First of all, in the United States, I have a master's degree in financial mathematics (except Dr. Princeton), and my main job search direction is quantification, including quantitative trading, quantitative finance and quantitative investment. However, in big banks such as JPMorgan Chase and Goldman Sachs, the quantitative jobs are saturated and the competition is fierce. Although it has advantages in derivative pricing, asset management and other financial aspects, it has disadvantages compared with CS in programming. So the prospect is generally felt.
Mathematical engineering, if you are a doctor of mathematics, you can go to the industry to do financial quantification, and you can retire to school to teach. It can be said that there is a decent job, but it usually takes five years to study for a doctorate, and the investment is not generally large. Engineering is generally a master's degree, because there is no financial background, programming may only be reflected in the project, and individuals are the least optimistic.
CS can be said to be the brightest development in the future. Machine learning and big data will be the mainstream of the future. Because of the programming background, quantification is very important, and it can usually compete with financial mathematics/financial engineering students in the financial field. You can also go to a technology company to write code, or you can go to a big company to do data analysis. It has a wide range of applications.
Say that finish the United States, China. China's quantification can be said to be just beginning, so financial mathematics is widely used. Compared with the big banks in the United States, almost every front office has a quantitative position to support and make decisions. The domestic front desk feels a bit "slapping the head" (not black, only a handful of domestic fund brokers have done a good job in quantification). So I think the prospect of financial mathematics in China is very good at least now, and it is very scarce.
Mathematics and engineering may be average in China, after all, there are cross-majors, and quantification has not yet taken shape, so the scope of application is not large. After all, to price derivatives, at least understand the mechanism.
CS is still optimistic. After all, big data and machine learning are the mainstream of the world.