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

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

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