. The CGMY process is defined directly by its L´evymeasure. We develop the CGMY process as a time changed Brownian motion with drift, where the law of the time change is absolutely continuous over finite time intervals with respect to the one-sided stable subordinator. Furthermore, in this chapter, we introduce a variant of the tempered stable distributions, called a tempered stable (CGMY) distribution and apply it to the GARCH option pricing model.

120 4.2 The Option Pricing Models

(a) GARCH (1,1) process

Definition 4.2.1 Let (Zn) be a sequence of i.i.d. random variables such that Zt ∼

N(0, 1). (Yt) is called the generalized autoregressive conditionally heteroskedastic or GARCH (q,p) process if

Yt = σtZt, t ∈ Z (4.2.1)

where (σt) is a nonnegative process such that

2 2 2 2 2 σt = α0 + α1 Yt−1 + ... + αqYt−q + βσt−1 + ... + βpσt−p, t ∈ Z (4.2.2) and

α0 > 0, αi ≥ 0 i = 1, ..., q βi ≥ 0 i = 1, ..., p. (4.2.3)

The condition 4.2.3 ensures the strong positivity of the conditional variance expres- sions of the equation 4.2.3 in terms of the lag-operator B since

2 2 2 σt = α0 + α(B)Yt + β(B)σt , (4.2.4) where

2 q α(B) = α1B + α2B + ... + αqB (4.2.5) and

p β(B) = β1B + β2B2 + ... + βpB . (4.2.6)

121 If the roots of the characteristic equation, i.e.

2 p 1 − β1Y − β2Y ... − βpY = 0 (4.2.7)

Since, (Yt) is a stationary process, then we can rewrite the equation 4.2.2 as follows;

α α(B) σ2 = 0 + Y 2 t 1 − β(1) 1 − β(B) t

∞ ∗ X 2 = α0 + δiYt−i (4.2.8) i=1 α where α∗ = 0 , and δ are coefficients of Bi in the expansion of α(B)[1 − 0 1 − β(1) i β(B)]−1.

2 The GARCH (1,1) process is stationary if the process (σt ) is stationary.

Theorem 4.2.2 Let (Yt) be the stochastic process satisfying the equation

Yt = Ct + DtYt−1, t ∈ N, (4.2.9) or explicitly t t t Y X Y Yt = Y0 Dj + Cn Dj, t ∈ N. (4.2.10) j=1 n=1 j=n+1

Suppose that Y0 is independent of the i.i.d. sequence (Ct,Dt)t≥1. Assume that

E ln+|C| < ∞ , and − ∞ ≤ E ln|D| < 0. (4.2.11)

Then

D (a) Yt → Y for some random variable Y such that it satisfies the identity in law Y = C + DY, where Y and (C,D) are independent.

122 (b) The solution for the above equation is given by

∞ n−1 D X Y Y = Cn Dj (4.2.12) n=1 j=1

The right hand side of 4.2.12 converges absolutely with probability 1.

D (c) If we choose Y0 = Y as in 4.2.12, then the process (Yt)t≥0 is strictly stationary. assuming the moment conditions E|A|p < ∞ and E|C|p < 1 for some p ∈ [1, ∞). (d) Then E|Y |p < ∞,and the series in 4.2.12 converges in p-th mean.

p (e) If E|Y0| < ∞, then

p p E|Yt| → E|Y | as t → ∞.

(f) The moments E[Y ]m are uniquely determined by the equations

m X m E[Y ]m = E(DkCm−k)EY k, m = 1, ...[p] k k=0 where [p] indicate the floor function.

2 Theorem 4.2.3 Let (σt ) be the conditional variance of GARCH (1,1) process. Also, assume that

2 E[ ln(α1Z0 + β1)] < 0 (4.2.13)

2 and that (σt ) is independent from (Zt).

2 (i) The process (σt ) is strictly stationary if

∞ 2 D Y 2 (σt ) = α0 (β1 + α1Zj−1) (4.2.14) m=1 and the series (5.2.14) converges absolutely with probability 1.

2 2 (ii) Assume that (σt ) is strictly stationary and let σ = σ0,Z = Z1. Let E(β1 +

123 2 p 2 m α1Z ) < 1 for some p ∈ [1, ∞). Then E(σ ) < ∞ for some 1 ≤ m ≤ [p]. For such integer m, it the folowing equation is satisfied:

m−1 X m E[σ2m] = [1 − E(β + α Z2)m]−1 E(α Z2 + β )kαm−k × E[σ2k] < ∞. 1 1 k 1 1 0 k=0

(b) Duan’s NGARCH This concept was introduced in [150].

Consider St to be the asset price at date t, and let ht be the conditional volatility of the logarithmic return over the period [t; t + 1], which is a day. We can express the logarithmic stock return as follows;

St+1 1 2 2 ln = rf + λht − ht + ht t+1, St 2

2 2 2 2 ht+1 = β0 + β1ht + β2ht (t+12 − γ) ,

where, rf and λ are the risk free rate and the unit risk premium respectively for the asset. t+1 ∼ Normal(0, 1) and γ is a nonnegative parameter that captures the negative correlation between return and volatility. Also, β0, β1 and β2 are positive which keep that the conditional volatility(ht) stays positive. The following formulas are expressed according to [67].

St+1 1 2 ln = rf − ht + htνt+1, St 2

2 2 2 2 ht + 1 = β0 + β1ht + β2ht [νt+1 − ω] .

These are the risk neutral probability measures under which discounted claims are

1 martingales, where ω = γ + λ + 2 , and νt+1 ∼ Normal (0, 1). The four parameters

β0, β1, β2 and ω need to be estimated, together with the initial volatility, h0. Also, the additional parameter λ can be identified, if time series information is to be

124 incorporated. (c) HNGARCH This concept was introduced in [68].

St+1 2 ln = rf + λht + htt+1 St

2 2 2 ht+1 = β0 + β1ht + β2(t+1 − γht)

2 When the β1 + β2γ < 1, then the above process is stationary with finite mean and variance. This dynamic is similar to the NGARCH model, but the β2 term is not multiplied by the local variance as NGARCH . That is, the last term is determined to a large degree by the normalized residual, rather than the residual.

St+1 1 2 ln = rf − ht + htνt+1, St 2

2 2 2 ht+1 = β0 + β1ht + β2(νt+1 − ωht) ,

1 where ω = γ + λ + 2 Also, the paper [68] shows that, for this dynamic, the moment generating function of the logarithmic price at date T takes on a log linear form. As a result risk neutral probabilities and European call option can be computed. (d) GARCH-JUMP This concept was introduced in [152]. Consider a discrete-time economy for a pe- riod of [0,T ] where uncertainty is defined on a complete filtered probability space

(Ω, F,P ) with filtration F = (Ft) ∈ {0, 1,,T } where F0 contains all P −null sets in F . Assume the pricing dymamics as follows;

P  mt  St−1 = E St |Ft−1 mt−1

125 where, St is the total payout, consisting of price and dividends and mt be the marginal utility of consumption at date t. The expectation is taken under the data generating measure, P, conditional on the information up to date t − 1. m The pricing kernel, t will defined as below: mt−1 Assume mt/mt−1, is given by:

m t = ea+bJt , mt−1

and

Nt (0) X (j) Jt = Xt + Xt , j=1 where

(0) Xt ∼ N(0, 1),

(j) 2 Xj ∼ N(µ, γ ),

for j = 1, 2, ..., and Nt is distributed as a Poisson random variable with parameter

(j) λ. The random variables Xt are independent for j = 0, 1, 2, ..., and t = 1, 2, ..., T. √ St ¯ αt+ htJt ¯ The asset price, St, is assumed to follow the process: = e where Jt is St−1 a standard normal random variable plus a Poisson random sum of normal random t variables. In particular:

Nt ¯ ¯ (0) X ¯ (j) Jt = Xt + Xt j=1 where ¯ (0) Xt ∼ N(0, 1)

126 ¯ (j) 2 Xt ∼ N(¯µ, γ¯ ) furthermore, t = 1, 2, ..., T : (i) ¯ (j) Corr(Xt , Xτ ) =

and Nt is the same Poisson random variable as in the pricing kernel. The local ¯ variance of the logarithmic returns for date t, viewed from date t−1 is ht V ar(Jt) = ht(1 + λγˆ2), where

γˆ2 =µ ¯2 +γ ¯2,

ht is the local scaling factor because it differs from the local variance by a constant. The dynamics of the asset price under the measure P can be derived as follows;

√ S ¯ t = eαt+ htJt St−1 where

ht p α = r − − h bρ + λκ(1 − K (1)) t 2 t t ¯ ht = F (ht−i, Jt−i; i = 1, 2, ...)

1 κ = exp(bµ + b2γ2) 2 p 1 K = exp(q h (¯µ + bργγ¯) + q2 h γ¯2). t t 2 t

127 4.3 GARCH-CGMY-jumps

4.3.1 CGMY Distributions we will derive basic tools and definitions for L´evyprocess, tempered stable and the CGMY process according to [154].

