Professor: 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.

DESCRIPTION: This course is an introduction to the mathematics of financial models at the graduate level. The aim is to provide students with an introduction to some basic models of finance and the associated mathematical machinery.

OUTLINE: 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.

TEXT: 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. The course 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.


  • 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.

    Background in Probability and Stochastic Calculus:

  • 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 Economics/Finance:
  • Investment Science, David G. Luenberger, Oxford University Press, 1998.
  • Financial Economics, H. H. Panjer (ed.), The 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, Blackwell, 1999.
  • 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, 1996.
  • A Course in Financial Calculus, A. Etheridge, Cambridge University Press, 2002.
  • 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 Press, 1998.
  • 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, Springer, 2004.
  • Essentials of Stochastic Finance, A. N. Shiryaev, World Scientific, 1999.
    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 theory
  • 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, Springer, 2001.
  • Credit Risk: Modeling, Valuation and Hedging, Bielecki and Rutkowski, Springer, 2002.


    Please direct any questions to Professor R. J. Williams, email: williams at math dot ucsd dot edu

    Last updated January 8, 2012.