This book explores recent topics in quantitative finance with an emphasis on applications and calibration to time-series. This last aspect is often neglected in the existing mathematical finance literature while it is crucial for risk management. The first part of this book focuses on switching regime processes that allow to model economic cycles in financial markets. After a presentation of their mathematical features and applications to stocks and interest rates, the estimation with the Hamilton filter and Markov Chain Monte-Carlo algorithm (MCMC) is detailed. A second part focuses on self-excited processes for modeling the clustering of shocks in financial markets. These processes recently receive a lot of attention from researchers and we focus here on its econometric estimation and its simulation. A chapter is dedicated to estimation of stochastic volatility models. Two chapters are dedicated to the fractional Brownian motion and Gaussian fields. After a summary of their features, we present applications for stock and interest rate modeling. Two chapters focuses on sub-diffusions that allows to replicate illiquidity in financial markets. This book targets undergraduate students who have followed a first course of stochastic finance and practitioners as quantitative analyst or actuaries working in risk management.
Author(s): Donatien Hainaut
Series: Bocconi & Springer Series: Mathematics, Statistics, Finance and Economics, 12
Publisher: Springer
Year: 2022
Language: English
Pages: 358
City: Cham
Preface
Structure of the Book
Reading Guide
Acknowledgments
Contents
About the Author
Notation
1 Switching Models: Properties and Estimation
1.1 A Hidden Markov Chain
1.2 A Modulated Asset Model
1.3 A Modified Hamilton Filter for Estimation
1.4 Numerical Illustration
1.5 Change of Measure
1.6 European Options Pricing
1.7 The Markov Switching Multifractal (MSM) Model
1.8 Numerical Illustrations
1.9 Hitting Time of a Regime Switching Model
1.10 Further Reading
References
2 Estimation of Continuous Time Processes by Markov Chain Monte Carlo
2.1 Markov Chains
2.2 MCMC
2.3 Bayesian Inference
2.4 Estimation of a Multivariate Switching Regime
2.5 Further Reading
References
3 Particle Filtering and Estimation
3.1 The Heston Model
3.2 Filtering of Stochastic Volatility
3.3 Estimation with a Rolling Window
3.4 Bayesian Estimation
3.5 Further Reading
References
4 Modeling of Spillover Effects in Stock Markets
4.1 The Self-exciting Jump-Diffusion (SEJD)
4.2 Likelihood of Inter-Arrival Times
4.3 Jump Detection with the ``Peak Over Threshold'' Method
4.4 Sampling of Self-excited Processes
4.5 Particle Filtering of the Hawkes Intensity and MCMC Estimation
4.6 Properties of Jump Intensity and Log-Return in the SEJD
4.7 Change of Measure
4.8 Further Reading
References
5 Non-Markov Models for Contagion and Spillover
5.1 The Multivariate Processes
5.2 Infinite-Dimensional Reformulation
5.3 Finite-Dimensional Approximation
5.4 Moment Generating Function
5.5 Cauchy Memory Kernels
5.6 Estimation
5.7 Probability Density Functions by Fast Fourier Transform
5.8 Further Reading
References
6 Fractional Brownian Motion
6.1 Definition and Properties
6.2 Estimation of H by Rescaled Range Analysis
6.3 Integrals of Deterministic Functions and the Wick Product
6.4 Fractional Integrals and Itô's Lemma
6.5 Options Pricing in a Fractional Setting
6.6 A Fractional Interest Rate Model
6.7 Further Reading
References
7 Gaussian Fields for Asset Prices
7.1 Conditional Gaussian Fields
7.2 Market Model
7.3 Choice of Autocovariance Functions
7.4 Simulation of a Conditional Field by Spectral Decomposition
7.5 Calendar Spread Exchange Options Pricing
7.6 Asian Calendar Spread Exchange Options, with Geometric Average
7.7 Numerical Illustration
7.8 Further Reading
References
8 Lévy Interest Rate Models with a Long Memory
8.1 A Lévy Model with an Exponential Memory Kernel
8.2 A Lévy Model with a Mittag-Leffler Kernel
8.3 Empirical Motivation
8.4 Alternative Formulation
8.5 Bond Prices and Forward Rates
8.6 Discretization Scheme
8.7 Pricing of Bond Options
8.8 Further Reading
Appendix
References
9 Affine Volterra Processes and Rough Models
9.1 Introduction
9.2 Convolution and Resolvent
9.3 Moments and the Moment Generating Function
9.4 The Volterra and Rough Heston Model
9.5 Filtering
9.6 Further Reading
References
10 Sub-diffusion for Illiquid Markets
10.1 The Stochastic Clock of Sub-diffusions
10.2 The Non-fractional Market
10.3 The Fractional Market
10.4 A Fractional Fokker–Planck Equation
10.5 Option Pricing in the Fractional Setting
10.6 A Particle Filter
10.7 Estimation of Parameters
10.8 Further Reading
References
11 A Fractional Dupire Equation for Jump-Diffusions
11.1 Non-fractional Jump-Diffusion Model
11.2 Subordinators
11.3 The Dzerbayshan–Caputo Derivatives
11.4 Fractional Financial Market
11.5 Numerical Framework
11.6 Numerical Illustration
11.7 Further Reading
References
