Graduate Program

Focuses on the pricing and use of financial derivative securities and their role in investment management and financial risk management.

This sequence covers general measure and integration theory, Lp theory, differentiation theory, and basic functional analysis.

Measure and integration, axiomatic foundations of probability theory, random variables, distributions and their characteristic functions, stable and infinitely divisible laws, limit theorems for sums of independent random variables, conditioning, and martingales.

Stock prices and foreign currency exchange rates are time series. This course covers applied statistical methodologies pertaining to financial time series, especially series of stock prices, equity returns, interest rates, and exchange rates, with an emphasis on model building and accurate prediction. The course introduces forecasting techniques and methods for incorporating model uncertainty into financial forecasts.

Coverage of important topics in modern mathematical finance at the graduate level. Emphasis is placed on general principles of modeling the price dynamics of financial assets, quantitative techniques, behavioral finance, market risk and other financial risks, volatility modeling, and the foundations of high-frequency arbitrage trading.

The mathematical foundation for understanding modern financial theory, beginning with general probability theory and leading to basic results in pricing exotic and American derivatives. Topics include filtrations and generalized conditional expectation, the Girsanov theorem and Radon-Nikodym process, martingales, Brownian motion, Ito integration and processes, the Black-Scholes formula, risk-neutral pricing, and the Feynman-Kac theorem. Applications to financial instruments are discussed throughout.

An introduction to the valuation of financial options through the numerical solution of partial differential equations. The course covers the principal concepts, models, methods, and results used in the numerical approach, beginning with one-dimensional financial PDEs such as the Black-Scholes equation and continuing to two-dimensional PDEs in finance.

This course introduces numerical analysis and its applications in quantitative finance. Topics include numerical algorithms used in options, simulation, fixed-income valuation, and financial optimization. Attention is given to both how numerical methods are used and the mathematical principles underlying them.

Essential C++ topics with applications to finance. The course focuses on numerical analysis and quantitative finance applications.

An introduction to probability theory and stochastic processes for financial applications. Topics include modeling financial markets with stochastic processes, the Black-Scholes-Merton model, predictability in investment portfolios and hedging strategies, martingales and martingale measures, market efficiency, absence of arbitrage, Lévy models, and the pricing of derivative securities.

Probability spaces, continuous and discrete distributions, functions of random variables, expectation, conditional expectation, central limit theorem, convergence concepts, order statistics, and sampling distributions.

Sufficiency and completeness, information, estimation, maximum likelihood, confidence intervals, uniformly most powerful tests, likelihood ratio tests, normal-based inference, and Bayesian inference.

Estimation and testing in linear regression, residual analysis, influence diagnostics, multicollinearity, logistic regression, and nonlinear regression.

Theory of estimation and tests of statistical hypotheses and sequential analysis.

Basics of computing, optimization methods, the EM algorithm, simulation of random variables, Monte Carlo methods, Markov Chain Monte Carlo, and additional topics as time permits.

A detailed overview of the theory of Bayesian methods and their applications to financial modeling.