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Mathematical finance is a collection of mathematical results that attempts
to provide quantitative explanations of the behavior of financial markets.
It could be considered a subdiscipline of mathematical economics, which is
broadly concerned with mathematical explanations of any "economic" behavior.
It is sometimes referred to as "computational finance", emphasizing the need
for specific numeric results and the use of computers to provide them; as such it relies on econometrics, the
measurement and statistical analysis of economic quantities.
Mathematical finance as a discipline has been strongly motivated by the
need to price derivative instruments, representing put or call options on
underlying securities, or combinations thereof. In general, as financial
market products become more complex, quantitative analysis becomes essential
for portfolio management and marketing.
Landmark results in mathematical finance include
mean-variance optimization
for portfolio asset allocation, which earned a Nobel prize for Harry Markowitz (shared with Merton Miller and William Sharpe), and the
Black-Scholes model
for option pricing, which earned one for Robert Merton and Myron Scholes
(Fischer Black had died before the prize was awarded).
Financial engineering applies mathematical finance results, along with
techniques such as statistical data analysis, mathematical optimization, stochastic
process modeling, and Monte Carlo simulation to the conduct of financial business.
Application areas include investment banking, equity and credit risk management,
portfolio management, primary and derivative securities valuation, financial product
development, corporate strategic planning, and many more.
We are particularly interested in the use of
Mathematica
as a tool for financial engineering. Mathematica can be useful in several contexts:
For the development of complete standalone products; see the
links below for examples.
As a rapid prototyping tool for evaluating algorithms prior to recoding them in
some other programming language.
As a computational server in an integrated application and delivered
using client/server or web technology. (See our
MathLink
page for more details.)
Links
General information
Risk management
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The term risk management refers to organizational processes
or specific methodologies used to assess exposure to various forms of
risk, and
evaluate alternatives to reduce risk to acceptable levels.
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GARP (Global Association of Risk Professionals) -
news, links, research, bookstore.
IFCI (International Financial Risk Institute) - risk
management concepts, glossary, methods of computing risk exposures, international regulatory documents.
Glyn Holton's Contingency Analysis -
links, glossary, tutorials, and research on financial risk management.
GloriaMundi.org has links and tutorials on
VaR (Value at Risk)
and other risk management methodologies.
BobsGuide risk management page.
Risk management glossaries:
from Contingency Analysis;
from the Treasury Management Association
of Canada.
Portfolio management
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A portfolio (collection of financial assets such as stocks, bonds, derivative
instruments, etc.) has properties derived from the asset collection, such as portfolio
risk and return. Management to meet specified goals overlaps risk management,
but includes other aspects as well.
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Mathematica in mathematical finance/financial engineering
Other mathematical tools
Books on Mathematica in finance and economics
Lewis, Alan L.
Option Valuation under Stochastic Volatility: With Mathematica Code.
Finance Press, 2000.
Fairly advanced mathematics. The book can be read without Mathematica
knowledge (Mathematica code is in appendices). Several sample chapters can be downloaded from
http://www.optioncity.net/publications.htm.
Huang, Cliff J. and Philip S. Crooke.
Mathematics and Mathematica for Economists.
Blackwell Publishers, 1997. A graduate or senior undergraduate
text on mathematical economics, using Mathematica (includes an introduction to Mathematica).
Mathematica code, including solutions to problems, can be downloaded from
http://www.math.vanderbilt.edu/~pscrooke/b.
Shaw, William T.
Modeling Financial Derivatives With Mathematica.
Cambridge University Press, 1998.
Includes an introduction to Mathematica and mathematical finance
preliminaries as well as a wide range of derivative modeling techniques.
Includes a disk with Mathematica code.
Stinespring, John Robert.
Mathematica for Microeconomics.
Springer Verlag, 2002.
A supplemental text for courses in advanced microeconomics, using
Mathematica as a computational tool.
Includes a disk with Mathematica code.
Stojanovic, Srdjan.
Computational Financial Mathematics using Mathematica.
