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

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

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.

This said: 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.

Regards,
    Luigi


On Mon, Sep 11, 2023 at 6:00 PM U.Mutlu <um...@mu...> wrote:

> Ioannis Rigopoulos wrote on 09/11/23 17:28:
> > Thank you U.Mutlu.
> >
> > The histograms indicate that the numbers represent indeed lognormal
> distributions.
> >
> > The average of all numbers in file GBM_ts1_A amounts to *104.5935229*
> >
> > The average of all numbers in file GBM_ts1_B amounts to *104.6047793*
> >
> > Both data sets exhibit the correct log standard deviation, very close to
> *0.3*.
>
> So far so good, and these are the most simplest things,
> but with the rest of your analysis I don't agree.
>
> What do other experts say on the following analysis of Ioannis? :
>
> > *Since you have assumed a drift of 0, the SDE represents a martingale and
> > therefore the average stock price at t=1 owes to be very close to 100.*
>  >
> > *This proves that the simulated numbers cannot represent the GBM
> diffusion of
> > a stock price starting at 100 and having drift = 0.*
> >
> > Furthermore, the quantity S(0)exp(0.5*sigma^2*t) equals *104.602786*,
> which is
> > extremely close to the above averages and indicates that the Ito term
> > 0.5*sigma^2*t is embedded in your results, albeit in a wrong fashion,
> since -
> > as mentioned above - it should not affect the average of S(t).
> >
> > It seems that you have set up your code in such a way that ln(S) is
> driftless
> > and therefore S acquires a positive Ito drift of 0.5*sigma^2.
> >
> > It also seems that the number of time steps are much greater than 1,
> otherwise
> > the distribution would appear as normal.
> >
> > At any case, since I cannot run your own code and verify the results, I
> will
> > finish my feedback with this email.
> >
> > Ioannis
>
> Oh oh, that's too much incorrect assumptions & interpretations by you.
> Nevertheless, thank you for attempting to analyse it.
>
> Maybe some other experts too will analyse the data and post their findings
> here.
>
> I claim that Ito's lemma is the culprit and that it's unnecessary
> both in GBM and BSM (Black-Scholes-Merton option pricing formula).
>
>
> > On 9/11/2023 4:19 PM, U.Mutlu wrote:
> >> Ioannis Rigopoulos wrote on 09/11/23 13:43:
> >>> Could you please send us a link to a file that contains the simulated
> values
> >>> produced by your program when you use one time step?
> >>>
> >>> Since you use 1,000,000 simulations, the file should contain 1,000,000
> >>> numbers.
> >>
> >> On the following web page you can find download links to even 2 such
> >> GBM data sets I just have created and uploaded to a server:
> >>
> https://www.elitetrader.com/et/threads/simulating-stock-prices-using-gbm.375533/page-5#post-5861992
> >>
> >>
> >> "
> >> 2 GBM generated test files in CSV format (with header line; see there),
> >> with 1,000,000 generated stock prices (S0=100, IV=30, t=1y, timeSteps=1,
> >> r=0, q=0),
> >> each about 29 MB uncompressed (about 12 MB zip-compressed).
> >>
> >> This GBM generator does not use Ito's lemma.
> >>
> >> ...
> >> "
> >>
> >>
> >>> On 9/11/2023 1:33 PM, U.Mutlu wrote:
> >>>> Ioannis Rigopoulos wrote on 09/11/23 11:47:
> >>>> > By the way, this is how my histogram of S(t) (for t = 1) simulated
> values
> >>>> > looks like when I used 100,000 time steps and 100,000 simulations.
> It is
> >>>> > clearly lognormal in shape.
> >>>>
> >>>> Yes, it's lognormally. My values are lognormlly as well,
> >>>> You just restate the obvious fact.
> >>>>
> >>>> > I am sure, if you do the same when time steps = 1,
> >>>> > you will get a normal distribution for S(t), which would prove
> >>>> > that time steps = 1 does not reproduce the theoretically expected
> >>>> > values for S(t).
