Math 586 Computational Finance
Fall 2008
Professor Emeritus F. B. Hanson (hanson at uic dot edu, 507 SEO, x3-3041)
Time Table:
16494 LCD 01:00PM - 01:50PM MWF 315 BSB Hanson, F
Lecturer: F. B. Hanson, 507 SEO, please use email
(X6-3041msg)
- E-Mail: hanson at uic dot edu
- Instructor Web Page:
http://www.math.uic.edu/~hanson/
- Class Web Page (this page):
http://www.math.uic.edu:~hanson/math586/
- Office Hours: Please make arrangements for
- Professor Hanson will usually answer any questions right after class
outside 315 BSB, as well as questions on the lecture in class or by email
anytime.
Spring 2008 (Tentative):
Catalog description:
The course will present current topics in computational finance
emphasizing the pricing of financial derivatives such as stock options
and fixed income derivatives. The stress will be on the construction and
computation of the derivative prices and the optimal portfolio problem.
This will involve the solution of
stochastic differential equations and their related partial differential
equations. Analytic methods will introdcued and used to construct
pricing formulas, if possible. Numerical methods, based on
MATLAB, will
be presented and used to analyze both data and analytically intractable
models. Both diffusion (normal randomness) market environments and
jump-diffusions (normal with occasional jumps, i.e., crashes
and buying frenzies, type of randomness) will be studied.
Prerequisites: Student background: students should be familiar
with basic probability, differential equations and elementary numerical
methods.
Approximate List of Topics: ---- Hours:
- Introduction to derivatives - interest rates, forward and future
contracts; European and American stock options, combinations of options,
replication of contracts, valuations and profit and loss curves,
arbitrage and the principle of non-arbitrage pricing.
- Random behavior of assets: historical data, return statistics.
- Stochastic Differential Equations - what are they, how to solve,
relationship to partial differential equations.
- Option Pricing Models - derivation, integral and differential
equation formulations, theories of Bachelier, Black-Scholes, and Merton.
- Brief review of partial differential equations; backward and forward
diffusion equations, analytic solution of Black-Scholes model; free
boundary value problem for pricing American options, similarity
solutions
- Numerical solution of partial differential equations arising in
pricing models; finite difference methods - explicit and
implicit, Crank-Nicholson, methods for European and American options.
- Interest rate models - yield curves, FRA, Swaps, short rate models,
bond pricing, calibration, caps, swaptions
- Option Pricing in a jump-diffusion market.
- Optimal portfolio and consumption problem for jump-diffusion models.
Texts and References:
Text:
- John C. Hull, Options, Futures and Other Derivatives, 6th Edition,
Prentice-Hall, 2005 (see
Amazon.com for less expensive used copies.)
Useful References (Optional):
-
John C. Hull, Students Solutions Manual for Options, Futures, and Other
Derivatives, Sixth Edition, Prentice-Hall, 2005, (see
Amazon.com for less expensive used copies.)
-
Desmond J. Higham, An Introduction to Financial Option Valuation,
Cambridge University Press, 2004. (Excellent computational reference.)
-
Desmond J. Higham and Nicolas J. Higham, MATLAB Guide, SIAM Books,
2nd Edition, 2005,
Order Code OT92.
(There is a 30% discount with SIAM student membership and
student membership is free with UIC academic membership. )
-
Floyd B. Hanson,
Applied Stochastic Processes and Control for Jump-Diffusions: Modeling,
Analysis, and Computation,
SIAM Books: Advances in Design and Control Series,
Order Code DC13 (Hanson[100]),
published 03 October 2007, 28 + 441 pages, plus online appendices and
sample codes.
(There is a sale price with 30% off for non-members until
June 30, 2008, so is the same as the member price.
There is always a 30% discount with SIAM student membership and
student membership is free with UIC academic membership. Chapter 12 is
on Application in Financial Engineering.)
Some online material is freely available:
-
Phelim P. Boyle and Feidhlim Boyle,
Derivatives: The Tools That Changed Finance, Risk Books,
2000.
