Statistics for Finance

Statistics for Finance

by Erik Lindström, Henrik Madsen, Jan Nygaard Nielsen
Released April 2015
Publisher(s): CRC Press
ISBN: 9781482229004

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Book description

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
    1. Erik Lindström
    2. Henrik Madsen
    3. Jan Nygaard Nielsen
  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
      1. 1.3.1 No autocorrelation in returns
      2. 1.3.2 Unconditional heavy tails
      3. 1.3.3 Gain/loss asymmetry
      4. 1.3.4 Aggregational Gaussianity
      5. 1.3.5 Volatility clustering
      6. 1.3.6 Conditional heavy tails
      7. 1.3.7 Significant autocorrelation for absolute returns
      8. 1.3.8 Leverage effects
    4. 1.4 Overview
      1. Figure 1.1
      2. Figure 1.2
      3. Figure 1.3
      4. Figure 1.4
      5. Figure 1.5
      6. Figure 1.6
      7. Figure 1.7
      8. Figure 1.8
      1. Table 1.1
  5. Chapter 2 Fundamentals
    1. 2.1 Interest rates
      1. 2.1.1 Future and present value of a single payment
      2. 2.1.2 Annuities
      3. 2.1.3 Future value of an annuity
      4. 2.1.4 Present value of a unit annuity
    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
      1. Figure 2.1
      2. Figure 2.2
      3. Figure 2.3
      4. Figure 2.4
      5. Figure 2.5
      1. Table 2.1
  6. Chapter 3 Discrete time finance
    1. 3.1 The binomial one-period model
    2. 3.2 One-period model
      1. 3.2.1 Risk-neutral probabilities
      2. 3.2.2 Complete and incomplete markets
    3. 3.3 Multiperiod model
      1. 3.3.1 σ-algebras and information sets
      2. 3.3.2 Financial multiperiod markets
      3. 3.3.3 Martingale measures
    4. 3.4 Notes
    5. 3.5 Problems
      1. Figure 3.1
      2. Figure 3.2
      3. Figure 3.3
      4. Figure 3.4
      5. Figure 3.5
  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
      1. 4.4.1 ARMA process
      2. 4.4.2 ARIMA process
      3. 4.4.3 Seasonal models
    5. 4.5 Autocovariance functions
      1. 4.5.1 Autocovariance function for ARMA processes
    6. 4.6 Prediction in linear processes
    7. 4.7 Problems
      1. Figure 4.1
      2. Figure 4.2
  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
      1. 5.3.1 Volterra series expansion
      2. 5.3.2 Generalized transfer functions
    4. 5.4 Parameter estimation
      1. 5.4.1 Maximum likelihood estimation
        1. 5.4.1.1 Cramér-Rao bound
        2. 5.4.1.2 The likelihood ratio test
      2. 5.4.2 Quasi-maximum likelihood
      3. 5.4.3 Generalized method of moments
        1. 5.4.3.1 GMM and moment restrictions
        2. 5.4.3.2 Standard error of the estimates
        3. 5.4.3.3 Estimation of the weight matrix
        4. 5.4.3.4 Nested tests for model reduction
    5. 5.5 Parametric models
      1. 5.5.1 Threshold and regime models
        1. 5.5.1.1 Self-exciting threshold AR (SETAR)
        2. 5.5.1.2 Self-exciting threshold ARMA (SETARMA)
        3. 5.5.1.3 Open loop threshold AR (TARSO)
        4. 5.5.1.4 Smooth threshold AR (STAR)
        5. 5.5.1.5 Hidden Markov models and related models
      2. 5.5.2 Models with conditional heteroscedasticity (ARCH)
        1. 5.5.2.1 ARCH regression model
        2. 5.5.2.2 GARCH model
        3. 5.5.2.3 EGARCH model
        4. 5.5.2.4 FIGARCH model
        5. 5.5.2.5 ARCH-M model
        6. 5.5.2.6 SW-ARCH model
        7. 5.5.2.7 General remarks on ARCH models
        8. 5.5.2.8 Multivariate GARCH models
      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
      1. 5.8.1 Electricity spot prices
