Mathematics in Finance

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Mathematics in Finance

• 12/12/2007
• Max Chen
• Department of Finance
• Ming Chuan University

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How to lose a billion dollars in stock market

• It is as difficult as making a billion dollars in
  stock market
• But its trivial in derivatives market !

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3 Major Fields

• Financial Mathematics
• Financial Engineering
• Financial Econometrics

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Tools you have at hand

• Probability
• Mathematical Statistics
• Stochastic Process
• Linear Algebra
• Real Analysis
• Optimization
• PDE
• Numerical Approximation

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• Any mathematicians make big fortunes?
• Yes, but not too many.

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James Harris Simons

• Renaissance Technologies is a hedge fund
  management company. Renaissance was started by
  James Simons in 1982. At March 31, 2007, it held
  some 27 billion in public equity positions
• Chern-Simons form (aka Chern-Simons invariants,
  or Chern-Simons theory )
• http//en.wikipedia.org/wiki/James_Harris_Simons

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David E. Shaw

• Founded by David E. Shaw. Having a Ph.d in
  Computer Science could not have hurt Shaw. He
  made a fortune building automated trading systems
  which exploited anomalies in the stock market.
  Fortune magazine referred to him as King Quant.
  The firm manages approximately US 29 billion in
  aggregate capital.
• Employment opportunities at D. E. Shaw are known
  to be extremely competitive. A notable past
  worker at D.E Shaw is Jeff Bezos, the founder of
  Amazon.

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http//www.deshaw.com/index.html

• Quantitative Analyst
• Quants apply mathematical techniques and write
  software to develop and analyze statistical
  models for our computerized financial trading
  strategies. Specific responsibilities range from
  examining trading data in an effort to increase
  profitability, decrease risk, and reduce
  transaction costs to conceiving new trading ideas
  and devising the simulations needed to test them.
  Successful quant candidates have traditionally
  been the top students in their respective math,
  physics, engineering, and computer science
  programs a considerable number have also
  competed successfully in the United States and
  International Math Olympiads as well as the
  Putnam Competition.
• http//en.wikipedia.org/wiki/David_E._Shaw

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• http//sparrow.yculblog.com/post.1969205.html

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E. Robert Fernholz

• Lead Manager since 28-Feb-03Fernholz is the
  director and executive vice president and chief
  investment officer with Enhanced Investments
  Technologies, LLC. He joined the firm in June of
  1987, and was formerly director of research at
  Metropolitan Securities. He has more than 26
  years of investment experience.

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http//finance.yahoo.com/q/pr?sjrmsx

• Profile  As of 31-Oct-07 for INTECH Risk-Mgd
  Stock Fund
• FUND OVERVIEW CategoryLarge Blend
• Fund FamilyJanus
• Net Assets524.00M
• Year-to-Date Return9.65
• Fund Inception Date28-Feb-03

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• INTECHEnhanced Investment Technologies LLC
  (INTECH) has managed institutional portfolios
  since 1987 establishing one of the industrys
  longest continuous records of mathematically
  driven equity investing strategies. INTECH is one
  of the fastest growing and most successful money
  managers in the U.S. and is available in Canada
  to retail mutual fund investors exclusively
  through AGF.

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• PhilosophyINTECHs unique investment process
  is based on a mathematical theorem that attempts
  to capitalize on the random nature of stock price
  movements. The goal is to achieve long-term
  returns that outperform the benchmark index,
  while controlling risks and trading costs.

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• Dr. Fernholz pioneered mathematical investing
  with the publication of Stochastic Portfolio
  Theory and Stock Market Equilibrium in 1982.
  Today it is the basis of INTECHs investment
  strategy. He has held various academic positions
  in Mathematics and Statistics at Princeton
  University, City University of New York,
  Universidad de Buenos Aires and the University of
  Washington. Dr. Fernholz speaks extensively
  around the world about his work in the field of
  mathematical finance and his research continues
  to advance new and innovative ideas.
• He received his Ph.D in Mathematics from Columbia
  University and holds an A.B. in Mathematics from
  Princeton University.

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• The Key Paper (also book)
• Stochastic Portfolio Theory An Overview ,
• Robert Fernholz, Ioannis Karatzas
• 11/24/2006
• https//ww3.intechjanus.com/Janus/Intech/intech?co
  mmandresearchListing

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• Anyone else?
• Yes, to name a few
• Merton, Nobel Prize winner
• Derman, My Life as a Quant
• Litterman, Research Head, Goldman Sachs
• Mulvey, Zenios, Dempster

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How Complex Math Formulas And "Quant Funds"
Failed Wall Street

• LTCM
• Long Term Capital Market was a hedge fund founded
  in 1994 by John Meriwether. It had Myron Scholes
  and Robert Merton on its board, two Nobel Prize
  Winners! At its peak, it made about 40 return
  for its investors. After being heavily leveraged,
  it went bankrupt.
• The LTCM example teaches you that no matter how
  smart you are, there are some things that are out
  of our reach. A good book on LTCM is "When Genius
  Failed The Rise and Fall of Long-Term Capital
  Management"

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• Who Makes Money in Wall Street and How? By
  Daniel Nathan

http//ezinearticles.com/?Who-Makes-Money-in-Wall-
Street-and-Howid588239
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How about max???

• What do I specialize?
• Secrets?
• Asset Management !
• and Asset/Liability Management.
• Science Arts

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Lets talk about Math !

• Radon-Nikodym derivative
• Feynman-Kac Theorem
• PDE
• Conditional Expectations
• All about no arbitrage and equilibrium
• No risk, no return !
• Benchmark

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How about Stat?

• Cointegration and Pairs Trading
• Parameter and Model Uncertainty

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Model Uncertainty

• Without Model Uncertainty
• With Model Uncertainty

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Money, Money, Money

• 2/20 rule in hedge fund industry

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Advanced Education

• Within the last few years, mathematics
  departments at several universities have
  introduced professional master's degree programs
  in financial mathematics. Joining forces in an
  effort to bring the new programs to the attention
  of both the industry and other universities,
  three of the programs--at Chicago, Columbia
  University, and the Courant Institute of
  Mathematical Sciences, New York University--have
  founded the Association of Financial Mathematics
  Programs.

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QA
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Reference

• Salih N. Neftci, 2000, 2nd edition, An
  Introduction to the Mathematics of Financial
  Derivatives. Academic Press.
• BAXTER, Martin and RENNIE, Andrew, Financial
  Calculus an introduction to derivative pricing,
  Cambridge University Press, 1998.
• Hull, J. (2003) Option, Futures, and other
  Derivatives, 5th edition.

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• Chi-Fu Huang and Robert H. Litzenberger (1988)
  Foundations for Financial Economics. Prentice
  Hall.
• Eric Zivot and Jiahui Wang, 2003, Modeling
  Financial Time Series with S-Plus, Springer.
• Tsay, Ruey S., (2002) , Analysis of Financial
  Time Series, John Wiley and Sons.

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• Musiela, M. and Rutkowski, M., 1998, Martingale
  Methods in Financial Modelling, 2nd Edition,
  Springer Finance.
• Thomas Bjork (1998) Arbitrage Theory in
  Continuous Time. Oxford University Press.
• WILMOTT, Paul, HOWISON, Sam and DEWYNE, Jeff, The
  Mathematics of Financial Derivatives a student
  introduction, Cambridge University Press, 1998.
• OKSENDAL, Bernt, Stochastic Differential
  Equations an introduction with applications,
  Springer-Verlag, 1998.