Statistics
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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 Models for Stock Prices 1/30 Random Walk Models for Stock Prices A Model for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball 30000 32500 0 1000000 60 800000 40 Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% Frequency Listing 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. Percent 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 2/30 Random Walk Models for Stock Prices Application 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 Listing 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. Percent 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 3/30 Random Walk Models for Stock Prices Using the Central Limit Theorem to Describe the Total P1 = Δ1 P2 = Δ1 + Δ2 P3 = Δ1 + Δ2 + Δ3 And so on… PT = Δ1 + Δ2 + Δ3 + … + ΔT 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 Listing Let PT = Δ1+ Δ2+…+ ΔT be the total of the changes (variables) from times (observations) 1 to T. The sequence is Percent 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 4/30 Random Walk Models for Stock Prices Summing If the individual Δs are each normally distributed with mean μ and standard deviation σ, then P1 = Δ1 = Normal [ μ, σ] P2 = Δ1 + Δ2 = Normal [2μ, σ√2] P3 = Δ1 + Δ2 + Δ3= Normal [3μ, σ√3] And so on… so that PT = N[Tμ, σ√T] 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 5/30 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… 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 6/30 Random Walk Models for Stock Prices This defines a Random Walk P1 = Δ1 P2 = Δ1 + Δ2 P3 = Δ1 + Δ2 + Δ3 And so on… PT = Δ1 + Δ2 + Δ3 + … + ΔT Pie Chart of Percent vs Type Meatball Garlic 5.0% 2.3% Pepperoni 21.8% Sausage 5.8% 900000 800000 800000 600000 500000 400000 Mushroom 16.2% Plain 32.5% Scatterplot of Listing vs IncomePC 900000 900000 Mean StDev N AD P-Value 95 90 400000 200000 100000 15000 60 50 40 30 20000 22500 25000 IncomePC 27500 30000 32500 800000 700000 Histogram of Listing e mc 6 200000 2 1 100000 15000 800000 1000000 17500 20000 22500 25000 IncomePC 27500 369687 156865 51 80 8 5 400000 600000 Listing Mean StDev N 10 4 200000 Normal 100 12 500000 300000 0 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 14 2 400000 10 17500 600000 70 20 300000 200000 369687 156865 51 0.994 0.012 80 500000 P1 = Δ1 P2 = P1 + Δ2 P3 = P2 + Δ3 And so on… PT = PT-1 + ΔT Scatterplot of Listing vs IncomePC Normal - 95% CI 600000 300000 100000 Probability Plot of Listing 99 700000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing Category Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball 30000 32500 Percent It follows that Frequency Listing Percent Mushroom and Onion 9.2% 0 1000000 60 800000 40 Listing The sequence is Listing 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 7/30 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). 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 8/30 Random Walk Models for Stock Prices Random Walk Simulations Pt = Pt-1 + Δt Example: P0= 10, Δt Normal with μ=0, σ=0.02 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 9/30 Random Walk Models for Stock Prices Uncertainty Expected Price = E[Pt] = P0+Tμ We 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. 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 10/30 Random Walk Models for Stock Prices Using the Empirical Rule to Formulate an Expected Range [P0 t] 2 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 11/30 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. P 40.25 $41.00 $40.25$ P[P25 $41] P 1.40 1.40 = P[Z > 0.54] = P[Z < -0.54] = 0.2496 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 12/30 Random Walk Models for Stock Prices Prediction Interval 900000 800000 800000 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Sausage 5.8% Scatterplot of Listing vs IncomePC 900000 700000 Listing Pepper and Onion 7.3% Boxplot of Listing Category Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball Percent Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Frequency Pie Chart of Percent vs Type Mushroom and Onion 9.2% Listing 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. Percent 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 13/30 Random Walk Models for Stock Prices Random Walk Model 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category Pepperoni Plain Mushroom Sausage Pepper and Onion Mushroom and Onion Garlic Meatball 30000 32500 0 1000000 60 800000 40 Listing Pepperoni 21.8% Listing Meatball Garlic 5.0% 2.3% Percent Pie Chart of Percent vs Type Mushroom and Onion 9.2% Frequency Listing 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 … Percent 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 14/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 15/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 16/30 Random Walk Models for Stock Prices 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 30 700000 17500 20000 22500 25000 IncomePC 27500 30000 32500 e mc 6 2 1 100000 15000 1000000 17500 20000 22500 25000 IncomePC 27500 Normal Mean StDev N 369687 156865 51 80 200000 800000 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 8 5 400000 600000 Listing 10.4 10 4 200000 9.6 12 500000 300000 0 8.0 8.8 logGDPC Histogram of Listing 400000 10 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 17/30 Random Walk Models for Stock Prices 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 30 700000 17500 20000 22500 25000 IncomePC 27500 30000 32500 e mc 6 2 1 100000 15000 1000000 17500 20000 22500 25000 IncomePC 27500 Normal Mean StDev N 369687 156865 51 80 200000 800000 Marginal Plot of Listing vs IncomePC Empirical CDF of Listing 100 8 5 400000 600000 Listing 8.4 10 4 200000 8.0 12 500000 300000 0 7.6 Histogram of Listing 400000 10 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 18/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 19/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 20/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 21/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 22/30 Random Walk Models for Stock Prices A Second Period P1 If P1 = P0 [1 + Δ1 ] then = [1 + Δ1 ] P0 Now, change for a second period If P2 = P1[1 + Δ 2 ], then P2 = P0 [1 + Δ1] [1 + Δ 2 ] so P2 = [1 + Δ1 ] [1 + Δ 2 ] P0 P2 log = log[1 + Δ1 ]+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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 23/30 Random Walk Models for Stock Prices 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] P 0 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc T-1 t=1 Δt 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 Δt 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 Category 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% t=1 PT-1 T T-1 log Δ t=1 t t=1 Δ t P0 = ΔT Percent PT log P0 T 20 600000 400000 0 0 200000 300000 400000 500000 600000 Listing 700000 800000 900000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 24/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 25/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 26/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 27/30 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 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 28/30 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 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 29/30 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. 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000 30/30 Random Walk Models for Stock Prices 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 30 20000 22500 25000 IncomePC 27500 30000 32500 e mc 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 Category 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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 20 30 40 50 60 70 80 90 Listing 200000 15000 20000 25000 IncomePC 30000
