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martiniirichard
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Name: Richard
Country: United States
Birthday: 6/10/1987
Gender: Male
Interests: chess, math, science, reading, economics, arguing
Expertise: chess, econ
Occupation: Other
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Member Since:
7/7/2005
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| Richard Martin
Professor Hastings
Math 399 Paper
The Behavior of Underlying Assets Modeled as Geometric Browian Motion
Introduction
The purpose of this project is to determine if the intervals of log stock prices follow geometric Brownian motion (GBM), a key assumption in the Black-Scholes paper for pricing European options. The four properties of GBM are tested using list plots, histograms, Chi Square tests, correlation tests, and One Way ANOVA tests. The results showed strong evidence that we cannot conclude stocks follow GBM, but I assumed before hand that the recent market crash would distort the results. So, I also gathered data that excluded the last 27 weeks from the original data set. The time line includes weekly closing prices from March 1, 2006 to March 1, 2009. When more recent data was taken out, stocks were more likely to follow GBM, but these results still leave one unconvinced that GBM is effective in explaining the behavior of stocks. Another paper (Ang and Peterson, An Emperical Study of the Diffusion Process of Securities and Portfolios) explored similar ideas in 1984 and reached a similiar conclusion. The results suggest fatal flaws exist in assuming log price changes follow geometric Brownian motion. A process is Brownian if it satisfies these four properties: 1) Subscript[X, t] is continuous everywhere, 2) Subscript[X, t] -Subscript[X, s] \[TildeTilde]Subscript[X, t-s], 3)Subscript[X, t+\[CapitalDelta]t]-Subscript[X, t] is independent of Subscript[X, r] for all r<=t, 4)Subscript[X, t+\[CapitalDelta]t]-Subscript[X, t]\[TildeTilde] N (\[Mu]*\[CapitalDelta]t,\[Sigma]^2*\[CapitalDelta]t). My paper explores each of these four properties through statistical tests with Mathematica commands. Properties 1 and 2 were tested visually with list plots and histograms. Properties 3 and four were tested with Chi Square and correlation tests. The key to getting this research problem rolling comes from the Black-Scholes paper, "The Pricing of Options and Corporate Liabilities". This paper states that "the distribution of possible stock prices at the end of any finite interval is log normal" (pg.640). Let Subscript[Y, t] be the hypothesized distribution of stock prices and Subscript[X, t] be the actual distribution of stock price changes. Thus,
Subscript[Y, t]=e^Subscript[X, t]
Log[Subscript[Y, t]]=Subscript[X, t]
Log[Subscript[Y, t-1]]-Log[Subscript[Y, t]]=Subscript[X, t]
The final line follows by breaking the random variable Subscript[Y, t] into equally spaced finite intervals that are log normal satisfying a key assumption of Black-Scholes. How Subscript[X, t] is collected will be explained in the next section. Section 1 will give back-ground on what the Black-Scholes model is, why it is important, and where it came from. Section 2 explores how the data was collected and what manipulations were done to make the stock prices easily testable. Section 3 goes into the back ground of the statistical tests used, so the reader better understands what is happening and why the tests are important. Section 4 shows the results for Chi Square Distribution and Chi Square Independence tests, while section 5 goes over Correlation Tests and list plots. Section 6 explores the results, makes some conclusions.
Section 1 Black-Scholes
The Black-Scholes model was created for the purpose of making investment and gambling less risky. The fundamental question that plagued investors for decades was how to maximize the return on one's investments, while minimizing one's risk with a certain risk aversion \[Alpha]. Ideally, you want no risk and the maximum possible return. Hedge funds, banks, and individuals gamble on whether or not an underlying asset (in this paper, stocks) will go up or down. Some studiess peg financial services at 40% of GDP for the United States. The impact that derivatives have on banking and financing for the world cannot be underestimated. Almost every bank anywhere delves into the mysterious world of option pricing. It starts with the following equation:
dS = \[Mu]Sdt + \[Sigma]SdB
This is a stochastic differential equation that models random price fluctuations over time(dS), where \[Mu] is the rate of a return for a particular asset, S is some future price, dt is making the equation differentiable over time, \[Sigma] is the risk of the asset (or volatility) that depends on \[Mu], and the stochastic process dB helps predict fluctuation of the price for an asset by assuming flucuations follow Brownian motion. What financial mathematicians want to do is make stock price fluctuations predictable. Assume that the varianc is 0 in the above expression. We want to model an asset so that the asset behaves as a bond compounded continuously.
