# Financial math

The Financial Mathematics is a discipline of applied mathematics that deals with topics in the field of financial services companies such as banks or insurance companies. In the narrower sense, financial mathematics is usually the best-known sub-discipline, valuation theory, i.e. H. the determination of the theoretical present values of financial products. A distinction must be made between financial mathematics and actuarial mathematics, both in terms of the type of transactions considered and the methodological basis . The latter deals with the evaluation of insurance services.

## Mathematical basics

Methodologically, financial mathematics is based on stochastics , the theory of stochastic processes and, with regard to the ( risk-neutral ) valuation of financial derivatives, on the theory of Martingales .

## history

The year 1900 is considered to be the birth of modern financial mathematics, when the Frenchman Louis Bachelier published his dissertation Théorie de la speculation . However, it was not widely used until over 50 years later, after it was translated into English. Many of the techniques commonly used today were described here for the first time, and in honor of Bachelier, the international financial mathematics society is now called the Bachelier Society .

The best-known result of financial mathematics is the Black-Scholes model established in the early 1970s . It quickly developed into the standard model for valuing options on stocks and was later expanded to include other classes of underlying transactions under the name Black'76. The model assumes that the probability distribution of shares for a point in time in the future corresponds to a logarithmic normal distribution and uses a Wiener process as a basis for the fluctuations in the share price .

To date, the field of financial mathematics has expanded significantly. This affects both the number of asset classes (i.e. the type of underlying transaction) and the number of models. The asset classes covered include stocks, exchange rates, interest rates, credit default risks (which are modeled differently depending on the model), but also prices of raw materials (e.g. crude oil, grain, coffee, sugar), electricity or weather-dependent parameters (e.g. Number of hours of sunshine over a certain period of time at a certain weather station). Combinations of different asset classes (hybrid products) and portfolios of assets are also dealt with. The most important models include jump processes ( jump diffusion ), stochastic and local volatility models and the group of interest structure models .

## Valuation of financial derivatives

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

Derivative financial products are those whose payments depend on other financial products, the underlyings . Examples of non-derivative financial products are traded stocks and bonds . Examples of derivative financial products are futures contracts and options . The price of a financial product that is traded in sufficient numbers (i.e. with sufficient liquidity ) is usually determined by supply and demand. If a financial product is not traded or traded with insufficient liquidity and if this financial product is a derivative financial product, the basic products of which are traded, a “fair value” can be determined and thus a price can be determined using financial mathematical methods. The basic principle of replication is used, which requires a mathematical model of the (traded) underlying assets.

The derivative financial products are differentiated according to the type of option and base value. The latter are historically divided into the asset classes share (equity), interest rate (interest rate), exchange rate (foreign exchange, FX for short) and creditworthiness . Accordingly, there is an extensive modeling theory for the respective asset class (e.g. share models and interest structure models ).