> >>>>
> >>>> Not true, simply from the fact that the set boundary is lognormally:
> >>>>   -1SD: 74.08182206817179
> >>>>   +1SD: 134.9858807576003
> >>>> Ie. it's not symmetric (Gaussian, normally distributed) around the
> >>>> initial mean (100), but is lognormally (and therefore unsymmetric).
> >>>>
> >>>> FYI: in my other test program (that does not use QuantLib)
> >>>> I get in such simulations exact results almost identical
> >>>> to the expected theoretical results! So, your math and arguing is
> wrong, IMO.
> >>>>
> >>>> You are too much fixated on timeSteps=1. Maybe you have
> >>>> a much different understanding of what timeSteps really
> >>>> means in practice. It's _not_ something like the number
> >>>> of simulations that you simply can change and expect
> >>>> to get the same result; no, it has a clearly defined context
> >>>> and gives different results for different timeSteps.
> >>>>
> >>>> And you haven't answered the question why you think it has
> >>>> to be 68.27% for all timeSteps (which IMO is wrong).
> >>>>
> >>>> For those interested, here are more details about the above said:
> >>>>
> https://www.elitetrader.com/et/threads/simulating-stock-prices-using-gbm.375533/page-3#post-5849548
> >>>>
> >>>>
> >>>>
> >>>> And, I today did a brief summary of these ongoing discussions on GBM
> here:
> >>>>
> https://www.elitetrader.com/et/threads/simulating-stock-prices-using-gbm.375533/page-5#post-5861919
> >>>>
> >>>>
> >>>>
> >>>>
> >>>> Ioannis Rigopoulos wrote on 09/11/23 11:47:
> >>>>> Our SDE for the evolution of S is:
> >>>>>
> >>>>> dS = μSdt + σSdw
> >>>>>
> >>>>> In your code you use:
> >>>>>
> >>>>>
> >>>>> PathGenerator<RandomSequenceGenerator<MersenneBoxMuller>>
> >>>>> gbmPathGenerator(gbm, t, timeSteps, gsg, false);
> >>>>> const Path& samplePath = gbmPathGenerator.next().value;
> >>>>>
> >>>>> The PathGenerator::next().value is designed to return an array of
> numbers
> >>>>> {S(T₁) , S(T₂) , ... , S(Tᵤ)}
> >>>>>
> >>>>> where
> >>>>>
> >>>>> u := timeSteps
> >>>>>
> >>>>> Δt := t/u
> >>>>>
> >>>>> Tᵢ := iΔt for i = 1,...,u
> >>>>>
> >>>>> It follows:
> >>>>>
> >>>>> Tᵤ = t = the final time when we are interested to know the simulated
> >>>>> values of
> >>>>> S (in your test you use t = 1).
> >>>>>
> >>>>> QuantLib builds the sequence {S(T₁) , S(T₂) , ... , S(Tᵤ)}
> incrementally
> >>>>> through a total number of u steps by adding the quantity ΔS := μSΔt +
> >>>>> σSΔw on
> >>>>> the previous member of the sequence, with Δw := random number drawn
> from
> >>>>> N~(0,Δt¹ᐟ²).
> >>>>>
> >>>>> The previous member of S(T₁) is assumed the T₀ = 0 and the initial
> value
> >>>>> S(T₀)
> >>>>> = S(0) is assumed deterministic and known.
> >>>>>
> >>>>> So, starting with the given S(T₀), QuantLib calculates the S(T₁)
> using the
> >>>>> formula:
> >>>>>
> >>>>> S(T₁) := S(T₀) + μS(T₀)Δt + σS(T₀)Δw
> >>>>>
> >>>>> Then given the S(T₁) as calculated above, it proceeds to calculate
> S(T₂) as:
> >>>>>
> >>>>> S(T₂) := S(T₁) + μS(T₁)Δt + σS(T₁)Δw
> >>>>>
> >>>>> For simplicity I have used the same notation for the Δw in the
> equations
> >>>>> above, but they are different.
