>tools such as martingales, stochastic calculus, Black-Scholes (and vast generalizations), Brownian motion, Stochastic differential Equations
These are all practically the same thing, i.e. the single field of stochastic processes. The "meat" of stochastic processes is the underlying topology, measure theory and probability upon which it's built, all of which an undergrad learns (e.g. I'd expect a good undergrad to be able to follow and understand the proof of Ito's lemma, and the proof of Ito's lemma is way more mathematically interesting and involved than that of Black-Scholes).
It also probably depends on whether the firm in question is HFT or not, and whether it trades options. It'd be perfectly possible to be a quant pricing futures at a HFT without even understanding stochastic calculus, as most of the logic ends up boiling down to just some variant of "oh shit, would you look at the size of that trade tick, better giddy up and follow it!".
These are all practically the same thing, i.e. the single field of stochastic processes. The "meat" of stochastic processes is the underlying topology, measure theory and probability upon which it's built, all of which an undergrad learns (e.g. I'd expect a good undergrad to be able to follow and understand the proof of Ito's lemma, and the proof of Ito's lemma is way more mathematically interesting and involved than that of Black-Scholes).
It also probably depends on whether the firm in question is HFT or not, and whether it trades options. It'd be perfectly possible to be a quant pricing futures at a HFT without even understanding stochastic calculus, as most of the logic ends up boiling down to just some variant of "oh shit, would you look at the size of that trade tick, better giddy up and follow it!".