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Re: Finance PhD Profile Evaluation
In your case not having real analysis will not prevent you from getting into top schools, though having it helps. While the some of the topics covered in a typical baby rudin level real analysis class are used in micro theory (which most finance PhD programs require) and in the core finance theory classes, most schools will have some sort of math camp or mathematical methods class or will integrate the material into the micro theory sequence. In the application process it is mostly used for signaling that you have the ability to handle higher level proof based math. In your case, the masters level stochastic calculus class, especially if it was taught at something like the level of Karatzas and Shreve and some of your other masters level classes would probably send a similar signal. That being said, I would still advise you to take it (possibly online) before starting the PhD program just to familiarize yourself with the material even if you don't have it at the time of application.
Conditional on having the necessary math, LOR's are by far the most important component, and yours appear to be very good.
Your profile, while good is not good enough for me to consider you a "lock" for a top 15 admit. Finance PhD programs typically matriculate only 2-4 students per year, with a few top 10's matriculating 6-8, so there are probably less than 80 total slots available at top 15's and the competition is tough. Apply to a few top 30 schools as well. If you look at job market outcomes in finance, the difference between top 30's and top 15's is not that big (top 5 is another story)