Math 649B

Math 649B Fall 2014
Prerequisites: None, but having taken 657 or 672 would give a head start.

Math 649B will run seminar-style, with participants taking turns presenting, and with detailed topics and meeting schedule depending on participants’ preparation and interests.
One possibility is to continue some work done by my last 4 Master’s students.
Pakravan and Saadat (Master of Financial Engineering project, 2013) calculated numerically the price of an American option that pays the automatic complexity deficiency (a concept invented by Hyde (Master of Arts project, Math, 2013)) of a sequence of up- and down-ticks for a stock or financial instrument, or simply a sequence of heads and tails of a coin.

See for an entertaining game based on their work.

Alikhani (Master of Arts project, Math, 2014) theoretically estimated the price corresponding to a weaker/stronger variant, nonnegative automatic complexity, corresponding to finite automata that are not allowed to move backward to previously visited states.

We may start by going through Sipser Chapter 1 and Shreve Volume I Chapters 1-4.
(Sipser, Theory of Computation, and Shreve, Stochastic Calculus for Finance Vol. I)

Another possibility is to study game-theoretical probability, a reinvention of the foundations of probability inspired by finance. The official textbook for the seminar is in fact:
Shafer and Vovk
“Probability and finance: it’s only a game!”
Wiley Series in Probability and Statistics, 2001