      2. 5.8.2 Comparing ARCH models
    9. 5.9 Problems
      1. Figure 5.1
      2. Figure 5.2
      3. Figure 5.3
      4. Figure 5.4
  9. Chapter 6 Kernel estimators in time series analysis
    1. 6.1 Non-parametric estimation
    2. 6.2 Kernel estimators for time series
      1. 6.2.1 Introduction
      2. 6.2.2 Kernel estimator
      3. 6.2.3 Central limit theorems
    3. 6.3 Kernel estimation for regression
      1. 6.3.1 Estimator for regression
      2. 6.3.2 Product kernel
      3. 6.3.3 Non-parametric estimation of the pdf
      4. 6.3.4 Non-parametric LS
      5. 6.3.5 Bandwidth
      6. 6.3.6 Selection of bandwidth — cross validation
      7. 6.3.7 Variance of the non-parametric estimates
    4. 6.4 Applications of kernel estimators
      1. 6.4.1 Non-parametric estimation of the conditional mean and variance
      2. 6.4.2 Non-parametric estimation of non-stationarity — an example
      3. 6.4.3 Non-parametric estimation of dependence on external variables — an example
      4. 6.4.4 Non-parametric GARCH models
    5. 6.5 Notes
      1. Figure 6.1
      2. Figure 6.2
      3. Figure 6.3
      4. Figure 6.4
      5. Figure 6.5
      6. Figure 6.6
      7. Figure 6.7
      8. Figure 6.8
  10. Chapter 7 Stochastic calculus
    1. 7.1 Dynamical systems
    2. 7.2 The Wiener process
    3. 7.3 Stochastic Integrals
    4. 7.4 Itō stochastic calculus
    5. 7.5 Extensions to jump processes
    6. 7.6 Problems
  11. Chapter 8 Stochastic differential equations
    1. 8.1 Stochastic Differential Equations
      1. 8.1.1 Existence and uniqueness
      2. 8.1.2 Itō formula
      3. 8.1.3 Multivariate SDEs
      4. 8.1.4 Stratonovitch SDE
    2. 8.2 Analytical solution methods
      1. 8.2.1 Linear, univariate SDEs
    3. 8.3 Feynman-Kac representation
    4. 8.4 Girsanov measure transformation
      1. 8.4.1 Measure theory
      2. 8.4.2 Radon–Nikodym theorem
      3. 8.4.3 Girsanov transformation
      4. 8.4.4 Maximum likelihood estimation for continuously observed diffusions
    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
        1. 9.2.2.1 Quadratic hedging
    3. 9.3 Modern approach using martingale measures
    4. 9.4 Pricing
    5. 9.5 Model extensions
    6. 9.6 Computational methods
      1. 9.6.1 Fourier methods
    7. 9.7 Problems
      1. Figure 9.1
      2. Figure 9.2
  13. Chapter 10 Stochastic interest rate models
    1. 10.1 Gaussian one-factor models
      1. 10.1.1 Merton model
      2. 10.1.2 Vasicek model
    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
        1. 10.3.3.1 CIR++model
    4. 10.4 Multifactor and stochastic volatility models
      1. 10.4.1 Stochastic volatility models
      2. 10.4.2 Affine Term Structure models
    5. 10.5 Notes
    6. 10.6 Problems
      1. Figure 10.1
      2. Figure 10.2
      1. Table 10.1
  14. Chapter 11 Term structure of interest rates
    1. 11.1 Basic concepts
      1. 11.1.1 Known interest rates
      2. 11.1.2 Discrete dividends
      3. 11.1.3 Yield curve
      4. 11.1.4 Stochastic interest rates
    2. 11.2 Classical approach
      1. 11.2.1 Exogenous specification of the market price of risk
      2. 11.2.2 Illustrative example
      3. 11.2.3 Modern approach
    3. 11.3 Term structure for specific models
      1. 11.3.1 Example 1: The Vasicek model
      2. 11.3.2 Example 2: The Ho–Lee model
      3. 11.3.3 Example 3: The Cox–Ingersoll–Ross model
      4. 11.3.4 Multifactor models
    4. 11.4 Heath–Jarrow–Morton framework
    5. 11.5 Credit models
      1. 11.5.1 Intensity models
    6. 11.6 Estimation of the term structure — curve-fitting
