*Denotes a course created specifically for this program.
ACMS 60690. Numerical Analysis I
A solid theoretical introduction to numerical analysis. Polynomial interpolation. Least squares and the basic theory of orthogonal functions. Numerical integration in one variable. Numerical linear algebra. Methods to solve systems of nonlinear equations. Numerical solution of ordinary differential equations. Solution of some simple partial differential equations by difference methods.
ACMS 60850. Applied Probability
A thorough introduction to probability theory. Elements of measure and integration theory. Basic setup of probability theory (sample spaces, independence). Random variables, the law of large numbers. Discrete random variables (including random walks), continuous random variables, the basic distributions and sums of random variables. Generating functions, branching processes, basic theory of characteristic functions, central limit theorems. Markov chains. Various stochastic processes, including Brownian motion, queues, and applications. Martingales. Other topics as time permits.
ACMS 60901. Mathematical Finance I*
An introduction to the contract terms, payoffs, and classical pricing theories for a variety of financial instruments: stocks, bonds, foreign exchange, and their derivatives: forwards, futures, options, swaps. Classical theory of options pricing. Put-call parity. The multiperiod binomial model. Discrete time pricing of American options. Continuous time models for stock prices and interest rates. Explicit derivation of the Black-Scholes option pricing formula. Limitations of the Black-Scholes model.
ACMS 60902. Mathematical Finance II*
An introduction to the more recent advances in the continuous modeling of asset prices and pricing contingent claims. Solution to Heston’s stochastic volatility model. Merton’s jump-diffusion model. Introduction to Lévy processes. Stochastic calculus for jump processes. Hedging in incomplete markets.
ACMS 60921. Financial Computing*
An introduction to using computer programming to solve problems in mathematical finance. Introduction to the rapid prototyping of financial systems using MATLAB and R computing environments, programming languages, and finance-oriented packages. Exposure to writing more efficient function libraries in C/C++. Emphasis will be placed on the practical implementation of numerical algorithms most useful to practitioners in quantitative finance.
ACMS 60932. Statistical Inference for Finance*
Review of general concepts of statistical estimation theory: bias, mean-squared error, consistency, maximum likelihood estimation, standard error, and interval estimation. Estimating stock price volatility and parameters in binomial pricing models. Overview of financial time series analysis including ARMA, GARCH, and stochastic volatility modeling and estimation. The loss distribution approach to financial risk management and estimating risk measures like value-at-risk and expected shortfall. Using extreme value theory to model tail behavior. Multivariate modeling, dimension reduction, and the use of copulas. Introduction to statistical inference for continuous time asset pricing models.
ACMS 60842. Time Series Analysis
This is an introductory and applied course in time series analysis. Popular time series models and computational techniques for model estimation, diagnostic and forecasting will be discussed. Although the book focuses on financial data sets, other data sets, such as climate data, earthquake data and biological data, will also be included and discussed within the same theoretical framework.
ACMS 60790. Numerical Analysis II
A solid introduction to numerical partial differential equations with an emphasis on finite difference methods for time-dependent equations and systems of equations. Interpolation. Stability and convergence of solutions in systems of PDE arising in science and engineering. High-order accurate difference methods and Fourier methods. Well-posed problems and general solutions for a variety of types of systems of equations.
ACMS 60882. Linear Statistical Models
A comprehensive treatment of the theory of multiple regression analysis and an introduction to the generalized linear model. Least squares estimators. The Gauss-Markov theorem. Maximum-likelihood estimation of parameters, confidence, and prediction intervals. Residuals analysis: detecting heteroskedasticity and autocorrelation. The Box-Cox transformation. Principal components regression. Models for a binomial response. Models for a categorical response. Analysis of contingency tables using generalized linear models.