In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other.
Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios.
French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical finance.
But mathematical finance emerged as a discipline in the 1970s, following the work of Fischer Black, Myron Scholes and Robert Merton on option pricing theory.
While trained economists use complex economic models that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input.
[7] The theory remained dormant until Fischer Black and Myron Scholes, along with fundamental contributions by Robert C. Merton, applied the second most influential process, the geometric Brownian motion, to option pricing.
The main quantitative tools necessary to handle continuous-time Q-processes are Itô's stochastic calculus, simulation and partial differential equations (PDEs).
[10] Risk and portfolio management aims to model the statistically derived probability distribution of the market prices of all the securities at a given future investment horizon.
Based on the P distribution, the buy-side community takes decisions on which securities to purchase in order to improve the prospective profit-and-loss profile of their positions considered as a portfolio.
[citation needed] While numerous empirical studies have examined the effectiveness of technical analysis, there remains no definitive consensus on its usefulness in forecasting financial markets.
[13] Over the years, increasingly sophisticated mathematical models and derivative pricing strategies have been developed, but their credibility was damaged by the financial crisis of 2007–2010.
Contemporary practice of mathematical finance has been subjected to criticism from figures within the field notably by Paul Wilmott, and by Nassim Nicholas Taleb, in his book The Black Swan.
[14] Perhaps more fundamental: though mathematical finance models may generate a profit in the short-run, this type of modeling is often in conflict with a central tenet of modern macroeconomics, the Lucas critique - or rational expectations - which states that observed relationships may not be structural in nature and thus may not be possible to exploit for public policy or for profit unless we have identified relationships using causal analysis and econometrics.