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Re: [Quantlib-users] QuantLib's GBM found flawed. What say the experts? (SOLVED)
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From: U.Mutlu <um...@mu...> - 2023-09-11 08:20:08
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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 >>> >> >> <https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient> >> >> Virus-free.www.avast.com >> <https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient> >> >> >> >> >> >> >> _______________________________________________ >> QuantLib-users mailing list >> Qua...@li... >> https://lists.sourceforge.net/lists/listinfo/quantlib-users >> > > |
