# Mathematics of finance

mathematics of finance is a discipline of applied mathematics, which concerns itself with topics from the range from Finanzdienstleistern, as for instance banks or insurance. Strictly speaking usually the most well-known Unterdisziplin, which designates evaluation theory, becomes i.e. with mathematics of finance. the determination of theoretical bar values of financial products. Bothmathematics of finance is to be differentiated from the kind of the regarded business and the methodical bases from the actuarial science to. The latter is concerned with the evaluation of insurance services.

#### mathematical bases

methodicalmathematics of finance is based on the stochastics, the theory of stochastic processes and concerning (risk-neutral) the evaluation of financial derivatives on the theory of the Martingale.

A most important axiom of mathematics of finance is that the arbitrage liberty, thus the absence of each possibility for the arbitrage. As consequence of the arbitrage libertythe theoretical bar value of a financial transaction is determined in such a manner that each financing strategy, which makes a replication the payment stream of the financial transaction accurately requires itself an initial investment at height of the bar value.

### history

as birth of modern mathematics of finance applies today the year 1900, in thatthe Frenchman Louis Bachelier its thesis Théorie de la spéculation published. Many of the today usual techniques were for the first time described here, and in honours Bacheliers carries the international financialmathematical society today for the names Bachelier Society.

The most well-known result of mathematics of finance is that beginningthe 70's set up Black Scholes model. Very fast the standard model for the evaluation of options on shares and was extended later under the name Black'76 to further classes by underlying transactions. The model assumes those the probability distribution of shares for oneTime in the future of a log-normal distribution corresponds and takes as a basis to the fluctuations of the share quotation a Viennese process .

Until today the area of mathematics of finance expanded strongly. This concerns both the number of the eating classes (therefore the kind of the underlying transactions) and the number of the models. Tooto the treated eating classes shares, rates of exchange, interest, credit loss risks (depending upon model to be differently modelled) belong, in addition, prices of raw goods (e.g. Oil), river or weather-dependent characteristics (e.g. Number of sun hours during a certain period of a certain Wetterstation). Also combinations of different eating classes (hybrid products) and haven foilsfrom Assets will treat. To the most important models Jump diffusion of processes, stochastic Volatilität and local Vola of models, as well as the group of the interest structured models belong.

### evaluation of financial derivatives

a goal of the evaluation theory is it to determine the bar value of a financial product.

Derivatives of financial products are such, of themPayments of other financial products, which Underlyings depend. Examples of non-derivatives of financial products are acted shares and loans. Examples of derivatives of financial products are Terminkontrakte and options. The price of a financial product, which in sufficient number of items (i.e. with sufficient liquidity) one acts, determines themselves usually overSupply and demand. If a financial product is not acted or with insufficient liquidity and if this financial product is derivative a financial product its basic products is acted, then the regulation one is possible „fair value “and thus a selling price calculation with financialmathematical methods. The basic principle of the Replikation comes toEmployment, which a mathematical model (acted) of the financial products, which are the basis for the derivative (Underlyings) necessarily. The central component of the evaluation theory is therefore the stochastic model of the which are the basis derivatives (Underlyings).

Derivatives the financial products according to kind of the Optionalität and Underlying are differentiated. The latters become historical inthe eating classes share (Equity), interest (Interest rate), rate of exchange (Foreign Exchange, briefly FX) and address loss (Credit) partitions. Accordingly an extensive modelling theory exists for the respective eating class.