Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical or numerical models suggested by financial economics. Thus, for example, while a financial economist might study the structural reasons why a company may have a certainshare price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the fair value of derivatives of the stock (see: Valuation of options).

In terms of practice, mathematical finance also overlaps heavily with the field of computational finance (also known as financial engineering). Arguably, these are largely synonymous, although the latter focuses on application, while the former focuses on modeling and derivation (see: Quantitative analyst). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance. Many universities around the world now offer degree and research programs in mathematical finance; see Master of Mathematical Finance.

Quantitative derivatives pricing was initiated by Louis Bachelier in The Theory of Speculation (published 1900), with the introduction of the most basic and most influential of processes, the Brownian motion, and its applications to the pricing of options. However, Bachelier’s work hardly caught any attention outside academia.

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. For this M. Scholes and R. Merton were awarded the 1997 Nobel Memorial Prize in Economic Sciences. Black was ineligible for the prize because of his death in 1995.

A process satisfying (1) is called a “martingale”. A martingale does not reward risk. Thus the probability of the normalized security price process is called “risk-neutral” and is typically denoted by the blackboard font letter ““.
The relationship (1) must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.
The quants who operate in the Q world of derivatives pricing are specialists with deep knowledge of the specific products they model.
Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature. Calibration is one of the main challenges of the Q world: once a continuous-time parametric process has been calibrated to a set of traded securities through a relationship such as (1), a similar relationship is used to define the price of new derivatives.
The main quantitative tools necessary to handle continuous-time Q-processes are Ito’s stochastic calculus and partial differential equations (PDE’s).