Download Mathematics in Finance

Mathematics in Finance
12/12/2007
Max Chen
Department of Finance
Ming Chuan University
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 Major Fields
• Financial Mathematics
• Financial Engineering
• Financial Econometrics
Tools you have at hand
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Probability
Mathematical Statistics
Stochastic Process
Linear Algebra
Real Analysis
Optimization
PDE
Numerical Approximation
• Any mathematicians make big fortunes?
• Yes, but not too many….
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
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.
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
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
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.
http://finance.yahoo.com/q/pr?s=jr
msx
• 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
• INTECH
Enhanced 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.
• 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.
• 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.
• The Key Paper (also book)
• Stochastic Portfolio Theory: An
Overview ,
• Robert Fernholz, Ioannis Karatzas
• 11/24/2006
• https://ww3.intechjanus.com/Janus/Intech/i
ntech?command=researchListing#
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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
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"
• Who Makes Money in Wall Street and How?
By Daniel Nathan
http://ezinearticles.com/?Who-Makes-Money-in-Wall-Street-and-How&id=588239
How about “max”???
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What do I specialize?
Secrets?
Asset Management !
and Asset/Liability Management….
Science + Arts
Let’s talk about Math !
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Radon-Nikodym derivative
Feynman-Kac Theorem
PDE
Conditional Expectations
All about “no arbitrage” and “equilibrium”
No risk, no return !
Benchmark
How about Stat?
• Cointegration and Pairs Trading
• Parameter and Model Uncertainty
Model Uncertainty
• Without Model Uncertainty
p( RT  K | D) 

 
p( j ,  j | D) p( RT  K |  j ,  j , D)d j d j
j, j
• With Model Uncertainty
p( RT  K | D)   j 1 P(  j | D)
2M

 
p( j ,  j |  j , D) p( RT  K |  j ,  j ,  j , D)d j d j
j, j

{(1   ' lN )exp(rf K )   'exp(rf KlN  RT  K )}1
(g)
*
 arg max 

RT  K
1 
 P( RT  K | D)dRT  K ,
1
1 G {(1   ' lN ) exp(rf K )   'exp(rf KlN  R T  K )}
[U (WT  K ( ))]}  
G g 1
1 
(g)
Money, Money, Money
• 2/20 rule in hedge fund industry
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.
Q&A
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.
• 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 SPlus, Springer.
• Tsay, Ruey S., (2002) , Analysis of
Financial Time Series, John Wiley and
Sons.
• 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.