Mathematical finance, also known as quantitative finance, is a field of applied mathematics, concerned with mathematical modeling of 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; Asset pricing). 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.

Mathematical finance also overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models (see: Quantitative analyst), while the former focuses, in addition to analysis, on building tools of implementation for the 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.

French mathematician Louis Bachelier is considered the author of the first scholarly work on mathematical finance, published in 1900. 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.

Today many universities offer degree and research programs in mathematical finance.

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Tulane University of Louisiana
Four or more years; Private not for profit; $48,920 average out-state tuition; $48,920 average in-state tuition |

University of Connecticut
Four or more years; Public; $34,066 average out-state tuition; $11,998 average in-state tuition |

Pepperdine University
Four or more years; Private not for profit; $51,740 average out-state tuition; $51,740 average in-state tuition |

Massachusetts Institute of Technology
Four or more years; Private not for profit; $49,580 average out-state tuition; $49,580 average in-state tuition |

University of Chicago
Four or more years; Private not for profit; $53,292 average out-state tuition; $53,292 average in-state tuition |

University of California-Los Angeles
Four or more years; Public; $39,516 average out-state tuition; $11,502 average in-state tuition |

Boston University
Four or more years; Private not for profit; $50,980 average out-state tuition; $50,980 average in-state tuition |

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