Movimiento browniano geométrico

El movimiento Browniano geométrico dado por

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

tiene una distribución log-normal con función de densidad dada por:

${\displaystyle f_{S_{t}}(s)={\frac {1}{s\sigma {\sqrt {2t\pi }}}}\exp \left(-{\frac {\left(\ln s-\ln S_{0}-\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right)^{2}}{2\sigma ^{2}t}}\right).}$

para ${\displaystyle s\in (0,\infty )}$.

La función de distribución acumulada está dada por

${\displaystyle F_{S_{t}}(s)=\Phi \left({\frac {\ln s-\ln S_{0}-\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t}{\sigma {\sqrt {t}}}}\right)}$

para ${\displaystyle s\in (0,\infty )}$.

Para hallar la media, varianza y covarianza del movimiento browniano geométrico, usaremos el hecho de que

${\displaystyle M_{S}(s)=\exp \left(\mu s+{\frac {1}{2}}\sigma ^{2}s^{2}\right)}$

es la función generadora de momentos de una distribución normal con parámetros ${\displaystyle \mu }$ y ${\displaystyle \sigma ^{2}}$.

La media del movimiento Browniano geométrico es

${\displaystyle \operatorname {E} [S_{t}]=S_{0}e^{\mu t}}$

pues

${\displaystyle {\begin{aligned}\operatorname {E} [S_{t}]&=\operatorname {E} \left[S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t}\right]\right]\\&=S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\operatorname {E} \left[e^{\sigma W_{t}}\right]\\&=S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]e^{\frac {t\sigma ^{2}}{2}}\\&=S_{0}e^{\mu t}\end{aligned}}}$

La varianza del movimiento Browniano geométrico es

${\displaystyle \operatorname {Var} (S_{t})=S_{0}^{2}e^{2\mu t}\left(e^{\sigma ^{2}t}-1\right)}$

pues

${\displaystyle {\begin{aligned}\operatorname {Var} [S_{t}]&=\operatorname {Var} \left[S_{0}\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t+\sigma W_{t}\right]\right]\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\operatorname {Var} \left[e^{\sigma W_{t}}\right]\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\left(\operatorname {E} [\exp(2\sigma W_{t})]-\operatorname {E} [\exp(\sigma W_{t})]^{2}\right)\\&=S_{0}^{2}\exp \left[2\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right]\left[\exp \left({\frac {t(2\sigma )^{2}}{2}}\right)-\exp \left({\frac {2t\sigma ^{2}}{2}}\right)\right]\\&=S_{0}^{2}e^{2\mu t}\left(e^{\sigma ^{2}t}-1\right)\end{aligned}}}$

La covarianza del movimiento Browniano geométrico es

${\displaystyle \operatorname {Cov} (S_{t},S_{s})=S_{0}^{2}e^{2\mu (s+t)}\left(e^{\sigma ^{2}s}-1\right)}$

Para ${\displaystyle n\in \mathbb {N} }$ y ${\displaystyle t\in [0,\infty )}$, el ${\displaystyle n}$-ésimo momento del proceso está dado por

${\displaystyle \operatorname {E} [S_{t}^{n}]=S_{0}\exp \left(\left[n\mu +{\frac {\sigma ^{2}}{2}}(n^{2}-n)\right]t\right)}$