Jun 20

Lognormal distribution. Introduction

by David Harper, CFA, FRM, CIPM

FRM | Quant | |

doodleTeacher

 

If a random variable X is log-normally distributed, we can write:


ln(X)~N(μ,σ2)


That says, the natural log of X is approximately normal with mean of mu and variance of sigma. This turns out to be an important distribution for modeling stock prices when we assume that returns are normally distributed. But the relationship between normal returns and lognormal prices can be confusing.

If the single-period return is normally distributed...

To understand this, start with a single daily (period = one day) return. The simple percentage return is approximately equal to the natural log of the ratio of prices (aka, the wealth relative or wealth ratio. We are assuming no dividend):

 

lognormal_return

 

Test of blockquote

For example, if today's price is $11 and yesterday's price is $10, then simple percentage return = +10% and the continuously compounded return is pretty close at ln(11/10) = +9.53%. Note that we can characterize this periodic return as normal. If today's price instead is $9, then ln(9/10) = -10.5%. See how the period return can go up or down, positive or negative with approximate symmetry, per normality?

...then the n-period price level is log-normally distributed

We can go directly from normal returns (I'll spare you the math) to the finding that future price levels are log-normally distributed. See the difference: price levels not periodic returns. Levels are not symmetrical. The worst case is a total loss, where the terminal price is zero. Hence, the lognormal distribution is non-zero. Further, it is skewed right. Think about compounding effects. If we are lucky enough to see consecutive, periodic gains of +10%, each jump is jumping from a larger base. Therefore, the lognormal has a skewed right tail; a small chance of a very large terminal price.

Lognormal illustrated

I have plotted a lognormal density function below (a density function is the probability that X is equal to [=] a value; the cumulative distribution function shows the probability that X is less than or equal to[<=]). I could have solved it by using the built-in =LOGNORMDIST() function, but that is a cumulative function. So I would have to transpose the cumulative into a density.

Instead, it is much easier to translate the x-axis (a standard normal variable: by definition, with a mean of zero and standard deviation of one) into EXP(X). Then plot that again a "normal" p.d.f. (=NORMSDIST). That's the mathematical equivalent of where we started in English: if the lognormal of a variable is normal, the variable is normal. And LN[EXP(X)] = X! You can open your own read/write version here.

EditGrid Spreadsheet by bt/admin.

Comments

  1. Chuck 24 Jun 2007

    If I remember my college stats correctly, sigma is the Greek for standard deviation, not variance, which is sigma squared. Please refer to your tutorial on Lognormal Distribution, an Introduction, for the reference. Thanks for the excellence in presenting abstruse material in an interesting way.

  2. David Harper, CFA, FRM 24 Jun 2007

    Chuck, you remember correctly. Thanks for the help. Appreciate your kind feedback.

  3. malou 10 Aug 2007

    this doesn’t state the more important part of the lognormal discussion, such as pricing derivatives with stock as the underlying asset which takes a lognormal distribution behaviour. Such as what if security’s payoff is the square of the stock price, or 2 times the stock price, or if the derivative is based on a portfolio with 2 different stocks, that are linearly independent. This cannot be answered because the properties of the lognormal such as adding 2 lognormal, or multiplying a constant to a lognormal, is not stated. What happens to its mean and standard deviation in every cases and more is important.

  4. Neuberg 14 Jan 2008

    Hi,
    First of all happy new year and congrats for your very nice site.

    But, your graph of the lognormal p.d.f. should be corrected.
    If Y=Ln X =N(0,1), you cant get the p.d.f. the way you used because the function log is not linear ... so the area under your curve is not 1 !
    If y=ln x, dy=dx/x, so you will get a corrected lognormal curve if you divide column D by x, i.e. by column C…
    If you intend to graph the p.d.f of LN(0,1), x between 0 and 5 is enough !!! 

    I would much appreciate if you can give me a free username and password to read the rest of your nice site.

    Your’s

    Gérard Neuberg
    Paris (FRANCE)

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