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strange problems with Random

  • To: mathgroup at
  • Subject: [mg47812] strange problems with Random
  • From: Andrzej Kozlowski <akoz at>
  • Date: Tue, 27 Apr 2004 04:48:05 -0400 (EDT)
  • Sender: owner-wri-mathgroup at

Recently Mark Coleman wrote a message with the subject "Differences in 
Random Numbers" in which he described his experience of getting 
statistically incorrect answers when using  
Random[UniformDistribution[0,1]] instead of simple Random[]. Jens Kuska 
and Bill Rowe pointed out that the package  ContinuousDistributions 
contains the code

UniformDistribution/: Random[UniformDistribution[min_:0, max_:1]] :=
Random[Real, {min, max}]

and that seemed to be it. However, I have now encountered a problem 
that makes me feel strongly that there is more to this problem than it 
seemed to me at first.

I have been doing some Monte Carlo computations of values of of 
financial options for the Internet based course on this subject I am 
teaching for Warsaw University.  In the case of the simplest options, 
the so called vanilla european options, an exact formula (due to black 
and Scholes) is known. Neverheless, computations with Mathematica using 
the function

Random[NormalDistribution[mu,sigma]] (for some fixed mu and sigma) have 
been giving answers which are consistently higher than the 
Black-Scholes value. On the other hand, when 
Random[NormalDistribution[mu,sigma]]  is replaced by the following code 
using a well known algorithm due to Marsaglia:

RandomNormal=Compile[{mu, sigma}, Module[{va = 1., vb, rad =
       2.0, den = 1.}, While[rad ³ 1.00, (va = 2.0*Random[] - 1.0;
             vb = 2.0*Random[] - 1.0;
             rad = va*va + vb*vb)];
         den = Sqrt[-2.0*Log[rad]/rad];
         mu + sigma*va*den]];

the answers agree very closely with the Black-Scholes ones.
I am not sure if this problem is related to the one mark mentioned but 
it seems quite likely.  One could in principle test it by replacing 
Random[] by Random[Uniformdistribution[0,1]] (after loading the 
Statistics`ContinuousDistributions`) package, but if one does that the 
code will no longer compile. That makes the function to slow for tests 
that are sufficiently precise, so I have not tested it.

In any case, I am sure that using Random[Normal  ] consistently gives 
wrong answers. I looked at the Statistics`NormalDistributionPackage` 
but everything looks O.K. to me. I now tend to believe that the problem 
lies with Random itself, that is that  Random called with any 
parameters or perhaps some particular parameters may not be behaving 
correctly. Random[] called with no parameters does not seem to  suffer 
from any such problems.

Andrzej Kozlowski
Chiba, Japan

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