Mathematical finance, also known as
quantitative finance, is a field of
applied mathematics, concerned with
financial markets. 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. Mathematical consistency is required, not compatibility with economic theory. 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 corresponding 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.