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

[Ioannis R. <qua...@de...>]

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
>>
>> On 9/9/2023 11:25 AM, U.Mutlu wrote:
>>> A short standalone test code in C++ with test results posted online at
>>> https://www.elitetrader.com/et/threads/simulating-stock-prices-using-gbm.375533/page-4#post-5861368 
>>>
>>>
>>> demonstrates that the GBM method in QuantLib
>>> is fatally buggy and has been so since start.
>>>
>>> Can the experts please check it and comment it.
>>> It's suspicious that such a flaw in a very important part of the 
>>> library
>>> (GBM and Monte Carlo)
>>> got undetected by the experts and the user community for more that 
>>> 20+ years.
>>>
>>> Can the experts verify / confirm / duplicate the findings made there?
>>> Or is maybe that test code itself buggy or the test method maybe 
>>> unscientific?
>

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