Financial mathematics

 

Financial mathematics is the field of applied mathematics concerned with financial markets; Louis Bachelier's doctoral thesis, defended in 1900, is considered to be the first scholarly work in this area. The field is largely focused on the modeling of derivatives—with much emphasis on interest rate- and credit risk modeling—while other important areas include insurance mathematics and quantitative portfolio management. Relatedly, the techniques developed are applied to pricing and hedging a wide range of asset-backed, government, and corporate-securities.

As above, in terms of practice, the field is referred to as quantitative finance and / or mathematical finance, and comprises primarily the three areas discussed. The main mathematical tools and techniques are, correspondingly:

for derivatives, Itô's stochastic calculus, simulation, and partial differential equations; see aside boxed discussion re the prototypical Black-Scholes model and the various numeric techniques now applied

for risk management, value at risk, stress testing and "sensitivities" analysis (applying the "greeks"); the underlying mathematics comprises mixture models, PCA, volatility clustering and copulas.

in both of these areas, and particularly for portfolio problems, quants employ sophisticated optimization techniques

Mathematically, these separate into two analytic branches: derivatives pricing uses risk-neutral probability (or arbitrage-pricing probability), denoted by "Q"; while risk and portfolio management generally use physical (or actual or actuarial) probability, denoted by "P". These are interrelated through the above "Fundamental theorem of asset pricing".

The subject has a close relationship with financial economics, which, as outlined, is concerned with much of the underlying theory that is involved in financial mathematics: generally, financial mathematics will derive and extend the mathematical models suggested. Computational finance is the branch of (applied) computer science that deals with problems of practical interest in finance, and especially emphasizes the numerical methods applied here.