Definition 4.3.1 Let h : R → R be a measurable function such that for every s, R |eisx − 1 − ish(x)|ν(dx) < ∞. The L´evy-Khintchinerepresentation with the truncation function h takes the form:

i sX tψ(s) E[e t ] = e , x ∈ R (4.3.1)

Z 1 isx ψ(s) = − sAs + iγhs + (e − 1 − ish(x))ν(dx). (4.3.2) 2 Rd

A and ν do not depend on the choice of h but γh depends on this choice. If γ is the value of γh for the standard truncation function h(x) = x1|x|≤1, γh for arbitary h it can be computed with the formula

Z ∞ γh = γ + (h(x) − x1|x|≤1)ν(dx) −∞

The value of γh corresponding to the truncation function h ≡ 0 (drift) will be denoted by γ0 and the value corresponding to h ≡ x (center) will be denoted by γc.

Furthermore, we will use the following results on exponential moments of L´evy processes.

Proposition 4.3.2 Let X be a L´evyprocess having characteristic triplet (A, ν, γh)

λ with respect to a truncation function h and let λ ∈ R. Then E[e Xt] < ∞ for some

128 R λx t > 0 or equivalently for all t > 0 if and only if |x|≥1 e ν(dx) < ∞. In this case, E[eλXt ] = etψ(−iλ).

4.3.2 Equivalent Measure Changes for L´evyProcesses

Proposition 4.3.3 This is according to [156].

0 Let (Xt,P ) and (Xt,P ) be two L´evyprocesses on R with characteristics triplets

0 0 0 0 (A, ν, γ) and (A , ν , γ ). Then P |Ft and P |Ft are equivalent for all t (or equivalently for one t > 0) if and only if the three following conditions are satisfied: 1. A = A0 2. The L´evymeasures are equivalent with

Z ∞ (eΦ(x)/2 − 1)2ν(dx) < ∞ −∞

0 dν where Φ(x) = ln( dν ) 3. If A = 0 then we must in addition have

∞ 0 Z 0 γh = γh + h(x)(ν − ν)(dx) −∞

When P and P 0 are equivalent, the Randon-Nikodym derivative is

dP | = eUt dP Ft with

2 Z c η At X Φ(x) Ut = ηXt − − ηγht + lim→0( Φ(∆Xs) − t (e − 1)) 2 |x|> |∆Xs|>

c Here (Xt ) is the continuous part of (Xt), i.e. the L´evyprocess with generating triplet

129 (A, 0, γ) and η is such that

∞ 0 Z 0 γh − γ + (h(x)(ν − ν)(dx) if A > 0 −∞

and zero if A = 0.Ut is a L´evyprocess with characteristic triplet (AU , νU , γU ) given by:

2 2 AU = η σ

Z ∞ 1 2 y −1 γU = − Aη − (e − 1 − h(y))(νΦ )(dy) 2 −∞

−1 νU = ν ◦ Φ .

4.3.3 Stable Processes

The L´evymeasure for the α− stable L´evyprocess with 0 < α < 2 and no continuous martingale part, can be formed as follows;

A A ν(s) = + 1 + − 1 . (4.3.3) |x|α+1 x>0 |x|α+1 x<0

If 1 < α < 2, the process has finite mean and its characteristic function has the following form:

Z isx Φt(s) = expt{iµs + (e − 1 − isx)ν(dx)} (4.3.4) R for some center R. In the case 0 < α < 1, the process has finite variation and we can write Z isx Φt(s) = expt{iµs + (e − 1)ν(dx)} (4.3.5) R

130 i.e., µ is now the drift. The characteristic function of a stable process may also be expressed as [156]:

  α α πα exp{−σ |s| t(1 − iβsgn(s tan 2 ) + iµst}, if α 6= 1 ΦXt (s) = (4.3.6)  α α πα exp{−σ |s| t(1 − iβsgn(s tan 2 ) + iµst}, if α = 1, where α ∈ (0, 2], σ ≥ 0, β ∈ [−1, 1] and µ ∈ R. stable law with parameters α, σ, β, ν is denoted by Sα (σ, β, ν). In the case α < 2 the two parameterizations are linked by the following relations:

 πα 1  2 [−(A+ + A−)Γ(−α) cos( 2 )] when α 6= 1 σ = (4.3.7)  π  2 (A+ + A−) when α = 1,

A − A β = + − . A+ + A−

Furthermore, when α 6= 1 and γ is the third parameter of the characteristic triplet of X for the standard truncation function h(x) = x 1|x|≤1 then

A − A µ = γ + − + . 1 − α

4.3.4 CGMY Process as Time Changed Brownian Motion

Tempered stable distribution is obtained by taking a symmetric β−stable distribu- tion and multiplying the L´evymeasure with exponential functions on each half of the real axis. It is defined in [152].

Definition 4.3.4 An infinitely divisible distribution is called a tempered stable (TS) distribution with parameter (A+,A−, δ+, δ, β), or β-tempered stable (β − TS), if its

131 L´evytriplet (σ2, ν, γ) is given by σ = 0, ν ∈ R and

dx ν(dx) = (A e−δ+x1 + A e−δ−x1 ) , (4.3.8) + x>0 − x<0 |x|β+1

Z E[Xt] = γ + xν(dx),, (4.3.9) |x|>1 where,

A+,A−, δ+, δ− > 0, β ∈ (0, 2) and γ ∈ R and E[Xt] = γt which is the drift param- eter.

This process was first constructed by Koponen [72] under the name truncated L´evy

flights. In particular, if A+ = A− = C > 0, then this distribution is called the CGMY distribution which has been used in [153], for financial modeling. In the above definition, A+ and A− give the tail decay rates, β describes the jumps near zero, and A+ and A− determine the arrival rate of jumps for a given size. This process was studied in [157] under the name CGMY process with L´evymeasure

e−Mx e−G|x| ν (x) = C 1 + 1 . (4.3.10) CGMY x1+Y x>0 x1+Y x<0

Because of the exponential tempering, in the case of the tempered stable process, big jumps need not be truncated and one can use the truncation function h(x) = x.

In general, the characteristic function φCGMY (s) of a tempered stable distribution is given by,

iu isβ+ φ (s) = exp[isγ + Γ(−β )δβ+ A {(1 − )β+ − 1 + } CGMY A + + + δ δ + + (4.3.11) iu isβ α+ β+ + + Γ(−β+)δ+ A+{(1 − ) − 1 + }] δ+ δ+

132 Furthermore, the characteristic exponent of the tempered stable process for 0 < α < 1 with h ≡ 0 as truncation function:

β β β β φCGMY (s) = exp (iγs + A+Γ(−β)((δ+ − is) − δ+ ) + A−Γ(−β)((δ− + is) − δ−)), (4.3.12) for some γ ∈ R , φCGMY can be extended to the region {s ∈ C : |Im(s)| < δ+ ∧ δ}. Using the characteristic function, we obtain cumulants

dm C (X) = log φ (s)| = 0 m dsm CGMY s for all orders. The proof can be found in [?].

Proposition 4.3.5 Let X be distributed as CGMY random variable whose charac- teristic function is given by (4.3.11). The cumulant cn(X) of X is given by:

β−n n β−n cn(X) = Γ(n − β)Cδ+ + (−1) Γ(n − β)Cδ− , for n ∈ N, n ≥ 2, β−1 β−1 and c1(X) = γ + Γ(1 − β)Cδ+ − Γ(1 − β)Cδ− .

We can perform the CGMY process as the form of the Tempered stable subordinator as follows:

XCGMY (t) = ATS(t) + B(TS(t)) where, the (TS(t), t ≥ 0) is an increasing time changing process (the β−Tempered stable subordinator (TS)). TS process is independent of the Brownian motion (B(t), t ≥ 0). Let take the CGMY process as time changed Brownian motion, we will express the following proposition as in [154].

Proposition 4.3.6 Let C > 0, β ∈ (0, 2), δ+ > 0 and δ− > 0 and let Z(t) be a subordinator with zero drift and L´evydensity

√ t A2− t B2 ce 2 4 D (W t) ν (t) = −α , Z tα/2+1

133 where D is the parabolic cylinder function([158]), A = (δ− − δ+/)2 and W =

(δ+ + δ−)/2. Then the Brownian subordination, is defined as follows;

X (t) = AZ(t) + B(Z(t)), where B(t) is the standard Brownian motion with variance t.