Subscript[S, t]=Subscript[S, o]*e^rt
Subscript[S, t]=Subscript[S, o]*e^\[Mu]t
Subscript[S, t]=Subscript[S, o]*e^\[Mu]t*e^(Subscript[\[Sigma]W, t]-\[Sigma]/2*t)
Subscript[S, t]=Subscript[S, o]*e^((\[Mu]-\[Sigma]/2)t+Subscript[\[Sigma]W, t])
The first line is the formula for a bond compounded continuously. We wish to model a stock the same way so that the stock behaves deterministically. This is not true in reality so an extra term is added that allows the stock price to go up or down at each time unit. The means in which the extra term, e^(Subscript[\[Sigma]W, t]-\[Sigma]/2*t), is added is explored more precisely in Stampfli and Gooman's book The Mathematics of Financing: Modeling and Hedging. We call this term a stochastic process that has standard normal properties. But first an important identity, from moment generating functions, will prove to be useful :
E[e^X]=e^(\[Mu]+1/2*\[Sigma]^2)
What we are interested in is the first moment for the standard normal distribution Z, which has mean 0 and variance \[Sigma]^2. Applying the m.g.f to a standard normal with coefficient c yields:
E[e^(c*Z)]=e^(0*c+1/2*1*c^2)=e^((c^2*1)/2)
The project is based on the following expectation :
E[Subscript[S, T]]=Subscript[S, o]*E[e^(Subscript[\[Sigma]W, t]+(u-\[Sigma]^2/2)*T)]
=Subscript[S, 0]*e^((u-\[Sigma]^2/2)T)*E[e^Subscript[\[Sigma]W, t]]
where Subscript[W, t ]\[TildeTilde]N(0,T), and \[Sigma]=t. Subscript[W, t] is the sum of all Subscript[Z, i] random variables that act stochastically at time i with mean 0 and variance 1. The properties of Brownian motion tells us that means Subscript[W, t] has mean 0 and variance t. Using what we know about Subscript[W, t] and the result in equation 2 yields:
Subscript[S, T]=Subscript[S, o]*e^((u-\[Sigma]^2/2)*T)*e^(1/2 T\[Sigma]^2)
=Subscript[S, 0]*e^(\[Mu]*T)
What this tells us is that we have found a way to take into account random fluctuations of the mean, so we can assume that assets behave a certain way, like Geometric Brownian motion. What you see above is the deterministic formula for an asset with fixed interest mean return \[Mu]. \[Mu]*t in my project is estimated by actual collected data and is equal to Subscript[\[Sigma]W, t]+(u-\[Sigma]^2/2)*T. Notice that when the log of both sides is tkaen, the resulting equation is of a line, where Log[Subscript[S, 0]] is the y intercept, (\[Mu] - \[Sigma]^2)/2 is the slope, and t is the x coordinate of the line with Subscript[\[Sigma]W, t] causing the line to look jaggid. For the sake of clarity, the following quote from the Black-Scholes paper contains all the assumptions of that go into option pricing.
1) the short term interest rate is known and is constant through time.
2)the stock price follows a random walk through time with a variance rate proportional to the square of the stock price. Thus, the distribution of possible stock prices at the end of any finite interval is lognormal. The variance rate of the return on the stock is constant.
3)The stock pays no dividends or other distributions.
4)The option is "European", that is, it can only be exercised at maturity.
5)There are no transaction costs in buying or selling the stock or the option
6) It is possible to borrow any fraction of the price of a security to buy it or hold it, at the short-term interest rate.
7)There are no penalties for short selling. (640)
Not included, but implicitly assumed, are that the stock satisfies the four characteristics of Brownian motion. The formula for European calls are below.
European Calls=S(t)N(d1)-X e^-r(T-t) N(d2)
d1=(ln(S(t)/X)+(r+\[Sigma]^2/2)(T-t))/(\[Sigma] Sqrt[T-t])=d1
d2 (ln(S(t)/X)+(r-\[Sigma]^2/2)(T-t))/(\[Sigma] Sqrt[T-t])=d2
S (t) is the stock price at time t, N(d1) is the value of the normal integral from -\[Infinity] to d1, X is the strike price, r is the risk-free interest rate, \[Sigma]^2 is the variance of the asset, T-t is the amount of time that transpired from start time t to expiration date T, and N(d2) is the normal integral value from -\[Infinity] to d2. d1 and d2 are shown above with the same variables.
