MATH 294: MATHEMATICS OF FINANCE (WINTER 2001)
Professor: Professor R. J. Williams
Time: Tu, Th 3.55-5.15 p.m.
This is the scheduled time for the class. The first and second classes will meet at this time, i.e., Tuesday, January 9, 2001 and Thursday, January 11, 2001, at 3.55 p.m. in York 3050B. However, after the first week, the time will change to Tu 11.50-1.20 p.m., Th 11.50-1 p.m. and the meeting place will be AP&M 6218. Lecture notes will be provided for the first few lectures.
Place: AP&M 6218.
Office Hours: Tu, Th 5-6 p.m.
Related Seminars: Click here.

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.

TEXT:

  • Risk-Neutral Valuation, N. H. Bingham and R. Kiesel, Springer.

    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.
  • 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: williams@math.ucsd.edu