Mathematical finance comprises the branches of applied mathematics concerned with the 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 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.In terms of practice, mathematical finance also overlaps heavily with the field of computational finance. Arguably, these are largely synonymous, although the latter focuses on application, while the former focuses on modeling and derivation.The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance.

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