Course Descriptions

MATH 227. Partial Differential Equations and Diffusion Processes. Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Prerequisites: Math 131 and Math 136/Stats 219, or equivalents.

MATH 236. Introduction to Stochastic Differential Equations. Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: 136 or equivalent and differential equations.

MATH 238/STATS 250. Mathematical Finance. Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk. Term structure models and interest rate derivatives. Optimal stopping and American options. Corequisites: MATH 236 and 227 or equivalent.

MATH 239. Computation and Simulation in Finance. Monte Carlo, finite difference, tree, and transformation methods for the numerical solution of partial differential equations in finance. Emphasis is on derivative security pricing. Prerequisite: 238 or equivalent.

MATH 240. Topics in Financial Mathematics: Fixed Income Models.

Introduction to continuous time models for arbitrage-free pricing of interest rate derivatives. Bonds, yields, and the construction of yield curves. Caps, floors, swaps, swaptions, and bond options. Short rate models. Yield curve models. Forward measures. Forward and futures. LIBOR and swap market models. Prerequisite: MATH 238.

MS&E 347. Credit Risk: Modeling and Management.

Credit risk modeling, valuation, and hedging emphasizing underlying economic, probabilistic, and statistical concepts. Point processes and their compensators. Structural, incomplete information and reduced form approaches. Single name products: corporate bonds, equity, equity options, credit and equity default swaps, forwards and swaptions. Multiname modeling: index and tranche swaps and options, collateralized debt obligations. Implementation, calibration and testing of models. Industry and market practice. Data and implementation driven group projects that focus on problems in the financial industry. Prerequisites: stochastic processes at the level of MSE 321, 322 or equivalent, and financial engineering at the level of MSE 342, MATH 180, MATH 240, FINANCE 622 or equivalent

STATS 241. Financial Modeling and Risk Management. Nonparametric regression and yield curve smoothing. Advanced time series modeling and forecasting. Market risk measures. Substantive and empirical modeling approaches in financial markets. Statistical trading strategies. Prerequisite: 240 or equivalent.

STATS 362. Monte Carlo Sampling. Fundamentals of Monte Carlo methods. Generating uniform and nonuniform variables, random vectors and processes. Monte Carlo integration and variance reduction. Quasi-Monte Carlo sampling. Markov chain Monte Carlo, including Gibbs sampling and Metropolis-Hastings. Examples, problems and motivations from Bayesian statistics, computational finance, computer graphics, physics.