Random Walk Models for Stock Prices Statistics and Data Analysis Professor William Greene Stern...

Random Walk Models for Stock Prices

Statistics and Data Analysis

Professor William Greene

Stern School of Business

Department of IOMS

Department of Economics


Statistics and Data Analysis

Random Walk Modelsfor Stock Prices

PepperoniPlainMushroomSausagePepper and OnionMushroom and OnionGarlicMeatball

CategoryMeatball

5.0%Garlic2.3%

Mushroom and Onion9.2%

Pepper and Onion7.3%

Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%

Pie Chart of Percent vs Type

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Boxplot of Listing

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Scatterplot of Listing vs IncomePC

Listing

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10000008000006000004000002000000

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Mean 369687StDev 156865N 51AD 0.994P-Value 0.012

Probability Plot of ListingNormal - 95% CI

IncomePC

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3250030000275002500022500200001750015000

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Histogram of Listing

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Mean 369687StDev 156865N 51

Empirical CDF of ListingNormal

IncomePC

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Marginal Plot of Listing vs IncomePC

2e mc


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.

1/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

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800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

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3250030000275002500022500200001750015000

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IncomePC

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3250030000275002500022500200001750015000

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Mean 369687StDev 156865N 51


IncomePC

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30000250002000015000

1000000

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2e mc


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.

2/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

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800000

700000

600000

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300000

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100000

Boxplot of Listing

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IncomePC

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3250030000275002500022500200001750015000

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800000

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Mean 369687StDev 156865N 51


IncomePC

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30000250002000015000

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2e mc


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

3/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

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300000

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100000

Boxplot of Listing

IncomePC

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Mean 369687StDev 156865N 51


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30000250002000015000

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2e mc


Summing

If the individual Δs are each normally distributed with mean μ and standard deviation σ, thenP1 = Δ1 = Normal [ μ, σ]

P2 = Δ1 + Δ2 = Normal [2μ, σ√2]

P3 = Δ1 + Δ2 + Δ3= Normal [3μ, σ√3]And so on… so thatPT = N[Tμ, σ√T]

4/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

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300000

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100000

Boxplot of Listing

IncomePC

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IncomePC

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3250030000275002500022500200001750015000

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Mean 369687StDev 156865N 51


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30000250002000015000

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2e mc


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…

5/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

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ing

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IncomePC

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Mean 369687StDev 156865N 51


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2e mc


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


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

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300000

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100000

Boxplot of Listing

IncomePC

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IncomePC

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Mean 369687StDev 156865N 51


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2e mc


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).

7/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

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800000

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600000

500000

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100000

Boxplot of Listing

IncomePC

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Listing

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IncomePC

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Mean 369687StDev 156865N 51


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2e mc


Random Walk Simulations Pt = Pt-1 + Δt

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

8/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

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800000

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600000

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Boxplot of Listing

IncomePC

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Mean 369687StDev 156865N 51


IncomePC

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30000250002000015000

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2e mc


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.

9/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


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3250030000275002500022500200001750015000

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Mean 369687StDev 156865N 51


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2e mc


Using the Empirical Rule to Formulate an Expected Range

10/30

0[P t ] 2 t


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


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3250030000275002500022500200001750015000

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Mean 369687StDev 156865N 51


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2e mc


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.

25

P 40.25 $41.00 $40.25$P[P $41] P

1.40 1.40

= P[Z > 0.54]

= P[Z < -0.54]

= 0.2496

11/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

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IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

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100000


Listing

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Mean 369687StDev 156865N 51


IncomePC

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2e mc


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.

12/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

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3250030000275002500022500200001750015000

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Listing

Perc

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10000008000006000004000002000000

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IncomePC

List

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3250030000275002500022500200001750015000

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800000

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100000


Listing

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Mean 369687StDev 156865N 51


IncomePC

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30000250002000015000

1000000

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2e mc


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 …

13/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

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800000

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Boxplot of Listing

IncomePC

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IncomePC

List

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3250030000275002500022500200001750015000

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100000


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900000800000700000600000500000400000300000200000

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Mean 369687StDev 156865N 51


IncomePC

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30000250002000015000

1000000

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2e mc


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.

