Is Part III Suitable for Quant?

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Hi.
For some background, im currently studying Maths at Warwick (finished second year) and im interested in trying to break into quant trading.
I'm currently looking at what masters programs to do and atm the plan is Cambridge Part III > Oxford OMMS > Imperial Pure/Applied Maths (though still not sure about this last choice) and stay at Warwick for masters if i don't get any of these.
Firstly, would doing a more pure Maths (like Part III) Masters than Mathematical Finance put me at a big disadvantage for getting a Quant trading role? My interests do cover Probability and Stochastics among other things so its not like I'm just doing abstract algebra.
Secondly, I'm also considering a PhD after this (although I would only do it if i really want to research a topic more in depth, wouldn't do it just to get a job). Is it possible to get quant research roles with just a Masters?
Also, how much does your institution matter in getting a role? Talking about say Warwick compared to Imperial for quant, as they're both similar for maths but ofc Imperial is a better name in general. In terms of experience, I did a (non convertible) spring week at GS in the Engineering division (didnt do much other than talks and networking though) so my CV isn't empty. Does university reputation only matter to the point of getting an interview, or is it still considered after interviewing and doing tests etc?
 
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I'm not sure what the course structure is like (how many you'd take, which ones you get to take, ect.) but there are some good courses on there, yeah. You could combine courses from many different sections and, if you're diligent about it, create a format not terribly unfamiliar to the field of financial mathematics. But a top MFE in the US would still be more specific and focused.
 
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