> >>>>>
> >>>>> After u steps, the final value S(t) = S(Tᵤ) is calculated as:
> >>>>>
> >>>>> S(t) = S(Tᵤ) := S(Tᵤ₋₁) + μS(Tᵤ₋₁)Δt + σS(Tᵤ₋₁)Δw
> >>>>>
> >>>>> All equations above hold only in an approximate sense when Δt is
> finite.
> >>>>>
> >>>>> They are accurate only in a limiting sense as Δt → 0.
> >>>>>
> >>>>> It follows that the value S(t) will be calculated correctly only if
> Δt → 0,
> >>>>> which is equivalent to u → ∞
> >>>>>
> >>>>> If you use u = 1 (i.e. one time step), you will still match the
> 68.27% rule
> >>>>> when the 1SD interval is defined in accordance with a normal
> distribution
> >>>>> because S(T₁) := S(T₀) + μS(T₀)Δt + σS(T₀)Δw corresponds to a normal
> >>>>> distribution and if you build a histogram of all simulated values
> you will
> >>>>> notice that S(T₁) is distributed normally, which is clearly not
> consistent
> >>>>> with the SDE that predicts lognormal distribution.
> >>>>>
> >>>>> The correct procedure is of course to use a very large u and define
> the 1SD
> >>>>> interval in accordance with a lognormal distribution, as I described
> in my
> >>>>> previous email.
> >>>>>
> >>>>> Then you will get convergence to the 68.27% rule.
> >>>>>
> >>>>> By the way, this is how my histogram of S(t) (for t = 1) simulated
> values
> >>>>> looks like when I used 100,000 time steps and 100,000 simulations.
> It is
> >>>>> clearly lognormal in shape. I am sure, if you do the same when time
> steps
> >>>>> = 1,
> >>>>> you will get a normal distribution for S(t), which would prove that
> time
> >>>>> steps
> >>>>> = 1 does not reproduce the theoretically expected values for S(t).
> >>>>>
> >>>>> Ioannis
> >>>>>
> >>>>> On 9/11/2023 10:19 AM, U.Mutlu wrote:
> >>>>>> Dear Ioannis Rigopoulos, I now analysed your table and
> >>>>>> must say that you seem to have a much different expectation to,
> >>>>>> and understanding of that table.
> >>>>>>
> >>>>>> The  values in your table for timeSteps > 1 can definitely
> >>>>>> not be correct, b/c why do you think it has to be 68.27 for all
> timeSteps?
> >>>>>> This is illogical b/c the more timeSteps the more the hitrate%
> >>>>>> must be, like in my table. It saturates at about 84.93%
> >>>>>>
> >>>>>> Sorry, I have to retract my initial enthusiastic judment
> >>>>>> of your work below, as it's fundamentally flawed & wrong,
> >>>>>> and I should have been more careful when I quickly & briefly
> >>>>>> inspected your posting in the early morning here.
> >>>>>>
> >>>>>> Since your results in the table for timeSteps > 1 are wrong,
> >>>>>> then your formulas must be wrong too, so I had to stop my analysis
> right
> >>>>>> there.
> >>>>>>
> >>>>>> Please just answer why you think it has to be 68.27% for all
> timeSteps?
> >>>>>>
> >>>>>>
> >>>>>> U.Mutlu wrote on 09/11/23 08:36:
> >>>>>>> Mr. Ioannis Rigopoulos, that's a brilliant work by you. Thank you
> very
> >>>>>>> much.
> >>>>>>> I need some time to study and verify the reults, but it looks very
> good
> >>>>>>> to solve the confusion and mystery I had experienced when I
> started this
> >>>>>>> research some weeks ago.
> >>>>>>> I'll try to replicate your results and will report later in detail
> here.
> >>>>>>>
> >>>>>>> Ioannis Rigopoulos wrote on 09/10/23 19:29:
> >>>>>>>  > On a different topic, I do not understand your statement "/I've
> never
> >>>>>>> seen
> >>>>>>>  > that Sx@-1SD and Sx@+1SD get computed with Ito's lemma
> applied/".
