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

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

I just mean that you're taking points at the t=1/12, 2/12 etc and seeing
whether they're inside the SD of the expected distribution at t=1.  That
looks to me like comparing apples to oranges.  If you have some use for
that, though, good.

Luigi


On Wed, Sep 13, 2023 at 2:55 AM U.Mutlu <um...@mu...> wrote:

> The said table is intended for this practical situation:
> You know (or will know) the stock price only at the end
> of the timesteps, and are interested to know from these
> datapoints the expectancy% at end for the pre-calculated
> range from -1SD to +1SD for t of the initial stock price.
>
> It's the same procedure like for timeSteps=1 (which is easier to
> understand).
>
> Normally you are interested in the outcome after t (ie. timeSteps=1 giving
> the
> usual 68.27%).
> But then you would also like to know the result if you
> watch more datapoints than just the one at the end,
> ie. in your example timeSteps=12 in t=1y.
>
> Your observation below is correct I think (except
> the sentence "But that makes no sense"), but that
> all is already counted for in the table, IMO.
>
> For timeSteps=12 in your example the expectancy is 83.5962% (ie.
> p=0.835962).
> Ie. 68.27% now suddenly becomes 83.5982% only b/c we watch the stock price
> more times (1 vs 12 times, at end of equal time intervals).
>
> Do you get a different value for the above said situation?
> Or asked differently: how should the table ideally be made up instead?
>
> I'll also create a cumulative version for 0 to t_i.
> Maybe you mean that one.
>
> This method can be important for bots who watch the
> stock price at such fixed intervals, like weeks or months,
> and need to know the probability... and/or for (auto-)trading
> systems based on such stochastics.
>
>
> Luigi Ballabio wrote on 09/11/23 19:01:
> > in your sample implementation, you generate paths from 0 to t using
> > a number of timesteps.  For the sake of example, let's say t=1 year and
> you
> > use 12 timesteps, so one step per month.  You calculate the price plus or
> > minus 1SD, where 1SD is the standard deviation of the expected
> distribution at
> > t=1.  Then you do the following to calculate percentage hits:
> >
> >      for (size_t j = 1; j <= timeSteps; ++j)
> >            {
> >              const auto Sx = samplePath.at(j);
> >              if (Sx >= m1SD && Sx <= p1SD) ++cHit;
> >            }
> >
> > that is, for every point in the path you check whether it's between S -
> 1SD
> > and S + 1SD.  But that makes no sense.  You can only check the point at
> the
> > end of the path, because it's the only point that belongs to the expected
> > distribution at t=1 year.  In the case of 12 timesteps, the point at j=1
> > belongs to the expected distribution at t=1 month, which has a
> different, much
> > smaller SD.  The point at j=2 belongs to the expected distribution at t=2
> > months, which has a different SD.  Of course your percentages increase
> with
> > the timesteps, because you're sampling points from all times between 0
> and 1
> > (which belong to much narrower distributions) and compare them with the
> SD at
> > t=1 year.  If you compare the points at each timestep with their correct
> > respective distributions, you'll find about the same percentage (the
> usual
> > 68%) falling within the respective SDs.
> >
> > Hope this helps clear this up—I won't be able to give much more time to
> this,
> > if at all.
>
>
>
>
>