Then the X (t) is a tempered stable process with the L´evydensity

dx ν(dx) = (Ce−δ+x1 + Ce−δ−x1 ) , x>0 x<0 |x|β+1 and the drift parameter;

Z −Ax γC = x(1 − e )ν(x)dx R the drift is zero when 0 < β < 1. According to [155], we can express the character- istic function of the X (t) as follows:

Z(t) s2 E[exp(iX (t))] = E[exp(i asZ(t) − s2)] = E[exp(−( − is a)Z(t)]. 2 2

Then the Laplace transform of the Z(t) is as follows:

E[exp(−λZ(t))] = exp(tCΓ(−Y )[(M − i s(λ))Y − M Y ) + (G + i s(λ))Y − GY ]) where, s(λ) is a solution of the equation

s2 λ = ( − isa). (4.3.13) 2

134 From the equation of 4.3.13 we get that

√ s = ia ± 2λ − a2.

We can see that a good choice for a, for sufficiently large λ where suppose that a2 < 2λ, is A = (G − M)/2 and in this case

s G + M G − M 2 M − is = + i 2λ − (4.3.14) 2 2 s G + M G − M 2 M + is = − i 2λ − . 2 2

Also, we can express the Laplace transform of the subordinator as

E[exp(−λZ(t))] = exp(tCΓ(−Y )[2rY cos(ηY ) − M Y − GY ]), where

√ r = 2λ + GM, (4.3.15) and q  (G−M) 2  2λ − ( 2 ) η = arctan G+M . (4.3.16) ( 2 ) When G = M, the Laplace transform of the subordinator is

√ 2λ E[exp(−λZ(t))] = exp(2tCΓ(−Y )[(2λ + M 2)Y/2cos(Y arctan( )) − M Y ]). M (4.3.17)

135 4.3.5 The CGMY-GARCH Option Pricing Model

This model is described accordingly to [159]. The CGMY-GARCH stock price model is defined over a filtered probability space (Ω, F, (Ft)t∈N , P) which is formed as follows: Let (t)t∈N of i.i.d. real random variables on a probability spaces (Ωt,Pt)t∈N, be such that t ∼ CGMY (α, λ+, λ−) which is infinitely divisible and distributed random variable with zero mean and unit variance on (Ωt,Pt), and assume that

E[ext ] < ∞ where x ∈ I for some real interval I containing zero. Next define

t Ω := Πt∈NΩt, Ft := ⊗k=1σ(k) ⊗ F0 ⊗ F0..., F := σ(∪t∈NFt), and P := ⊗t∈NPt where F0 = ∅, Ω and σ(k) means the σ− generated by k on Ωk. The CGMY-GARCH model for the stock price dynamics as the following

St log = rt − dt + δtσt − L(σt; α, δ+, δ−) + σtt, t ∈ N, (4.3.18) St−1

where St denote the stock price of the underlying asset at time t, rt, and dt de- note the risk free rate and dividend rate for the period [t − 1, t], and δt is a

Ft−1 measurable random variable. S0 is the present observed price. The function L(x; α, δ+, δ−) is the characteristic exponent of the Laplace transform for the dis-

xt tribution CGMY (α, δ+, δ−), i.e. L(x; α, δ+, δ−) = log(EPt [e ]), and defined on the interval (−a, b). The function g(x; α, δ+, δ−) is defined if x ∈ (δ, δ+) and its value can be obtained from |x| ≥ δ+ ∧λ−, and by numerical calculation if x ∈ (δ, δ+)|x|δ+ ∧λ−.

2 The one period ahead conditional variance σt at time t − 1 follows a GARCH(1,1) process with a restriction 0 < σt < b, i.e.

2 2 2 2 σt = (α0 + α1σt−1t−1 + β1σt−1) ∧ ρ, t ∈ N, 0 = 0

where the coefficients α0, α1 and β1 are non-negative, α1 + β1 < 1, α0 > 0 and 0 <

136 2 ρ < b . Clearly σt is Ft−1-measurable and hence the process (σt) ∈ N is predictable. ˆ ˆ Moreover, the conditional expectation E[St/St1|Ft−1] equals exp(rt + λtσt) where ˆ Pt St = Stexp( k=1 dk) is the stock price considering re-investment of the dividends, thus λt can be interpreted as the market price of risk.

Remarks: If εt equals the standard normal distributed random variable for all t ∈ N, g is to be the Laplace transform of εt and we ignore the restriction σt < δ+, then the model becomes the normal GARCH model introduced by Duan [67].

Definition 4.3.7 Let T ∈ N be the time horizon. Define a new measure Q on dQ FT quivalent to measure P, with a Radon-Nikodym derivative = ZT where the dP density process (Zt)0≤t≤T is defined according to

Z0 ≡ 1

d(P1 ⊗ ... ⊗ Pt−1 ⊗ Qt ⊗ Pt+1 ⊗ ... ⊗ PT ) Zt := Zt−1, t = 1, 2, ..., T. dP

Lemma 4.3.8 The measure Q satisfies the following requirements: rt ˆ (a) The discount asset price process (e St)1≤ttT is a Q−martingale w.r.t. the fil- tration (Ft)1≤t≤T . (b) We have,

St St V arQ(log |Ft−1) = V arP(log |Ft−1), 1 ≤ t ≤ T St−1 St−1

(c) The expression of the stock dynamic under Q as the following;

St log = rt − dt + δtσt − L(σt; α, δ+, δ−) + σtt, 1 ≤ t ≤ T St−1 ˆ ˆ where t1≥t≥T is a sequence of real random variable on Ωt. The t ∼ CGMY (α, δ−, δ+)

137 under Qt for 1 ≤ t ≤ T. The volatility process under Q has the following form;

2 2 2 2 2 σt = (α0 + α1σt−1(t−1 − k) + β1σt−1) ∧ (δ+(1 − )), t ∈ N, 0 = 0

The stock price dynamics under Q, is described according to the above Lemma .The

CGMY-GARCH is the risk neutral price process for the stock price under Q mea- sure. The arbitrage free price of a call option with strike price K and maturity T can be expressed as;

T X + Ct = exp(− rk)EQ[(ST − K) |Ft] (4.3.19) k=t+1 where the stock price t at time T is given by

T X ˆ ˆ ST = Stexp( ((rk − dk) − L(σt; α, δ+, δ−) + σkk)). (4.3.20) k=t+1

Here ST is the stock price at time T.

138 Conclusions

In chapter 1, we build the necessary background to model financial data using the L´evyprocesses such as α−stable and tempered stable processes. Also, it has pro- vided all the theoretical proofs and simulation methods to model the financial mar- ket data. In chapter 2, we have examined the TSB process and compared the main characteristic of the system, like Laplace transform, simulation methods as well as the ensemble averaged MSD can be used as a tool to compare the TSB with other L´evyprocesses. Also, we described the two-parameter estimation procedures such as LS and MME and validated them using the statistical simulation method. We can confirm that the characteristic function method provides the best precision esti- mating parameters for the subordinated processes. Also, in chapter 3, we provided the essential numerical results using applications to real stock data. That is given to illustrate the flexibility of TSB distribution for fitting a variety of skewed (left or right), and long-tailed as well as high peak (leptokurtic) data sets. Since the state variables are fully determined by the path of prices and once the parameters are estimated, the option prices can be computed by the inverse Fourier transformation available in R-packages. Finally, we can recommend the empirical characteristic function with the method of the least square as a suitable approach for modeling log-returns of financial market data using the TSB process. Furthermore, In Chapter 4, we see that the CGMY and TS distributions have sim- ilar properties. In future work, the parameter for the CGMY-GARCH model will be estimated from empirical stock data sets such as S&P (SPX) index, Amazon stock (AMZN), Apple stock( APPLE) and WMK stock data within the specific period. Moreover, we will investigate the CGMY-GARCH time series model for stock returns and explain the volatility clustering phenomenon, the leverage effect, conditional skewness, and conditional leptokurtosis.

139 Appendix A

L´evy Processes and Infinite Divisibility

L´evyProcess A process {Xtt≥0 } defined on a probability space (Ω, F, P), is said to be a L´evyprocess if it possesses the following properties:

(i) The paths of X are P almost surely right continuous with left limits.

(ii) P(X0 = 0) = 1.

(iii) For 0 ≤ s ≤ t, Xt − Xs is equal in distribution to Xt−s.

(iv) For 0 ≤ s ≤ t, Xt − Xs is independent of {Xuu≤s }.

A real-valued random variable X has an infinitely divisible distribution if for any integer n ≥ 2, there exist a sequence of i.i.d. random variables X1,X2,...,Xn, such d that X = X1 + X2 + ... + Xn. where =d is equality in distribution. Another way, we could have expressed this relation in terms of probability laws. The law µ of a real-valued random variable is infinitely divisible if for each n = 1, 2, ...

n there exists another law µ of a real valued random variable such that µ = µn. (Here

n µn denotes the n− fold convolution of µn). According to the definition of a L´evy process, for any t > 0,Xt is a random variable of the class of infinitely divisible

140 distributions. This conclude that for any n = 1, 2, ...,

Xt = Xt/n + (X2t/n − Xt/n) + ... + (Xt − X(n−1)t/n) that X has stationary independent increments. Suppose now that we define for all u ∈ R, t ≥ 0,

iuXt Ψt(u) = log E(e ) then,

mΨ1(u) = Ψm(u) = nΨm/n(u) where, m, n are positive constants. Hence, for any rational t > 0,

Ψt(u) = tΨ1(u).