> >>>>>>>  > What do the Sx@-1SD and Sx@+1SD mean? I suppose m -1SD and m
> +1SD,
> >>>>>>> where
> >>>>>>>
> >>>>>>> Sorry for the consfusion, it just means the stock price (Sx after
> time t,
> >>>>>>> ie. S(t) or S_t) for the lower boundary at -1SD and for the upper
> >>>>>>> boundary at
> >>>>>>> +1SD,
> >>>>>>> ie. your "min of S(t)" and "max of S(t)" in your other posting.
> >>>>>>> In future I'll use a more common term like yours, sorry.
> >>>>>>>
> >>>>>>> Thx
> >>>>>>>
> >>>>>>>
> >>>>>>> Ioannis Rigopoulos wrote on 09/10/23 19:29:
> >>>>>>>> Yes, I mean mu * t - 0.5 * sigma * sigma * t
> >>>>>>>>
> >>>>>>>> Are you sure, you also adjusted the boundaries corresponding to
> plus and
> >>>>>>>> minus
> >>>>>>>> 1 SD?
> >>>>>>>>
> >>>>>>>> I ran an Excel spreadsheet by modifying a spreadsheet I had
> published
> >>>>>>>> several
> >>>>>>>> years ago at
> https://blog.deriscope.com/index.php/en/excel-value-at-risk
> >>>>>>>> that
> >>>>>>>> uses the GeometricBrownianMotionProcess , albeit with the
> >>>>>>>> PseudoRandom::rsg_type generator.
> >>>>>>>>
> >>>>>>>> At any case, I got convergence to 68.27% as I increased the
> number of
> >>>>>>>> time
> >>>>>>>> steps and keeping the number of simulations fixed to 100,000,
> provided I
> >>>>>>>> used
> >>>>>>>> the correct (Ito-adjusted) means, as below:
> >>>>>>>>
> >>>>>>>> As you see above, the "No Ito-adjusted" boundaries are 74.08 and
> 134.99,
> >>>>>>>> just
> >>>>>>>> like those you use.
> >>>>>>>>
> >>>>>>>> The Ito-adjusted boundaries are 70.82 and 129.05 produced by
> shifting the
> >>>>>>>> no-adjusted boundaries - in log terms - to the left by
> 0.5*sigma^2*t.
> >>>>>>>>
> >>>>>>>> The simulation results (percentage of values being within the min
> and max
> >>>>>>>> boundaries) are below:
> >>>>>>>>
> >>>>>>>> As you see, the Ito-adjusted percentage reaches 68.30% for
> 100,000 time
> >>>>>>>> steps,
> >>>>>>>> which is close to the expected 68.27%.
> >>>>>>>>
> >>>>>>>> The No Ito-adjusted percentage seems instead to converge to a
> different
> >>>>>>>> value
> >>>>>>>> around 67.73%
> >>>>>>>>
> >>>>>>>> On a different topic, I do not understand your statement "/I've
> never
> >>>>>>>> seen
> >>>>>>>> that Sx@-1SD and Sx@+1SD get computed with Ito's lemma applied/".
> >>>>>>>>
> >>>>>>>> What do the Sx@-1SD and Sx@+1SD mean? I suppose m -1SD and m
> +1SD, where
> >>>>>>>> m is
> >>>>>>>> the mean of a normally distributed variable. This means that one
> needs to
> >>>>>>>> calculate m in order to do the mentioned computation.
> >>>>>>>>
> >>>>>>>> If your starting point is an SDE like dS = S*mu*dt + S*sigma*dw,
> then
> >>>>>>>> ln(S(t))
> >>>>>>>> at any time t will be normally distributed with mean m = ln(S(0))
> +
> >>>>>>>> mu*t -
> >>>>>>>> 0.5*sigma^2*t , where the last term is a consequence of the Ito
> lemma.
> >>>>>>>>
> >>>>>>>> Do you have any reason to believe that the Ito lemma is somehow
> not
> >>>>>>>> valid so
> >>>>>>>> that m = ln(S(0)) + mu*t?
