May
13

## Log Normal Distribution

A guest post by Sidharth Mallik.

The lognormal distribution is used extensively as an approxmiation to the price of a financial asset after time t given a known price at t=0. However there is very little known about the lognormal distribution and the resources are even more limited. I tried finding books about the lognormal distribution and came up with the following 2 (only 2) :

1. Aitchison J and Brown JAC, 1957. The lognormal distribution, Cambridge University Press, Cambridge UK.

2. Crow EL and Shimizu K Eds, 1988. Lognormal Distributions: Theory and Application, Dekker, New York.

among all the books on statistics and millions of resources on normal distribution the exponential counterpart only has two genuine reference books.

However my job is to give you some light in the matter. Lets start with the basic properties of the lognormal distribution :

parameters: sigma^2 > 0 — squared scale (real),

mu an element of R — location

support: x =(0, +Inf)

pdf: \frac{1}{x\sqrt{2\pi\sigma^2}}\, e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}

cdf: \frac12 + \frac12\,\mathrm{erf}\Big[\frac{\ln x-\mu}{\sqrt{2\sigma^2}}\Big]

mean: e^{\mu+\sigma^2/2}

median: e^{\mu}\,

mode: e^{\mu-\sigma^2}

variance: (e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2}

skewness: (e^{\sigma^2}\!\!+2) \sqrt{e^{\sigma^2}\!\!-1}

kurtosis: e^{4\sigma^2}\!\! + 2e^{3\sigma^2}\!\! + 3e^{2\sigma^2}\!\! - 3

entropy: \frac12 + \frac12 \ln(2\pi\sigma^2) + \mu

mgf: (defined only on the negative half-axis, see text)

cf: representation \sum_{n=0}^{\infty}\frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2} is asymptotically divergent but sufficient for numerical purposes

Fisher information: \begin{pmatrix}1/\sigma^2&0\\0&1/(2\sigma^4)\end{pmatrix}

As you may guess the same information is available at wikipedia at http://en.wikipedia.org/wiki/Log-normal_distribution.

now for the estimation properties. well conviniently you would want to estimate the mean and the variance of the corresponding logarithmic distribution which happens to be normal and then may be use the above formulas to find the lognormal's parameters. but there is error associated with this.

An example of the error is :

say you decide on a parameter m as the mean of the corresponding normal distribution and sigma^2 as the variance. the you may say that the corresponding mean of the lognormal distribution is simply

mean of lognormal distribution = exp(m+sigma^2/2)

However using some math i will show you that this is not an unbiased estimator. Let E[.] denote the expectation of the quantity within the brackets and let M denote the mean of the lognormal distribution.Then if the above estimator was unbiased, we should have

E[M] = E[exp(m+sigma^2/2)]

But,

E[exp(m+sigma^2/2)] => exp(E[m+sigma^2/2])

according to the Jensen inequality. for those of you who don't know what this is refer to the http://en.wikipedia.org/wiki/Jensen_inequality.

Hence there are problems with this form of the estimation. So i would redirect you to this page check out some other ways how this can be done

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1285465

Now having resolved the univariate case we turn our attention to the bivariate case.

The parameters in this case are the two means, the two standard deviations and the correlation coefficients. Now the natural question is how to calculate the correlation coefficient.

The expression for the covariance is

cov(x1,x2) = exp((sigma1*sigma2)-1)*exp(m1+m2+(sigma1^2+sigma2^2)/2)

The expression for the correlation coefficient is

rho = [exp(sigma1*sigma2)-1]/sqrt(exp(sigma1^2-1)*exp(sigma2^2-1))

For more such references and also to get a table for actual values check the following out

http://www.stuart.iit.edu/shared/shared_stuartfaculty/whitepapers/thomopoulos_some.pdf

I will want to finish off with some intuition as to where can you apply the lognormal distribution. Now first identify if the variable under study is throughout positive or not like a stock price. Next identify the fact that the variable contains kind of multiplicative factors in the sense that say there are levels to the quantity in multiplicative terms. What I mean to say is that the variable has let us say a level r on normal days.Then when you expect the variable to go up it goes upto 1.5 times r then even higher means that it goes to say 5 times r. Similarly for the lower side. If your variable is indicative of this then it suggests a lognormal distribution.

