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

[U.Mutlu <um...@mu...>]

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.