Mathematical finance

Mathematical finance is a field of applied mathematics, concerned with mathematical modeling and analysis 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 a key concept in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.

History

Mathematical finance has a history of study dating back to the 17th century. While early developments were considered in the field of probability, it was not until the 20th century that mathematical finance began to be formalized. The works of Louis Bachelier, Harry Markowitz and Robert Merton were particularly influential in the development of this field.

Key Concepts

Financial Markets

Financial markets are a broad term describing any marketplace where buyers and sellers participate in the trade of assets such as equities, bonds, currencies and derivatives. They are one of the most important areas in which mathematical finance is applied.

Stochastic Calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used for defining the time evolution of complex systems, and is central to the Black-Scholes model of option pricing.

Derivatives Pricing

Derivatives pricing, such as the pricing of options, is a major application of mathematical finance. Models here include the Black-Scholes model and the binomial options pricing model.

Applications

Mathematical finance has wide applications in the financial industry. Financial institutions, such as banks and insurance companies, as well as investment funds, employ mathematicians, statisticians, financial engineers, and other professionals with a knowledge of mathematical finance to develop and implement mathematical models for pricing, hedging, and risk management.

See Also

• Financial mathematics
• Quantitative finance
• Computational finance
• Econometrics
• Financial engineering
• Financial economics