Mathematical finance — Safekipedia Discoverer

Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics that uses math to solve problems in the financial field. It helps people understand how to price things like options and other complex financial products, and 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 make smart decisions about money. This field overlaps with computational finance and financial engineering, which also use math and computers to solve financial problems.

The roots of mathematical finance go back to 1900 when French mathematician Louis Bachelier wrote a thesis about it. But the field really grew in the 1970s thanks to the work of Fischer Black, Myron Scholes, and Robert Merton, who developed important theories about pricing options. Today, many universities have special programs where 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 two areas use different kinds of probabilities. In derivatives pricing, we use something called "risk-neutral probability," shown as "Q." This helps figure out fair prices for things like options and bonds by comparing them to other easily traded items. This idea started with Louis Bachelier in 1900, who used a concept called Brownian motion to model stock prices.

In risk and portfolio management, we use "actual probability," shown as "P." This helps us understand how different securities might change in value over time, so investors can decide which ones to buy to improve their overall results. This area of finance was advanced by researchers like Markowitz and Sharpe, who 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})}

**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

As mathematical finance has grown more complex, it has also faced 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 pointed out that simple models may not capture the true behavior of financial markets. They argue that these models can sometimes be misleading.

Some believe that traditional models do not account for how people’s emotions and reactions can affect the economy. For example, panic can lead to events like bank runs, which models often fail to predict. These criticisms suggest that while mathematical finance is useful, it has limits and should be used carefully.