Hi Ron: I'm sorry if I wasn't clear. The problem is that I don't remember
what you had
for GBM and RW in your previous email.

I think it will be more useful for you to read stefano's book or hamilton
than listen to me
but I'll give you a short summary on it.

brownian motion and geometric brownian motion are continous stochastic
processes.  So, comparing GBM and RW is like comparing apples and oranges
because a random walk is the discrete version of BM. As I said earlier, the
RW for log prices is log(S_t) = log(S_t-1) + epsilon_t. That's not a
continuous stochastic process because S_t is sampled at integer
values of t.

In chapter 17 of  Hamilton, there is a beautiful explanation of how RW
converges
to brownian motion if you fix the time window, T,  but sample more and more
frequently in
that fixed time window. Clearly, I'd be doing Hamilton an injustice if I
even
tried to explain it here. The rough idea behind the convergence is that,  if
you sample an RW more finely and more finely, the the sum of the
epsilon_t's, scaled by the correct T, converges to normal(0,1).

The other thing I should say is that GBM for prices is the SAME PROCESS as
BM for log prices. It's just that one descibes the trajectory of price and
the other is describing the trajectory of log(price). Definitely check out
Hamilton's text or Stefano's or both and it will all
be come much clearer. Actually, they complement each other so I suggest
checking out
both of them.  But I hope this helped a little.


Mark




























On Mon, Apr 18, 2011 at 5:46 PM, Ron Michael <ron_michael70@yahoo.com>wrote:

> Thanks Horace and Mark for replying on my query. However I believe that my
> question was not properly understood or may be I could not understand your
> statement. Whatever it happened, I believe there is some miscommunication.
>
> I want to understand that: between these 2 candidates GBM & RW (form of
> them I already mentioned in my previous mail), which could be better. I
> understand that both are very simplistic in modelling any price, however if
> I have to chose anyone from them then what should I chose? GBM or RW.
>
> Many previous mails seems to me that, they are in support of GBM, where I
> believe that there are no fundamental difference between them. Or perhaps I
> am missing something?
>
> Thanks,
>
> --- On Tue, 19/4/11, Horace Tso <Horace.Tso@pgn.com> wrote:
>
> > From: Horace Tso <Horace.Tso@pgn.com>
> > Subject: RE: [R-SIG-Finance] Artificial price series
> > To: "Ron Michael" <ron_michael70@yahoo.com>, "
> r-sig-finance@r-project.org" <r-sig-finance@r-project.org>
> > Date: Tuesday, 19 April, 2011, 1:44 AM
> > Ron & others,
> >
> > I venture to give my 2 cent worth of opinion here. Many
> > (the Professors on the list please weigh in) have more
> > insight than I do.
> >
> > Here is what I find curious. As Worik pointed out in his
> > original post, if you use any of the R simulation functions
> > to generate a random walk or a fancier brownian motion
> > series, plot it with some real price data, let's say daily
> > closing level of the Dow Jones Industrial Average, you could
> > tell they are different. But it's hard to say exactly what
> > makes them look different to the human eyes.
> >
> > Not too long ago I played this trick on a friend of mine,
> > who claimed he had no prior knowledge of finance nor
> > familiar with securities price patterns. I sampled a number
> > of time series from Bloomberg. I think I picked daily
> > S&P, minute ticks of gold, nymex natural gas, the
> > Shanghai Composite stock index, and other unrelated real
> > prices. Then I simulated equal number of artificial data
> > series using different stochastic models. A few were just
> > straightforward random walk with varying sigma, others were
> > jump diffusion (EMJumpDiffusion) with different jump prob
> > and size. I plotted them along side and I was careful to
> > remove the axis labels so he couldn't see the scale of
> > things.
> >
> > I asked my friend to tell me which were real and which were
> > artificial data. Amazingly he got many of them right. (OK,
> > full disclosure here: this occurred in a bar after he had a
> > couple of beers.)
> >
> > I could see how even the most sophisticated simulation
> > models couldn't quite imitate reality. First, many stock
> > data have trend. The buy-and-hold crowd is right. Secondly,
> > the unobserved volatility parameter is in itself a
> > stochastic process (huge literature on this). Third, the
> > jump processes in simulation models are probably too
> > simplistic. Just plot Dow Jones from July to Dec, 1987.
> >
> > Horace
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> > -----Original Message-----
> > From: r-sig-finance-bounces@r-project.org
> > [mailto:r-sig-finance-bounces@r-project.org]
> > On Behalf Of Ron Michael
> > Sent: Monday, April 18, 2011 8:05 AM
> > To: r-sig-finance@r-project.org
> > Subject: [R-SIG-Finance] Artificial price series
> >
> > 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,
> >
> > _______________________________________________
> > R-SIG-Finance@r-project.org
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> > first.
> > -- Also note that this is not the r-help list where general
> > R questions should go.
> >
>
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