Función de densidad
El movimiento Browniano geométrico dado por
tiene una distribución log-normal con función de densidad dada por:
para 
.
Función de distribución
La función de distribución acumulada está dada por
para .
Estadísticas
Para hallar la media, varianza y covarianza del movimiento browniano geométrico, usaremos el hecho de que
es la función generadora de momentos de una distribución normal con parámetros 
y 
.
La media del movimiento Browniano geométrico es
![{\displaystyle \operatorname {E} [S_{t}]=S_{0}e^{\mu t}}](//wikimedia.org/api/rest_v1/media/math/render/svg/09117b33454896ce34e6ee8419bfb4fe86bb18ea)
pues
![{\displaystyle {\begin{aligned}\operatorname {E} [S_{t}]&=\operatorname {E} \left[S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t}\right]\right]\\&=S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\operatorname {E} \left[e^{\sigma W_{t}}\right]\\&=S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]e^{\frac {t\sigma ^{2}}{2}}\\&=S_{0}e^{\mu t}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6c389aaa3b6423c307c74234167370cc6d11d241)
Varianza
La varianza del movimiento Browniano geométrico es
pues
![{\displaystyle {\begin{aligned}\operatorname {Var} [S_{t}]&=\operatorname {Var} \left[S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t}\right]\right]\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\operatorname {Var} \left[e^{\sigma W_{t}}\right]\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\left(\operatorname {E} [\exp(2\sigma W_{t})]-\operatorname {E} [\exp(\sigma W_{t})]^{2}\right)\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\left[\exp \left({\frac {t(2\sigma )^{2}}{2}}\right)-\exp \left({\frac {2t\sigma ^{2}}{2}}\right)\right]\\&=S_{0}^{2}e^{2\mu t}\left(e^{\sigma ^{2}t}-1\right)\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/fa3247531495d00fc7564208ab15ea94e2a5e4ff)
Covarianza
La covarianza del movimiento Browniano geométrico es
![{\displaystyle \operatorname {E} [S_{t}]=S_{0}e^{\mu t}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/09117b33454896ce34e6ee8419bfb4fe86bb18ea)
![{\displaystyle {\begin{aligned}\operatorname {E} [S_{t}]&=\operatorname {E} \left[S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t}\right]\right]\\&=S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\operatorname {E} \left[e^{\sigma W_{t}}\right]\\&=S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]e^{\frac {t\sigma ^{2}}{2}}\\&=S_{0}e^{\mu t}\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6c389aaa3b6423c307c74234167370cc6d11d241)
![{\displaystyle {\begin{aligned}\operatorname {Var} [S_{t}]&=\operatorname {Var} \left[S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t}\right]\right]\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\operatorname {Var} \left[e^{\sigma W_{t}}\right]\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\left(\operatorname {E} [\exp(2\sigma W_{t})]-\operatorname {E} [\exp(\sigma W_{t})]^{2}\right)\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\left[\exp \left({\frac {t(2\sigma )^{2}}{2}}\right)-\exp \left({\frac {2t\sigma ^{2}}{2}}\right)\right]\\&=S_{0}^{2}e^{2\mu t}\left(e^{\sigma ^{2}t}-1\right)\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/fa3247531495d00fc7564208ab15ea94e2a5e4ff)
![{\displaystyle \operatorname {E} [S_{t}^{n}]=S_{0}\exp \left(\left[n\mu +{\frac {\sigma ^{2}}{2}}(n^{2}-n)\right]t\right)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/820054141bf5b5df83ea626143be342136603b46)