ABSTRACT
Stock prices can become negative in this model. A more accurate model that removes this unhappy possibility is geometric Brownian motion:
dS Sdt SdWt= +µ σ (20.2)
Here
S(t) is stock price versus time
S(0) is stock price today
dW is random increment of a Wiener process
σ is volatility
μ is dri
We know by denition that
E dW Std dW dtt t( ) , ( )= =0
What exactly does dS Sdt VSdWt= +µ mean? Suppose we have
dx a x t dt b x t dWt= +( , ) ( , )
x(0) = A
en, by integrating both sides (at rst proceeding rather formally), we have
x t A a x s ds b x s dW
s( ) ( , ) ( , )= + +∫ ∫ 0 0
(20.3)
e second integral in Equation 18.3 is what we term a “stochastic integral.”
