El movimiento browniano geométrico (GBM) (también conocido como movimiento browniano exponencial ) es un modelo de amplio uso en finanzas y sirve para representar el precio de algunos bienes que fluctúan siguiendo los vaivenes de los mercados financieros, en particular, es utilizado en matemáticas financieras para modelar precios en el modelo de Black-Scholes .
El movimiento Browniano geométrico dado por
S
t
=
S
0
exp
(
(
μ
−
σ
2
2
)
t
+
σ
W
t
)
,
t
≥
0
{\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:
f
S
t
(
s
)
=
1
s
σ
2
t
π
exp
(
−
(
ln
s
−
ln
S
0
−
(
μ
−
1
2
σ
2
)
t
)
2
2
σ
2
t
)
.
{\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
s
∈
(
0
,
∞
)
{\displaystyle s\in (0,\infty )}
editar
La función de distribución acumulada está dada por
F
S
t
(
s
)
=
Φ
(
ln
s
−
ln
S
0
−
(
μ
−
σ
2
2
)
t
σ
t
)
{\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
s
∈
(
0
,
∞
)
{\displaystyle s\in (0,\infty )}
Para hallar la media , varianza y covarianza del movimiento browniano geométrico, usaremos el hecho de que
M
S
(
s
)
=
exp
(
μ
s
+
1
2
σ
2
s
2
)
{\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
μ
{\displaystyle \mu }
σ
2
{\displaystyle \sigma ^{2}}
La media del movimiento Browniano geométrico es
E
[
S
t
]
=
S
0
e
μ
t
{\displaystyle \operatorname {E} [S_{t}]=S_{0}e^{\mu t}}
pues
E
[
S
t
]
=
E
[
S
0
exp
[
(
μ
−
1
2
σ
2
)
t
+
σ
W
t
]
]
=
S
0
exp
[
(
μ
−
1
2
σ
2
)
t
]
E
[
e
σ
W
t
]
=
S
0
exp
[
(
μ
−
1
2
σ
2
)
t
]
e
t
σ
2
2
=
S
0
e
μ
t
{\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
Var
(
S
t
)
=
S
0
2
e
2
μ
t
(
e
σ
2
t
−
1
)
{\displaystyle \operatorname {Var} (S_{t})=S_{0}^{2}e^{2\mu t}\left(e^{\sigma ^{2}t}-1\right)}
pues
Var
[
S
t
]
=
Var
[
S
0
exp
[
(
μ
−
1
2
σ
2
)
t
+
σ
W
t
]
]
=
S
0
2
exp
[
2
(
μ
−
1
2
σ
2
)
t
]
Var
[
e
σ
W
t
]
=
S
0
2
exp
[
2
(
μ
−
1
2
σ
2
)
t
]
(
E
[
exp
(
2
σ
W
t
)
]
−
E
[
exp
(
σ
W
t
)
]
2
)
=
S
0
2
exp
[
2
(
μ
−
1
2
σ
2
)
t
]
[
exp
(
t
(
2
σ
)
2
2
)
−
exp
(
2
t
σ
2
2
)
]
=
S
0
2
e
2
μ
t
(
e
σ
2
t
−
1
)
{\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
Cov
(
S
t
,
S
s
)
=
S
0
2
e
2
μ
(
s
+
t
)
(
e
σ
2
s
−
1
)
{\displaystyle \operatorname {Cov} (S_{t},S_{s})=S_{0}^{2}e^{2\mu (s+t)}\left(e^{\sigma ^{2}s}-1\right)}
n
{\displaystyle n}
editar
Para
n
∈
N
{\displaystyle n\in \mathbb {N} }
t
∈
[
0
,
∞
)
{\displaystyle t\in [0,\infty )}
n
{\displaystyle n}
E
[
S
t
n
]
=
S
0
exp
(
[
n
μ
+
σ
2
2
(
n
2
−
n
)
]
t
)
{\displaystyle \operatorname {E} [S_{t}^{n}]=S_{0}\exp \left(\left[n\mu +{\frac {\sigma ^{2}}{2}}(n^{2}-n)\right]t\right)}
![{\displaystyle \operatorname {E} [S_{t}]=S_{0}e^{\mu t}}](https://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}}}](https://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}}}](https://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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/820054141bf5b5df83ea626143be342136603b46)
Datos: Q1503307
Multimedia: Brownian motion / Q1503307
