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
Introduction to financial derivatives
Financial derivatives—what’s the big deal?
Continuously compounded interest rates
Interest rate options: caps and floors
The binomial one period model
The one period model
The multi period model
Linear Time Series Models
Linear systems in the time domain
Linear stochastic processes
Linear processes with a rational transfer function
Prediction in linear processes
Non-Linear Time Series Models
The aim of model building
Qualitative properties of the models
Prediction in non-linear models
Applications of non-linear models
Kernel Estimators in Time Series Analysis
Kernel estimators for time series
Kernel estimation for regression
Applications of kernel estimators
The Wiener process
Itō stochastic calculus
Extensions to jump processes
Stochastic Differential Equations
Stochastic differential equations
Analytical solution methods
Girsanov measure transformation
Continuous-Time Security Markets
From discrete to continuous time
Classical arbitrage theory
Modern approach using martingale measures
Stochastic Interest Rate Models
Gaussian one-factor models
A general class of one-factor models
Multifactor and stochastic volatility models
The Term Structure of Interest Rates
The classical approach
The term structure for specific models
Estimation of the term structure—curve-fitting
Stochastic Taylor expansion
Multilevel Monte Carlo
Simulation of SDEs
Parameter Estimation in Discretely Observed SDEs
High frequency methods
Approximate methods for linear and non-linear models
State dependent diffusion term
MLE for non-linear diffusions
Generalized method of moments (GMM)
Model validation for discretely observed SDEs
Inference in Partially Observed Processes
Conditional moment estimators
State filtering and prediction
The unscented Kalman filter
A maximum likelihood method
Sequential Monte Carlo filters
Application of non-linear filters
Appendix A: Projections in Hilbert Spaces
Appendix B: Probability Theory
Problems appear at the end of each chapter.
Erik Lindström is an associate professor in the Centre for Mathematical Sciences at Lund University. His research ranges from statistical methodology (primarily time series analysis in discrete and continuous time) to financial mathematics as well as problems related to energy markets. He earned a PhD in mathematical statistics from Lund Institute of Technology/Lund University.
Henrik Madsen is a professor and head of the Section for Dynamical Systems in the Department for Applied Mathematics and Computer Sciences at the Technical University of Denmark. An elected member of the ISI and IEEE, he has authored or co-authored 480 papers and 11 books in areas including mathematical statistics, time series analysis, and the integration of renewables in electricity markets. He earned a PhD in statistics from the Technical University of Denmark.
Jan Nygaard Nielsen is a principal architect at Netcompany, a Danish IT and business consulting firm. He earned a PhD from the Technical University of Denmark.
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