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

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

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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