      1. 11.6.1 Polynomial methods
      2. 11.6.2 Decay functions
      3. 11.6.3 Nelson–Siegel method
    7. 11.7 Notes
    8. 11.8 Problems
      1. Figure 11.1
      2. Figure 11.2
      3. Figure 11.3
      4. Figure 11.4
      5. Figure 11.5
      1. Table 11.1
      2. Table 11.2
  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
        1. 12.3.1.1 Explicit Euler scheme
        2. 12.3.1.2 Milstein scheme
        3. 12.3.1.3 The order 1.5 strong Taylor scheme
      2. 12.3.2 Weak Taylor approximations
        1. 12.3.2.1 The order 2.0 weak Taylor scheme
      3. 12.3.3 Exponential approximation
    4. 12.4 Multilevel Monte Carlo
    5. 12.5 Simulation of SDEs
      1. Figure 12.1
      2. Figure 12.2
      3. Figure 12.3
  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
      1. 13.4.1 A transformation approach
    5. 13.5 MLE for non-linear diffusions
      1. 13.5.1 Simulation-based estimators
        1. 13.5.1.1 Jump diffusions
      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
      1. 13.6.1 GMM and moment restrictions
    7. 13.7 Model validation for discretely observed SDEs
      1. 13.7.1 Generalized Gaussian residuals
        1. 13.7.1.1 Case study
    8. 13.8 Problems
      1. Figure 13.1
      2. Figure 13.2
      3. Figure 13.3
      4. Figure 13.4
      5. Figure 13.5
  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
        1. 14.3.1.1 Scalar case
        2. 14.3.1.2 General case
      2. 14.3.2 Updating
    4. 14.4 Conditional moment estimators
      1. 14.4.1 Prediction and updating
    5. 14.5 Kalman filter
    6. 14.6 Approximate filters
      1. 14.6.1 Truncated second order filter
      2. 14.6.2 Linearized Kalman filter
      3. 14.6.3 Extended Kalman filter
      4. 14.6.4 Statistically linearized filter
      5. 14.6.5 Non-linear models
      6. 14.6.6 Linear time-varying models
      7. 14.6.7 Linear time-invariant models
      8. 14.6.8 Case: Affine term structure models
    7. 14.7 State filtering and prediction
      1. 14.7.1 Linear models
        1. 14.7.1.1 Linear time-varying models
        2. 14.7.1.2 Linear time-invariant 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
      1. 14.10.1 Optimal filtering
      2. 14.10.2 Bootstrap filter
      3. 14.10.3 Parameter estimation
    11. 14.11 Application of non-linear filters
      1. 14.11.1 Sequential calibration of options
      2. 14.11.2 Computing Value at Risk in a stochastic volatility model
      3. 14.11.3 Extended Kalman filtering applied to bonds
        1. Data description
      4. 14.11.4 Case 1: A Wiener process
      5. 14.11.5 Case 2: The Vasicek model
    12. 14.12 Problems
      1. Figure 14.1
      2. Figure 14.2
      3. Figure 14.3
      4. Figure 14.4
      5. Figure 14.5
      6. Figure 14.6
      7. Figure 14.7
      8. Figure 14.8
      1. Table 14.1
      2. Table 14.2
      3. Table 14.3
  18. Appendix A Projections in Hilbert spaces
    1. A.1 Introduction
    2. A.2 Hilbert spaces
    3. A.3 The projection theorem
      1. A.3.1 Prediction equations
    4. A.4 Conditional expectation and linear projections
    5. A.5 Kalman filter
    6. A.6 Projections in ℝn
      1. Figure A.1
  19. Appendix B Probability theory
    1. B.1 Measures and σ-algebras
    2. B.2 Partitions and information
    3. B.3 Conditional expectation
    4. B.4 Notes
  20. Bibliography

Product information

  • Title: Statistics for Finance
  • Author(s): Erik Lindström, Henrik Madsen, Jan Nygaard Nielsen
  • Release date: April 2015
  • Publisher(s): CRC Press
  • ISBN: 9781482229004