Statistics and Data Analysis

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Statistics and Data Analysis. Professor William Greene Stern School of Business Department of IOMS Department of Economics. Statistics and Data Analysis. Random Walk Models for Stock Prices. 1/30. A Model for Stock Prices. Preliminary:

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Statistics and Data Analysis

Professor William Greene

Department of IOMS

Department of Economics

Statistics and Data Analysis

Random Walk Modelsfor Stock Prices

1/30

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.

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.

3/30

Using the Central Limit Theorem to Describe the Total
• Let PT = Δ1+ Δ2+…+ ΔTbe 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

4/30

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

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…
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

6/30

This defines a Random Walk

7/30

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).

8/30

Random Walk Simulations

Pt = Pt-1 + Δt

Example: P0= 10, Δt Normal with μ=0, σ=0.02

9/30

Uncertainty
• Expected Price = E[Pt] = P0+TμWe 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.

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] = 40 + 25(\$.01) = \$40.25. The standard deviation will be \$0.28√25=\$1.40.

12/30

Prediction Interval
• From the normal distribution,P[μt - 1.96σt< X <μt + 1.96σt] = 95%
• This range can provide a “prediction interval, where μt = P0 + tμ and σt = σ√t.

13/30

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 …

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.

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]

18/30

Lognormal Variable Exhibits Skewness

The mean is to the right of the median.

19/30

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(1 + 0.04).
• (Price ratio) If P1 = P0(1 + 0.04) then P1/P0 = (1 + 0.04).
• (Math fact) For smallish Δ, log(1 + Δ) ≈ ΔExample, if Δ = 0.04, log(1 + 0.04) = 0.39221.

27/30

Application
• Suppose P0 = 40, μ=0 and σ=0.02. What is the probabiity that P25, the price of the stock after 25 days, will exceed 45?
• logP25 has mean log40 + 25μ =log40 =3.6889 and standard deviation σ√25 = 5(.02)=.1. It will be at least approximately normally distributed.
• P[P25 > 45] = P[logP25 > log45] = P[logP25 > 3.8066]
• P[logP25 > 3.8066] =P[(logP25-3.6889)/0.1 > (3.8066-3.6889)/0.1)]=P[Z > 1.177] = P[Z < -1.177] = 0.119598

28/30

Prediction Interval

We are 95% certain that logP25 is in the intervallogP0 + μ25 - 1.96σ25 to logP0 + μ25 + 1.96σ25. Continue to assume μ=0 so μ25 = 25(0)=0 and σ=0.02 so σ25 = 0.02(√25)=0.1Then, the interval is 3.6889 -1.96(0.1) to 3.6889 + 1.96(0.1)or 3.4929 to 3.8849.This means that we are 95% confident that P0 is in the rangee3.4929 = 32.88 and e3.8849 = 48.66

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.

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 40.77 > 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.