I would ask a sort of "philosophical" question. Is there any author who gave a scientific definition of this subject? Is it considered a branch of mathematics or does it primarily concern economics? Thank you in advance.
Excellent information from Matthew Gun. I am just going to throw out what I think could be a reasonable definition of Quantitative Finance;
A field of applied mathematics that uses concepts from statistics, economics, econometrics, game theory, time series analysis, AI theory and other areas to provide a better understanding and competitive advantage within financial markets.
I am not sure if that is quite what you are looking for, but it should be a decent start.
I am happy to update this if people have some suggestions that would help.
There are recurring questions as to what kind of academic discipline finance is. Is it applied math? A subfield of economics? Both? Neither? You'll find a multitude of opinions over time.
Was it? Perhaps the most famous definition of economics comes from Lionel Robbins, "Economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses." Friedman agreed with Robbins, writing in Price Theory, "an economic problem exists whenever scarce means are used to satisfy alternative ends." The phrase, "alternative ends" captures some notion that preferences over multiple outcome variables are at the heart of economics and distinguishes an economic problem from an engineering problem. Building the fastest car from limited resources is an engineering problem. Building the best car would be an economic problem.
Is portfolio optimization an engineering problem? Or an economic problem that involves preferences? The subsequent decades have put portfolio choice squarely within the realm of economic problems.
No arbitrage pricing, Black Scholes, and interest rate models
The 1970s, 1980s brought substantial developments in finance theory that arguably fall outside the traditional boundaries of economics.
Fischer Black and Myron Scholes found that continuous trading of a stock and a risk free bond could replicate the payoff of an option on that stock. Under the assumptions of Black-Scholes, a stock option from a mathematical standpoint is a redundant security. The late 1970s and subsequent decades also saw the development of term structure models, (eg. Vasicek model, Heath-Jarrow-Morton framework).
Much of the work here has been picked up by mathematics and financial math departments. And I've heard Myron Scholes express his opinion that finance isn't just a subfield of economics.
Is quantitative finance just applied math? That's a problematic position as well.
Economics has also been broadening the boundaries of what's considered economics in other areas. For example, Alvin Roth originally worked in operations research, but his matching theory has been been brought under the banner of economics.