Lognormal Random Walks for Stock Prices
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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 Part 11A – Lognormal Random Walks Lognormal Random Walk Pepperoni 21.8% Sausage 5.8% 900000 800000 800000 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC 900000 Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 700000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Meatball Garlic 5.0% 2.3% 30000 32500 0 1000000 60 800000 40 Listing Pie Chart of Percent vs Type Mushroom and Onion 9.2% Percent Frequency Listing The lognormal model remedies some of the shortcomings of the linear (normal) model. Somewhat more realistic. Equally controversial. Description follows for those interested. Percent 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 30/46 Lognormal Variable 2 1 1 logx -μ f(x) = exp - , 0 < x < + xσ 2π 2 σ Histogram of Wage Lognormal 120 Loc Scale N 100 6.951 0.4384 595 If the log of a variable has a normal distribution, then the variable has a lognormal distribution. 60 Mean =Exp[μ+σ2/2] > 40 20 Median = Exp[μ] Sausage 5.8% 900000 800000 800000 600000 500000 400000 Mushroom 16.2% Plain 32.5% Probability Plot of Listing Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 99 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 4800 Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball 4000 30000 32500 0 1000000 60 800000 40 Listing Pepperoni 21.8% 2400 3200 Wage Listing Meatball Garlic 5.0% 2.3% 1600 Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 800 Listing 0 Percent 0 Frequency Frequency 80 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 31/46 Lognormality – Country Per Capita Gross Domestic Product Data Histogram of GDPC Histogram of logGDPC Normal Normal 70 Mean StDev N 60 16 6609 7165 191 14 Frequency 30 800000 800000 Probability Plot of Listing 900000 Mean StDev N AD P-Value 95 90 400000 100000 15000 60 50 40 700000 17500 20000 22500 25000 IncomePC 27500 30000 32500 e mc 6 2 1 100000 15000 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Normal Mean StDev N 369687 156865 51 80 200000 400000 600000 Listing Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 8 5 200000 10.4 10 4 0 9.6 12 500000 300000 10 8.0 8.8 logGDPC Histogram of Listing 400000 30 7.2 14 2 600000 70 20 300000 200000 800000 Listing Listing 500000 200000 369687 156865 51 0.994 0.012 80 600000 6.4 Scatterplot of Listing vs IncomePC Normal - 95% CI 99 700000 300000 100000 0 30000 30000 32500 0 1000000 60 800000 40 Listing 900000 500000 24000 Scatterplot of Listing vs IncomePC 900000 600000 18000 Frequency 12000 GDPC 700000 Listing Plain 32.5% 6000 Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball 400000 Mushroom 16.2% 0 Percent -6000 Pie Chart of Percent vs Type Sausage 5.8% 6 2 0 Pepper and Onion 7.3% 8 4 10 Pepperoni 21.8% 10 Percent Frequency 40 20 Meatball Garlic 5.0% 2.3% 8.248 1.060 191 12 50 Mushroom and Onion 9.2% Mean StDev N 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 32/46 Lognormality – Earnings in a Large Cross Section Histogram of Wage Normal 120 Mean StDev N 100 1148 531.1 595 Frequency 80 Histogram of LogWage Normal 60 80 70 40 6.951 0.4384 595 60 0 800 1600 2400 3200 Wage 4000 Frequency 20 0 Mean StDev N 4800 50 40 30 20 10 0 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 60 50 40 700000 17500 20000 22500 25000 IncomePC 27500 30000 32500 e mc 6 2 1 100000 15000 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Normal Mean StDev N 369687 156865 51 80 200000 400000 600000 Listing Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 8 5 200000 8.4 10 4 0 8.0 12 500000 300000 10 7.6 Histogram of Listing 400000 30 7.2 LogWage 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 6.8 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 6.4 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 6.0 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 33/46 Lognormal Variable Exhibits Skewness Histogram of Wage Lognormal 120 Loc Scale N 100 6.951 0.4384 595 Frequency 80 The mean is to the right of the median. 60 40 20 800000 800000 600000 500000 400000 Mushroom 16.2% Plain 32.5% 900000 Mean StDev N AD P-Value 95 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 4800 Scatterplot of Listing vs IncomePC Normal - 95% CI 99 700000 300000 100000 Probability Plot of Listing 4000 30000 32500 Percent 900000 2400 3200 Wage 0 1000000 60 800000 40 Listing Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball 1600 Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% 800 Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 0 Frequency 0 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 34/46 Lognormal Distribution for Price Changes 500000 Plain 32.5% 900000 Mean StDev N AD P-Value 95 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Scatterplot of Listing vs IncomePC Normal - 95% CI 700000 700000 600000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 400000 Mushroom 16.2% Scatterplot of Listing vs IncomePC 900000 Frequency Sausage 5.8% (Math fact) For smallish Δ, log(1 + Δ) ≈ Δ Example, if Δ = 0.04, log(1 + 0.04) = 0.39221. Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepper and Onion 7.3% Listing Pepperoni 21.8% (Price ratio) If P1 = P0(1 + 0.04) then P1/P0 = (1 + 0.04). Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 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). 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 35/46 Collecting Math Facts Pt If Pt = Pt-1[1 + Δ ] then = [1 + Δ ] Pt-1 Pt log = log[1 + Δ ] Pt-1 Δ 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 36/46 Building a Model Slightly change the assumptions. Suppose Δ isn't a constant, but can be different each period. Pt If Pt = Pt-1[1 + Δ t ] then = [1 + Δ t ] Pt-1 Pt log = log[1 + Δ t ] Pt-1 Δt I.e., prices change by different amounts in different periods. 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 37/46 A Second Period P1 If P1 = P0 [1 + Δ 1 ] then = [1 + Δ 1] P0 Now, change for a second period If P2 = P1[1 + Δ 2 ], then P2 = P0 [1 + Δ 1 ] [1 + Δ 2 ] so P2 = [1 + Δ 1 ] [1 + Δ 2 ] P0 P2 log = log[1 + Δ 1 ]+log[1 + Δ 2 ] P0 Δ1 Δ 2 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 38/46 What Does It Imply? For T periods P log T = log[1 + Δ 1 ]+log[1 + Δ 2 ]+...+log[1 + Δ T ] P0 For T-1 periods PT-1 log = log[1 + Δ 1 ]+log[1 + Δ 2 ]+...+log[1 + Δ T-1 ] P0 By subtraction 800000 800000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc t=1 Δt 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 Percent 900000 600000 T-1 0 1000000 60 800000 40 Listing Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Frequency Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Listing Pie Chart of Percent vs Type Mushroom and Onion 9.2% Δt t=1 PT-1 T T-1 log Δ t=1 t t=1 Δ t P0 = ΔT Percent P log T P0 T 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 39/46 Random Walk in Logs By subtraction P log T P0 But PT-1 T-1 T Δ log t=1 t t=1 Δ t = Δ T P0 P log T P0 so, PT-1 log logPT logP0 logPT 1 logP0 P0 logPT logPT 1 Δ T This is the same random walk we had before, but now it is in logs, rather than in prices. 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 40/46 Lognormal Model for Prices PT log P0 = log[1 + Δ 1 ]+log[1 + Δ 2 ]+ ...+log[1 + Δ T ] Δ 1 Δ 2 ... Δ T so, logPT logP0 t 1 Δ t T If the period to period changes Δ t are normally distributed with mean and standard deviation , then logPT has a normal distribution with mean logP0 +T and standard deviation T. 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 41/46 Lognormal Random Walk If logPT logP0 t 1 Δ t T Then t 1 t PT = P0 e T which looks like the present value result, VT V0 erT for T periods and constant growth rate per period, r. 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 42/46 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 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 43/46 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 range e3.4929 = 32.88 and e3.8849 = 48.66 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 44/46 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. 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 700000 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 30000 32500 0 1000000 60 800000 40 Listing 800000 800000 Percent 900000 Frequency Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 45/46 Observations - 2 Pepperoni 21.8% Sausage 5.8% 900000 800000 800000 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC 900000 Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 700000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball 30000 32500 0 1000000 60 800000 40 Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% Frequency Listing Percent 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. Listing 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 46/46 Summary Normal distribution approximation to binomial Approximate with a normal with same mean and standard deviation Continuity correction Pie Chart of Percent vs Type Pepperoni 21.8% Sausage 5.8% 900000 800000 800000 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC 900000 Scatterplot of Listing vs IncomePC Normal - 95% CI 900000 Mean StDev N AD P-Value 95 90 500000 400000 200000 100000 15000 800000 700000 60 50 40 20000 22500 25000 IncomePC 27500 30000 32500 e mc 30 6 5 200000 2 1 100000 15000 400000 600000 Listing 800000 1000000 17500 20000 22500 25000 IncomePC 27500 Mean StDev N 369687 156865 51 80 8 4 200000 Normal 10 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 12 500000 400000 10 17500 Histogram of Listing 14 2 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 600000 300000 100000 Probability Plot of Listing 99 700000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing C ategory Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Listing Meatball Garlic 5.0% 2.3% Mushroom and Onion 9.2% 30000 32500 0 1000000 60 800000 40 Listing Percent Frequency Sums and central limit theorem Random walk model for stock prices Lognormal variables Alternative random walk model using logs Listing Percent 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 1 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000