(There is a free chapter download at the above link.
Dr. Phelim Boyle wrote the pioneering and award winning paper on the
formulation of Monte Carlo simulation for financial applications
(Options: A Monte Carlo Approach, J. Fin. Econ., vol. 4, 1977,
pp. 323-338.) and his son Feidhlim Boyle runs a hedge fund.)
Useful Online Documents:
-
Roy Davies,
Gambling on Derivatives: Hedging Risk or Courting Disaster?,
University of Exeter (retired), UK, January 2008. Good, brief summary of
financial derivative history and disasters. Some good links to other
documentation too.
-
David Gauthier-Villars and Carrick Mollenkamp,
How to Lose $7.2 Billion: A Trader's Tale (Kerviel Cooked Books,
Skipped His Holidays; Calling in a Doctor),
Wall Street Journal, p. A1, 02 February 2008.
Well told story of
Jerome Kerviel, a "nut and bolts" trader,
who bet the whole Société Générale bank
and lost only US$7.2 billion, the most ever by a single trader.
-
Global Derivatives,
-
Floyd B. Hanson,
Stochastic Processes and Control for
Jump-Diffusions, under revision, 44 pages, 22 October 2007.
IISc (Bangalore, INDIA) Stochastics Workshop Notes, February 2007.
(This is a brief tutorial on the main topics of Prof. Hanson's book, but
more from the view of generalizations of ordinary differential equations
to stochastic differential equations in stages, with applications. This
version is very appropriate for Math 586 Spring 2008)
-
Floyd B. Hanson,
Publications in Computational Finance and Bioeconomics with
Abstracts,
for example, you can download any of recent preprints:
- Zongwu Zhu
and
Floyd B. Hanson,
"
A Monte-Carlo Option-Pricing Algorithm for Log-Uniform Jump-Diffusion
Model
,
Proceedings of Joint 44nd IEEE Conference on Decision
and Control and European Control Conference, pp. 1-6, 12 December
2005.
- Guoqing Yan
and
Floyd B. Hanson,
"
Option Pricing for a Stochastic-Volatility Jump-Diffusion Model with
Log-Uniform Jump-Amplitudes,"
Proceedings of American Control Conference, pp. 2989-2994,
14 June 2006.
- Zongwu Zhu and Floyd B. Hanson,
"
Optimal Portfolio Application with Double-Uniform Jump
Model,"
Stochastic Processes, Optimization, and Control Theory:Applications
in Financial Engineering, Queueing Networks
and Manufacturing Systems/A Volume in Honor of Suresh Sethi,
International Series in Operations Research & Management Sciences,
Vol. 94, H. Yan, G. Yin, Q. Zhang (Eds.), Springer Verlag, New York,
pp. 331-358, Invited chapter, June 2006.
- Floyd B. Hanson
and
Guoqing Yan,
"
American Put Option Pricing for Stochastic-Volatility, Jump-Diffusion
Models,"
Proceedings of 2007 American Control Conference, pp. 384-389,
11 September 2007.
- Floyd B. Hanson,
"Optimal Portfolio Problem for Stochastic-Volatility,
Jump-Diffusion Models with Jump-Bankruptcy Condition: Practical
Theory,"
Fifth World Congress of Bachelier Finance Society, 2008,22 pages,
submitted 03 January 2008.
-
Floyd B. Hanson,
Math 586 Spring 2008 - Quantitative Finance References
and Related References.
A more extensive list of references with added descriptions.
-
Floyd B. Hanson,
Help for MATLAB Matrix Numeric Tools.
Many links to MATLAB Help, including Professor Hanson's pages of Format
and Functions.
-
John C. Hull,
John Hull's Web Site, Rotman School of Management,
University of Toronto.
-
John C. Hull,
John Hull's Technical Notes for Options, Futures, and
Other Derivatives, Sixth Edition, Rotman School of Management,
University of Toronto.
-
Technical Note No. 4:
Exact Procedure for Valuing American Calls on Dividend-Paying Stocks.