In conclusion, any L´evyprocess has the property that for all t ≥ 0

E[eiuXt ] = e−tΨ(u),

where, Ψ(u) := Ψ1(u) is the characteristic exponent of X1, which has an infinitely divisible distribution.

141 Appendix B

Proof of Proposition 1.6.2. Proof . If 0 < α < 1, then

Z 0 −δx Z ∞ δx isx e isx e Ψ(s) = iγ s + a1A (1 − e + isxI|x|<1) α+1 dx + Aa2 (1 − e + isxI|x|<1) α+1 dx −∞ x 0 x Z 0 Z 1 −α−1 isx −δx −δx −α = iγ s + a1A( x (1 − e )e dx + ise x dx) −∞ 0 Z 0 Z 0 −α−1 isx δx δx −α + a2A( (−x) (1 − e )e dx − ise (−x) dx) −∞ −1 Z δ α α α −α −x = iγ s + Aa1(Γ(−α)(δ + (δ − is) ) + δ isx e dx) 0 Z δ α α α −α −x + Aa2(Γ(−α)(δ + (δ + is) ) − δ isx e dx) 0 α α α = isγ + Aδ Γ(−α)[a1((1 − s/δ) + 1) + a2((1 + s/δ) + 1)], (B.0.1) where

Z δ α −α −x γ = γ + δ A(a1 − a2) x e dx 0 α = γ + δ A(a1 − a2)(Γ(1 − α) − Γ(1 − α, δ))

142 R ∞ α−1 −y and Γ(α, x) = x y e dy is the ”upper” Gamma function. The integral of e−δxx−α diverges, when 1 < α < 2, then we can find the characteristic exponent (Ψ(s)) as the follows:

Z 0 −δx Z ∞ δx isx e isx e Ψ(s) = iγ s + a1A (1 − e + isxI|x|<1) α+1 dx + Aa2 (1 − e + isxI|x|<1) α+1 dx −∞ x 0 x Z 0 −δx Z ∞ δx isx e isx e = iγ s + a1A (1 − e + isx) α+1 dx + Aa2 (1 − e + isx) α+1 dx −∞ x 0 x

α α α−1 α α α−1 = iγ s + Aa1(Γ(−α)(δ + (δ − is) + δ αis) + Aa2(Γ(−α)(δ + (δ + is) − δ αis)

α α α = isγ + Aδ Γ(−α)[a1((1 − is/δ) + 1 + iαs/δ) + a2((1 + s/δ) + 1 − iαs/δ)], where Z ∞ −α−1 −δx −α−1 γ = m − A(a1 − a2) isx e x dx. 1

Let us consider n Ψ(s) = Ψα as the characteristis exponent of the tempered stable with α 6= 1, then, Ψα(s) → Ψ1 as α → 1 [96]. Therefore,

α α lim Ψ(s) = isγ + A(lim δ Γ(−α)[a1((1 − is/δ) (B.0.2) α→1 α→1

α 1 + iαs/δ) + a2((1 + s/δ) + 1 − iαs/δ)]), (B.0.3)

143 We can observe that the quantity of,

α α Γ(2 − α) α lim δ Γ(−α)(a1((1 − is/δ) + 1)) = lim( ((1 − is/δ) + 1)) α→1 α→1 α(α − 1)

α α Γ(2 − α) α lim δ Γ(−α)(a1((1 − is/δ) + 1)) = lim( ((1 − is/δ) + 1)) α→1 α→1 α(α − 1) Γ(2 − α) (1 − is/δ)α + 1 Γ(2 − α) (1 − is/δ)α + 1 = lim
= lim
α→1 α (α − 1) α→1 (α − 1) (α) Γ(2 − α) (1 − is/δ)αlog(1 − is/δ) = lim lim
α→1 (α − 1) α→1 (1) = (1 − is/δ)log(1 − is/δ).

Therefore,

α lim Ψα(s) = Ψ1(s) = isγ + A(δ[a1((δ − is/δ) log(1 − is/δ) + is/δ) α→1

+ a2((δ + is/δ) log(1 + is/δ) − is/δ)].

Proof of Proposition 1.6.4 Proof . We can derive density of the distribution from the inverse Laplace transfor- mation. Let us consider an strictly increasing α−stable distribution for characteristic exponent Ψ(s) = AΓ(−α)(is)α, Then the Laplace transformation of the density is Gˆ(s) = exp(xΓ(−α)(−s)α), finding the density using the inverse Laplace trasforma- tion,

144 content... (B.0.4)

1 Z γ+i∞ g(x) = esxGˆ(s)ds 2πi γ−i∞

1 Z γ+i∞ g(x) = esxexp(AΓ(−α(−s)α))ds. 2πi γ−i∞

Also, the density of TSα(A, 1, δ) can be formed as the following,

1 Z γ+i∞ f(x) = esxfˆ(s)ds 2πi γ−i∞ 1 Z γ+i∞ = esxexp(Γ(−α)A(δ + s)α − δα))ds 2πi γ−i∞ exp[−Γ(−α)Aδα] Z γ1+∞ (B.0.5) = e−(y+δ)xexp(AΓ(−α)(−y)αdy 2πi γ1−i∞ = exp[−Γ(−α)Aδα]e−δxg(x)

= Ce−δxg(x)

where, γ1 is the contour in the complex plane. We can obtain the γ using the transformation of the γ1 = γ + δ. Proof of Proposition 1.6.6 Proof . We show that the Φ(u) has an analytic extension to the lower half plane.

Z ∞ 2 2 −iux iux Φ(u) = −γiu − σ u + (e − 1 + 2 )ν(dx) (B.0.6) 0 1 + x

For (z) < 0, let

Z 2 2 1 + x −izx izx x Ψ(z) = 2 (e − 1 + 2 ) 2 ν(dx). (B.0.7) (0,∞) x 1 + x 1 + x

145 where x2 ν is a L´evymeasure, R ν(dx) < ∞. [0,∞] 1 + x2

1 + x2  izx  e−izx − 1 + +izx e−ikx − 1 + = + e−izx − 1 x2 1 + x2 x2   2 (B.0.8) |z |exp(|z|x) + 2 x < 1 <  2 (2 + |z|x)/x + 2 x ≥ 1.

Therefore, Ψ is well defined, and show for complex derivative,

Z 2 2 1 1 + x −i(z+w)x i(z + w)x −ikx izx  x lim 2 (e − 1 + 2 e − 1 + 2 ) 2 ν(dx) w→0 w [0,∞] x 1 + x 1 + x 1 + x

Z −izx −iwx 2 1 e (e − 1) + iwx −izx −iwx  x = lim ( 2 + e (e − 1) 2 ν(dx) w→0 w [0,∞] x 1 + x Z −izx−1 2 e −izx x = −i + xe 2 ν(dx) [0,∞] x 1 + x

exists by the Dominated Convergence Theorem as;

1 e−izx(e−iwx − 1) + iwx (e−iwx − 1) 1 e−izx(e−iwx − 1) + iwx e−iwx − 1 +e−izx | < | |+|e−izx w x2 w w x2 w (B.0.9) The second term is bounded by

|e−izx|xew|x| < xe(σ(z)+|w|)x < M

146 for σ(z)+|w| < 0. For x > 1, applying the same bounds to the first term, For x < 1, a Taylor expansion shows that

1 e−izx(e−iwx − 1) + iwx < M w x2 and

Z 2 2 −ikx−δx −ikx δx Ψ(k − iδ) − Φ(k) − γik − σ k = e − e − 2 ν(dx) [0,∞] 1 + x Z (B.0.10) −ikx −δx δx = e (e − 1) − 2 ν(dx) [0,∞] 1 + x tends to zero as δ → 0+ by another dominated convergence argument, similar to 1 (B.0.9) without the . Hence Ψ(z) − γiz − σ2z2 is the analytic extension of Ψ. w Particularly,

Z ∞ 2 2 −(ik+δ)x (ik + δ) Φ(k −iλ) = −γi(k −iδ)−σ (k −iδ) + (e −1+ 2 )ν(dx) (B.0.11) 0 1 + x is the unique extension of Φ.

R ∞  −ikx ikx  −δx 2 2 2 0 e − 1 − 1+x2 e ν(dx) = ν(k − iδ) − Φ(−iδ) + γik + σ k − 2ikδσ + bik

R ∞ −δx x for b = 0 (e − 1) 1+x2 ν(dx) ∈ R Assume, the H denotes the probability distribution of the infinitely divisible process with L´evyrepresentation [γ, σ, νδ], and then its Fourier transform, Hˆ(k, t) = R e−ikxH(dx, t) can express as the follows, Hˆ(k, t) = exp[t(Φ(k − iδ) − Φ(−iδ) − 2kiδσ2] + bik]. Hˆ is infinitely differentiable, and all moments are finite.

