Geometric Brownian motion

Three (dependent) geometric Brownian movements with drift μ = 0.8 and volatility σ = 0.4 (blue) , σ = 0.25 (red) and σ = 0.1 (yellow)

The geometric Brownian movement is a stochastic process that is derived from the Wiener process (also called Brownian movement ). It is mainly used in financial mathematics .

definition

Let be a standard Brownian motion, i.e. H. a Viennese trial . So is ${\displaystyle W_{t}}$

${\displaystyle S_{t}=a\exp \left[\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}\right]}$

a geometric Brownian movement.

Derivation

Three independent geometric Brownian movements with volatility 0.2 and drift 0.7 (green), 0.2 (blue) and −0.7 (red)

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

${\displaystyle \mathrm {d} S_{t}=\mu S_{t}\,\mathrm {d} t+\sigma S_{t}\,\mathrm {d} W_{t},\quad t\geq 0,\quad S_{0}=a}$

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 . ${\displaystyle \mu }$ ${\displaystyle \mu >0}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle \mu =0}$ ${\displaystyle S}$

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 ${\displaystyle \sigma }$ ${\displaystyle S}$ ${\displaystyle \sigma =0}$

${\displaystyle \mathrm {d} S(t)=\mu \,S(t)\,\mathrm {d} t,\;S(0)=a}$,

which has the exponential function as a solution. Therefore one can understand the geometric Brownian movement as a stochastic counterpart to the exponential function. ${\displaystyle S(t)=ae^{\mu t}}$

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 : ${\displaystyle S_{t}=e^{X_{t}}}$ ${\displaystyle X_{t}=\ln(S_{t})}$

${\displaystyle {\begin{aligned}\mathrm {d} X_{t}&=\left(\mu S_{t}{\frac {\partial X_{t}}{\partial S_{t}}}+{\frac {1}{2}}(\sigma S_{t})^{2}{\frac {\partial ^{2}X_{t}}{\partial S_{t}^{2}}}\right)\mathrm {d} t&&+\sigma S_{t}{\frac {\partial X_{t}}{\partial S_{t}}}\mathrm {d} W_{t}\\&=\left(\mu S_{t}{\frac {1}{S_{t}}}-{\frac {1}{2}}(\sigma S_{t})^{2}{\frac {1}{S_{t}^{2}}}\right)\mathrm {d} t&&+\sigma S_{t}{\frac {1}{S_{t}}}\mathrm {d} W_{t}\\&=\left(\mu -{\frac {\sigma ^{2}}{2}}\right)\mathrm {d} t&&+\sigma \mathrm {d} W_{t}\end{aligned}}}$

So it turns out

${\displaystyle \mathrm {d} X_{t}=\left(\mu -{\frac {\sigma ^{2}}{2}}\right)\mathrm {d} t+\sigma \mathrm {d} W_{t}\,,}$

and consequently after integration

${\displaystyle \ln(S_{t})=\ln(S_{0})+\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}\,.}$

Subsequent exponentiation results in the formula given in the definition.

Another way to find the solution is to use the stochastic exponential : With applies . ${\displaystyle Z_{t}=\mu t+\sigma W_{t}}$ ${\displaystyle S_{t}={\mathcal {E}}(Z)_{t}}$

properties

• Expected value : for all applies: ${\displaystyle t\geq 0}$ ${\displaystyle \mathbb {E} (S_{t})=a\mathrm {e} ^{\mu t}}$

• Covariance : The following applies to all : ${\displaystyle s,t\geq 0}$ ${\displaystyle \mathrm {Cov} (S_{s},S_{t})=a^{2}\mathrm {e} ^{\mu (t+s)}(\mathrm {e} ^{\sigma ^{2}\min(t,s)}-1)}$

In particular, then . ${\displaystyle \mathrm {Var} (S_{t})=a^{2}\mathrm {e} ^{2\mu t}(\mathrm {e} ^{\sigma ^{2}t}-1)}$

• The geometric Brownian motion has independent multiplicative increments; h., for all are ${\displaystyle 0\leq t_{1}\leq t_{2}\leq \ldots \leq t_{n}}$

${\displaystyle S_{t_{1}},{\frac {S_{t_{2}}}{S_{t_{1}}}},\ldots ,{\frac {S_{t_{n}}}{S_{t_{n-1}}}}}$ independent .

• Distribution function : is logarithmically normally distributed with parameters and . ${\displaystyle {\tfrac {S_{t}}{a}}}$ ${\displaystyle (\mu -{\tfrac {1}{2}}\sigma ^{2})t}$ ${\displaystyle \sigma ^{2}t}$

application

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 .

Web links

Commons : Geometric Brownian Movement  - collection of images, videos and audio files