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Financial Modeling with L´Evy Processes and Applying L

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Financial Modeling with L´Evy Processes and Applying L 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 p 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 T SB (β = 0:8; γ = 2; δ = 10) using MLE and Laplace transformation. 76 2.13 MSD of 1000 trajectories of the T SB (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.
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