Mathematics in Finance

1. Mathematics in Finance 12/12/2007 Max Chen Department of Finance Ming Chuan University

2. How to lose a billion dollars in stock market • It is as difficult as making a billion dollars in stock market • But it’s trivial in derivatives market !

3. 3 Major Fields • Financial Mathematics • Financial Engineering • Financial Econometrics

4. Tools you have at hand • Probability • Mathematical Statistics • Stochastic Process • Linear Algebra • Real Analysis • Optimization • PDE • Numerical Approximation

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

6. 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

7. 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.

8. 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

9. 江平 • 江平年入逾亿：入选华尔街百位顶尖交易者(ZZ) • http://sparrow.yculblog.com/post.1969205.html

10. 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.

11. http://finance.yahoo.com/q/pr?s=jrmsx • Profile  As of 31-Oct-07 for: INTECH Risk-Mgd Stock Fund • FUND OVERVIEW Category:Large Blend • Fund Family:Janus • Net Assets:524.00M • Year-to-Date Return:9.65% • Fund Inception Date:28-Feb-03

12. INTECHEnhanced Investment Technologies LLC (INTECH) has managed institutional portfolios since 1987 – establishing one of the industry’s 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.

13. Philosophy:INTECH’s 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.

14. Dr. Fernholz pioneered mathematical investing with the publication of Stochastic Portfolio Theory and Stock Market Equilibrium in 1982. Today it is the basis of INTECH’s 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.

15. The Key Paper (also book) • Stochastic Portfolio Theory: An Overview , • Robert Fernholz, Ioannis Karatzas • 11/24/2006 • https://ww3.intechjanus.com/Janus/Intech/intech?command=researchListing#

16. 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

17. 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"

18. Who Makes Money in Wall Street and How? By Daniel Nathan http://ezinearticles.com/?Who-Makes-Money-in-Wall-Street-and-How&id=588239

19. How about “max”??? • What do I specialize? • Secrets? • Asset Management ! • and Asset/Liability Management…. • Science + Arts

20. Let’s talk about Math ! • Radon-Nikodym derivative • Feynman-Kac Theorem • PDE • Conditional Expectations • All about “no arbitrage” and “equilibrium” • No risk, no return ! • Benchmark

21. How about Stat? • Cointegration and Pairs Trading • Parameter and Model Uncertainty

22. Model Uncertainty • Without Model Uncertainty • With Model Uncertainty

23. Money, Money, Money • 2/20 rule in hedge fund industry

24. 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.

25. Q&A

26. 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.

27. 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.

28. 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.