Mathematical finance

Mathematical finance, also called quantitative finance and financial mathematics, is a part of applied mathematics. It uses math to solve problems in the financial field. It helps people learn how to price things like options and other complex financial products. It also shows how to manage risk when investing money.

There are two main areas in mathematical finance: derivatives pricing and risk and portfolio management. These areas use advanced math and computer models to help make smart decisions about money. This field overlaps with computational finance and financial engineering. These also use math and computers to solve financial problems.

The roots of mathematical finance go back to 1900. A French mathematician named Louis Bachelier wrote about it in a thesis. The field grew a lot in the 1970s. This was because of the work of Fischer Black, Myron Scholes, and Robert Merton. They created important theories about pricing options. Today, many universities have special programs. Students can study mathematical finance and learn how to use math to make better financial decisions.

History: Q versus P

Main article: Risk-neutral measure

Further information: Black–Scholes model, Brownian model of financial markets, Martingale pricing, and Quantitative analysis (finance) § History

Mathematical finance has two main parts: pricing derivatives and managing risk and portfolios. These parts use different kinds of probabilities.

In pricing derivatives, we use "risk-neutral probability," shown as "Q." This helps find fair prices for things like options and bonds. We compare them to other items that are easy to trade. This idea began with Louis Bachelier in 1900. He used Brownian motion to model how stock prices change.

In managing risk and portfolios, we use "actual probability," shown as "P." This helps us see how different securities might change over time. Investors can use this to choose which ones to buy to get better results. Researchers like Markowitz and Sharpe improved this area of finance. They won a Nobel Prize for their work.

The Q world

Goal	"extrapolate the present"
Environment	risk-neutral probability Q {\displaystyle \mathbb {Q} }
Processes	continuous-time martingales
Dimension	low
Tools	Itō calculus, PDEs
Challenges	calibration
Business	sell-side

P 0 = E 0 ( P t ) {\displaystyle P_{0}=\mathbf {E} _{0}(P_{t})}

1

The P world

Goal	"model the future"
Environment	real-world probability P {\displaystyle \mathbb {P} }
Processes	discrete-time series
Dimension	large
Tools	multivariate statistics
Challenges	estimation
Business	buy-side

Criticism

Further information: Financial economics § Challenges and criticism, and Financial engineering § Criticisms

See also: Financial models with long-tailed distributions and volatility clustering

As mathematical finance has become more complex, it has also faced some criticism. The 2008 financial crisis showed that some of the models used were not always reliable. Experts like Paul Wilmott and Nassim Nicholas Taleb have said that simple models might not fully show how financial markets really work. They think these models can sometimes be misleading.

Some people believe that traditional models do not think about how feelings and reactions can change the economy. For example, panic can cause events like bank runs, which models often cannot predict. These ideas show that while mathematical finance is helpful, it has limits and should be used with care.