∂ ˆ 2 The mean, tµ = i ∂k H(0, t) = itΦ(−iδ) + 2tδσ − bt.

Setting d = a−µ yields an infinitely divisible process Sδ(t) with L´evyrepresentation

147 [d, σ, νδ] satisfying E[Sδ(t) = 0] for all t > 0.

Its probability distribution Pδ(fc, y) has Fourier transform

ˆ Pδ(k, t) = exp[t(Φ(k − iδ) − Φ(−iδ) − ikΦ(−iδ))]. (B.0.12)

∞ n ∂n ˆ The moments sequence {µn0 } with µn = i ∂kn Pδ(0, t) is positive (in the sense of [24]).

ˆ n dn Let q(s) = P(−is, t), then µn = (−1) dsn lq(0) and hence by[24], the integral,

ˆ R ∞ −sx q(s) = Pδ(−is, t) = −∞ e Pδ(dx, t) exists for some measure Pδ and all R(s) > −δ. Therefore;

ˆ R ∞ −ikx −vx Pδ(k − iv, t) = −∞ e e Pδ(dx, t) for all k ∈ R and v > −δ. Also, we can obtain the following expression by equation (1.53),

ˆ 0 lim Pδ(k − iv, t) = exp[t(Φ(k) − Φ(−iδ) + (ik − δ)iΦ (−iδ))] v→−δ+ and we can obtain the inverse Fourier transforms as the same as using the shift formula for the Fourier transform i.e;

0 δx −φ(−iδ)t−iδφ (−iδ)t e Pδ(dx, t) = e P(dx + iΦ(−δ)t, t). Proof of Theorem 1.6.6 Proof . This proof is according to [24].

−1 Assume, w = P (X ≤ Y − a) and compute Fa(x) = P(XNa ≤ x) = w P(X ≤ x, X ≤ Y − a). we consider, F a(x) = w−1P(X ≤ x) for x < −a, and , for x ≥ −a we have,

−1 R x+a −δy −δ(x+a) Fa = w ( 0 P(X ≤ y − a)δe dy + P(X ≤ x)e )

148 −1 Then the density of XNa is fa(x) = w f(x) for x < −a and, by the same argument

−1 −δ(x+a) as (1.49), fa(x) = w f(x)e for x ≥ −a. Also, we can compute w,

Z x+a Z ∞ −δy −δa −δy w = P(X ≤ y − a)δe dy = e P (X ≤ y)δe dy 0 −a (B.0.13) Z −a Z ∞ = f(y)dy + e−δa f(y)e−δydy. −∞ −a

δa R −a R −a −δy Let, ha = e −∞ f(y) dy and ga = −∞ e f(y)dy. −δa 1 Then, w = e (I − ga + ha), 0 ≤ ha ≤ ga → 0 and the L distance,

Z ∞ Z −a e−δxf(x) f(x) Z inf e−δxf(x) f(x)e−δ(x+a) |fδ(x) − fa(x)|dx = | − |dx + | − |dx −∞ −∞ I q −a I q Z −a e−δxf(x) eδaf(x) Z ∞ e−δxf(x) f(x)e−δ(x) = | − |dx + | − |dx −∞ I I − ga + ha −a I I − ga + ha 2ga I 1 3ga ≤ + (I − ga)( − ) ≤ → 0 I − ga + ha I − ga + ha I I − ga + ha (B.0.14) Equivalent changes of measure are important for option pricing in financial mathematics. we can describe the all locally equivalent measure changes under which Y remains tempered stable. Assume that Y = Y+ − Y can be decomposed as the difference of two independent subordinators. Also, we assume that ω = D(R+) is the space of cadlag functions equipped with the natural filtration

Ft = σ(Ys : s ∈ [0, t]) and the σ− algebra F = σ(Yt : t ≥ 0), where Y denotes the process Yt(ω) = ω(t). Consider P be a probability measure on(Ω, F) such that

+ + + − − − Y ∼ TS(C1 , β1 , δ1 ; C1 β1 , δ1 ) is a tempered stable process. Furthermore, let Q be another probability measure on (Ω, F) such that

+ + + − − − Y ∼ TS(C2 , β2 , δ2 ; C2 β2 , δ2 )

149 Proposition B.0.1 The following statement are equivalent:

(1) The measures N and Q are locally equivalent. + + − − + + − − (2) We have C1 = C2 ,C1 = C2 , β1 = β2 and β1 = β2 .

Proof . The Radon-Nikodym derivative Φ = dF1 of the L´evymeasures as follows; dF2

+ − C + − + + C + − − − 2 β1 −β1 −(δ2 −δ1 ) 2 β2 −β2 −(δ2 −δ1 ) Φ(x) = + x e x1(0,∞)(x) + − |x| e |x|1(0,∞)(x), x ∈ R C1 C1 (B.0.15) according to[ [156],Thm 33.1] the measures P and Q are locally equivalent if and only if Z p 2 (1 − Φ(x)) F1(dx) < ∞ (B.0.16) R using above expression,

s Z Z ∞ + + p 2 C2 β+−β−/2 −(δ+−δ+)x/2 2 C1 −δ+x (1 − Φ(x)) F1(dx) = (1 − x 1 1 e 2 1 ) e 1 dx + 1+β+ R 0 C1 x 1 s Z 0 − + C2 (β+−β−)/2 −(δ−−δ−)|x|/2 2 C2 −δ+|x| + (1 − |x| 2 2 e 2 2 ) e 2 dx − 1+β+ ∞ C1 x 2 ∞ Z q + + q + + + −(1+β1 )/2 −(δ1 /2)x + −(1+β2 )/2 −(δ2 /2)x 2 = ( C1 x e − C2 x e ) dx 0 ∞ Z q − − q − − − −(1+β1 )/2 −(δ1 /2)x − −(1+β2 )/2 −(δ2 /2)x 2 + ( C1 x e − C2 x e ) dx 0 (B.0.17) The above condition is satisfied if and only if

+ + − − + + − − C1 = C2 ,C1 = C2 , β1 = β2 and β1 = β2 .

+ + − − + + − − Suppose that , C1 = C2 ,C1 = C2 , β1 = β2 and β1 = β2 ,

dQ we can determine the Random-Nikodym derivatives |F for t ≥ 0, when decom- dP t pose Y = Y + − Y − as the difference of two independent one-sided tempered stable subordinator. Let denote those by Ψ+, Ψ− are cumulant generating functions re-

150 spectively.

Proposition B.0.2 The Radon-Nikodym derivatives of the measure transformation for Y can be derived as follows;

dQ + + + + + + − − − − − − |Ft = exp((δ1 −δ2 )Yt −Ψ (δ1 −δ2 )t)×exp((δ1 −δ2 )Yt −Ψ (δ1 −δ2 )t) , t ≥ 0. dP

Proof . The Radon-Nikodym derivative Φ = dF1 of the L evy measures as follows: dF2

dF1 −(δ+−δ+)x −(δ−−δ−)x Φ(x) = = e 2 1 1(0,∞)(x) + e 2 1 1(−∞,0)(x) dF2 also, the we can show that

+ + + − − − X X −(δ −δ )∆Xs X −(δ −δ )∆Xs ln Φ(∆Xs) = ln e 2 1 + ln e 2 1 s≤t s≤t s≤t

+ + X + − − X − (δ1 − δ2 ) ∆Xs + (δ1 − δ2 ) ∆Xs , t ≥ 0 s≤t d≤t Furthermore,

Z Z −(δ+−δ+)x −(δ−−δ−)x (Φ(x) − 1)F1(dx) = (e 2 1 1(0,∞)(x) + e 2 1 1(−∞,0)(x) − 1)F1(dx) R R + + − − Z ∞ −δ2 x −δ1 x Z ∞ −δ2 x −δ1 x + e − e − e − e = C 1+β dx + C 1+β dx 0 x 0 x Z + + Z − − (δ2 −δ1 )x + (δ2 −δ1 )x − = (e − 1)F1 (dx) + (e − 1)F1 (dx) R R + + + − − − = Ψ (δ1 − δ2 ) + Ψ (δ1 − δ2 ). (B.0.18)

151 According to [156], the Radon-Nikodym derivatives can be expressed as follows: d X Z Q| = exp( ln Φ(∆X ) − t (Φ(x) − 1)F (dx)) d Ft s 1 P s≤t R

+ + + + + + − − − − − − = exp((δ1 − δ2 )Yt − Ψ (δ1 − δ2 )t) × exp((δ1 − δ2 )Yt − Ψ (δ1 − δ2 )t) , t ≥ 0. (B.0.19)

152 BIBLIOGRAPHY

[1] R. Weron, On the Chambers-Mallows-Stuck method for simulating skewed stable random variables, Statistics & probability letters, vol. 28, no. 2, pp. 165 171, 1996.