Now that the background of what we are testing is clear, let us proceed with how the data was collected.
Section 2 Data Gathering and Manipulation
The Mathematica output below allows the data to be analyzed and manipulated for statistical tests.
stockbyweek1 = FinancialData["AA", {"March 1,2006", "March 1,2009", "Week"}];
logpbyweek1 = Log[Transpose[stockbyweek1][[2]]];
prchbyweek1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 2, 157}];
prchbyweekcut1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 2, 120}];
ChiTablea1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 2, 32}];
ChiTableb1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 33, 64}];
ChiTablec1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 65, 96}];
ChiTabled1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 97, 128}];
ChiTablee1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 129, 157}];
ChiTableacut1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 2, 25}];
ChiTablebcut1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 26, 50}];
ChiTableccut1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 51, 75}];
ChiTabledcut1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 76, 100}];
ChiTableecut1 =
Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 101, 120}];
muweek1 = Mean[prchbyweek1];
sigweek1 = StandardDeviation[prchbyweek1];
muweekcut1 = Mean[prchbyweekcut1];
sigweekcut1 = StandardDeviation[prchbyweekcut1];
hist1 = Histogram[prchbyweek1, Automatic, "ProbabilityDensity",
PlotRange -> All, AxesLabel -> {bins, density}, BaseStyle -> {24},
PlotLabel -> Full Set]
histcut1 =
Histogram[prchbyweekcut1, Automatic, "ProbabilityDensity", PlotRange -> All,
AxesLabel -> {bins, density}, BaseStyle -> {24}, PlotLabel -> Shorter Set]
Sort[prchbyweek1]
muweek1 - (3*sigweek1)
muweek1 + (3*sigweek1)
The data of the stocks were collected using a command in Mathematica 6 called Financial Data. This command allows one to reach back for as long as a stock was in existence, for any stock publicly traded, and does so in daily, weekly, monthly, or yearly intervals. I tested all 30 DOW Jones stocks to see how well behaved they are overall. The first stock looked at is Alcoa, or AA. I wanted a wide range of data to make statistical testing easier, thus I collected from March 1, 2006 through March 1, 2009. The option "week" at the end of the command returns all the weekly closing prices. Next, it is required to transform the data to satisfy the key assumption explained earlier. Logbyweek1 takes the second element of the nx2 matrix generated by stockbyweek1 and pulls out only closing weekly stock prices. I found this to be a convenient time to take the log of all these prices. prchbyweek1 takes the difference between successive intervals to give us data that allows us to test whether the stock follows the four assumption of GBM. This is done in table form. prchbyweekcut1 represents a cut in how much of the original data is being collected in a table. There are 156 observations in the full data set and 119 in the second data set, which is reprented by prchbyweekcut1. What follows are Chi Square Tables for each type of collected data, breaking up the long tables into five separate categories of tables through time. These separate tables are going to consitute elements of a factor A that will be compared to elements in another Factor B for the purpose of testing independents. Mathematica conveniently calculates mean and standard deviation to be later used to show the normal curve with the histogram of the collected data.
The first histogram shows the density for each bin of the full data set. As you can see there are extreme outliers that leave a left skew of the histogram. But the general shape of the histogram appears to be normal.
\!\(\*
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For your convenience I listed all the data in order from least to greatest and found the critical x values that correspond to the boundaries of plus or minus 3 standard deviations away from the mean. Two observations are extreme on the negative side (-.265819,.246598), supporting my hypothesis that outliers from the market crisis would distort the behavior of stocks; possibly enough to break down the assumptions of how the stocks should behave.
