I am only now reading into Mathematical Finance, I understand the derivation of the BS equation with vanilla European options. On the next page of my book it starts to delve into obtaining exact solutions for the BS equation for Euro-options, and the introductory chapter on numerical methods has this to say:
[...] There are many examples (particularly of multi-factor models) where it is not feasible or even not possible to reduce the problem to a constant coefficient diffusion equation; in this case there is little choice but to use finite differences on the BS equation [...]"
There are no links to these models or relevant chapters in the book. I have googled "multi-factor black scholes" and I am not getting anything digestible.
Could someone take me through an instance of the BS equation using European options that cannot be solved analytically? Possible some references to derivation?
On first thought I thought it was something to do with the type of option (European over American) but it seems as though you can get solutions to American options too.