[2] R. Cont, Financial modelling with jump processes, Chapman and Hall, 2004.

[3] B. Baeumer and M. M. Meerschaert, Tempered stable L´evymotion and tran- sient super-diffusion, Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 24382448, 2010.

[4] C. L. Dym, A. M. Agogino, O. Eris, D. D. Frey, and L. J. Leifer, Engineering design thinking, teaching, and learning, Journal of Engineering Education, vol. 94, no. 1, pp. 103120, 2005.

[5] J. Rosinski, Tempering stable Processes, Stochastic Processes and Their Applications, vol. 117, no. 6, pp. 677707, 2007.

[6] S. T. Rachev, Y. S. Kim, M. L. Bianchi, and F. J. Fabozzi, Financial models with L´evyprocesses and volatility clustering, vol. 187. John Wiley & Sons, 2011.

153 [7] I. Koponen, Analytic approach to the problem of convergence of truncated L´evyflights towards the Gaussian stochastic process, Physical Review E, vol. 52, no. 1, pp. 1197, 1995.

[8] Y. S. Kim, S. T. Rachev, M. L. Bianchi, and F. J. Fabozzi, Tempered sta- ble and tempered infinitely divisible garch models, Journal of Banking & Finance, vol. 34, no. 9, pp. 20962109, 2010.

[9] A. W. Janicki, I. Popova, P. Ritchken, and W. Woyczynski, Option pric- ing bounds in an α−stable security market, Stochastic Models, vol. 13, pp. 817839, 1997.

[10] A. Wylomanska, The tempered stable process with infinitely divisible inverse subordinators, Journal of Statistical Mechanics: Theory and Experiment, vol. 2013, no. 10, pp. P10011, 2013.

[11] M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, L´evyflights and related topics in physics, Lecture notes in physics, vol. 450, pp. 52, 1995.

[12] J.H.McCulloch, Continuous Time Processes With Stable Increments, J. Business, vol. 51 no 4, 601619, 1978(a).

[13] J.H.McCulloch, The Pricing of ShortLived Options When Price Uncertainty is LogSymmetric Stable, Working Paper 89, Boston College, Department of Economics, Boston, MA,1978(b).

[14] S. Mittnik, and S.T. Rachev, Stable distributions for asset returns, Appl. Math. Lett. vol. 213, pp. 301-304, 1989.

[15] E.E. Peters, Fractal Market Analysis, J. Wiley, New York, 1994.

154 [16] J.H. McCulloch, Toward Numerical Approximation of the Skew-Stable Dis- tributions and Densities, preprint, 1996.

[17] S.T. Rachev, G. Samorodnitsky, Option pricing formulae for speculative prices modelled by subordinated stochastic processes, pre print, 1995.

[18] W.T. Ziemba, Choosing investment portfolios when the returns have sta- ble distributions, Mathematical Programming in Theory and Practice, P.L. Hammer and G. Zoutendijk, Eds., North Holand, pp. 443-482, 1974.

[19] J.H McCulloch, Foreign exchange option pricing with log-stable uncertainty, in Recent Developments in International Banking and Finance, Vol. I, S.J. Khoury and A. Ghosh, Eds., Lexington Books, pp. 231-245, 1987.

[20] R. N. Mantegna and H. E. Stanley, Stochastic process with ultra slow con- vergence to a gaussian: the truncated l´evyflight, Physical Review Letters, vol. 73, no. 22, pp. 2946, 1994.

[21] R. Metzler and J. Klafter, The random walks guide to anomalous diffusion: a fractional dynamics approach, Physics reports, vol. 339, no. 1, pp. 177, 2000.

[22] V. M. Zolotarev, One-dimensional stable distributions, American Mathe- matical Soc., vol. 65, 1986.

[23] M. Kanter, Stable densities under change of scale and total variation in- equalties, The Annals of Probability, pp. 697707, 1975.

[24] B. Baeumer and M. M. Meerschaert, Tempered stable l´evymotion and tran- sient super-diffusion, Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 24382448, 2010.

155 [25] J. M. Chambers, C. L. Mallows, and B. Stuck, A method for simulating stable random variables, Journal of the American statistical association, vol. 71, no. 354, pp. 340344, 1976.

[26] M. Magdziarz and J. Gajda, Anomalous dynamics of black-scholes model time-changed by inverse subordinators, Acta Physica Polonica B, vol. 43, no. 5, 2012.

[27] R. Weron, Levy-stable distributions revisited: tail index¿2 does not exclude the levy-stable regime, International Journal of Modern Physics C, vol. 12, no. 02, pp. 209223, 2001.

[28] A. Janicki and A. Weron, Simulation and chaotic behavior of -stable stochas- tic processes. New York: Marcel Dekker Inc, 1995. 41

[29] E. Abdel-Rehim and R. Gorenflo, Simulation of the continuous time random walk of the space-fractional diffusion equations, Journal of Computational and Applied Mathematics, vol. 222, no. 2, pp. 274283, 2008.

[30] P. Chauss e et. al., Computing generalized method of moments and general- ized empirical likelihood with r, Journal of Statistical Software, vol. 34, no. 11, pp. 135, 2010.

[31] L. Devroye, Nonuniform random variate generation, Handbooks in opera- tions research and management science, vol. 13, pp. 83121, 2006.

[32] J. Fox, Nonlinear regression and nonlinear least squares, 2002.

[33] X. Gong and X. Zhuang, Pricing foreign equity option under stochastic volatility tempered stable L´evyprocesses,Physica A: Statistical Mechanics and its Applications, vol. 483, pp. 8393, 2017.

156 [34] A. Janicki and A. Weron, Simulation and chaotic behavior of alpha-stable stochastic processes, CRC Press, vol. 178, 1993.

[35] P. Jelonek et. al., Generating tempered stable random variates from mixture representation, tech. rep., 2012.

[36] R. Kawai and H. Masuda, On simulation of tempered stable random variates, Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 28732887, 2011.

[37] M. Magdziarz, A. Weron, and K. Weron, Fractional fokker-planck dynamics: Stochastic representation and computer simulation, Physical Review E, vol. 75, no. 1, pp. 016708, 2007. 42

[38] M. Magdziarz, A. Weron, and J. Klafter, Equivalence of the fractional fokker- planck and subordinated langevin equations: the case of a time-dependent force, Physical review letters, vol. 101, no. 21, pp. 210601, 2008.

[39] A. Piryatinska, A. Saichev, and W. Woyczynski, Models of anomalous diffu- sion: the subdiffusive case, Physica A: Statistical Mechanics and its Appli- cations, vol. 349, no. 3-4, pp. 375420, 2005.

[40] T. H. Rydberg and N. Shephard, Dynamics of trade-by-trade price move- ments: decomposition and models, Journal of Financial Econometrics, vol. 1, no. 1, pp. 225, 2003.

[41] A. Saichev and D. Sornette, Generation-by-generation dissection of the re- sponse function in long memory epidemic processes, The European Physical Journal B, vol. 75, no. 3, pp. 343355, 2010.

157 [42] A. Stanislavsky, Long-term memory contribution as applied to the motion of discrete dynamical systems, Chaos: An Interdisciplinary Journal of Non- linear Science, vol. 16, no. 4, pp. 043105, 2006.

[43] A. Stanislavsky, K. Weron, and A. Weron, Diffusion and relaxation con- trolled by tempered -stable processes, Physical Review E, vol. 78, no. 5, pp. 051106, 2008.

[44] R. Weron et al., Correction to: on the chambersmallowsstuck method for simulating skewed stable random variables, tech. rep., University Library of Munich, Germany, 2010.

[45] A. Wylomanska, Arithmetic brownian motion subordinated by tempered sta- ble and inverse tempered stable processes, Physica A: Statistical Mechanics and its Applications, vol. 391, no. 22, pp. 56855696, 2012. 43

[46] E.E. Peters, Fractal Market Analysis, J. Wiley, New York, 1994.

[47] D. Applebaum, L´evyprocesses and stochastic calculus. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2004.

[48] E.F. Fama, Mandelbrot and the stable paretian hypothesis, The Journal of Business, vol. 36, pp. 420429, 1963.

[49] E.F. Fama and R. Roll, Parameter estimates for symmetric stable distribu- tions, Journal of the American Statistical Association, vol. 66, pp. 331338, 1971.

[50] J.H. McCulloch, 13 financial applications of stable distributions, Statistical Methods in Finance, vol. 14, pp. 393425, 1996.

158 [51] S. Mittnik, et.al., Maximum likelihood estimation of stable paretian models, Mathematical and Computer Modeling, vol. 9, pp. 275293, 2001.