{-0.536531, -0.278203, -0.249965, -0.226229, -0.20175, -0.173779, -0.141183, \
-0.136922, -0.129386, -0.12704, -0.123643, -0.116293, -0.113667, -0.103986, \
-0.101783, -0.0944564, -0.0941611, -0.0867696, -0.0859799, -0.0847689, \
-0.0763465, -0.0763355, -0.0704094, -0.06965, -0.067829, -0.0674127, \
-0.0608177, -0.0582203, -0.0556107, -0.0552296, -0.0537254, -0.0477047, \
-0.0466029, -0.0464612, -0.0449346, -0.0443584, -0.0424506, -0.042165, \
-0.0411529, -0.0407631, -0.0402204, -0.0393947, -0.0392351, -0.0387721, \
-0.0383268, -0.0345494, -0.03248, -0.0316305, -0.028862, -0.0244138, \
-0.0229716, -0.0218571, -0.0212713, -0.0194403, -0.0185087, -0.0170803, \
-0.0160134, -0.0151651, -0.0148024, -0.0136428, -0.0116346, -0.0108697, \
-0.00989801, -0.00961546, -0.0094822, -0.00848128, -0.00843198, -0.00840472, \
-0.00834603, -0.00652105, -0.00451614, -0.00446762, -0.00434091, -0.00288913, \
-0.00242592, -0.000366099, 0.000755858, 0.00163425, 0.00263418, 0.00288913, \
0.00298954, 0.00524752, 0.00572849, 0.00587373, 0.00610648, 0.00736458, \
0.00799309, 0.0083414, 0.0083879, 0.00852608, 0.00857323, 0.00981894, \
0.0101258, 0.0117895, 0.0131293, 0.0132172, 0.0134404, 0.0141233, 0.0145893, \
0.0149729, 0.0152576, 0.0153424, 0.0159502, 0.016033, 0.0166227, 0.0171717, \
0.0173655, 0.0193644, 0.0198876, 0.0199882, 0.0211799, 0.0213459, 0.0232366, \
0.0242546, 0.0261358, 0.0274558, 0.0290411, 0.029842, 0.0300203, 0.0302936, \
0.0311607, 0.0321514, 0.0339316, 0.0359953, 0.0363196, 0.0406283, 0.0427483, \
0.0438066, 0.0448704, 0.0462229, 0.0477784, 0.0478748, 0.0479463, 0.0487284, \
0.0499621, 0.0529347, 0.0533036, 0.0551565, 0.0562081, 0.061718, 0.0644452, \
0.0670838, 0.0682699, 0.0699781, 0.0768857, 0.0778603, 0.0825889, 0.0972757, \
0.0976997, 0.100211, 0.110538, 0.128119, 0.20037, 0.213284, 0.2143, 0.242375}
-0.265819
0.246598
Section 3 Statistics Background
Hypothesis testing is the means by with which I have tested the last two properties of Brownian motion (Subscript[X, t+\[CapitalDelta]t]-Subscript[X, t] is independent of Subscript[X, r] for all r<=t and Subscript[X, t]\[TildeTilde] N (\[Mu]*\[CapitalDelta]t,\[Sigma]^2*\[CapitalDelta]t)). A hypothesis test starts by assuming that you believe something is true. In this case I will assume that F=Subscript[F, o] where F is the actual distribution that describes how weekly log price incements behave. Subscript[F, o] represents a hypothetical distribution that in this case is normal. This is called the null hypothesis. The alternative hypothesis is F!=Subscript[F, o], which is to say we can conclude that F is following a distribution that is not Subscript[F, o]. We have \[Alpha] which tells us how likely type one errors occur. If \[Alpha] is .05 (as in this paper) this means we have only a five percent chance of rejecting the null hypothesis, even though it is true. The test we will use is Chi Square, which is a distribution test that allows one to test for what kind of distribution collected data follows. The Chi square value is determined by the number of observations in the data set (n), and the number of degrees of freedom (r). The degrees of freedom are determined by taking the total number of categories, subtracting how many parameters have been estimated (\[Mu] and \[Sigma] in this paper) minus 1. The number of categories is determined by the size of the sample and by personal preferences. It is important to have enough categories to catch discrepancies, but not so many as have too few observations falling within each category making the test useless. Chi square tests require that at least five observances theoretically fit within each artificially constructed category. If each category's value (Subscript[X, 1],...Subscript[X, 10] below) summed is greater than the value determined from the Chi Square Distribution table, we reject Subscript[H, o]. In this paper, we do not want to reject Subscript[H, o] because the Black-Scholes Model greatly depends on log price changes follow the normal distribution. There is a p value associated with each Chi Square value that tells us how likely a type I error has occurred. If a p value is .05, that means there is a 5% chance that Subscript[H, o] was rejected wrongly or that the null hypothesis is true.To determine Q, which will be used to compare to the Chi Square value in the back of a statistics book:
Q=\!\(
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The following table gives a more visual view of what is happening
TableForm[{{Subscript[X, 1], Subscript[X, 2], Subscript[X, 3], Subscript[X,
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Chi Square Distribution Table
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The formula changes to check for independence, but the same principles apply. This independence test will construct two different sets of categories. One set, called Factor A, will represent the data as different segements of time, the other set, called Factor B, will be intervals of the normal distribution that should contain a certain percentage of data from each category it is compared against on the factor A side. The formula is below:
\!\(
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\*FractionBox[
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Chi Square Independence Table
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Things get more complicated because each element in category A is compared to each element in Factor B, to see if each possible entry as shown in the table above, has the same number of observations. The Q value is calculated by subtracting 1 from the total categories h from Factor A multiplied by k-1 categories from Factor B. Summing for all values i,j >0 will give us value to compare to the calculated test statistic. Remember that if calculated Q is greater than the chi square value Subscript[H, o] must be rejected, which would imply correlation or some abnormal factor influencing the distribution of log stock prices changes. This would violate assumption four of Brownian motion.
