Professor: Professor R. J. Williams
Time: M, W 5-6.20 p.m.
Place: AP&M 6438.
Office Hours: M, W 3-4 p.m., AP&M 6121.
DESCRIPTION: This course is an introduction to the mathematics of financial models. 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 will be discussed. As time allows, additional topics will be discussed, possibly including models of the interest rate market.

TEXT: No text is required, though the books by J. Hull, and M. Musiela and M. Rutkowski (see below), provide useful background material. Lecture notes will be available for many of the lectures.

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.

HOMEWORK: Click here.

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.
  • 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.
  • Options, Futures and other Derivative Securities, J. Hull, Prentice Hall, Fifth Edition.
    Mathematics of Finance: Stochastic Approaches
  • S. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell, 1999.
  • H. Follmer and A. Schied, Stochastic Finance -- an Introduction in Discrete Time, de Gruyter, 2002.
  • Financial calculus, Martin Baxter and Andrew Rennie, Cambridge University Press, 1996.
  • A. Etheridge, A Course in Financial Calculus, 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.
  • Steven Shreve's Lectures on Stochastic Calculus and Finance, Prepared by P. Chalasani and S. Jha.
  • Risk-Neutral Valuation, N. H. Bingham and R. Kiesel, Springer.
  • Martingale methods in financial modeling, M. Musiela and M. Rutkowski, Springer, 1998.
  • Mathematics of Financial Markets, R. J. Elliott and P. E. Kopp, Springer, 1999.
  • 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.
    Mathematics of Finance: more advanced stochastic theory
  • Methods of mathematical finance, I. Karatzas and S. Shreve, Springer, 1998.
  • Derivatives in Financial Markets with Stochastic Volatility, J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Cambridge University Press, 2000.

    LINKS TO RELATED WEB SITES (under construction):

    Please direct any questions to Professor Ruth J. Williams, email:

    Last updated January 1, 2003.