What is an advanced quantitative science and why do investment banks like you to have a background in this?

It's mathematics applied to finance.

For several decades, advanced math such as stochastics, Brownian motion, and that sort of thing, has been used to model financial markets. For example, the pricing of options is done using the Black-Scholes formula which was developed in the 1970s.

To learn something about this, check out:

"Paul Wilmott Introduces Quantitative Finance" by Paul Wilmott.

In order to work in this field, you must have a graduate degree in math or physics with an understanding of stochastic processes.

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