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

2. Statistics and Data Analysis
Random Walk Models for Stock Prices

3. A Model for Stock Prices
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A Model for Stock Prices
Preliminary:
Consider a sequence of T random outcomes, independent from one to the next, Δ1, Δ2,…, ΔT. (Δ is a standard symbol for “change” which will be appropriate for what we are doing here. And, we’ll use “t” instead of “i” to signify something to do with “time.”)
Δt comes from a normal distribution with mean μ and standard deviation σ.

4. 2/30
Application
Suppose P is sales of a store. The accounting period starts with total sales = 0
On any given day, sales are random, normally distributed with mean μ and standard deviation σ. For example, mean $100,000 with standard deviation $10,000
Sales on any given day, day t, are denoted Δt
Δ1 = sales on day 1,
Δ2 = sales on day 2,
Total sales after T days will be Δ1+ Δ2+…+ ΔT
Therefore, each Δt is the change in the total that occurs on day t.

5. Using the Central Limit Theorem to Describe the Total
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Using the Central Limit Theorem to Describe the Total
Let PT = Δ1+ Δ2+…+ ΔT be the total of the changes (variables) from times (observations) 1 to T.
The sequence is
P1 = Δ1
P2 = Δ1 + Δ2
P3 = Δ1 + Δ2 + Δ3
And so on…
PT = Δ1 + Δ2 + Δ3 + … + ΔT

6. 4/30
Summing
If the individual Δs are each normally distributed with mean μ and standard deviation σ, then
P1 = Δ = Normal [ μ, σ]
P2 = Δ1 + Δ = Normal [2μ, σ√2]
P3 = Δ1 + Δ2 + Δ3= Normal [3μ, σ√3]
And so on… so that
PT = N[Tμ, σ√T]

7. 5/30
Application
Suppose P is accumulated sales of a store. The accounting period starts with total sales = 0
Δ1 = sales on day 1,
Δ2 = sales on day 2
Accumulated sales after day 2 = Δ1+ Δ2
And so on…

8. This defines a Random Walk
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This defines a Random Walk
The sequence is
P1 = Δ1
P2 = Δ1 + Δ2
P3 = Δ1 + Δ2 + Δ3
And so on…
PT = Δ1 + Δ2 + Δ3 + … + ΔT
It follows that
P1 = Δ1
P2 = P1 + Δ2
P3 = P2 + Δ3
And so on…
PT = PT-1 + ΔT

9. A Model for Stock Prices
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A Model for Stock Prices
Random Walk Model: Today’s price = yesterday’s price + a change that is independent of all previous information. (It’s a model, and a very controversial one at that.)
Start at some known P0 so P1 = P0 + Δ1 and so on.
Assume μ = 0 (no systematic drift in the stock price).

10. Random Walk Simulations
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Random Walk Simulations
Pt = Pt-1 + Δt
Example: P0= 10, Δt Normal with μ=0, σ=0.02

11. 9/30
Uncertainty
Expected Price = E[Pt] = P0+Tμ We have used μ = 0 (no systematic upward or downward drift).
Standard deviation = σ√T reflects uncertainty.
Looking forward from “now” = time t=0, the uncertainty increases the farther out we look to the future.

12. Using the Empirical Rule to Formulate an Expected Range
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Using the Empirical Rule to Formulate an Expected Range

13. 11/30
Application
Using the random walk model, with P0 = $40, say μ =$0.01, σ=$0.28, what is the probability that the stock will exceed $41 after 25 days?
E[P25] = ($.01) = $ The standard deviation will be $0.28√25=$1.40.

14. 12/30
Prediction Interval
From the normal distribution, P[μt σt < X < μt σt] = 95%
This range can provide a “prediction interval, where μt = P0 + tμ and σt = σ√t.

15. Random Walk Model Controversial – many assumptions
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Random Walk Model
Controversial – many assumptions
Normality is inessential – we are summing, so after 25 periods or so, we can invoke the CLT.
The assumption of period to period independence is at least debatable.
The assumption of unchanging mean and variance is certainly debatable.
The additive model allows negative prices. (Ouch!)
The model when applied is usually based on logs and the lognormal model. To be continued …

16. 14/30
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.

17. 15/30
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]

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

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

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

21. 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( ) =

22. 20/30
Collecting Math Facts

23. 21/30
Building a Model

24. 22/30
A Second Period

25. 23/30
What Does It Imply?

26. 24/30
Random Walk in Logs

27. Lognormal Model for Prices
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Lognormal Model for Prices

28. 26/30
Lognormal Random Walk

29. 27/30
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 < ] =

30. 28/30
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

31. 29/30
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.

32. 30/30
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.