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Geometric Brownian motionƒ ƒ 228

One realization of a geometric Brownian motion

()

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()

dt

S

dt

S

N

dS

dt

dt

N

dS S

dt

dW

V

dW

dt

dW

V

dt

V

dW

dt

V

dS S

V

dt

dt

dW

E

dt

dW

E

dt

E

dW

dt

E

dS S

E

dt

N

dW

where

dW

dt

dS S

t

t

t

t t

t

t

t

t

t t

t

t

t

t t

t

t

t t

2 ² , ~ & ² , ~

²

0

²

0

,

cov

2

0

, 0

~

,

σ

μ

σ

μ

σ σ σ μ σ μ σ μ

μ μ σ μ σ μ σ

σ μ

μ

⇒

= + + = + + = + =

⎤ ⎥ ⎦

⎡ ⎢ ⎣

= + = + = + = + =

⎤ ⎥ ⎦

⎡ ⎢ ⎣

+

=

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