Birkhauser, 2002. Mathematica techniques for pricing and modeling
cash account evolution, stock price evolution, European and American options,
implied option volatility, optimal portfolio rules,
and trading strategies.
Varian, Hal (editor).
Computational Economics and Finance: Modeling and Analysis With Mathematica.
Telos, 1996. A collection of chapters by various authors on implementing
fundamental techniques such as financial data analysis and optimization methods.
Includes a disk with Mathematica code.
General books
Links in this list are to information on the books at Amazon.com:
Alexander, Carol.
Market Models: A Guide to Financial Data Analysis.
John Wiley & Sons, 2001.
A comprehensive
treatise on the use of market data to develop models for financial analysis.
Mathematical prerequisites include statistics and calculus. Includes a disk
with Excel spreadsheet models.
Anson, Mark J. P.
Credit Derivatives. John Wiley & Sons, 1999.
An overview of credit risk and the use of derivatives to manage it.
Assumes knowledge of the credit market.
Downes, John, and Jordan Elliot Goodman.
Barron’s Dictionary of Finance and Investment Terms.
6th Ed., 2003. Comprehensive, in-depth entries defining finance and investment terms and concepts.
Convenient small size, durable binding. There is considerable overlap between this and other books
in the Barron’s Business Dictionary series. This and Fitch are particularly
recommended, and will probably cover the needs of most people, whether managers or professionals.
Downes, John, and Jordan Elliot Goodman.
Barron’s Finance and Investment Handbook.
5th Ed., 1998. A comprehensive (1400 pp.) directory and reference book for investors. Includes
a 550 page dictionary of finance and investment, comparable to
Barron’s Dictionary of Finance and Investment Terms. A useful desk reference.
Fitch, Thomas.
Barron’s Dictionary of Banking Terms.
4th Ed., 2000. Comprehensive, in-depth entries on banking and related investment terms and concepts.
Convenient small size, durable binding. There is considerable overlap between this and other books
in the Barron’s Business Dictionary series. See also Downes & Goodman.
Miller, Ross M.
Computer-Aided Financial Analysis. Addison-Wesley, 1990. A broad
survey of computational financial analysis techniques. Code examples are
given in Lisp, a disadvantage for many readers.
Michaud, Richard O.
Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation.
Oxford University Press, 1998. Deals with
Markowitz mean-variance optimization:
pros and cons, variants, and related methods.
Neftci, Salih N.
An Introduction to the Mathematics of Financial Derivatives.
Academic Press, 2000 (2nd edition).
A very well-written book on the topic. Though the author insists that
"a strong background in calculus or stochastic processes is not needed",
the reader without some knowledge of stochastic processes and partial
differential equations will find the book hard going.
Ross, Sheldon M.
An Elementary Introduction to Mathematical Finance: Options and other Topics.
Cambridge University Press, 2002 (2nd edition).
Includes well-written treatments of arbitrage, Brownian notion, and derivation
of the Black-Scholes
formula. The reader should have some knowledge of probability and statistics,
but the relevant concepts are carefully reviewed.
Shim, Jae K., and Joel G. Siegel.
Barron’s Dictionary of International Investment Terms.
2001. A useful supplement to Downes & Goodman
in the Barron’s Business Dictionary series for those involved in
international investment and finance.
Tavakoli, Janet M.
Credit Derivatives: A Guide to Instruments and Applications.
John Wiley & Sons, 2001 (2nd edition).
A comprehensive guide, dense with information; pays lots of attention to
fundamental issues of risk. Not for the beginner - assumes a knowledge of
the credit market.
Example products
Algorithmics - risk measurement and management
tools based on their "Mark-to-Future" framework.
FinancialCAD offers Fincad XL, a set of integrated
financial modeling and risk management products for many types of instruments, and Fincad Developer,
a toolkit for developing custom applications.
Financial Engineering Associates -
developers of risk analytics software.
NumeriX - analytic tools for
structured products pricing and risk management.
SunGard Trading and Risk Systems
provides integrated solutions for trading, risk management,
ALM, and financial planning and forecasting.
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