University of Michigan Department of Mathematics

Math 623 - Computational Finance

Page Updated: 4/29/2009

Prerequisites: A solid background in probability theory, differential equations, mathematical finance and computer programming.

Credit: 3 credit hours

Recommended Texts: Interest Rate Models: Theory and Practice by Brigo and Mercurio, Springer 2007.
The Mathematics of Financial Derivatives by Wilmott, Howison and Dewynne, Cambridge 1998.
Monte Carlo Methods in Finance by Jaeckel, Wiley 2002.

Background: Sophisticated mathematical methods are becoming ever more important in the financial industry. This trend started with the advent of exchange traded options in the 1970's and the fundamental research by Black, Merton and Scholes, later awarded the Nobel Prize. Today, Financial Mathematics is used not only by investment banks and hedge funds on Wall Street, but also by energy companies and large corporations in need of managing financial risk.

Goals: Using Financial Mathematics (like many branches of applied mathematics) in practice involves two tasks. First, one has to develop mathematical models that accurately describe the ``real'' objects that one wishes to study. In our case, this typically means finding models, based on probability theory, for the evolution of stock prices, interest rates and other underlying financial quantities. It also means to derive theoretical equations or formulas for prices of derivative securities (options, caps, etc).
The second task is to get actual numbers out of these equations. Doing so involves both calibration, ie finding the parameters in the models (eg the volatility of a stock) from financial data, and numerically solving the equations that were obtained theoretically.
While the two tasks cannot, in practice, be separated from each other, we will in this course emphasize the second of them. Namely, the students are assumed to be familiar with the more theoretical parts of mathematical finance (at the level of Math 542/IOE 552) and we will focus on the numerical implementing and calibration of financial models.

Contents: . The course has four components. In the first part we study finite difference methods. This is a technique for solving partial differential equations (PDEs) numerically. We will apply this to the Black-Scholes equation as well as PDEs appearing in fixed income. Both explicit and implicit schemes will be discussed, as will concepts such as stability and convergence.
After that we will turn to Monte Carlo simulations a quite general method for computing expected values numerically. The importance of this method stems from risk-neutral pricing, an important principle saying that prices of derivative securities can often be expressed as a the discounted expected payoff under a risk-neutral measure. In addition to ``vanilla'' Monte Carlo, which is normally quite easy to understand and implement, we will study variance reduction techniques, which are often necessary for obtaining accurate results.
The third part involves lattice and tree methods. The idea behind this method is to approximate a continuous-time stochastic process by discrete time process and then compute prices using this approximate model. Alternatively, lattice models can be used as financial models in their own right.
Finally, we shall discuss calibration, that is, finding parameters in the models from available financial data. As we shall see, calibration to stock prices or options data is done quite differently.
In the homework, which forms an integral part of the class, you will implement many of the models yourself in a computer language of your choosing.

Grading: The grade for the course will be determined from performances on 6 homeworks (45%) and a final exam (55%).
- My hand written lecture notes.
- Additional hand written lecture notes.
- Interest rate derivative lecture notes 2008.
- Notes on using MATLAB. postscript
- Numerical Methods for PDE. Math 572
- Review article on interest rate models. Review
- Article on the BDT interest rate model. BDT Model
- Homework I.
- Homework I Solutions.
- Homework II.
- Homework II Solutions.
- Homework III.
- Homework III Solutions.
- Homework IV.
- Homework IV Solutions.
- Homework V.
- Homework V Solutions.
- Homework VI.
- Homework VI Solutions.
- Winter 2009 final exam: Friday April 24, 4.00-6.00 pm in East Hall 1360.
  The final is a closed book exam. You may bring up to ten one sided A4 pages of notes to the exam. You may also use a calculator but not its memory function.
- Fall 2003 final exam.