Mathematical Finance

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 without necessarily establishing a link to financial theory, taking observed market prices as input. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share 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; Financial modeling). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.

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), often by help of stochastic asset models. 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 hand. These are discussed below.

Many universities around the world now offer degree and research programs in mathematical finance; see Master of Mathematical Finance.