Another check on independence is a correlation test. This tests takes two different sets of data and calculates the T-value and whether it is significant using the formula below. If significant, this implies some general behavior is determinable about how log stock price changes are behaving. The critical value is calculated with the following formula:
T=(Sqrt[n-2]*R)/Sqrt[1-R^2]
n is the number of observations in the data sample, and R is the correlation value between the two sets of data.
To check and see if there are irregularities between different segments of time for mean and standard deviation, a One-Way ANOVA test, was performed to check whether there is reason to reject the null hypothesis that Subscript[\[Mu], 1] =Subscript[\[Mu], 2] =Subscript[\[Mu], k] , where k is the category of data tested. Essentially, for each category with Subscript[X, 11] , Subscript[X, 12] , Subscript[X, 1n1] averaged, one expects that they follow N(\[Mu], \[Sigma]^2). This should hold true for each Subscript[X, k,nk] data set we work with. If not, this implies that the mean and variance has changed for the stock we are working with. How exactly an Anova value is calculated is below and any statistics text will be able to help explain the derivation.
SST=\!\(
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F=MSF/MSE=(SSF/(k-1))/(SSE/(N-k))
Section 4 Results for Chi Square Tests
Chi Square Distribution Tests For Full Data Set
The following tests will see if the data follows a normal distribution; an important component needed to price derivatives. The results for the full data set are below.
catsweek1 = CategoryCutoffs[NormalDistribution[muweek1, sigweek1], 10];
freqsweek1 = BinCounts[prchbyweek1, {catsweek1}]
N[MultinomialGOF[
freqsweek1, {1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10}, 7,
ShowTables -> True]]
plot1 = Plot[PDF[NormalDistribution[muweek1, sigweek1], x], {x, -.3, .3}];
Show[hist1, plot1]
{11, 9, 10, 18, 16, 30, 29, 17, 9, 7}
Goodness of fit test to multinomial probabilities:
Value of chi-square test statistic: 39.
Critical value: 14.0671
Reject H0 at level 0.05
p-value of test: 1.95447*10^-6
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The command CategoryCutOffs is creates categories for which we will check to see how many observations fall into each created category. The categories were picked by picking critical x values that should have intervals that contain precisely 10 % of the data. BinCounts tells how often observations fall into a given category. Multinomial GOF is the ChiSquare command that requires the frequency of occurrences for each category, the weight which you expect each category to have, and the degrees of freedom. The expected number is 15.6 (156/10) and the degrees of freedom is 7 because of 2 estimated parameters \[Mu], \[Sigma] subtracted from 10 minus 1. We reject Ho here, which means that the underlying distribution explaining stock price changes is not normal. Because 40.9231 > 14.0671 (chi square value) with p value near 0. Assuming \[Alpha] of .05 this means 95 % of the random samples that could be picked from this stock are certain to not follow a normal distribution in the defined categories.
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The histogram with the plot of the normal distribution with mean and variance calculated from the data shows that too much weight is towards the mean and not enough is in the categories away from the mean. The 6th and 7th intervals contain the most differences between expected and observed values. But with even a slight difference on the tails, the data vary wildly from normal. There are very significant observations on the left tail that may affect the variance of the normal curve and prevent the histogram from meeting the normal curve.
