- Models: Black-Scholes-Merton (BSM), CEV, Heston, etc.
- Numerical methods applicable to the aforementioned models: closed-form expression (as in the BSM model), numerical PDE/PIDE solution methods (finite difference methods, finite element methods), Fourier Transform (either using one of the direct numerical integration algorithms, or one of the wide variety of FFT algorithms), lattices & trees (like binomial, trinomial), Monte Carlo (involving SDE simulation using one of the various discretization algorithms).
Recommendation: Take a look at Chapter 7 (preferably up to & including--I think you'll need to anyway if you want to proceed with a project in quant finance) of Joshi's book:
http://www.markjoshi.com/concepts/
DD, perhaps you're right. I'm honestly unsure which of the paths will be "easier" (more suitable) for the OP, since it strongly depends on the individual's background. Arguably, with a good background in probability theory (and at least stochastic processes intro), the MC way may be the way to go (I still think it'd be somewhat underwhelming not to do parallelization in the 21st century). OTOH, with a background in Fourier analysis, real & complex analysis (which I expect to be a part of a standard pure math program at the undergrad level), using FT with direct integration (perhaps not FFT, that'd be a good fit for a typical CS or EE background) should be within grasp. In any case, numerical analysis & programming will come in handy either way, and I'd expect the time budget for the BSc project to include time for learning these skills as part of of the work on the dissertation. That, and the usual
persistence, diligence, patience, and willingness to learn from failure ;-)
It also depends on the goals: If the OP wants to do the Heston model, then the (semi-)closed-form solution for the price in that _is_ essentially an FT, and so may be unavoidable (whereas writing, say, a _good_ PDE solver for that problem definitely won't be any easier, IMHO; MC (preferably non-naive; read: higher-order SDE discretization schemes, variance reduction and/or QMC) may be good for the V&V, though).
