tag:blogger.com,1999:blog-1352688022620133175.post1994241734843955736..comments2014-11-12T05:08:08.435ZComments on The NAG Blog: Easter egg or not?Katie O'Harehttp://www.blogger.com/profile/09366741271809330805noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-1352688022620133175.post-35397241380755251492010-11-29T11:15:46.213Z2010-11-29T11:15:46.213ZHi Feyn,
Thanks for your interest in this blog.
...Hi Feyn,<br /><br />Thanks for your interest in this blog.<br /><br />In both second and third graphs I plot the prices of Asian options, but only the one where I use pseudorandoms shows a proper distribution. What you say about lognormal distributions is right, but note that I don't really plot an average of MC simulations, but for every simulation I compute average as in Asian options and this is proper way to do it (as seen on the 3rd pic).<br /><br />I don't need to fill the bits inbetween with numbers from BB, they are not required. In the first example I move straight from S0 to ST and I don't need what's between them. If I calculate the prices between, then I have the 3rd example.<br /><br />I'm not sure what you mean by projecting the problem to the 1st axis, could you elaborate more on that please?<br /><br />And then 252 is not a special dimension of the NAG Sobol generator. It can be an arbitrary number. I tried it with 50 steps, with 252 steps and the diestribution in example 2 (using quasirandoms) is still hairy.Marcin Krzysztofikhttp://www.blogger.com/profile/18261558624689201066noreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-24491220581590283042010-11-25T10:21:14.000Z2010-11-25T10:21:14.000ZHi,
If in the second graph, you plot the average o...Hi,<br />If in the second graph, you plot the average of Spot, then it is not a log normal distribution (it is well know that sum of log normal random variable is not a log normal random variable).<br /><br />Now suppose that for each path you generate 252 quasi-random number and take the first one to contruct the S_T (the other is constructed with Brownian Bridge). It is like you consider a 252 dimension problem and consider the projection to the 1st axis.<br /><br />I guess then 252 is a special dimension of your Sobol manager?<br /><br />FeynJekyllhttp://www.blogger.com/profile/15068334311443161654noreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-75850541209456743032010-04-12T10:21:01.292+01:002010-04-12T10:21:01.292+01:00Jeremy, Mike and Kai, thank you for your interest ...Jeremy, Mike and Kai, thank you for your interest in this post of mine. This wasn't an Easter egg, the problem I encountered was real.<br /><br />Jeremy, you're right that quasirandoms are more evenly distributed, but when I perform the Monte Carlo using one single step the results are fine. The problem arises in the stepwise MC.<br /><br />Mike, the numbers are indeed not statistically independent, you're right. Do you have an idea how this could be solved? You can use pseudorandoms of course, but is there a way to this properly with quasirandoms? I actually have an idea how this can be done, but need to try it out. The answer is to use a scrambled sequence...<br /><br />Kai, I agree with you about the higher dimensions, but in this case I've taken numbers just from the first dimension. There are 20k simulations and 50 steps in each, altogether 1m quasirandom numbers from the first dimension!Marcin Krzysztofikhttp://www.blogger.com/profile/18261558624689201066noreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-79585429210170057152010-04-07T12:02:38.920+01:002010-04-07T12:02:38.920+01:00This could due to the poor quality of quasi-random...This could due to the poor quality of quasi-random number beyond a certain dimension (say 100). Even if NAG's sobol generator is able to generate up to 50,000 dimensions, there is no guarantee that projection of any two of them is evenly distributed over the unit square. <br /><br />The fact is the convergence of low discrepancy number is of order n*(log n)^d where d is dimension. With a large d, it barely has any advantage of pseudo-random numbers. This has not taken into account the reliability of Sobol' numbers in high dimension.Kai Zhangnoreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-28779377928757652132010-04-07T10:56:07.320+01:002010-04-07T10:56:07.320+01:00Is it something to do with the fact that quasi ran...Is it something to do with the fact that quasi random numbers are not statistically independent from each other?Mike Croucherhttp://www.walkingrandomly.comnoreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-57389847066530908722010-04-06T15:31:42.669+01:002010-04-06T15:31:42.669+01:00Well, let me take a crack at this egg. The main (...Well, let me take a crack at this egg. The main (only?) difference between pseudo and quasi random numbers seems to be that the latter are more evenly distributed across the interval of interest than the former (there are more words about this, along with a rather nice graphic which illustrates the point <a href="http://www.nag.co.uk/industryarticles/usingtoolboxmatlabpart3.asp#g05randomnumber" rel="nofollow">here</a>). Perhaps this has something to do with why their distribution is closer-to-normal than that of the pseudo-random sequence?Jeremy Waltonhttp://www.blogger.com/profile/10917026591452126254noreply@blogger.com