Post on 30-Mar-2015
Random Walk Models for Stock Prices
Statistics and Data Analysis
Professor William Greene
Stern School of Business
Department of IOMS
Department of Economics
Random Walk Models for Stock Prices
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
Random Walk Simulations Pt = Pt-1 + Δt
Example: P0= 10, Δt Normal with μ=0, σ=0.02
8/30
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
Using the Empirical Rule to Formulate an Expected Range
10/30
0[P t ] 2 t
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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
28/30
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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.
29/30
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
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
Scatterplot of Listing vs IncomePC
Listing
Perc
ent
10000008000006000004000002000000
99
95
90
80
70
605040
30
20
10
5
1
Mean 369687StDev 156865N 51AD 0.994P-Value 0.012
Probability Plot of ListingNormal - 95% CI
IncomePC
List
ing
3250030000275002500022500200001750015000
900000
800000
700000
600000
500000
400000
300000
200000
100000
Scatterplot of Listing vs IncomePC
Listing
Frequency
900000800000700000600000500000400000300000200000
14
12
10
8
6
4
2
0
Histogram of Listing
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
Empirical CDF of ListingNormal
IncomePC
List
ing
30000250002000015000
1000000
800000
600000
400000
200000
Marginal Plot of Listing vs IncomePC
2e mc
Random Walk Models for Stock Prices
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.
30/30