Re: Is Ito's Lemma correct?

David Bernier wrote:
David Bernier wrote:
Ralf wrote:
David Bernier wrote:
Ralf wrote:
David Bernier wrote:
Ralf wrote:
David Bernier wrote:
Ralf wrote:
Ralf wrote:
In Geometric Brownian Motion calculations Ito's Lemma gets used.
Can someone who has studied Ito's Lemma confirm that it's correct?

My experiments show that Ito's lemma cannot be correct.
OTOH I wonder how it come that so many authors and researchers blindly use it,
especially in the financial industry. Maybe they deserve it! :-)
Even the famous Black-Scholes option pricing model does use it,
and therefore it too cannot be correct as well.
My conclusion is: ANOTHER BIG FRAUD IN WALL STREET!

Fascinating. Did you use Monte Carlo simulations in
your calculations ? Do you have any algorithms?
Any source code? What's the evidence that Ito's lemma
is wrong ?

Did you compute confidence intervals? Compute some
statistics? Where did you start from?

Of course did I make many MC simulations (GBM) for my tests.
[...]

What algorithm did you use for your random number generator?

Normal distributed random numbers of course as
was implemented in the following source (C/C++).

* FileName : rvgs.c (Random Variate GeneratorS)
* Author : Steve Park & Dave Geyer
* Language : ANSI C * Latest Revision : 10-28-98

double Normal(double m, double s)
/* ========================================================================

* Returns a normal (Gaussian) distributed real number.
* NOTE: use s > 0.0
*
* Uses a very accurate approximation of the normal idf due to
* Odeh & Evans, J. Applied Statistics, 1974, vol 23, pp 96-97. ...

Yes, I see. I found the C source code here:

https://lyosvn.in2p3.fr/ilc/browser/xdaq/trunk/TriDAS/daq/myrics/frlgen/src-rgs/rvgs.c




Here's the original location and distribution of Mr. Steve Park:
http://www.cs.wm.edu/~va/software/park/park.html

What about stochastic calculus? What were you doing with your Monte
Carlo simulations that led you to believe Ito's lemma was incorrect?

Mean and StdDev of course. But it's hard to prove it with MC alone
due to its nature (ie. stochastic / random processes).
Actually I got the difference (ie. the deviation) when I was writing my own so called Value-at-Risk (VaR) calculation model.
I succeeded but the result deviated a little bit from other methods.
Then I studied the other methods and located Ito's lemma in their methods
as the culprit. The given relation
ln(St) = ln(S0) + z * s * sqrt(dt) + (u - s^2 / 2) * dt made me confident that my method is the correct method as it satisfies this relation and doesn't make use of Ito's lemma.

I wish to present another theory:

I don't know much about stochastic calculus. But I do
know how to check for formulas that give a Normal distribution
random variable, say Z, given an input uniformly
distributed variable, say U from 0 to 1.

[...]
168 u = Random();
169 if (u < 0.5)
170 t = sqrt(-2.0 * log(u));
171 else
172 t = sqrt(-2.0 * log(1.0 - u));
173 p = p0 + t * (p1 + t * (p2 + t * (p3 + t * p4)));
174 q = q0 + t * (q1 + t * (q2 + t * (q3 + t * q4)));
175 if (u < 0.5)
176 z = (p / q) - t;
177 else
178 z = t - (p / q);
179 return (m + s * z);


From the link I found, which I hope matches what you were using:
https://lyosvn.in2p3.fr/ilc/browser/xdaq/trunk/TriDAS/daq/myrics/frlgen/src-rgs/rvgs.c


So i u< 0.5,
z = (p / q) - t;
p, q being polynomials in t.

It's characterized as "a very accurate approximation of the normal idf".
I can imagine that it's very accurate. But in generating
Geometric Brownian Motion, I'd rather use the
Box–Muller transform, or something else which generates
exactly normal random variates, rather than approximations.

Cf.:
http://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform

I don't believe the algorithm coded above
generates r.v.s that are exactly standard Normal variates ...

Believing isn't neccessarily knowing, isn't it? :-)
Does there any such routine exist which gives "exactly" standard Normal variates? I doubt. They can all be just approximations only, but one can measure their
statistical quality.

The article in an ACM publication is from 1988 or so:

http://www.eng.auburn.edu/cse/classes/comp8700/papers/park_RNG.pdf

The uniform RNG has a period of about 2^31 - 1, and is
a linear congruential generator. I think today fifteen minutes
or less of simulation in compiled code on a decent machine
could lead to a loop in the RNG. I don't know about
the Normal variable approximation. It might be pretty good.
So if indeed Geometric Monte Carlo was based on an RNG
with a period of 2^31 or so, passed some point
the same pseudo-random numbers would be used again.

The actual implementation in the said library has a period of 2,147,483,646:

"The generator used in this library is a so-called 'Lehmer random number generator'
which returns a pseudo-random number uniformly distributed [between] 0.0 and 1.0. The period is (m - 1) where m = 2,147,483,647 and the smallest and largest possible values are (1 / m) and 1 - (1 / m) respectively." (--> http://www.cs.wm.edu/~va/software/park/park.html )

And, I had tried them all: Box-Mueller, Polar method, and the above one.
But as said in the prev. posting, the rng-stuff (and MC and GBM) actually wasn't strictly neccessary to prove Ito's lemma wrong
(but it is possible to show the discrepancy even with MC if one chooses a big drift, ie. u, and a small s,
and this can be repeated over and over again!...)

OTOH, analytically one can take straight for example z=-1 or +1 (ie. +/- 1SD), or any other value,
and verify whether it satisfies the said relation:
ln(St) = ln(S0) + z * s * sqrt(dt) + (u - s^2 / 2) * dt
QED
.

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