Talk:Geometric Brownian motion

the notation is a bit dodgy for example: does 'exp' mean " e" or does it stand for expected value. I'm all for clearing this up to avoid any sort of confusion! --ToyotaPanasonic (talk) 06:56, 24 April 2008 (UTC)

exp is the exponential function defined by exp(x)=e^x. Expectations are ususally denothed E[x]. I don't think this notation will cause undue confusion but in general I definitely agree that this article could use some expansion and cleaning up. I hope to do a little work here soon.Wik-e-wik (talk) 07:02, 10 May 2008 (UTC)

Mean of log(St/S0)

The article presents an explicit solution:

$S_{t}=S_{0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}\right),$

It follows (surely?) that

$\log(S_{t}/S_{0})=\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}$

Since the expectation of a Brownian motion is 0, the expectation of the rhs is

$\langle \log(S_{t}/S_{0})\rangle =\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t$

However, the article goes on to say that the random variable log(S_t/S₀) is normally distributed with mean $\mu t$. This looks wrong to me, but a recent edit was reverted. Could someone/anyone explain?LeBofSportif (talk) 17:34, 8 August 2010 (UTC)

You are right, the mean is certainly not $\mu t$ 131.159.68.131 (talk) 12:45, 12 August 2010 (UTC)

I've added the expert template - hopefully a competent mathematician can sort it out, and clarify the notation for SDEs used, since it is ambiguous without choice in the Ito/Stratonovitch dilemma. LeBofSportif (talk) 17:33, 20 August 2010 (UTC)

The mean should be $\mu t$ and is easy to see if you look at the SDE. I cannot see the flaw in the above argument at the moment though.Zfeinst (talk) 19:09, 29 August 2010 (UTC)

I was wrong, the mean is $\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t$. However, this does not change the rest of the article since $\mathbb {E} (\log(S_{t}/S_{0}))\neq \log \left[\mathbb {E} (S_{t}/S_{0})\right]$. See http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-GBM.pdf Zfeinst (talk) 19:33, 29 August 2010 (UTC)

"Exp"

"Exp" in this context is NOT dodgy - it conforms to standard notation in most any financial mathematics text. It is completely correct. —Preceding unsigned comment added by 65.221.3.17 (talk) 18:57, 30 August 2010 (UTC)

Expert attention tag

The two different values for the expected value of the same thing cannot be right... presumably there should be two different "solutions" S(t) corresponding to the two different types of calculus. This needs to be set out in an accurate way. Melcombe (talk) 16:34, 24 June 2011 (UTC)

This problem has been resolved as far as I can tell, so I have removed the 'expert' tag. Zfeinst (talk) 23:41, 9 October 2011 (UTC)

Kurtosis of stock price

At the end of the article, it is said that GMB is not realistic as it does not model the fat tail of stock prices. This seems wrong to me. The stock price according to the GMB model is log-normally distributed. The log-normal distribution does have a fat, long tail of large positive values. 152.3.59.111 (talk) 18:55, 7 August 2013 (UTC)

Error in the graphics

I think the graphic with two examples of geometric brownian motion has an error. It says $\sigma =20$ and $\sigma =50$, but there is no way that a geometric brownian motion with those parameters result in that graphic. I think $\sigma =.20$ and $\sigma =.50$ makes more sense. I made several simulations in R and the graphics with the corrected parameters looks much more like the original image. — Preceding unsigned comment added by 132.247.249.242 (talk) 21:10, 27 April 2017 (UTC)

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Mean and variance

There has been some back and forth (myself involved) the past few days on the mean and variance of the GBM. As I am looking to avoid an edit war, I want to move the discussion to the talk page. This problem seems to be related to the question of the parameters of the lognormal distribution that the GBM follows at time t. As provided on this page (and, for example, in Karatzas and Shreve (1998) "Brownian Motion and Stochastic Calculus") the GBM is defined by

$dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t}$

which results in the closed form solution

$S_{t}=S_{0}\exp([\mu -\sigma ^{2}/2]t+\sigma W_{t}).$

That is $S_{t}=S_{0}\exp(X(t))$ for $X(t)\sim N([\mu -\sigma ^{2}/2]t,\sigma ^{2}t)$. Therefore the mean and variance should be defined via

$\operatorname {E} (S_{t})=S_{0}e^{\mu t},$

$\operatorname {Var} (S_{t})=S_{0}^{2}e^{2\mu t}\left(e^{\sigma ^{2}t}-1\right).$

I want to find a consensus on this so as to avoid an edit war. And of course if there is a mistake in my reasoning then please correct it. Zfeinst (talk) 09:48, 29 June 2020 (UTC)

Apologies, you are correct. Parklane79 (talk) 12:51, 29 June 2020 (UTC)

Python code and main graphic

I think the code and main graphic (GBM2.png) are incorrect. In particular, the cumprod used does not translate the random variable into a Weiner process. Tbh, I am not exactly sure what it is doing. I've written an alternative in a Jupyter Notebook which also demonstrates how the solution also ties back to the underlying process (using a Monte Carlo approach). I've kept variable names as close to article as possible. I can update entry if no objections. The notebook can be found here https://github.com/bsdz/docs/blob/master/notebooks/Geometric%20Brownian%20Motion%20-%20Wikipedia.ipynb --Blair Azzopardi (talk) 07:51, 21 December 2021 (UTC)

A few edits

1. I added Arithmetic Brownian Motions (ABM), also known as Bachelier Model, because ABM was nowhere else in Wikipedia as far as I could see. The Bachelier model has a page, but I decided to put ABM here because GBM is simply an exponential of an ABM and ABM comes as a logarithm of a GBM.

2. I removed the tag for lack of references and added a reference to the multivariate GBM, although the reference refers to a slightly different formulation, which I added.

3. On extensions I added the examples of the lognormal mixture dynamics SDE, as one of the models extending GBM and least departing from it in terms of law (mixture of lognormals instead of single lognormal), and the Heston model as an example of stochastic volatility model close to GBM (again this is like a GBM but with a stochastic volatility term related to a square root process). — Preceding unsigned comment added by DamianoBrigo2 (talk • contribs) 16:24, 6 June 2023 (UTC)

Properties of GBM: Incorporating the formula for the Mode and Median values

The Mode (statistics) of the GBM is given by:

${\text{Mode}}(S_{t})=S_{0}e^{\left(\mu -{\frac {3}{2}}\sigma ^{2}\right)t}$

The Median of the GBM is given by:

${\text{Median}}(S_{t})=S_{0}e^{\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t}$

Please some super user incorporate it into the formulas.

Its interesting because shows the non-ergodicity of the GBM process: you could have at $\mu -{\frac {1}{2}}\sigma ^{2}=0$ a constant median value of 1 so 50% of the values are over and 50% below, but while the mean value will be growing exponentially, the most likely values to find (given by the Mode) will be decreasing faster to zero. 181.163.14.232 (talk) 03:38, 18 October 2024 (UTC)