The University of Chicago
Department of Statistics
Autumn Quarter 1996
Stochastic Calculus and Finance I
Per A. Mykland
This course is an introduction to stochastic calculus as it is relevant
to the pricing and hedging of options and other derivative securities.
It is offered in collaboration with the master's program in Mathematical
At the end of the course you should be able to use Ito's lemma, Girsanov's
theorem, martingale representation and martingale limit theory to
evaluate concrete derivatives.
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
- Billingsley, P. (1995).
Probability and Measure (3rd ed.).
- Dothan, M.U. (1990).
Prices in Financial Markets.
Oxford University Press.
- Duffie, D. (1996).
Dynamic Asset Pricing Theory (2nd ed.).
Princeton University Press.
- Jacod, J., and Shiryaev, A. N. (1987).
Limit Theory for Stochastic Processes
- Karatzas, I. and Shreve, S.E. (1987).
Brownian Motion and Stochastic Calculus.
- Protter, P. (1980).
Stochastic Integration and Differential Equations: A New Approach
- Recent research articles distributed during the course.