What follows is the test for the shorter data set.
catsweekcut1 =
CategoryCutoffs[NormalDistribution[muweekcut1, sigweekcut1], 10];
freqsweekcut1 = BinCounts[prchbyweekcut1, {catsweekcut1}];
N[MultinomialGOF[
freqsweekcut1, {1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10, 1/10},
7, ShowTables -> True]]
plotcut1 =
Plot[PDF[NormalDistribution[muweekcut1, sigweekcut1], x], {x, -.2, .2}];
Show[histcut1, plotcut1]
Goodness of fit test to multinomial probabilities:
Value of chi-square test statistic: 11.1681
Critical value: 14.0671
Fail to reject HO at level 0.05
p-value of test: 0.131454
\!\(\*
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We do not reject Ho here, which means that the underlying distribution explaining stock price changes could be normal. The table shows the greatest discrepancy in the third interval, but all the intervals are fine collectively.
\!\(\*
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AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->{True, True},
AxesLabel->{
FormBox["bins", TraditionalForm],
FormBox["density", TraditionalForm]},
AxesOrigin->NCache[{
Rational[-4, 25], 0}, {-0.16, 0}],
BaseStyle->{24},
FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
PlotLabel->FormBox[
RowBox[{"Set", " ", "Shorter"}], TraditionalForm],
PlotRangePadding->{{
Scaled[0.02],
Scaled[0.02]}, {
Scaled[0.02],
Scaled[0.1]}},
Ticks->{Automatic, Automatic}]\)
This histogram is far more pleasing and fits the curve well.
Chi Square Independence Results
What follows is a test to check for independence for the full data set.
catschiweek1 = CategoryCutoffs[NormalDistribution[muweek1, sigweek1], 5];
freqschiweeka1 = BinCounts[ChiTablea1, {catschiweek1}];
freqschiweekb1 = BinCounts[ChiTableb1, {catschiweek1}];
freqschiweekc1 = BinCounts[ChiTablec1, {catschiweek1}];
freqschiweekd1 = BinCounts[ChiTabled1, {catschiweek1}];
freqschiweeke1 = BinCounts[ChiTablee1, {catschiweek1}];
list1 = List[{catschiweek1, freqschiweeka1, freqschiweekb1, freqschiweekc1,
freqschiweekd1, freqschiweeke1}]
chitablelist1 = {{2, 5, 14, 9, 1}, {0, 4, 12, 12, 4}, {2, 8, 7, 13, 2}, {4, 7,
8, 9, 4}, {12, 4, 5, 3, 5}};
ChiSquareIndependenceTest[chitablelist1]
{{{-\[Infinity], -0.0814873, -0.031247, 0.0120261,
0.0622663, \[Infinity]}, {2, 5, 14, 9, 1}, {0, 4, 12, 12, 4}, {2, 8, 7, 13,
2}, {4, 7, 8, 9, 4}, {12, 4, 5, 3, 5}}}
Independence test:
Value of chi-square test statistic: 41.5985
Critical value: 26.2962
Reject H0 at level 0.05
p-value of test: 0.000452792
We reject Ho here, which means that the underlying distribution explaining stock price changes is not normal. This is due to 41.5985>26.2962 (chi square value) and a p value of 0.000452792. Assuming \[Alpha] of .05 this means 95% of the random samples that could be picked from this stock are certain to not be independent from another.
The same is done for the shorter data set.
catschiweekcut1 =
CategoryCutoffs[NormalDistribution[muweekcut1, sigweekcut1], 5];
freqschiweekacut1 = BinCounts[ChiTableacut1, {catschiweekcut1}];
freqschiweekbcut1 = BinCounts[ChiTablebcut1, {catschiweekcut1}];
freqschiweekccut1 = BinCounts[ChiTableccut1, {catschiweekcut1}];
freqschiweekdcut1 = BinCounts[ChiTabledcut1, {catschiweekcut1}];
freqschiweekecut1 = BinCounts[ChiTableecut1, {catschiweekcut1}];
listcut1 =
List[{catschiweekcut1, freqschiweekacut1, freqschiweekbcut1,
freqschiweekccut1, freqschiweekdcut1, freqschiweekecut1}]
chitablelistcut1 = {{7, 0, 9, 4, 4}, {3, 4, 8, 8, 2}, {5, 2, 6, 6, 6}, {8, 5,
3, 4, 5}, {3, 4, 2, 5, 6}};
ChiSquareIndependenceTest[chitablelistcut1];
{{{-\[Infinity], -0.0374752, -0.00948125, 0.0146306,
0.0426245, \[Infinity]}, {7, 1, 8, 4, 4}, {3, 4, 8, 8, 2}, {5, 2, 6, 6,
6}, {8, 5, 3, 4, 5}, {3, 4, 2, 5, 6}}}
Independence test:
Value of chi-square test statistic: 19.8085
Critical value: 26.2962
Fail to reject HO at level 0.05
p-value of test: 0.228969
We reject Ho here, which means that the underlying distribution explaining stock price changes is not normal.
