You certainly don't need a math PhD to become a finance quant, whatever that may mean these days. And, vice versa, most math PhDs are entirely unsuited to be quants.
Sorry to piggy back, bearish, but this is a good point to start on.
OP, when you see requirements like a PhD on a job spec understand that it's indicative. In theory a PhD would be someone that has worked on a problem closely linked to finance and therefore more suited than someone that learnt finance in a masters course. For example a college classmate did a PhD on properties of matter where the processes he used were linked to how market microstructure works.
Firstly that has changed drastically as what quants do has changed. Secondly in reality a lot of PhDs are unsuited for quant finance as they don't take to not working constantly with "pretty models". It's a frigging job FFS.
And many PhDs I've known are often clowns panting as they struggle to finish off their PhD thesis. I knew one who was finishing off a Physics PhD in Imperial but could tell from the garbage she came out with that she was hopelessly mismatched for being a quant and out of her depth. I mean, I expect some naivety but sheesh!!! Last I knew she was doing a job leafleting after just about finishing her thesis off. Completely no gumption at all. I don't know if it's most PhDs that are like this or if most MSc aren't but there is an issue definitely and employers will spot it if you have it.
At least that's the generalism. In reality there are many competent PhDs and many incompetent MSc grads. Main thing to realise is that the only difference is the time taken, so I'd only do a PhD if you actually wanted to spend those 5 years doing the research, don't do it for the sake of "getting a job".
It's a question of what can you do. Publish as others have said and/or even just post something USEFUL on github that could actually be used - this is what developers in CS/programming areas do anyway and if you are a math/physics oriented person your portfolio will be easily 1000 times better than some of the basic "hello world" shite I see on most people's githubs.
Some employers put in PhD requirements for the sake of it (there is an ego thing in certain banks), which means they'll ignore your application if you are an MSc, but what that meant for me is I ended up starting off at an employer which knew what they were doing and which helped me in my first few years in finance.