I'll teach an introductory course on mathematical finance in the near future. The course is intended to entertain and broaden some well-prepared advanced undergrad mathematics majors, some physics majors may attend as well. It need not cater the sort of poorly prepared students one typically finds in quantitate finance masters programs.
Does anyone have recommendations for textbooks, or other materials, that provides one or more of the following :
- A reasonably rigorous treatment of Itō calculous, Stochastic processes, and Black-Scholes.
It should use the Riemann–Stieltjes integral, not the Lebesgue integral. I found Baxter and Rennie very pleasant, but it isn't rigorous enough, and doesn't work as a textbook. Björk assumes too much background. I dislike Shreve because the mathematical point frequently gets lost amid numerical calculations. Any opinions?
- A good introduction to the Black-Scholes PDE.
There isn't imho nearly enough discussion of PDE approaches in most introductory texts on Itō calculous and Black-Scholes. Any opinions about texts that do this well?
- Draws parallels with physical processes.
You can easily show that the Black-Scholes PDE is a Heat equation evolving backwards in time, but I wish to see how a physicist would discuss this. Any entertaining treatments of other physical analogs are helpful as well.
- Treats American put options using free boundary problem techniques.
I'm content with almost any simplifying assumptions here, including doing only perpetual American put options. I cannot discuss Sobolev spaces, but some elementary calculous of variations works.
I considered making this into multiple questions, since there won't be a book that does exactly what I want, but asking all points simultaneously has advantages too.