14/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

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200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Lognormal Variable

02

1 1 logx -μf(x) = exp - , < x < +

2 σxσ 2π

Wage

Frequency

480040003200240016008000

120

100

80

60

40

20

0

Loc 6.951Scale 0.4384N 595

Histogram of WageLognormal

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]

15/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Lognormality – Country Per Capita Gross Domestic Product Data

GDPC

Frequency

3000024000180001200060000-6000

70

60

50

40

30

20

10

0

Mean 6609StDev 7165N 191

Histogram of GDPCNormal

logGDPC

Frequency

10.49.68.88.07.26.4

16

14

12

10

8

6

4

2

0

Mean 8.248StDev 1.060N 191

Histogram of logGDPCNormal

16/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Lognormality – Earnings in a Large Cross Section

Wage

Frequency

480040003200240016008000

120

100

80

60

40

20

0

Mean 1148StDev 531.1N 595

Histogram of WageNormal

LogWage

Frequency

8.48.07.67.26.86.46.0

80

70

60

50

40

30

20

10

0

Mean 6.951StDev 0.4384N 595

Histogram of LogWageNormal

17/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Lognormal Variable Exhibits Skewness

Wage

Frequency

480040003200240016008000

120

100

80

60

40

20

0

Loc 6.951Scale 0.4384N 595

Histogram of WageLognormal

The mean is to the right of the median.

18/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


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.

19/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Collecting Math Facts

tt t-1

t-1

t

t-1

PIf P = P [1 + ] then = [1 + ]

P

P log = log[1 + ]

P

Δ Δ

Δ

Δ

20/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Building a Model

tt t-1

t-1

t

t-1

Slightly change the assumptions. Suppose

isn't a constant, but can be different each

period.

PIf P = P [1 + ] then = [1 + ]

P

P log = log[1 + ]

P

t t

t

Δ

Δ Δ

Δ

I.e., prices change by different amounts in

different periods.

tΔ

21/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


A Second Period

11 0

0

2 1 2 0

2

0

PIf P = P [1 + ] then = [1 + ]

P

Now, change for a second period

If P = P [1 + ], then P = P [1 + ] [1 + ] so

P = [1 + ] [1 + ]

P

1 1

2 1 2

1 2

Δ Δ

Δ Δ Δ

Δ Δ

2

0

P log = log[1 + ]+log[1 + ]

P

1 2

1 2

Δ Δ

Δ Δ

22/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


What Does It Imply?

TTt=1

0

T-1T-1t=1

0

T T-1

0 0

For T periods

Plog = log[1 + ]+log[1 + ]+...+log[1 + ]

P

For T-1 periods

Plog = log[1 + ]+log[1 + ]+...+log[1 + ]

P

By subtraction

P Plog log

P P

1 2 T t

1 2 T-1 t

Δ Δ Δ Δ

Δ Δ Δ Δ

T T-1

t=1 t=1

=

t t

T

Δ Δ

Δ

23/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Random Walk in Logs

T T-1T T-1t=1 t=1

0 0

T T-1T 0 T 1 0

0 0

T T 1

By subtraction

P Plog log =

P P

But

P Plog log logP logP logP logP

P P

so,

logP logP

This is the same random walk we had before, but now

it i

t t T

T

Δ Δ Δ

Δ

s in logs, rather than in prices.

24/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Lognormal Model for Prices

T

0

T

T 0 t 1

Plog = log[1 + ]+log[1 + ]+ ...+log[1 + ]

P

...

so,

logP logP

If the period to period changes are normally distributed with

mean and standard deviation ,

1 2 T

1 2 T

t

t

Δ Δ Δ

Δ Δ Δ

Δ

Δ

T

0

then logP has a normal

distribution with mean logP +T and standard deviation T.

25/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


Lognormal Random Walk

Ttt 1

T

T 0 t 1

T 0

rTT 0

If

logP logP

Then

P = P e

which looks like the present value result, V V e

for T periods and constant growth rate per period, r.

tΔ

26/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

00

6000

00

5000

00

4000

00

3000

00

2000

00

1000

000

100

80

60

40

20

0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


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

27/30


CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


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Perc

ent

9000

00

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00

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000

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0

Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

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2e mc


Prediction Interval

We are 95% certain that logP25 is in the interval

logP0 + 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)=0.1 Then, 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

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CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

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Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


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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CategoryMeatball

5.0%Garlic2.3%



Sausage5.8%

Mushroom16.2%

Plain32.5%

Pepperoni21.8%


List

ing

900000

800000

700000

600000

500000

400000

300000

200000

100000

Boxplot of Listing

IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Perc

ent

10000008000006000004000002000000

99

95

90

80

70

605040

30

20

10

5

1



IncomePC

List

ing

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000


Listing

Frequency

900000800000700000600000500000400000300000200000

14

12

10

8

6

4

2

0


Listing

Perc

ent

9000

00

8000

00

7000

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Mean 369687StDev 156865N 51


IncomePC

List

ing

30000250002000015000

1000000

800000

600000

400000

200000


2e mc


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 + ½σT

2) 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.

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Random Walk Models for Stock Prices Statistics and Data Analysis Professor William Greene Stern...

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Transcript of Random Walk Models for Stock Prices Statistics and Data Analysis Professor William Greene Stern...