This is the Roll, Geske, and Whaley (RGW) formula according to Hull, but likely Technical
Note No. 5 will be needed to approximate the bivariate normal distribution
M(a,b,rho) = Phi(a,b;rho) in the formula by Hermite Gaussian quadtature
approximation.
-
Technical Note No. 5:
Calculation of Cumulative Probability in Bivariate Normal Distribution
.
This is helpful for Roll, Geske, and Whaley (RGW) formula for calculating
the bivariate normal distribution in Technical Note 4 using the
Hermite Gaussian quadrature approxmation of fourth order in 7 significant
digits of Drezner paper cited in this technical note. Note that the
standard univariate normal distribution N(x) = Phi(x), so
the robust MATLAB erfc function can be used:
N(x)=Phi(x) = 0.5*erfc(-x/sqrt(2));
or the univariate approximate version of Hull's note with his weights A(i) and
nodes B(i) for i=1:4 could be used
with f(u,x1) = exp(x1*(2*u-x1)); x1=x/sqrt(2);
if x<=0:
N(x)=Phi(x) = sum_{i=1:4}A(i)*f(B(i),x1);
if x>=0:
N(x)=Phi(x) = 1-Phi(-x) = sum_{i=1:4}A(i)*f(B(i),-x1);
Caution: The formulas for N(x) and M(x,y,rho) in this technical notes,
as in others technical notes or papers, should be verified using some known
test examples like when x = 0 or x >> 1.
-
Technical Note No. 6:
Differential Equation for Price of a Derivative on a Stock Providing a
Known Dividend Yield
.
This is another note related to the dividend problem,
but when the dividend yield is constant.
-
Andrew W. Lo,
How To Tell If You Might Be A Quant, MIT Laboratory for
Financial Engineering, Cambridge, MA, September 2007.
Professor Lo's amusing list of questions for determining your
"Quant" qualifications. See also his main webpage:
Andrew Lo's Homepage.
-
Mathworks,
The MathWorks Store for buying the Student Version of
MATLAB, $99?
Includes: MATLAB, Simulink, Control System Toolbox, Signal Processing Toolbox,
Signal Processing Blockset, Statistics Toolbox, Optimization Toolbox,
Image Processing Toolbox, Symbolic math functions. A great bargain,
since the total of list prices would be $2K-$3K for a non-student.
-
Peter A. McKay,
Old and New Secure: A Place at Options Table,
Wall Street Journal, Tracking the Numbers: Street Sleuth Blog,
January 24, 2006, Page C3.
Describes the transformation of The Chicago Board of Options Exchange
to electronic trading of options.
-
Numa Financial Systems, Ltd.,
Numa: The Internet Resource Center For Financial Derivatives.
Lots of useful links for References, Calculators, Indexs and more.
-
Bernt Arne Ødegaard,
Financial Numerical Recipes in C++,
Department of Financial Economics, BI Norwegian School of Management,
Oslo, Norway, October 2003.
Nicely designed webpage of financial numerical recipies with descriptions and
code from Ødegaard. Check it out, but verify as with all codes.
-
Options Clearing Corporation (OCC),
The Equity Options Strategy Guide,
The Options Industry Council (Options Education), January 2007.
Good options information documentations that clearly describes the
profits and losses of many types of options by words and graphs. It
also has explanations of may option related terms.
Highly recommended for Math 586.
-
Jan Verschelde,
MCS 320: Introduction to Symbolic Computation,
Help for Maple and MATLAB.
Other Texts:
-
Paul Wilmott, Sam Howison and Jeff Devine, Mathematics of Financial
Derivatives, A Student Introduction, Cambridge University Press, 1995.
(One of the oldest Math 586 texts. Wilmott has a long series of much
larger texts that he updates every several year under different titles.)
-
Stanley R. Pliska, Introduction to Mathematical Finance: Discrete Time
Models, Blackwell, 1997. (Professor Pliska is a co-founder of
the
Computational Finance Track with Professors Hanson and Tier. He
usually used this discrete-time finance book in one of the track main
core courses, Fin 551 Financial Decision Making. Math 586 is the second
main core course, but emphasizes continuous-time finance.)
