Applications of Lévy processes / edited by Oleg Kudryavtsev.

Format:

Book

Contributor:

Kudryavtsev, Oleg, editor.

Series:

Mathematics Research Developments

Language:

English

Subjects (All):

Lévy processes.

Physical Description:

1 online resource (272 pages)

Place of Publication:

New York : Nova Science Publishers, [2021]

Summary:

"Lévy processes have found applications in various fields, including physics, chemistry, long-term climate change, telephone communication, and finance. The most famous Lévy process in finance is the Black-Scholes model. This book presents important financial applications of Lévy processes. The Editors consider jump-diffusion and pure non-Gaussian Lévy processes, the multi-dimensional Black-Scholes model, and regime-switching Lévy models. This book is comprised of seven chapters that focus on different approaches to solving applied problems under Lévy processes: Monte Carlo simulations, machine learning, the frame projection method, dynamic programming, the Fourier cosine series expansion, finite difference schemes, and the Wiener-Hopf factorization. Various numerical examples are carefully presented in tables and figures to illustrate the methods designed in the book"-- Provided by publisher.

Contents:

Intro

APPLICATIONS OFLÉVY PROCESSES

CONTENTS

PREFACE

Chapter 1VARIANCE REDUCTION APPLIED TOMACHINE LEARNING FOR PRICINGBERMUDAN/AMERICAN OPTIONSIN HIGH DIMENSION

Abstract

1. INTRODUCTION

2. AMERICAN OPTIONS IN THE MULTI-DIMENSIONAL BLACK-SCHOLES MODEL

3. MACHINE LEARNING FOR AMERICAN OPTIONSIN THE MULTI-DIMENSIONAL BLACK-SCHOLESMODEL

3.1. Gaussian Process Regression

3.2. Machine Learning Exact Integration for European Options

3.3. Machine Learning Control Variate Algorithm for AmericanOptions

3.3.1. The GPR Monte CarloMethod

3.3.2. The GPR Monte Carlo Control Variate Method

3.3.3. The Control Variate for GPR-Tree and GRP-EI

4. NUMERICAL RESULTS

4.1. Geometric and Arithmetic Basket Put Options

4.2. Call on theMaximum Option

4.3. Variance Reduction

CONCLUSION

REFERENCES

Chapter 2A MACHINE LEARNING APPROACH TOOPTION PRICING UNDER L´E VY PROCESSES

1.1. Machine Learning in Finance

advance.1.2.

