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

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

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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