Mathematical finance

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Mathematical finance, 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.


There are two separate branches of finance that require advanced quantitative techniques:
derivatives pricing, and risk and portfolio management.

One of the main differences is that they use different probabilities such as
the risk-neutral probability (or arbitrage-pricing probability), denoted by "Q",
and the actual (or actuarial) probability, denoted by "P".

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In mathematical finance, a risk-neutral measure
(also called an equilibrium measure, or equivalent martingale measure)
is a probability measure such that each share price is exactly equal to
the discounted expectation of the share price under this measure.

This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing,
which implies that in a complete market a derivative's price is the discounted expected value of
the future payoff under the unique risk-neutral measure.[1]

Such a measure exists if and only if the market is arbitrage-free.

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Financial mathematics describes the application of mathematics and
mathematical modeling to solve financial problems.
It is sometimes referred to as quantitative finance, financial engineering, and computational finance.
The discipline combines tools from statistics, probability, and stochastic processes and
combines it with economic theory.

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