Advanced analysis topics from current problems of interest. May be repeated for credit if different topics are covered.

Sample Offerings:

Stochastic Calculus for Finance

This special topics course is intended as an introduction to some basic ideas for modeling and simulation in finance. The course begins with a discussion of simple random walks and the analysis of certain gambling games. These topics are used to motivate the theory of martingales and continuous time stochastic processes. The course will then take up the Ito integral and enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution. The foundations for the martingale theory of arbitrage pricing are then prefaced by a well-motivated development of the martingale representation theorems and Girsanov theory.

Prerequisites: Some analysis beyond calculus, an introduction to linear algebra, and basic methods from probability and statistics.

Computational Finance

This special topics course is intended provide hands on familiarity with simulation of financial models. This will be a "soft" computing course, i.e., we will not prove convergence of the approximations used in the simulations. The course will make use of Maple and MatLab programs from the literature: e.g.,

• D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review 43 (2001), 525-546.
• D. J. Higham and P. E. Kloeden, Maple and MatLab for stochastic differential equations in finance, research report.
• D. J. Higham, Nine ways to implement the binomial method of option valuation in MatLab, research report.
• S. Cygnowski, L. Grüne, and P. E. Kloeden, Maple for stochastic differential equations, research report.

Prerequisites: Basic concepts from probability and stochastic processes. Familiarity with martingales, MatLab, and Maple would be helpful but not necessary.