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Statistics for Finance develops students’ professional skills in statistics with applications in finance. Developed from the authors’ courses at the Technical University of Denmark and Lund University, the text bridges the gap between classical, rigorous treatments of financial mathematics that rarely connect concepts to data and books on econometrics and time series analysis that do not cover specific problems related to option valuation.

The book discusses applications of financial derivatives pertaining to risk assessment and elimination. The authors cover various statistical and mathematical techniques, including linear and nonlinear time series analysis, stochastic calculus models, stochastic differential equations, Itō’s formula, the Black–Scholes model, the generalized method-of-moments, and the Kalman filter. They explain how these tools are used to price financial derivatives, identify interest rate models, value bonds, estimate parameters, and much more.

This textbook will help students understand and manage empirical research in financial engineering. It includes examples of how the statistical tools can be used to improve value-at-risk calculations and other issues. In addition, end-of-chapter exercises develop students’ financial reasoning skills.

## Table of Contents

1. Preliminaries
2. Preface
3. Author biographies
4. Chapter 1 Introduction
1. 1.1 Introduction to financial derivatives
2. 1.2 Financial derivatives—what's the big deal?
3. 1.3 Stylized facts
4. 1.4 Overview
5. Chapter 2 Fundamentals
1. 2.1 Interest rates
2. 2.2 Cash flows
3. 2.3 Continuously compounded interest rates
4. 2.4 Interest rate options: caps and floors
5. 2.5 Notes
6. 2.6 Problems
6. Chapter 3 Discrete time finance
1. 3.1 The binomial one-period model
2. 3.2 One-period model
3. 3.3 Multiperiod model
4. 3.4 Notes
5. 3.5 Problems
7. Chapter 4 Linear time series models
1. 4.1 Introduction
2. 4.2 Linear systems in the time domain
3. 4.3 Linear stochastic processes
4. 4.4 Linear processes with a rational transfer function
5. 4.5 Autocovariance functions
6. 4.6 Prediction in linear processes
7. 4.7 Problems
8. Chapter 5 Nonlinear time series models
1. 5.1 Introduction
2. 5.2 Aim of model building
3. 5.3 Qualitative properties of the models
4. 5.4 Parameter estimation
1. 5.4.1 Maximum likelihood estimation
2. 5.4.2 Quasi-maximum likelihood
3. 5.4.3 Generalized method of moments
5. 5.5 Parametric models
1. 5.5.1 Threshold and regime models
2. 5.5.2 Models with conditional heteroscedasticity (ARCH)
3. 5.5.3 Stochastic volatility models
6. 5.6 Model identification
7. 5.7 Prediction in nonlinear models
8. 5.8 Applications of nonlinear models
9. 5.9 Problems
9. Chapter 6 Kernel estimators in time series analysis
1. 6.1 Non-parametric estimation
2. 6.2 Kernel estimators for time series
3. 6.3 Kernel estimation for regression
4. 6.4 Applications of kernel estimators
5. 6.5 Notes
10. Chapter 7 Stochastic calculus
11. Chapter 8 Stochastic differential equations
1. 8.1 Stochastic Differential Equations
2. 8.2 Analytical solution methods
3. 8.3 Feynman-Kac representation
4. 8.4 Girsanov measure transformation
5. 8.5 Notes
6. 8.6 Problems
12. Chapter 9 Continuous-time security markets
1. 9.1 From discrete to continuous time
2. 9.2 Classical arbitrage theory
1. 9.2.1 Black-Scholes formula
2. 9.2.2 Hedging strategies
3. 9.3 Modern approach using martingale measures
4. 9.4 Pricing
5. 9.5 Model extensions
6. 9.6 Computational methods
7. 9.7 Problems
13. Chapter 10 Stochastic interest rate models
1. 10.1 Gaussian one-factor models
2. 10.2 A general class of one-factor models
3. 10.3 Time-dependent models
1. 10.3.1 Ho–Lee
2. 10.3.2 Black–Derman–Toy
3. 10.3.3 Hull–White
4. 10.4 Multifactor and stochastic volatility models
5. 10.5 Notes
6. 10.6 Problems
14. Chapter 11 Term structure of interest rates
1. 11.1 Basic concepts
2. 11.2 Classical approach
3. 11.3 Term structure for specific models
4. 11.4 Heath–Jarrow–Morton framework
5. 11.5 Credit models
6. 11.6 Estimation of the term structure — curve-fitting
7. 11.7 Notes
8. 11.8 Problems
15. Chapter 12 Discrete time approximations
1. 12.1 Stochastic Taylor expansion
2. 12.2 Convergence
3. 12.3 Discretization schemes
1. 12.3.1 Strong Taylor approximations
2. 12.3.2 Weak Taylor approximations
3. 12.3.3 Exponential approximation
4. 12.4 Multilevel Monte Carlo
5. 12.5 Simulation of SDEs
16. Chapter 13 Parameter estimation in discretely observed SDEs
1. 13.1 Introduction
2. 13.2 High frequency methods
3. 13.3 Approximate methods for linear and non-linear models
4. 13.4 State dependent diffusion term
5. 13.5 MLE for non-linear diffusions
1. 13.5.1 Simulation-based estimators
2. 13.5.2 Numerical methods for the Fokker-Planck equation 273
3. 13.5.3 Series expansion
6. 13.6 Generalized method of moments
7. 13.7 Model validation for discretely observed SDEs
1. 13.7.1 Generalized Gaussian residuals
8. 13.8 Problems
17. Chapter 14 Inference in partially observed processes
1. 14.1 Introduction
2. 14.2 Model
3. 14.3 Exact filtering
1. 14.3.1 Prediction
2. 14.3.2 Updating
4. 14.4 Conditional moment estimators
5. 14.5 Kalman filter
6. 14.6 Approximate filters
7. 14.7 State filtering and prediction
1. 14.7.1 Linear models
2. 14.7.2 The system equation in discrete time
3. 14.7.3 Non-linear models
8. 14.8 Unscented Kalman Filter
9. 14.9 A maximum likelihood method
10. 14.10 Sequential Monte Carlo filters
11. 14.11 Application of non-linear filters
12. 14.12 Problems
18. Appendix A Projections in Hilbert spaces
1. A.1 Introduction
2. A.2 Hilbert spaces
3. A.3 The projection theorem
4. A.4 Conditional expectation and linear projections
5. A.5 Kalman filter
6. A.6 Projections in ℝn
19. Appendix B Probability theory
20. Bibliography