I'm reading Peter Carr's "A Practitioner’s Guide to Mathematical Finance". When talking about the math used in mathematical finance, he mentions Lie groups, differential geometry, string theory. Can anyone explain (either in an understandable or informal way is ok), and with examples if possible, how these 3 can be applied in MF?
MF is linked with physics mostly because it solves the same PDEs (Black-Scholes equation is a certain type of Schrödinger equation for instance). As for the specific links you mentioned :
- Lie Algebra : Magnus expansion (to build fast approximation of time dependent ODEs like those arising in credit risk)
- Differential geometry : link with Varadhan approximation (used for instance in Avellaneda's SABR)