1. Statistics and Data Analysis
Professor William Greene
Stern School of Business
Department of IOMS
Department of Economics

2. Statistics and Data Analysis
Part 11A – Lognormal Random Walks

3. Lognormal Random Walk
The lognormal model remedies some of the shortcomings of the linear (normal) model.
Somewhat more realistic.
Equally controversial.
Description follows for those interested.

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Lognormal Variable
If the log of a variable has a normal distribution, then the variable has a lognormal distribution.
Mean =Exp[μ+σ2/2] >
Median = Exp[μ]

5. Lognormality – Country Per Capita Gross Domestic Product Data
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Lognormality – Country Per Capita Gross Domestic Product Data

6. Lognormality – Earnings in a Large Cross Section
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Lognormality – Earnings in a Large Cross Section

7. Lognormal Variable Exhibits Skewness
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Lognormal Variable Exhibits Skewness
The mean is to the right of the median.

8. Lognormal Distribution for Price Changes
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Lognormal Distribution for Price Changes
Math preliminaries:
(Growth) If price is P0 at time 0 and the price grows by 100Δ% from period 0 to period 1, then the price at period 1 is P0(1 + Δ). For example, P0=40; Δ = 0.04 (4% per period); P1 = P0( ).
(Price ratio) If P1 = P0( ) then P1/P0 = ( ).
(Math fact) For smallish Δ, log(1 + Δ) ≈ Δ Example, if Δ = 0.04, log( ) =

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Collecting Math Facts

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Building a Model

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A Second Period

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What Does It Imply?

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Random Walk in Logs

14. Lognormal Model for Prices
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Lognormal Model for Prices

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Lognormal Random Walk

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Application
Suppose P0 = 40, μ=0 and σ= What is the probabiity that P25, the price of the stock after 25 days, will exceed 45?
logP25 has mean log μ =log40 = and standard deviation σ√25 = 5(.02)=.1. It will be at least approximately normally distributed.
P[P25 > 45] = P[logP25 > log45] = P[logP25 > ]
P[logP25 > ] = P[(logP )/0.1 > ( )/0.1)]= P[Z > 1.177] = P[Z < ] =

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Prediction Interval
We are 95% certain that logP25 is in the interval logP0 + μ σ25 to logP0 + μ σ Continue to assume μ=0 so μ25 = 25(0)=0 and σ=0.02 so σ25 = 0.02(√25)=0.1 Then, the interval is (0.1) to (0.1) or to This means that we are 95% confident that P0 is in the range e = and e = 48.66

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Observations - 1
The lognormal model (lognormal random walk) predicts that the price will always take the form PT = P0eΣΔt
This will always be positive, so this overcomes the problem of the first model we looked at.

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Observations - 2
The lognormal model has a quirk of its own. Note that when we formed the prediction interval for P25 based on P0 = 40, the interval is [32.88,48.66] which has center at > 40, even though μ = 0. It looks like free money.
Why does this happen? A feature of the lognormal model is that E[PT] = P0exp(μT + ½σT2) which is greater than P0 even if μ = 0.
Philosophically, we can interpret this as the expected return to undertaking risk (compared to no risk – a risk “premium”).
On the other hand, this is a model. It has virtues and flaws. This is one of the flaws.

20. Summary Normal distribution approximation to binomial
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Summary
Normal distribution approximation to binomial
Approximate with a normal with same mean and standard deviation
Continuity correction
Sums and central limit theorem
Random walk model for stock prices
Lognormal variables
Alternative random walk model using logs