2. OPTION PRICING

2.1. The Applications in Option Pricing

2.2. L´evy Processes

3. MACHINE LEARNING APPROACH

4. CGMY MODEL CALIBRATION WITH GPR

5. ARTIFICIAL NEURAL NETWORKS

5.1. Feedforward ANN

5.2. Recurrent NN

5.3. Long/Short Term

5.4. Gated Recurrent Units

5.5. Bidirectional Recurrent Neural Networks

5.6. BoltzmannMachines

5.7. Restricted BoltzmannMachines

5.8. Convolutional Networks

6. ACTIVATION FUNCTIONS

6.1. Step Function

6.2. Linear Activation Function

6.3. Sigmoid Activation Function

6.4. Hyperbolic Tangent Activation Function

6.5. Softsign Activation Function

6.6. Basic Rectified Linear Unit (ReLU)The

6.7. Leaky (

6.8. Modified Rectifiers (MELU)Numerous attempts have

6.9. Softplus Activation Function.

7. APPLYING A FF ANN TO SOLVE THE MODELCALIBRATION PROBLEM

7.1. Historical Data Preparation

7.2. Synthetic Data

7.3. Training the Network

7.4. Market States ClassificationFinancial markets

8. PRICING OPTIONS IN THE CGMY MODEL VIA AFF ANN

ACKNOWLEDGMENT

Chapter 3ON SWING OPTION PRICINGUNDER L´E VY PROCESS DYNAMICS

2. SWING OPTIONS

2.1. Policy Constraints

2.1.1. Volume Penalties

2.1.2. Ramping Constraints

2.2. Cash Flows

2.2.1. The Locally Constrained Case

2.3. Swing Rights and Recovery

3. MODELS FOR THE UNDERLYING

3.1. Exponential L´evy Dynamics

3.2. Mean-Reverting

4. PRICING METHODS

4.1. A Discrete Time Formulation

4.1.1. Value Functions

4.1.2. Optimal Swing Policies

4.2. Trees and Grids

4.3. Monte Carlo

4.4. PROJ Method

4.4.1. Value Functions

4.4.2. Pure Fixed Rights

4.4.3. Numerical Examples: Fixed Rights

4.5. A Continuous Time Formulation

4.5.1. Variational Inequalities

4.6. COSMethod

4.7. PROJ: American Contracts

4.7.1. Algorithm Structure

4.7.2. Numerical Example: Constant Recovery

4.7.3. Numerical Examples: Linear Recovery

Chapter 4FOURIER-COSINE EXPANSION METHODFOR PRICING EQUITY-INDEXED ANNUITIESUNDER L´E VY MODELS

2. MODEL DESCRIPTION AND RATCHET EIACONTRACTS

2.1. L´evy Models and Their Characteristic Functions

2.2. EIAs Contracts and Their Payoffs

3. VALUATION OF EIAS USING FOURIER-COSINEEXPANSION METHOD

3.1. Fourier-Cosine ExpansionMethod

3.2. Coefficients Ak for Ratchet EIAs

4. NUMERICAL TEST

Chapter 5THE MULTILEVEL MONTE CARLO METHODFOR JUMP L´E VY MODELS:CENTRAL LIMIT THEOREM

2. GENERAL FRAMEWORK AND PRELIMINARYRESULTS.

3. MAIN RESULTS

3.1. A Functional Limit Theorem

3.2. Central Limit Theorem

3.3. The Time Complexity

A. APPENDIX

A.1. Convergence of Infinitely Divisible Distributions

A.2. Lindeberg-Feller Central Limit Theorem

A.3. A Useful Lemma from the Paper of Cohen and Rosi ´nski

A.4. Tools Used for the Exponential L´evy Model Setting

Chapter 6OPTIMAL RESOURCE EXTRACTION INREGIME SWITCHING L´E VY MARKETS

2. PROBLEM FORMULATION

3. CHARACTERIZATION OF THE VALUE FUNCTION

3.1. Optimal Extraction and Stopping Strategies

4. NUMERICAL APPROXIMATION

5. NUMERICAL EXAMPLE

Chapter 7NUMERICAL METHODS FOR PRICINGOPTIONS IN L´E VY PROCESSES: THE APPROXIMATE WIENER-HOPFFACTORIZATION TECHNIQUES

1.1. Monte Carlo Methods

1.2. Semi-Analytical Numerical Methods

1.3. Numerical Methods

1.4. Enhanced Approximate Wiener-Hopf Factorization Methods

2. L´E VY PROCESSES AND THE REGIME STRUCTURE

2.1. L´evy Processes: General Definitions

2.2. Regular L´evy Processes of Exponential Type

2.3. The Wiener-Hopf factorization

2.4. Regime-Switching L´evy Processes

2.5. The System of the Generalized Black-Scholes Equations

3. LAPLACE TRANSFORM IN THE CONTEXT OF THE AWHF-METHODS

3.1. Numerical Laplace Transform Inversion: An Overview

3.2. The Fast Wiener-Hopf Factorization Method

3.3. The Gaver-Stehfest Algorithm and the FWHF-Method

3.4. The Post-Widder Formula or Carr's Randomization

3.5. The "Simple Wiener-Hopf Factorization Method"

4. PRICING OF BARRIER OPTIONS UNDER REGIMESWITCHING L´EVY MODELS

5. APPROXIMATE WIENER-HOPF FACTORIZATIONAND MONTE CARLO METHODS

5.1. Approximate Wiener-Hopf Factorization Monte Carlo Method.

5.2. General Approximate Wiener-Hopf Factorization MonteCarlo Method

6. NUMERICAL EXAMPLES

6.1. Pricing Options without Regime Switching

6.2. Pricing Options with Regime Switching

EDITORS' CONTACT INFORMATION

INDEX

Blank Page.

Notes:

Description based on print version record.

Includes bibliographical references and index.

ISBN:

1-5361-9849-8