> >>>>>>>>
> >>>>>>>> Ioannis
> >>>>>>>>
> >>>>>>>> On 9/10/2023 3:00 PM, U.Mutlu wrote:
> >>>>>>>>
> >>>>>>>>> Ioannis Rigopoulos wrote on 09/10/23 13:06:
> >>>>>>>>> > new GeometricBrownianMotionProcess(S, mu - 0.5*sigma^2*t,
> sigma)
> >>>>>>>>>
> >>>>>>>>> Thank you for your analysis.
> >>>>>>>>> I guess you mean "mu * t - 0.5 * sigma * sigma * t",
> >>>>>>>>> though in our example with t=1 it doesn't make any difference.
> >>>>>>>>> It gives for timeSteps=1 this result:
> >>>>>>>>>
> >>>>>>>>> $ ./Testing_GBM_of_QuantLib.exe 1
> >>>>>>>>> timeSteps=1 nRuns=1000000 seed=1694350548 S=100.000000
> mu=0.000000
> >>>>>>>>> sigma=0.300000 t=1.000000 : nGenAll=1000000 USE_ITO=1 :
> -1SD=70.822035
> >>>>>>>>> +1SD=129.046162
> >>>>>>>>> cHit=663067/1000000(66.307%)
> >>>>>>>>>
> >>>>>>>>> This is far from the expected value of 68.2689% for timeStep=1.
> >>>>>>>>> So, there must be a bug somewhere. IMO it's our suspect friend
> Ito :-)
> >>>>>>>>>
> >>>>>>>>> Cf. also the following showing similar results for GBM,
> >>>>>>>>> which then was compared to the standard lognormal calculation:
> >>>>>>>>>
> https://www.elitetrader.com/et/threads/simulating-stock-prices-using-gbm.375533/page-3#post-5849548
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>> And, honestly, I've never seen that Sx@-1SD and Sx@+1SD get
> computed
> >>>>>>>>> with
> >>>>>>>>> Ito's lemma applied.
> >>>>>>>>>
> >>>>>>>>>
> >>>>>>>>> Ioannis Rigopoulos wrote on 09/10/23 13:06:
> >>>>>>>>>> Thank you for testing the BoxMullerGaussianRng code, which - as
> I
> >>>>>>>>>> wrote -
> >>>>>>>>>> does
> >>>>>>>>>> not seem to be used in other areas of the standard part of
> QuantLib.
> >>>>>>>>>>
> >>>>>>>>>> Your numerical results below ...
> >>>>>>>>>>
> >>>>>>>>>> indicate that the simulated values for ln(S(t)) produced with
> >>>>>>>>>> timeSteps = 1
> >>>>>>>>>> are very likely normally distributed with mean (I use your
> notation)
> >>>>>>>>>> ln(S(0))
> >>>>>>>>>> + mu*t and standard deviation sigma * sqrt(t)
> >>>>>>>>>>
> >>>>>>>>>> This result _*is consistent*_ with:
> >>>>>>>>>>
> >>>>>>>>>> a) the SDE: dS(t) = S(t)*mu*dt + S(t)*sigma*dw
> >>>>>>>>>>
> >>>>>>>>>> _*and*_
> >>>>>>>>>>
> >>>>>>>>>> b) the fact that in QuantLib the PathGenerator.next().value
> returns the
> >>>>>>>>>> result
> >>>>>>>>>> of the SDE expression _*without *_applying the ITO correction
> >>>>>>>>>> associated
> >>>>>>>>>> with
> >>>>>>>>>> the fact that dt is not infinitesimal.
