The parameter is called drift and describes the deterministic tendency of the process. If , the value of in expectation increases , if it is negative, it tends to decrease . For is a martingale .
The parameter describes the volatility and controls the influence of chance on the process . If , then the diffusion term in the above differential equation vanishes , the ordinary differential equation remains
,
which has the exponential function as a solution. Therefore one can understand the geometric Brownian movement as a stochastic counterpart to the exponential function.
The stochastic differential equation of the geometric Brownian motion can be solved with the exponential approach. With the help of the Itō formula we get :
So it turns out
and consequently after integration
Subsequent exponentiation results in the formula given in the definition.
In the Black-Scholes model , the simplest and most widespread (continuous) financial mathematical model for valuing options , the geometric Brownian movement is used as an approximation for the price process of an underlying asset (for example, a share). This led to the simplifying assumption that the percentage rate of return is independent and normally distributed over disjoint time intervals. µ plays the role of the risk-free interest rate , σ represents the risk of fluctuations on the stock market. The martingale property mentioned above plays a central role here.
literature
Bernt Øksendal: Stochastic Differential Equations: An Introduction with Applications. Springer, 2003, ISBN 3-540-04758-1 .
Steven E. Shreve: Stochastic Calculus for Finance II: Continuous-Time Models. Springer, 2004, ISBN 0-387-40101-6 .