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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

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Random Walk Models for Stock Prices Statistics and Data Analysis Random Walk Models for Stock Prices

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A Model for Stock Prices Preliminary: Consider a sequence of T random outcomes, independent from one to the next, Δ 1, Δ 2,…, Δ T. (Δ is a standard symbol for change which will be appropriate for what we are doing here. And, well use t instead of i to signify something to do with time.) Δ t comes from a normal distribution with mean μ and standard deviation σ. 1/30

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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 Δ t Δ 1 = sales on day 1, Δ 2 = sales on day 2, Total sales after T days will be Δ 1 + Δ 2 +…+ Δ T Therefore, each Δ t is the change in the total that occurs on day t. 2/30

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Random Walk Models for Stock Prices Using the Central Limit Theorem to Describe the Total Let P T = Δ 1 + Δ 2 +…+ Δ T be the total of the changes (variables) from times (observations) 1 to T. The sequence is P 1 = Δ 1 P 2 = Δ 1 + Δ 2 P 3 = Δ 1 + Δ 2 + Δ 3 And so on… P T = Δ 1 + Δ 2 + Δ 3 + … + Δ T 3/30

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Random Walk Models for Stock Prices Summing If the individual Δs are each normally distributed with mean μ and standard deviation σ, then P 1 = Δ 1 = Normal [ μ, σ] P 2 = Δ 1 + Δ 2 = Normal [2μ, σ2] P 3 = Δ 1 + Δ 2 + Δ 3 = Normal [3μ, σ3] And so on… so that P T = N[Tμ, σT] 4/30

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Random Walk Models for Stock Prices Application Suppose P is accumulated sales of a store. The accounting period starts with total sales = 0 Δ 1 = sales on day 1, Δ 2 = sales on day 2 Accumulated sales after day 2 = Δ 1 + Δ 2 And so on… 5/30

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Random Walk Models for Stock Prices This defines a Random Walk The sequence is P 1 = Δ 1 P 2 = Δ 1 + Δ 2 P 3 = Δ 1 + Δ 2 + Δ 3 And so on… P T = Δ 1 + Δ 2 + Δ 3 + … + Δ T It follows that P 1 = Δ 1 P 2 = P 1 + Δ 2 P 3 = P 2 + Δ 3 And so on… P T = P T-1 + Δ T 6/30

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Random Walk Models for Stock Prices A Model for Stock Prices Random Walk Model: Todays price = yesterdays price + a change that is independent of all previous information. (Its a model, and a very controversial one at that.) Start at some known P 0 so P 1 = P 0 + Δ 1 and so on. Assume μ = 0 (no systematic drift in the stock price). 7/30

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Random Walk Models for Stock Prices Random Walk Simulations P t = P t-1 + Δ t Example: P 0 = 10, Δ t Normal with μ=0, σ=0.02 8/30

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Random Walk Models for Stock Prices Uncertainty Expected Price = E[P t ] = P 0 +Tμ We have used μ = 0 (no systematic upward or downward drift). Standard deviation = σT reflects uncertainty. Looking forward from now = time t=0, the uncertainty increases the farther out we look to the future. 9/30

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Random Walk Models for Stock Prices Using the Empirical Rule to Formulate an Expected Range 10/30

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Random Walk Models for Stock Prices Application Using the random walk model, with P 0 = $40, say μ =$0.01, σ=$0.28, what is the probability that the stock will exceed $41 after 25 days? E[P 25 ] = 40 + 25($.01) = $40.25. The standard deviation will be $0.2825=$1.40. 11/30

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Random Walk Models for Stock Prices Prediction Interval From the normal distribution, P[μ t - 1.96σ t < X < μ t + 1.96σ t ] = 95% This range can provide a prediction interval, where μ t = P 0 + tμ and σ t = σt. 12/30

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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

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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

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Random Walk Models for Stock Prices Lognormal Variable If the log of a variable has a normal distribution, then the variable has a lognormal distribution. Mean =Exp[μ+σ 2 /2] > Median = Exp[μ] 15/30

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Random Walk Models for Stock Prices Lognormality – Country Per Capita Gross Domestic Product Data 16/30

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Random Walk Models for Stock Prices Lognormality – Earnings in a Large Cross Section 17/30

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Random Walk Models for Stock Prices Lognormal Variable Exhibits Skewness The mean is to the right of the median. 18/30

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Random Walk Models for Stock Prices Lognormal Distribution for Price Changes Math preliminaries: (Growth) If price is P 0 at time 0 and the price grows by 100Δ% from period 0 to period 1, then the price at period 1 is P 0 (1 + Δ). For example, P 0 =40; Δ = 0.04 (4% per period); P 1 = P 0 (1 + 0.04). (Price ratio) If P 1 = P 0 (1 + 0.04) then P 1 /P 0 = (1 + 0.04). (Math fact) For smallish Δ, log(1 + Δ) Δ Example, if Δ = 0.04, log(1 + 0.04) = 0.39221. 19/30

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Random Walk Models for Stock Prices Collecting Math Facts 20/30

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Random Walk Models for Stock Prices Building a Model 21/30

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Random Walk Models for Stock Prices A Second Period 22/30

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Random Walk Models for Stock Prices What Does It Imply? 23/30

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Random Walk Models for Stock Prices Random Walk in Logs 24/30

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Random Walk Models for Stock Prices Lognormal Model for Prices 25/30

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Random Walk Models for Stock Prices Lognormal Random Walk 26/30

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Random Walk Models for Stock Prices Application Suppose P 0 = 40, μ=0 and σ=0.02. What is the probabiity that P 25, the price of the stock after 25 days, will exceed 45? logP 25 has mean log40 + 25μ =log40 =3.6889 and standard deviation σ25 = 5(.02)=.1. It will be at least approximately normally distributed. P[P 25 > 45] = P[logP 25 > log45] = P[logP 25 > 3.8066] P[logP 25 > 3.8066] = P[(logP 25 -3.6889)/0.1 > (3.8066-3.6889)/0.1)]= P[Z > 1.177] = P[Z < -1.177] = 0.119598 27/30

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Random Walk Models for Stock Prices Prediction Interval We are 95% certain that logP 25 is in the interval logP 0 + μ 25 - 1.96σ 25 to logP 0 + μ 25 + 1.96σ 25. Continue to assume μ=0 so μ 25 = 25(0)=0 and σ=0.02 so σ 25 = 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 P 0 is in the range e 3.4929 = 32.88 and e 3.8849 = 48.66 28/30

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Random Walk Models for Stock Prices Observations - 1 The lognormal model (lognormal random walk) predicts that the price will always take the form P T = P 0 e ΣΔ t This will always be positive, so this overcomes the problem of the first model we looked at. 29/30

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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 P 25 based on P 0 = 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[P T ] = P 0 exp(μ T + ½σ T 2 ) which is greater than P 0 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

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