The course starts out with a discrete stochastic calculus. We then go to continuous space, with notions of sigma-fields, conditional expectations, and Radon-Nikodym derivatives. After this, we treat continuous martingale theory: Brownian motion, martingales, semimartingales, predictability, stochastic integrals, Levy's theorem, Doob-Meyer decomposition, and quadratic variation. Finally, we look at the exotic option problem -- self financing strategies, and the three problems to be solved: numeraire invariance, Girsanov's theorem, and martingale representation. Application to American, Russian, and Asian options and to Caps. Explicit pricing of the Russian and Cap options. No arbitrage and the existence of equivalent martingale measures

As far as prerequisites in Mathematics and Probability are concerned, you need to have a solid working knowledge of (ordinary) multivariate calculus, and you need a practical understanding of concepts of probability, for example as in Ch. 1-6 of Rice, J. (1995). Mathematical Statistics and Data Analysis (2nd ed). (Duxbury Press).

Prerequisites:

• Consent of instructor.

Texts:

• Billingsley, P. (1995). Probability and Measure (3rd ed.). Wiley. (Required.)
  
  
• Dothan, M.U. (1990). Prices in Financial Markets. Oxford University Press. (Required.)
  
  
• Duffie, D. (1996). Dynamic Asset Pricing Theory (2nd ed.). Princeton University Press. (Required.)
  
  
• Jacod, J., and Shiryaev, A. N. (1987). Limit Theory for Stochastic Processes Springer-Verlag. (Optional).
  
  
• Karatzas, I. and Shreve, S.E. (1987). Brownian Motion and Stochastic Calculus. Springer-Verlag. (Required.)
  
  
• Protter, P. (1980). Stochastic Integration and Differential Equations: A New Approach Springer-Verlag. (Optional).
  
  
• Recent research articles distributed during the course.