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The lognormal distribution is used extensively as an approxmiation to the price of a financial asset after time t given a known price at t=0. However there is very little known about the lognormal distribution and the resources are even more limited. I tried finding books about the lognormal distribution and came up with the following 2 (only 2) :

1. Aitchison J and Brown JAC, 1957. The lognormal distribution, Cambridge University Press, Cambridge UK.

2. Crow EL and Shimizu K Eds, 1988. Lognormal Distributions: Theory and Application, Dekker, New York.

among all the books on statistics and millions of resources on normal distribution the exponential counterpart only has two genuine reference books.

However my job is to give you some light in the matter. Lets start with the basic properties of the lognormal distribution :

parameters: sigma^2 > 0 — squared scale (real),

mu an element of R — location

support: x =(0, +Inf)

pdf: \frac{1}{x\sqrt{2\pi\sigma^2}}\, e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}

cdf: \frac12 + \frac12\,\mathrm{erf}\Big[\frac{\ln x-\mu}{\sqrt{2\sigma^2}}\Big]

mean: e^{\mu+\sigma^2/2}

median: e^{\mu}\,

mode: e^{\mu-\sigma^2}

variance: (e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2}

skewness: (e^{\sigma^2}\!\!+2) \sqrt{e^{\sigma^2}\!\!-1}

kurtosis: e^{4\sigma^2}\!\! + 2e^{3\sigma^2}\!\! + 3e^{2\sigma^2}\!\! - 3

entropy: \frac12 + \frac12 \ln(2\pi\sigma^2) + \mu

mgf: (defined only on the negative half-axis, see text)

cf: representation \sum_{n=0}^{\infty}\frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2} is asymptotically divergent but sufficient for numerical purposes

Fisher information: \begin{pmatrix}1/\sigma^2&0\\0&1/(2\sigma^4)\end{pmatrix}

As you may guess the same information is available at wikipedia at http://en.wikipedia.org/wiki/Log-normal_distribution.

now for the estimation properties. well conviniently you would want to estimate the mean and the variance of the corresponding logarithmic distribution which happens to be normal and then may be use the above formulas to find the lognormal's parameters. but there is error associated with this.

An example of the error is :

say you decide on a parameter m as the mean of the corresponding normal distribution and sigma^2 as the variance. the you may say that the corresponding mean of the lognormal distribution is simply

mean of lognormal distribution = exp(m+sigma^2/2)

However using some math i will show you that this is not an unbiased estimator. Let E[.] denote the expectation of the quantity within the brackets and let M denote the mean of the lognormal distribution.Then if the above estimator was unbiased, we should have

E[M] = E[exp(m+sigma^2/2)]

But,

E[exp(m+sigma^2/2)] => exp(E[m+sigma^2/2])

according to the Jensen inequality. for those of you who don't know what this is refer to the http://en.wikipedia.org/wiki/Jensen_inequality.

Hence there are problems with this form of the estimation. So i would redirect you to this page check out some other ways how this can be done

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1285465

Now having resolved the univariate case we turn our attention to the bivariate case.

The parameters in this case are the two means, the two standard deviations and the correlation coefficients. Now the natural question is how to calculate the correlation coefficient.

The expression for the covariance is

cov(x1,x2) = exp((sigma1*sigma2)-1)*exp(m1+m2+(sigma1^2+sigma2^2)/2)

The expression for the correlation coefficient is

rho = [exp(sigma1*sigma2)-1]/sqrt(exp(sigma1^2-1)*exp(sigma2^2-1))

For more such references and also to get a table for actual values check the following out

http://www.stuart.iit.edu/shared/shared_stuartfaculty/whitepapers/thomopoulos_some.pdf

I will want to finish off with some intuition as to where can you apply the lognormal distribution. Now first identify if the variable under study is throughout positive or not like a stock price. Next identify the fact that the variable contains kind of multiplicative factors in the sense that say there are levels to the quantity in multiplicative terms. What I mean to say is that the variable has let us say a level r on normal days.Then when you expect the variable to go up it goes upto 1.5 times r then even higher means that it goes to say 5 times r. Similarly for the lower side. If your variable is indicative of this then it suggests a lognormal distribution.

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This link is also very useful for intuitive understanding

http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

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