-
Thomas Mikosch, Elementary Stochastic Calculus with Finance in
View, World Scientific, 1998. (Clearly written and short continuous-time
stochastic diffusion text.)
-
Claudio Albanese and Giuseppe Campolieti, Advanced Derivative Pricing and
Risk Management: Theory, Tools, and Hands-On Programming
Applications, Elsevier/Academic Press, 2006. (This text was used
by Tier for Math 586 Fall 2007 and has extensive financial analytical
and computational material.)
-
Alexander Lipton, Mathematical Methods for Foreign Exchange,
World Scientific, 2001. (Former Professor in MSCS, UIC. He is not
at Merrill Lynch in London and previously was at Citadel in Chicago,
but has worked at many financial institutions worldwide. This book is
more general than the foreign exchange topic in the title.)
-
Salih N. Neftci, An Introduction to the Mathematics of Financial
Derivatives, Academic Press, 2000. (This text was used at least
once by Professor Yau.)
-
Martin Baxter and Andrew Rennie, Financial Calculus: An Introduction to
Derivative Pricing, Cambridge University Press, 1996.
-
Robert C. Merton, Continuous-Time Finance, Blackwell, 1990.
(Mostly a collection of reprinted papers by one of the giants of mathematical
finance.)
Math 586 Homework:
Math 586 Spring 2008 Demos*:
-
Federal Funds Rates (%) r(t) and betahat(t), 1988-2004,
Time-Dependent Interest and Discount Rates Demonstration,
14 Jan 2008 (Hanson[97]).
-
S&P500 Stock Index Log-Returns, dLog(SP), 1995-2001.5,
Randomness Demonstration, 16 Jan 2008 (Hanson[80]).
-
Histogram of Empirical S&P500 Stock Index Log-Returns, frequency f^(SP),
for 2000 ,
Sampled Stock Random Distribution Demonstration, 16 Jan 2008 (Hanson[97]).
-
Histogram of Fitted S&P500 Log-Returns, frequency f^(fitSP),
for 2000,
Fitted Stock Random Distribution Demonstration for Jump-Diffusion,
16? Jan 2008 (Hanson[97]).
-
Diffusion Volatility and Drift Coefficients, S&P500, 1988-2004,
Time-Dependent Coefficient Demonstration, especially of volatility,
16 Jan 2008 (Hanson[97]).
-
Linear-Diffusion Stock Price, S(t),
Stock Price Linear-Diffusion Simulation Demonstration,
23 Jan 2008 (Math586Spr2008).
-
Integral of (dW)2(t) Simulations,
Computer Motivation for Fundamental Result, (dW)2(t) = dt,
25 Jan 2008 (Math586Spr2008).
- Normal and Lognormal Histograms, 01 Feb 2008:
* Notation like "(Hanson[97])" denotes Demo is from
number "[97]" on
Professor Hanson's publication list.
Math 586 Spring 2008 Codes:
-
Linear-Diffusion Stock Price, S(t),
Stock Price Linear-Diffusion Simulation (MATLAB) Code,
18 Jan 2008 (Math586Spr2008).
-
Hanson[100] Book Fig. 2.1 Example MATLAB code for integral of (dW)^2,
Computer Motivation for Fundamental Result, (dW)2(t) = dt,
25 Jan 2008 (Math586Spr2008).
- RGW approximation for early-exercised American call options codes
from Global Derivatives (Sivakumar Batthala, GSB, UofC):
-
RGW. See the Readme.m for information and rogewhaley.m
for the main m-file, which needs bisect.m, normcdfM.m, bivnormcdf.m
and bsprice.m function files.
Prior Math 586 Computational Finance, Professor Tier, Spring
2007.
Jobs in Quantitative Analysis.
Web Source: http://www.math.uic.edu/~hanson/math586/
Email Comments or Questions to
hanson at dot edu