# Mathematical finance

The Financial Mathematics is a discipline of applied mathematics that deals with topics from the field of financial services providers, such as banks or insurance companies, employs. In a narrower sense, with financial mathematics usually the most well-known sub- discipline, evaluation theory, referred to, namely the determination of theoretical present value of financial products. Both the type of the considered business as well as the methodological basis is financial mathematics to distinguish from the insurance mathematics. The latter deals with the measurement of insurance services.

## Mathematical Foundations

Methodologically based on the Stochastic financial mathematics, the theory of stochastic processes and with respect to the ( risk-neutral ) valuation of financial derivatives on the theory of martingales.

## History

As birth of modern financial mathematics is now the year 1900, when the Frenchman Louis Bachelier published his thesis Théorie de la spéculation. However, she received little recognition at the time and was not until more than 50 years later dissemination, after it had been translated into English. Many of today's conventional techniques described here for the first time, and in honor of Bachelier's carries the international actuarial company known today as the Bachelier Society.

The best-known result of financial mathematics is the beginning of the 1970s established the Black-Scholes model. It developed very quickly becoming the standard model for the valuation of options on shares and was later expanded under the name Black'76 to other classes of the underlying transactions. The model assumes that the probability distribution of shares corresponds to a point in the future a lognormal distribution, and lays the fluctuations in share price based on a Wiener process.

To date, the field of financial mathematics has greatly expanded. This affects both the number of asset classes ( ie the type of underlying transactions) as well as the number of models. Among the subjects asset classes include stocks, exchange rates, interest rates, credit risks ( which are modeled differently depending on the model ), but also prices of raw materials (eg petroleum, grain, coffee, sugar ), electricity or weather-dependent parameters ( eg number of hours of sunshine over a period of time at a particular weather station ). Combinations of different asset classes ( hybrid products ) and portfolios of assets are treated. The main models include jump processes (jump diffusion), stochastic and local volatility models and the group of term structure models.

## Valuation of financial derivatives

The aim of evaluation theory is to determine the present value of a financial product.

Derivative financial products are those whose payments of other financial products, the underlying assets ( underlying ) depend. Examples of non -derivative financial products are traded stocks and bonds. Examples of derivative financial products are futures and options. The price of a financial product which (that is, with sufficient liquidity) is traded in sufficient quantity, usually is determined by supply and demand. If a financial product traded or not with insufficient liquidity, and this financial product, a derivative financial product whose primary products are traded, the determination of a "fair value " and thus pricing with actuarial methods is possible. The basic principle of replication is used, which is a mathematical model of the ( traded ) requires underlyings.

Derivative financial products are distinguished by type of optionality and underlying. The latter are divided historically into the equity asset classes (equity), interest rate (interest rate ), exchange ( Foreign Exchange, FX short ), and credit (credit ). Accordingly, there exists for each asset class, an extensive modeling theory (eg, share models and term structure models).