This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.
Author(s): Jaya P. N. Bishwal
Publisher: Springer
Year: 2022
Language: English
Pages: 633
City: Cham
Contents
Basic Notations
Preface
Introduction
Chapter 1 Stochastic Volatility Models: Methods of Pricing, Hedgingand Estimation
1.1 Introduction
1.2 Stochastic Derivative and Malliavin Calculus for Stochastic Volatility Models
1.3 Pricing European Options
1.4 Hedging and Greek Estimation
1.5 Bootstrap Method for Volatility Estimation
1.6 Stochastic Gradient Descent Algorithm for American Option
1.7 Peacock Process and Indian Option
1.8 Brownian Excursion and Volatility Estimation inLimit Order Book
1.9 Rate of Cauchy Convergence of Brownian Winding and Asian Options
1.10 Bond Pricing for the Fractional Vasicek Model
1.11 Extreme Value Theory in Finance
1.12 Epsilon-Markov Processes
Chapter 2 Sequential Monte Carlo Methods
2.1 Introduction
2.2 Stochastic Volatility Models and SMC Methods
2.3 Spline Method in Volatility Estimation
2.4 Multilevel Monte Carlo Method for Nonlinear SPDE
2.5 Third Order Composition Scheme for Diffusions
Chapter 3 Parameter Estimation in the Heston Model
3.1 Introduction
3.2 Continuous Observation
3.3 Discrete Observations
3.4 Estimators in the Supercritical CIR Process
3.5 Sequential Estimation in CIR Process
3.6 Berry–Esseen Bound for Heston Model
3.7 Method of Moments Estimation in Heston Model
Chapter 4 Fractional Ornstein–Uhlenbeck Processes,Levy–Ornstein–UhlenbeckProcesses, and FractionalLevy–Ornstein–Uhlenbeck Processes
4.1 Introduction
4.2 Continuous Sampling
4.3 Discrete Sampling
4.4 Ornstein–Uhlenbeck–Gamma Process
4.5 Fractional Levy–Ornstein–Uhlenbeck Process
4.6 FIECOGARCH Process
4.7 Fractional Gamma and Fractional Inverse Gaussian Ornstein–Uhlenbeck Process
Chapter 5 Inference for General Semimartingales and Self-similarProcesses
5.1 Introduction
5.2 Continuous Sampling of Semimartingales
5.3 Discrete Sampling of Semimartingales
5.4 Asymptotics of the Log-likelihood Function
5.5 Random Observation Period: Sequential Inference
5.6 Fractional and Sub-Fractional Levy O-U Process
Chapter 6 Estimation in Gamma-Ornstein–Uhlenbeck StochasticVolatility Model
6.1 Introduction
6.2 Gamma-Ornstein–Uhlenbeck Model
6.3 Method of Moments Estimators
6.4 Fractional MS-OU and OU-MS Processes
6.5 Fractional Hermite OU Processes
Chapter 7 Berry–Esseen Inequalities for the FunctionalOrnstein–Uhlenbeck-Inverse-Gamma Process
7.1 Introduction
7.2 Approximate Maximum Likelihood Estimators
7.3 Berry–Esseen Bounds for AMLE1
7.4 Berry–Esseen Bounds for AMLE2
7.5 Berry–Esseen Bounds for AMCEs
7.6 Geometric Mean Reversion Process: Black-Karasinski Model
Chapter 8 Maximum Quasi-Likelihood Estimation in FractionalLevy Stochastic Volatility Model
8.1 Introduction
8.2 Fractional Levy Process
8.3 Quasi-Maximum Likelihood Estimator
8.4 Conclusion
Chapter 9 Estimation in Barndorff Nielsen-ShephardOrnstein–Uhlenbeck Stochastic Volatility Models
9.1 Introduction
9.2 Modified Tempered Stable Models
9.3 Method of Moments Estimators
9.4 Robust Estimation in Inverse Gaussian Ornstein- Uhlenbeck Stochastic Volatility Model
9.5 Quasi-Maximum Likelihood Estimation in Stable-OU Process
9.6 Empirical Characteristic Function Estimator
Chapter 10 Parameter Estimation in Student Ornstein–UhlenbeckProcess
10.1 Introduction
10.2 Student O–U Process
10.3 Speed of Convergence for the Cauchy Approximation
10.4 Berry–Esseen Type Bounds
Chapter 11 Berry–Esseen Asymptotics for Pearson Diffusions
11.1 Introduction
11.2 Pearson Diffusions
11.3 Fractional Student Ornstein–Uhlenbeck Process
11.4 Estimators
Chapter 12 Bayesian Maximum Likelihood Estimation in FractionalStochastic Volatility Model
12.1 Introduction
12.2 Fractional Stochastic Volatility Model
12.3 Bayesian Maximum Likelihood Estimation
12.4 Fractional Heston Model
Chapter 13 Berry–Esseen–Stein–Malliavin Theory for FractionalOrnstein–Uhlenbeck Process
13.1 Introduction
13.2 Exact Berry–Esseen Bounds
13.3 Continuous Sampling
13.4 Discrete Sampling
13.5 Approximate Minimum Contrast Estimator
13.6 Higher Order Estimators
13.7 Fractional Discrete Sampling
Chapter 14 Approximate Maximum Likelihood Estimationin Sub-fractional Hybrid Stochastic Volatility Model
14.1 Introduction
14.2 Term Structure Models and Derivative Pricing
14.3 Newton–Cotes Distribution and Drift Estimators
14.4 Test Function Estimator of Elasticity of Volatility and Stochastic Elasticity Model
14.5 Conclusion
Appendix
A Time Series Regression and Discrete Financial Models
B Stochastic Calculus
Bibliography
Index