[52] J.P. Nolan, Maximum likelihood estimation and diagnostics for stable dis- tributions, In: Barndorff-Nielsen O.E., Resnick S.I., Mikosch T. (eds) Lvy Processes. Birkhuser, Boston, MA, pp. 379400, 2001.

[53] Y. Yang, Option pricing with non-gaussian distribution-numerical approach (Technical Report), Stony Brook University, Department of Applied Math- ematics and Statistics, New York, NY, 1986.

[54] V. M. Zolotarev, Statistical estimates of the parameters of stable laws, Ba- nach Center Publications, vol. 6, pp. 359376, 1986.

[55] V. Zolotarev, One-dimensional Stable Distributions: Translations of mathe- matical monographs, Providence, RI: American Mathematical Society, 1986.

[56] O.E. Barndorff-Nielsen, T. Mikosch, and S.I.Resnick, L´evyprocesses Theory and Applications, Springer Science New York, 2001.

[57] M.F. Shlesinger, Fractal time in condence matter, Annu. Rev. Phys. Chem. vol. 39, pp. 269, 1988.

[58] D. Applebaum, L´evyprocesses from probability to finance and quantum groups, Notices of the American Mathematical Society, vol. 51, pp.12, 2004.

[59] L.Bachelier, Theory of speculation, Ann. c. Norm. Super, vol.17, pp 21, 1900.

[60] W.Ebeling, Active brownian motion of pairs and swarms of particles, Acta- Phys.Polon.B38(5),pp. 1657, 2007.

[61] J.LuczkaJ, B.Zaborek, ActaPhys.Polon.B35(9), pp. 2151, 2004.

159 [62] C. Vignat, P.W. Lamberti, Physica A 391(3), pp. 544, 2012.

[63] A. Janicki, and A. Weron, Simulation and Chaotic Behavior of Stable Stochastic Processes, Marcel Dekker, New York, 1994b.

[64] E.F. Fama, The Behavior of Stock Market Prices, J. Business, 38, pp. 34105, 1965.

[65] W. Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, 2000.

[66] J. Pitman and M. Yor, Infinitely Divisible Laws associated with Hyperbolic Functions, Canadian Journal of Mathematics, vol. 55, no. 2, pp. 292-330, 2003.

[67] J.C. Duan, The Garch option pricing model, Mathematical Finance, vol. 5, pp. 1332, 1995.

[68] S. L. Heston and S. Nandi, A closed-form Garch option valuation model, Review of Financial Studies, vol. 13, pp. 585625, 2000.

[69] C. Menn and S. Rachev, A Garch option pricing model with α?-stable in- novations, European Journal of Operational Research, vol. 163,no 02, pp. 201209, 2005.

[70] B. Bottcher, Feller processes: The next generation in modeling, brownian motion, L´evyprocesses and beyond, PLoS ONE, vol. 5, pp. e15102, 2010.

[71] B. Grigelionis, Processes of meixner type, Lithuanian Mathematical Journal, vol. 39, pp. 3341, 1999.

160 [72] I. Koponen, Analytic approach to the problem of convergence of truncated L´evyflights towards the gaussian stochastic process, Physical Review E, vol. 52, no. 1, pp. 11971199, 1995.

[73] B.B. Mandelbrot, New methods in statistical economics, Journal of Political Economy,vol. 71, pp. 421-440, 1963a.

[74] B.B. Mandelbrot, The Variation of Certain Speculative Prices, Journal of Business, vol. 36, pp. 394-419, 1963b.

[75] U. Kuchler, and S. Tappe, Bilateral Gamma distributions and processes in financial mathematics, Stochastic Processes and their Applications vol. 118 no 2, pp. 261283, 2008.

[76] U. Kuchler, and S. Tappe, Option pricing in bilateral Gamma stock models, Statistics and Decisions vol. 27, pp. 281307, 2009.

[77] I. Berkes, L. Horvath, and P. Kokoszka, GARCH process: structure and estimation. Bernoulli, vol. 9, pp. 201-227, 2003.

[78] S. Asmussen and J. Rosinski, Approximations of small jumps of L´evypro- cesses with a view towards simulation, J. Appl. Probab., vol. 38, pp. 482493, 2001.

[79] P. Carr and D. Madan, Option valuation using the fast Fourier transform, vol. 2, pp. 6173, 1998.

[80] C. Menn, and S.T.Rachev, Smoothly Truncated Stable Distri- butions, GARCH-Models,and Option Pricing, Technical Report (http://www.statistik.uni-karlsruhe.de/technical reports/sts-option.pdf), 2005b.

161 [81] D. Applebaum, L´evyprocesses and stochastic calculus, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2004.

[82] U. Kuchler, and S. Tappe, Tempered stable distribution and applications to financial mathematics, Stochastic Processes and their Applications, 2012.

[83] D. Hsu, R. Miller and D. Wichern, On the Stable Paretian Behavior of Stock Market Prices, J. American Statistical Assn., vol. 69, pp. 108113, 1974.

[84] R. Haggeman, More Evidence on the Distribution of Security Returns, J. Finance, vol. 33, pp. 12131221, 1978.

[85] R. Blattberg and N. Gonedes, A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices, J. Business, vol. 47, pp. 244280, 1974.

[86] B.B. Mandelbrot, The Variation of Certain Speculative Prices, Journal of Business, vol. 36, pp. 394-419, 1963b.

[87] S. Kwapien and W. Woyczynski, Random Series and Stochastic Integrals Single and Multiple, Birkha user, Boston, 1992.

[88] O. Barndorff-Nielsen, N. Shepardt, J.Roy, Statist.Soc.Ser., B63, 1, 2001.

[89] Y.S.Kim, et.al., Probab.Math.Statist. vol. 30 no 2, pp. 223, 2010.

[90] Janczura, S.Orze, A.Wyoman, ska, Physica, A390, pp. 4379, 2011.

[91] S.Bochner, Proc.Nat.Acad.Sci USA, vol. 35,pp. 368, 1949.

[92] S.Bochner,Harmonic Analysis and the Theory of Probability, Uni. California Press, 1955.

162 [93] P. L. Butzer and R. J. Nessel, Fourier analysis and approximation, vol. 1, Reviews in Group Representation Theory, Part A (Pure and Applied Mathematics Series, Vol. 7, 1971.

[94] G. Samoradnitsky, Stable non-Gaussian random processes: stochastic mod- els with infinite variance, Routledge, 2017.

[95] D.B. Widder, The Laplace Transformation, 2nd ed., in: Princeton Mathe- matical Series, vol.6, Princeton Universituy Press, Princeton, NJ, 1946.

[96] V. M. Zolotarev, One-dimensional stable distributions, vol. 65. American Mathematical Soc., 1986.

[97] M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, L´evyflights and related topics in physics, Lecture notes in physics, vol. 450, pp. 52, 1995.

[98] R. N. Mantegna and H. E. Stanley, Scaling behavior in the dynamics of an economic index, Nature, vol. 376, no. 6535, pp. 46, 1995.

[99] A. Cartea and D. del Castillo-Negrete, Fluid limit of the continuous-time random walk with general L´evyjump distribution functions, Physical Review E, vol. 76, no. 4, pp. 041105, 2007.

[100] F. Black, and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, vol. 81, no. 3, pp. 637-654, 1973.

[101] K.I. Sato, L´evyProcesses and Infinitely Divisible Distributions, Cambridge University Press, Phi. Mag. vol 35(1868) pp. 129,185, 1999.

[102] A. I. Saichev and W. A. Woyczynski, Distributions in the Physical and Engineering Sciences. Vol. I. Springer, 1997.

163 [103] M. Magdziarz and J. Gajda, Anomalous dynamics of black-scholes model time-changed by inverse subordinators., Acta Physica Polonica B, vol. 43, no. 5, 2012.

[104] A. Wylomanska, The tempered stable process with infinitely divisible in- verse subordinators, Journal of Statistical Mechanics: Theory and Experi- ment, vol. 2013, no. 10, pp. P10011, 2013.

[105] J. Nolan, Stable distributions: models for heavy-tailed data, Birkhauser New York, 2003.

[106] B. Mandelbrot, Variables et processus stochastiques de pareto-Levy et la repartition des revenus, Comptes Rendus de lAcademie des Sciences, vol. 249, pp. 21532155, 1959.

[107] B. Mandelbrot, Paretian distributions and income maximization, The Quar- terly Journal of Economics, vol. 76, pp. 5785, 1962.

[108] B. Mandelbrot, The variation of certain speculative prices. The Journal of Business, vol. 36, pp. 394419, 1963.

[109] L. Bachelier, Theorie de la speculation. Annales scienti ques de lE.N.S. 3e serie, vol. 17, pp. 2186, 1900.

[110] V.M. Zolotarev, On the representation of stable laws by integrals. Trudy Matematicheskogo Instituta imeni VA Steklova, vol. 71, pp. 4650, 1964.

