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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

Statistics and Data Analysis

Professor William Greene

Stern School of Business

Department of IOMS

Department of Economics

statistics and data analysis1

Statistics and Data Analysis

Random Walk Modelsfor Stock Prices

a model for 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.
application

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.
using the central limit theorem to describe the total

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
summing

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]
application1

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…
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

6/30

This defines a Random Walk
a model for stock prices1

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).
random walk simulations

8/30

Random Walk Simulations

Pt = Pt-1 + Δt

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

uncertainty

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.
application2

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.
prediction interval

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.
random walk model

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 …
lognormal random walk

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.
lognormal variable

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]

lognormal variable exhibits skewness

18/30

Lognormal Variable Exhibits Skewness

The mean is to the right of the median.

lognormal distribution for price changes

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.
application3

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
prediction interval1

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

observations 1

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.
observations 2

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.