Computer-aided [Financial] Mathematics Monographic Seminar

Andrzej Kozlowski

Michal Szurek

 
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The seminar will consist of two largely independent groups, one of them (led by professor Michal Szurek) devoted to general mathematics, the other one led by me, to Mathematica and financial mathematics. The seminar will begin with a 2-week intensive general introduction, given by professor Szurek, to doing mathematics with computer algebra systes, including particular Mathematica. After this introduction the groups will continue to work together in the computer lab, under the general direction of professor Szurek, but students taking my class will communicate with me by Internet and receive course materials and exercises to solve with the help of Mathematica. In second term two independent seminars will be held in different computer labs, one conducted by professor Szurek. and the other by myself. 




For more information about the general mathematics course please contact professor Szurek.

Financial Mathematics with Mathematica

Procedure

Students wishing to take the financial mathematics with Mathematica course should, in addition to attending the seminar lead by professor Szurek, also join a mailing list, through which they will communicate with me and will obtain access to an ftp server. The server will contain additional course materials, in the form of Mathematica notebooks and pdf files, explaining various methods of efficient Mathematica programming.  In the second term I will lead the seminar in person. This part of the course will be concerned with topics in mathematical finance. After my return to Japan, the course will contain through the Internet and will be concerned with solving a number of problems in mathematical finance, on which the assessment of the course will be based.

Course contents

First Part: Programming in Mathematica

The first part of the course, conducted by Internet, will be devoted to learning the essentials of the Mathematica programming language, which are vital in financial applications, and also on the new features of Mathematica 6, particularly new random number generation and the use of the new Dynamic features of the Mathematica Front End. Examples, mainly from finance (but not only) will also be included and also problems for testing student's understanding and skills. Students will be expected to learn not just to use Mathematica as a symbolic calculator, but also to program efficiently, to understand Mathematica's native programming language (quite different from that of other computer algebra systems), Mathematica's unique system of automatic precision control in extended precision computations, and to use the new abilities of the Mathematica 6.0 Front End.
  1. Interacting with Mathematica: the Kernel, The Front End, Input and Output Formats,The Help Browser
  2. The structure of a Mathematica expression
  3. Typesetting, Manipulating Notebooks, Dynamic Features etc.
  4. Using Add Ons - Standard and external.
  5. Functions: Built-in and user defined.
  6. Graphics as examples of expressions
  7. Functional programming in Mathematica
  8. Pattern based programing in Mathematica
  9. UpValues, DownValues etc.
  10. Functional Iteration
  11. Procedural Programing and Compiling procedural code
  12. Numerical Mathematics in Mathematica: Precision, Accuracy, Exact and Inexact numbers
  13. Interpolation functions and Numerical Soltutions of Algebraic and Differential Equations
  14. Radnom number generation in Mathematica and Monte-Carlo simulation.
  15. Designing and Making a Mathematica 6.0 Demonstration


Literature for the First Part

The basic textbook is   the Mathematica 6.0 Online Guide. Other  materials on specific topic will be provided by me to the course participants through the ftp server. 

There are many very good books on Mathematica and its programming language. They tend to refer to earlier versions of Mathematica and so do not cover everything that will be needed, but they still can be very useful One strongly recommended source are Michael Trott's Mathematica Guidebooks.



Second Part: Financial Mathematics

The financial mathematics course will be concerned primarily with the pricing of derivative instruments. Some of the following topics will be covered:
  1. An introduction to numerical and symbolic stochastic processes in m-dimensions, including simulation of continuous stochastic processes (Wiener process, Brownian Bridge, jump diffusion etc) as well as using a symbolic implementation of the Ito formula.
  2. The Black-Scholes PDE and the Black-Scholes formulas for the price of the European put and call options. Other options that can be priced with explicit formulas (based on Shaw's book).
  3. Pricing european options with the help of NIntegrate and NDSolve.
  4. Pricing of exotic options by means of the Monte-Carlo method.
  5. Pricing of American options by means of binomial trees and by other methods.
  6. Working with stock market statistics.

Prerequisites and Literature for the Second Part

Students should have a general background in probability theory and statistics and have some acquaintance with the basic notions of modern finance.

For general Mathematica usage and programing the main literature will be Mathematica notebooks written by me, will be provided to the course participants. Some published articles in PDF format will also be provided.

The official textbook for the financial part of the course is: William Shaw, Derivative Pricing with Mathematica. This book is basically a collection of formulas and computer code without much explanation. Note that this book was written for Mathematica v. 3 and later revised for Mathematica v. 4. Updated notebooks which work well with version 6 of Mathematica will be provided on the server (only the chapters that will actually be used in the course).

The above is the only literature that is actually required. However, the following books are highly recommended (PDF files of some selected sections of these books will be provided on the server).

First, there is the classic book:

John. C. Hull - Options, Futures and Other Derivatives, Fourth Edition, Prentice Hall (2000)

An excellent book, very relevant to the course is: Computational Financial Mathematics using Mathematica by Srdjan Stojanovic, Birkhauser 2003.

Another very useful book in Polish, which provides all the necessary non-computational background in mathematical finance is Matematyka Finansowa. Instrumenty Pochodne by Jakubowski, Palczewski, Rutkowski and Stettner (wydawnictwo Naukowo Techniczne, Warszawa 2003).