Geometric Brownian motion

The geometric Brownian motion is a stochastic process (also called Brownian motion) of the Wiener process is derived. It takes place mainly in financial mathematics using.

Definition

Let be a standard Brownian motion, that is a Wiener process. so is

A geometric Brownian motion.

Derivation

The geometric Brownian motion is the solution of the stochastic differential equation

The parameter is called makes drift and describes the deterministic trend of the process. If so, the value of growing in expectation, it is negative, is likely to fall. For a martingale.

The parameter describes the volatility and controls the influence of chance on the process. If so the diffusion term, the ordinary differential equation vanishes in the above differential equation remains

Having the exponential function as a solution. Therefore, one can interpret the geometric Brownian motion as a stochastic counterpart of the exponential function.

The stochastic differential equation of the geometric Brownian motion can be solved with the Exponentialansatz. With the help of Ito 's formula is obtained for:

Provides insertion of the differential equation

So

Exponentiation results specified in the definition formula.

Another possibility to determine the solution is to use the stochastic Exponentials With applies.

Properties

• Expectation: for all cases:

• Covariance: For all true:

• The geometric Brownian motion has independent multiplicative growth, that is, are for all

• Distribution function: is log normally distributed with parameters and.

Application

In the Black- Scholes model, the simplest and most common ( time-continuous ) mathematical model to price options, the geometric Brownian motion is used as an approximation for the price of an underlying process (eg a share). These led the simplifying assumption that the percentage return is independent and normally distributed over disjoint time intervals. μ plays the role of the risk-free rate, σ represents the risk of changes in the stock market. The above-mentioned martingale plays a central role here.