I gave a talk recently at the Mathematical Finance Days, organized in HEC Montréal Monday and Tuesday, on Advanced methods in trees with (as mentioned in the subtitle of the first slide) a some thoughts on teaching mathematical finance. It is mainly a survey on advanced tools, based on the idea expressed in Price (1996),
The paper that showed that European option pricing could be put on a rational mathematical basis was Black and Scholes published in 1973. It was so revolutionary that the authors had to submit it to a number of journals before it was accepted. Although there are now numerous approaches to the result, they mostly require specialized methods, including Ito calculus and partial differential equations, and perhaps Girsanov theory and Feynman-Kac methods. But it is the binomial method due initially to Sharpe and substantially extended by Cox, Ross, and Rubinstein that made the theory of option pricing accessible to everyone with limited mathematical background. Even though it requires only routine algebraic manipulations, the method is still able to elucidate many of the ideas behind the full theory. Furthermore, all the surprising results mentioned in the opening can be located in this approach. For these reasons it is usually the first method presented in text books and finance courses; we shall follow this trend and step through it. The binomial method is, however, much more than a pedagogical breakthrough, since it allows for the development of numerical approximation methods for a wide range of options for which there are no known analytic solutions.
Some recent results, obtained in work in progress with colleagues in combinatorial analysis are mentioned at the end of the talk (slides can be downloaded in a pdf format, with animations)