Hi Ron: when i said that returns follow brownian motion, I meant that , in
discrete time

log(S_t+1) = log(S_t) + epsilon_t

where epsilon_t is N(0,delta_t)

Sorry, for not being clear on that. Definitely this is the assumption for
black sholes derivation but as many have pointed out, it's not realistic. (
why is maybe why you get the volatility smile ?  ).

I'm not sure about your terminology below. The term geometric brownian
motion
means that the log of prices are brownian motion which means that prices are
geometric
brownian motion. So, GBM refers to the process of S_t and BM refers to the
process of log(S_t).   If you have stefano's book , it's all in there quite
nicely.  and
it should be in Hull or Steele or any other well known text but his is the
best text
as far as using R code to generate these processes.

 Exact title is

Simulation and Infewrence for Stochastic Differential Equations, Springer
Series
in Statistics.


good luck.





On Mon, Apr 18, 2011 at 11:04 AM, Ron Michael <ron_michael70@yahoo.com>wrote:

> Hi Worik, I have been following this thread in "
> http://r.789695.n4.nabble.com/Artificial-price-series-td3443230.html", for
> quite a time now, however could not convince myself in one aspect. You said
> that simple RW model is not quite satisfactory (Horace Tso:: evidence is
> clear that financial prices are anything but brownian motion) hence, many
> people suggested GBM for that. But I could not understand why they are
> essentially different. I have:
>
> for vanilla RW:   log(S[t+1]) = log(S[t]) + epsilon~N(.,.)
> for vanilla GBM:  log(S[t+1]) = log(S[t]) + (mu - 0.5sigma^2) +
> epsilon~N(.,.)
>
> Of course hare I am comparing both **vanilla** type and if I want to
> incorporate other features like jump, heavy tail etc., then I can
> incorporate those features in either case. Therefore driven by some common
> sense, why those 2 models would be fundamentally different? Only difference
> I see that, I generally do not include Intercept in RW, because including an
> Intercept signifies some deterministic trend in the underlying price, which
> also makes sense.
>
> Additionally Mark says, "returns follow brownian motion", did he mean to
> say that **price** follows brownian motion?
>
> Any clarification would be highly appreciated.
>
> Thanks,
>
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