MATH 294: MATHEMATICS OF FINANCE (WINTER 2012)
Professor R. J. Williams, AP&M 6121.
Class Time: M 4-5.30pm, W 4-5pm.
Class Meeting Place: AP&M B412.
Professor Office Hours: M 3-3.45pm, W 2-3pm.
This course is an introduction to the mathematics of financial models at the
The aim is to provide students with an introduction to some basic
models of finance and the associated mathematical machinery.
The course will begin with the development of the basic ideas of hedging and
pricing by arbitrage
in the discrete time setting of binomial tree models.
Key probabilistic concepts of conditional
expectation, martingale, change of measure, and representation,
will all be introduced first in this
simple framework as a bridge to the continuous model setting.
Mathematical fundamentals for the development and analysis of continous time
models will be covered, including Brownian motion, stochastic calculus, change
of measure, martingale representation theorem. These will then be combined
to develop the Black-Scholes option pricing formula.
Pricing and hedging for European and American call
options may be discussed.
As time allows, additional topics will be discussed, possibly
including models of the interest rate market.
Introduction to the Mathematics of Finance, R. J. Williams,
American Mathematical Society, 2006.
AMS members receive a discount if they buy the book directly
from the AMS.
PREREQUISITES: A course in probability or consent of instructor.
A possible probability course is Math 280AB (Graduate Probability).
However, other probability courses may be used in place of this with the
consent of the instructor. Some knowledge of conditional
expectation and martingales is an asset.
For background reading, students may wish to look at
the books below by Billingsley or Chung.
Math 286 (Stochastic Differential Equations) is a very useful
complement to Math 294 and students may find it helpful to take Math 286
before or after Math 294.
HANDOUT: Click here for the course handout.
HOMEWORK: Click here for homework.
INTERESTING WEB LINKS:
Frontline note relating to Proctor and Gamble's swap agreement with Banker's Trust.
Darrel Duffie's website
with articles on various financial market policy issues.
Article by Jeremy Kress in Harvard Journal on Legislation, 2011, related to clearinghouses for swaps.
New York Times Article about current use
of credit default swaps, January 30, 2012.
Probability and Measure, P. Billingsley, Wiley.
A Course in Probability Theory, K. L. Chung, revised second edition, Academic Press.
Probability: Theory and Examples, R. Durrett, Third edition, Duxbury Press.
Introduction to Stochastic Integration, K. L. Chung and R. J. Williams,
Birkhauser, Boston, Second Edition, 1990.
Continuous Martingales and Brownian Motion, D. Revuz and M. Yor,
Springer, Third Edition, 1999.
Background in Probability and Stochastic Calculus:
Background in Economics/Finance:
Investment Science, David G. Luenberger, Oxford University Press, 1998.
Financial Economics, H. H. Panjer (ed.),
Actuarial Foundation, Schaumburg, Illinois, 1998.
The Oxford Guide to Financial Modeling, T. S. Y. Ho and S. B. Lee, Oxford, 2004.
Options, Futures and other Derivative Securities, J. Hull, Prentice Hall, Fifth Edition.
Mathematics of Finance: Stochastic Approaches
An Elementary Introduction to Mathematical Finance, S. M. Ross,
Second Edition, Cambridge, 2003.
The Mathematics of Finance: Modeling and Hedging, J. Stampfli
and V. Goodman, Brooks/Cole, 2001.
Introduction to Mathematical Finance: Discrete Time Models, S. Pliska,
Stochastic Finance -- an Introduction in
Discrete Time, H. Follmer and A. Schied, de Gruyter, 2002.
Economics and Mathematics of Financial Markets, J. Cvitanic and
F. Zapatero, MIT Press, 2004.
Stochastic Calculus for Finance: Vol I and II, S. Shreve, Springer, 2004.
Financial calculus, Martin Baxter and Andrew Rennie, Cambridge University Press,
A Course in Financial Calculus, A. Etheridge, Cambridge University Press,
Introduction to Stochastic Calculus Applied
to Finance, D. Lamberton and B. Lapeyre,
Chapman and Hall, 1996.
Arbitrage Theory in Continuous Time, T. Bjork, Oxford University
An Introduction to the Mathematics of Financial Derivatives,
Salih N. Neftci, Academic Press, 1996.
Stochastic Calculus and Financial Applications, J. M. Steele, Springer, 2001.
Risk-Neutral Valuation, N. H. Bingham and R. Kiesel, Springer, 1998.
Financial Markets in Continuous Time, R.-A. Dana and
M. Jeanblanc, Springer, 2003.
Martingale methods in financial modeling, M. Musiela and M. Rutkowski, Second edition, Springer, 2005 (currently out of print).
Mathematics of Financial Markets, R. J. Elliott and P. E. Kopp,
Essentials of Stochastic Finance, A. N. Shiryaev, World Scientific,
Mathematics of Finance: PDE Approach
The Mathematics of Financial Derivatives: A student introduction, Paul Wilmott, et al., Cambridge
University Press, 1995.
Numerical Methods in Finance
Numerical methods in finance, L. C. G. Rogers and D. Talay, Cambridge
University Press, 1997.
Monte Carlo methods in financial engineering, P. Glasserman, Springer, 2004
Tools for Computational Finance, R. Seydel, Springer, 2004.
Mathematics of Finance: more advanced stochastic
Methods of mathematical finance, I. Karatzas
and S. Shreve, Springer, 1998.
Financial Derivatives in Theory and Practice, P. J. Hunt and J. E. Kennedy
, Wiley, 2000.
Derivatives in Financial Markets with Stochastic Volatility,
J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Cambridge University Press, 2000.
Interest Rate Models -- Theory and Practice, D. Brigo and F. Mercurio,
Credit Risk: Modeling, Valuation and Hedging, Bielecki and Rutkowski, Springer, 2002.
LINKS TO RELATED WEB SITES
Please direct any questions to
Professor R. J. Williams, email:
williams at math dot ucsd dot edu
January 8, 2012.