Statistics for Finance

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, Ito’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.

IntroductionIntroduction to financial derivatives Financial derivatives—what’s the big deal? Stylized factsOverview Fundamentals Interest rates Cash flows Continuously compounded interest rates Interest rate options: caps and floors Discrete-Time Finance The binomial one period model The one period model The multi period model Linear Time Series Models Introduction Linear systems in the time domain Linear stochastic processes Linear processes with a rational transfer functionAutocovariance functions Prediction in linear processes Non-Linear Time Series Models Introduction The aim of model buildingQualitative properties of the models Parameter estimationParametric models Model identification Prediction in non-linear models Applications of non-linear models Kernel Estimators in Time Series Analysis Non-parametric estimation Kernel estimators for time series Kernel estimation for regression Applications of kernel estimators Stochastic Calculus Dynamical systems The Wiener process Stochastic Integrals Ito stochastic calculus Extensions to jump processes Stochastic Differential Equations Stochastic differential equations Analytical solution methods Feynman–Kac representation Girsanov measure transformation Continuous-Time Security Markets From discrete to continuous time Classical arbitrage theoryModern approach using martingale measures Pricing Model extensions Computational methods Stochastic Interest Rate Models Gaussian one-factor models A general class of one-factor models Time-dependent models Multifactor and stochastic volatility models The Term Structure of Interest Rates Basic concepts The classical approach The term structure for specific models Heath–Jarrow–Morton framework Credit models Estimation of the term structure—curve-fitting Discrete-Time Approximations Stochastic Taylor expansionConvergence Discretization schemes Multilevel Monte Carlo Simulation of SDEs Parameter Estimation in Discretely Observed SDEsIntroduction High frequency methods Approximate methods for linear and non-linear modelsState dependent diffusion term MLE for non-linear diffusionsGeneralized method of moments (GMM) Model validation for discretely observed SDEs Inference in Partially Observed Processes IntroductionThe model Exact filtering Conditional moment estimators Kalman filter Approximate filters State filtering and predictionThe 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 Bibliography 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.