> >>>>>>>>>>
> >>>>>>>>>> The b) is also responsible for the lack of convergence in your
> output
> >>>>>>>>>> towards
> >>>>>>>>>> the theoretical target of 68.27%
> >>>>>>>>>>
> >>>>>>>>>> You would get the correct convergence if you modified your code
> by
> >>>>>>>>>> using
> >>>>>>>>>> the
> >>>>>>>>>> expressions below:
> >>>>>>>>>>
> >>>>>>>>>>      const double m1SD = S * exp(mu * t - 0.5*sigma^2*t + sigma
> *
> >>>>>>>>>> sqrt(t) *
> >>>>>>>>>> -1.0);     // Sx at -1SD
> >>>>>>>>>>      const double p1SD = S * exp(mu * t - 0.5*sigma^2*t + sigma
> *
> >>>>>>>>>> sqrt(t) *
> >>>>>>>>>> 1.0);     // Sx at +1SD
> >>>>>>>>>>
> >>>>>>>>>>      new GeometricBrownianMotionProcess(S, mu - 0.5*sigma^2*t,
> >>>>>>>>>> sigma) (as
> >>>>>>>>>> Peter also pointed out)
> >>>>>>>>>>
> >>>>>>>>>> Again, due to b) one can produce the correct simulated values
> of a GBM
> >>>>>>>>>> diffused quantity (such as a stock price in the GBM model) by
> using N
> >>>>>>>>>> time
> >>>>>>>>>> steps, with N very large. Using N = 1 (like in your example),
> the
> >>>>>>>>>> simulated
> >>>>>>>>>> values will still be lognormally distributed (whence your good
> result
> >>>>>>>>>> with
> >>>>>>>>>> N =
> >>>>>>>>>> 1), but will be centered at a wrong mean and thus will _*fail
> *_to
> >>>>>>>>>> represent
> >>>>>>>>>> the correct values expected by the GBM SDE.
> >>>>>>>>>>
> >>>>>>>>>> Ioannis
> >>>>>>>>>>
> >>>>>>>>>> On 9/10/2023 11:29 AM, U.Mutlu wrote:
> >>>>>>>>>>> As said in other posting here, after fixing the test program
> >>>>>>>>>>> by skipping the first item (the initial price) in the
> >>>>>>>>>>> generated sample path, BoxMullerGaussianRng now passes the
> said test.
> >>>>>>>>>>>
> >>>>>>>>>>> The bugfixed test code and the new test results can be found
> here:
> >>>>>>>>>>>
> https://www.elitetrader.com/et/threads/simulating-stock-prices-using-gbm.375533/page-4#post-5861666
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>> That important fact that the generated sample path contains
> >>>>>>>>>>> not timeSteps elements but 1 + timeSteps elements needs
> >>>>>>>>>>> to be documented in the library doc.
> >>>>>>>>>>> For example on this API doc page one normally would expect
> >>>>>>>>>>> to find such an important information, but it's missing:
> >>>>>>>>>>>
> https://www.quantlib.org/reference/class_quant_lib_1_1_path_generator.html
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>> If you or someone else can change/extend the test program by
> using
> >>>>>>>>>>> the suggested alternative(s) to BoxMullerGaussianRng, I would
> be happy
> >>>>>>>>>>> to hear about it. Thx.
> >>>>>>>>>>>
> >>>>>>>>>>>
> >>>>>>>>>>> Ioannis Rigopoulos wrote on 09/09/23 16:28:
> >>>>>>>>>>>> If you search within the QuantLib code for
> BoxMullerGaussianRng, you
> >>>>>>>>>>>> will
> >>>>>>>>>>>> see
> >>>>>>>>>>>> it is used only in the experimental folder. It is therefore
> not
> >>>>>>>>>>>> surprising if
> >>>>>>>>>>>> it doesn't produce the expected results.
> >>>>>>>>>>>>
> >>>>>>>>>>>> I use myself the MultiPathGenerator with
> PseudoRandom::rsg_type,
> >>>>>>>>>>>> which is
> >>>>>>>>>>>> used
> >>>>>>>>>>>> extensively in other areas of QuantLib.
> >>>>>>>>>>>>
> >>>>>>>>>>>> This type expands to InverseCumulativeRsg<
> RandomSequenceGenerator<
> >>>>>>>>>>>> MersenneTwisterUniformRng > , InverseCumulativeNormal > and
> gives me
> >>>>>>>>>>>> good
> >>>>>>>>>>>> results.
> >>>>>>>>>>>>
> >>>>>>>>>>>> Ioannis Rigopoulos, founder of deriscope.com
> >>
> >>
> >
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