Re: [Quantlib-users] QuantLib's GBM found flawed. What say the experts? (SOLVED)

[Luigi B. <lui...@gm...>]

I had seen your table, but please, read again the part of my email where I
explain that you're calculating it by taking points from different expected
distributions.

Luigi

On Mon, Sep 11, 2023 at 8:29 PM U.Mutlu <um...@mu...> wrote:

> U.Mutlu wrote on 09/11/23 19:41:
> > Luigi, your below posting still has not made to my mail client,
> > maybe the mailman takes longer to send it out to the list members,
> > or again your mail server could not deliver to my mail server.
> > I just saw your posting only in the archive.
> >
> > Luigi B. <lui...@gm...> - 2023-09-11 17:01:32
> >  >  for (size_t j = 1; j < timeSteps; ++j)
> >
> > Attention! The above loop is incorrect!
> > It must be
> >    for (size_t j = 1; j <= timeSteps; ++j)
> > That's very important!
> >
> > I think many QuantLib users even use this buggy one:
> >    for (size_t j = 0; j < timeSteps; ++j)
> > but that one is totally wrong!
> >
> >
> > I'll reply later to the rest of your posting.
>
> I created the table below that shows the expected values
> for the various timeSteps.
> It's correctness was also confirmed by simulations.
> If that table does not answer your questions, let me know and I'll try to
> clarify.
>
> "
> Expected HitRate% for 1SD around initial stock price for various GBM
> timeSteps
> This table is an extension of
> https://en.wikipedia.org/wiki/68–95–99.7_rule as
> there only timeStep=1 is given.
> These numbers are exact values, not the result of simulations.
> Params used: S=100 rPct=0 qPct=0 IV=30(s=0.3) DIY=365.00 DTE=365(t=1 dt=1
> dd=365) zFm=-1(SxFm=74.081822) zTo=+1(SxTo=134.985881)
>     timeSteps   Expected_HitRate
>            1       68.2689%   cf. the above wiki link
>            2       76.2695%
>            3       79.2918%
>            4       80.7919%
>            5       81.6648%
>            6       82.2304%
>            7       82.6265%
>            8       82.9197%
>            9       83.1461%
>           10       83.3265%
>           15       83.8653%
>           20       84.1337%
>           25       84.2942%
>           30       84.4010%
>           35       84.4771%
>           40       84.5341%
>           45       84.5785%
>           50       84.6139%
>           60       84.6671%
>           70       84.7050%
>           80       84.7334%
>           90       84.7555%
>          100       84.7732%
>          500       84.9003%
>         1000       84.9162%
>        10000       84.9305%
>       100000       84.9319%
>      1000000       84.9320%
>     10000000       84.9320%
> "
>
> Regarding Ito's lemma you write:.
>  > You talk as if Ito's lemma was something that one chooses to apply
> because
>  > one likes it.  It's not.  Just as there is a rule for performing a
> change
>  > of variable in an integral or a derivative, there is also a rule for
>  > performing a change of variable in a stochastic differential equation,
> and
>  > that's Ito's lemma.  If you change variables from x to log(x) or
> whatever
>  > else, you have to use it.  If you don't change variables, you don't
> have to
>  > use it.
>
> As you say, I indeed don't need to change any variables,
> so then I don't need this Ito's lemma stuff.
> I'm a practitioner from the field, not a theoretician.
> I just know this, and can even prove it: Ito's lemma
> is definitely not needed, neither in GBM nor in Black-Scholes-Merton (BSM).
> At the moment I can't say anything about the possible
> use of Ito's lemma in any other field beyond these two, if any.
>
>
>
>