[111] A. Weron, R. Weron, Computer simulation of Levy alpha-stable variables and processes. Lecture Notes in Physics, vol. 457, pp. 379392, 1995.

[112] V. Zolotarev, One-dimensional Stable Distributions. Translations of mathe- matical monographs. Providence, RI: American Mathematical Society, 1986.

164 [113] J. Janczura and A. Wylomanska, Anomalous diffusion models: different types of subordinator distribution, arXiv preprint arXiv: 1110.2868, 2011.

[114] L. P. Hansen, Large sample properties of generalized method of moments estimators, Econometrica: Journal of the Econometric Society, pp. 10291054, 1982.

[115] J. Yu, Empirical characteristic function estimation and its applications, Econometric reviews, vol. 23, no. 2, pp. 93123, 2004.

[116] J. P. Nolan, Numerical calculation of stable densities and distribution func- tions, Communications in statistics. Stochastic models, vol. 13, no. 4, pp. 759774, 1997.

[117] A. Wylomanska, Arithmetic brownian motion subordinated by tempered sta- ble and inverse tempered stable processes, Physica A: Statistical Mechanics and its Applications, vol. 391, no. 22, pp. 56855696, 2012.

[118] G. Terdik and W. A. Woyczynski, Rosinski measures for tempered stable and related Ornstein-Uhlenbeck processes, Probability and Mathematical Statistics, vol. 26, no. 2, pp. 213, 2006.

[119] R. Garcia, E. Renault, D. Veredas, Estimation of Stable Distribution by Indirect Inference, Working Paper: UCL and CORE., 2006.

[120] M. Carrasco, J.P. Florens, Efficient GMM Estimation Using the Empirical Characteristic Function, Working Paper, Institut dE conomie Industrielle, Toulouse, 2002.

165 [121] P. Chauss e et. al., Computing generalized method of moments and general- ized empirical likelihood with r, Journal of Statistical Software, vol. 34, no. 11, pp. 135, 2010.

[122] Computing Generalized Method of Moments and Generalized Empirical Likelihood with R, 2010.

[123] A. I. Saichev and W. A. Woyczynski, Distributions in the Physical and Engineering Sciences. Volume III. Springer, 2018.

[124] Financial Models with Levy Processes and Volatility Clustering. John Wiley and Sons Ltd, 2011.

[125] Ait-Sahalia, Y. & Lo, A., Nonparametric estimation of state-price densities implicit in financial asset prices. Journal of Finance vol. 52, pp. 49, 2000.

[126] F. Black and My. Scholes, The Journal of Political Economy, Vol. 81, No. 3, pp. 637-654, 1973

[127] Corcuera, J., Guillaume, F., Leoni, P., Schoutens, W.,Implied L´evyvolatil- ity. Quantitative Finance, vol. 9 no. 4, pp. 383393, 2009.

[128] P. Ritchken, Derivative Markets: Theory, Strategy, and Applications, 1996

[129] Estimation of affine asset pricing models using the empirical characteristic function, Journal of Econometrics, Elsevier, vol. 102 no. 1, pp.111-141, 2001.

[130] The Pricing of Options and Corporate Liabilities, The Journal of Political Economy, Vol. 81, No. 3, pp. 637-654, 1973

[131] K.R. Jackson, S. Jaimungal, V. Surkov, Fourier space time-stepping for op- tion pricing with Levy models. Journal of Computational Finance, vol. 12 no. 2, pp. 129, 2008.

166 [132] E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatil- ity: An analytic approach, The Review of Financial Studies, vol. 4, no. 4, pp. 727752, 1991.

[133] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, vol. 6, pp. 327343, 1993.

[134] D. Duffie, J. Pan, and K. Singleton, Transform analysis and asset pricing for affine jump diffusion. Econometrica, vol. 68 no. 6, pp. 13431376, 2000.

[135] K. S. Leung, H. Y. Wong, and Y. K. Kwok, Efficient options pricing using the fast Fourier transform, SSRN Electronic Journal, 2010.

[136] A.L. Lewis, A simple option formula for general jump-diffusion and other exponential Levy processes. Working paper of Envision Financial Systems and OptionsCity.net, Newport

[137] P. Carr and D. Madan, Option valuation using the fast Fourier transform, The Journal of Computational Finance, vol. 2, no. 4, pp. 6173, 1999.

[138] E. Ziegel, Kendalls advanced theory of statistics, vol. 1: Distribution theory, Technometrics, vol. 31, pp. 128128, 1989.

[139] P. Carra, L. Wu,Time-changed L´evyprocesses and option pricing, JFE, vol 17 no 1, pp 113141,2002, 2004.

[140] P. Chauss et. al., Computing generalized method of moments and generalized empirical likelihood with r, Journal of Statistical Software, vol. 34, no. 11, pp. 135, 2010.

167 [141] L.P. Hansen, Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, vol. 50, pp. 10291054, 1982.

[142] D.W.K. Andrews, Heteroskedasticity and Autocorrelation Consistent Co- variance Matrix Estimation, Econometrica, vol. 59, pp. 817858, 1991.

[143] L.P. Hansen, Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, vol. 50, pp. 10291054, 1982.

[144] L.P. Hansen, J. Heato, A. Yaron, Finit-Sample Properties of Some Alterna- tive GMM Estimators, Journal of Business and Economic Statistics, vol. 14, pp. 262280, 1996.

[145] P. Chauss, Computing Generalized Method of Moments and Generalized Empirical Likelihood with R, Journal of Statistical Software, vol. 34 no. 11, pp. 135, URL http://www. jstatsoft.org/v34/i11/, 2010

[146] D.W.K. Andrews, Heteroskedasticity and Autocorrelation Consistent Co- variance Matrix Estimation, Econometrica, vol. 59, pp. 817858, 1991.

[147] R. Garcia, E. Renault and D. Veredas, Estimation of Stable Distribution by Indirect Inference, Working Paper: UCL and CORE, 2006.

[148] M. Carrasco, J.P. Florens, Efficient GMM Estimation Using the Empirical Characteristic Function, Working Paper, Institut dE conomie Industrielle, Toulouse, 2002

[149] P. Posedel, Properties and estimation of garch(1,1) model, Metodoloski zvezki, Vol. 2, No. 2, pp. 243-257, 2005.

[150] K. C. Hsieh and P. Ritchken, An empirical comparison of Garch option pricing models, Review of Derivatives Research, vol. 8, pp. 129150, 2006.

168 [151] J.C. Duan, P. H. Ritchken, and Z. Sun, Jump starting Garch: Pricing and hedging options with jumps in returns and volatilities, 2006.

[152] Y. S. Kim, S.T. Rachev, Bianchi, D. M. Chung, The Modified Tempered Stable Distribution, GARCH Models and Option Pricing, Probability and Mathematical Statistics, 2007

[153] P. Carr and L. Wu, Time-changed L´evyprocesses and option pricing, Journal of Financial Economics, vol. 71, no. 1, pp. 113141, 2004.

[154] J. Poirot and P. Tankov, Monte carlo option pricing for tempered stable (cgmy) processes, Asia-Pacific Financial Markets, vol. 13, pp. 327344, 2007.

[155] M. Yor and D. B. Madan, CGMY and Meixner subordinators are absolutely continuous with respect to one sided stable subordinators, Prepublication du Laboratoire de Probabilites et Mod‘eles Aleatoires, 2005.

[156] K. Sato, L´evyProcesses and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, UK, 1999.

[157] P. Carr, H. Geman, D. Madan, and M. Vor, The fine structure of asset returns: An empirical investigation, Journal of Business, vol. 75, no 4, pp. 305332, 2002.

[158] I. Gradshetyn and I. Ryzhik, Table of Integrals, Series and Products, Aca- demic Press, 1995.

[159] Y. S. Kim, S. T. Rachev, M. L. Bianchi, and F. J. Fabozzi, Tempered stable and tempered infinitely divisible garch models, Journal of Banking Finance, vol. 34, no. 9, pp. 20962109, 2010.

169 [160] S.I. Boyarchenko and Levendorskii, Generalizations of the Black-Scholes equation for Truncated L´evy processes, Working paper, 1999.

[161] S.I. Boyarchenko and Levendorskii, Option pricing for Truncated L´evypro- cesses, International Journal for Theory and Applications in Finance, vol. 3, pp. 549-552, 2000.

[162] D.B. Nelson, Stationarity and persistance in the GARCH (1,1) model, Econometric Theory, vol. 6, pp. 318-334, 1990.

[163] S.I. Boyarchenko and S. Z. Levendorski, Option pricing for truncated L´evy processes, International Journal of Theoretical and Applied Finance, vol. 3, no. 3, pp. 549-552, 2002.

[164] S.L. Heston and S. Nandi, A Closed-Form GARCH Option Valuation Model, The Review of Financial Studies, vol. 13, pp. 585-625, 2000.

170