MATH 294: MATHEMATICS OF FINANCE (WINTER 2000)
Professor: Professor R. J. Williams
Time: MW, 4.40-6 p.m. Place: AP&M 6218.
Office Hours: Monday noon-1 p.m., Wednesday 1-2 p.m.

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. If time allows, attention will then turn to models of the interest rate market. Various models may be discussed, including the Heath-Jarrow-Morton and Cox-Ingersoll-Ross models.

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. 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.

COMPUTER MODULES: These will complement the theoretical material presented in the course.

TEXT: Martingale methods in financial modelling, M. Musiela and M. Rutkowski, Springer, 1998.

REFERENCES:
Background in Probability and Stochastic Calculus:

  • Probability and Measure, P. Billingsley, Wiley.
  • 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, Fourth Edition, 2000.
    Mathematics of Finance: Stochastic Approaches
  • Financial calculus, Martin Baxter and Andrew Rennie, Cambridge University Press, 1996.
  • 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.
  • 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.

    LINKS TO RELATED WEB SITES (under construction):

    Please direct any questions to Professor Ruth J. Williams, email: williams@math.ucsd.edu