Section 5 Correlation Tests and List Plots
Correlation Tests
Correlation Tests for Full Data Set
Below are two list plots (one of just closing weekly prices, the other of stock price increments) plotted together along with a correlation test for the full weekly data set.
lpweek1 = ListPlot[prchbyweek1, Joined -> True,
AxesLabel -> {time, price in change}, BaseStyle -> {20}];
lpprweek1 =
ListPlot[weekp1, Joined -> True, AxesLabel -> {time, closing weekly prices},
BaseStyle -> {20}];
Show[lpweek1, lpprweek1]
CorrelationTest[weekp1, prchbyweek1]
\!\(\*
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"]]}},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox["time", TraditionalForm],
FormBox[
RowBox[{"change", " ", "in", " ", "price"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
BaseStyle->{20},
PlotRange->Automatic,
PlotRangeClipping->True]\)
Hypothesis test of H0: rho=0
1-sided alternative
Correlation TStat D.F. p-value
0.087881 1.09481 154 0.137654
Accept H0 of no correlation at level 0.05
The test claims the data are correlated with each other with p value .137288. Assuming \[Alpha] of .05, this means 95% of the random samples that could be picked from this stock are certain to not be correlated with each other.
Next, tests were run to see how correlated the stocks were by comparing X2-X1 increments with X3-X2 increments all the way down to Xn-Xn-1 with Xn-1-Xn-2. This is slightly different than above because all of the data are intervals. Notice that another set of tables were needed.
prchbyweekcutoff1 =
Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 2, 156}];
prchadjbyweek1 = Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 3, 157}];
lpadjweek1 = ListPlot[prchadjbyweek1];
lpweekcutoff1 = ListPlot[prchbyweekcutoff1];
Show[lpweekcutoff1, lpadjweek1]
CorrelationTest[prchbyweekcutoff1, prchadjbyweek1]
\!\(\*
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Hypothesis test of H0: rho=0
1-sided alternative
Correlation TStat D.F. p-value
-0.107051 -1.3318 153 0.0924531
Accept H0 of no correlation at level 0.05
The test claims the data are not correlated with each other with p a value .0943034. Assuming \[Alpha] of .05 this means 95 % of the random samples that could be picked from this stock are certain to not be correlated with each other.
Correlation Tests For Short Data Set
Below are the two list plots for the short data set plotted together along with a correlation test.
lpweekcut1 =
ListPlot[prchbyweekcut1, Joined -> True, AxesLabel -> {time, price change},
BaseStyle -> {20}];
lpprweekcut1 =
ListPlot[weekpcut1, Joined -> True, AxesLabel -> {time, price change},
BaseStyle -> {20}];
Show[lpweekcut1, lpprweekcut1]
CorrelationTest[weekpcut1, prchbyweekcut1]
\!\(\*
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AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox["time", TraditionalForm],
FormBox[
RowBox[{"change", " ", "price"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
BaseStyle->{20},
PlotRange->Automatic,
PlotRangeClipping->True]\)
Hypothesis test of H0: rho=0
1-sided alternative
Correlation TStat D.F. p-value
-0.172728 -1.89685 117 0.0301569
Reject H0 of no correlation at level 0.05
The test claims the data are correlated with each other with p value .0302588. Assuming \[Alpha] of .05, this means 95% of the random samples that could be picked from this stock are certain to be correlated with each other.
Next, the same test was run as before with the full data set.
prchbyweekcutoffcut1 =
Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 2, 118}];
prchadjbyweekcut1 =
Table[(logpbyweek1[[i]] - logpbyweek1[[i - 1]]), {i, 3, 119}];
lpadjweekcut1 = ListPlot[prchadjbyweekcut1];
lpweekcutoffcut1 = ListPlot[prchbyweekcutoffcut1];
Show[lpweekcutoffcut1, lpadjweekcut1]
CorrelationTest[prchbyweekcutoffcut1, prchadjbyweekcut1]
\!\(\*
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AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesOrigin->{0, Automatic},
PlotRange->Automatic,
PlotRangeClipping->True]\)
Hypothesis test of H0: rho=0
1-sided alternative
Correlation TStat D.F. p-value
-0.00290854 -0.0311907 115 0.487586
Accept H0 of no correlation at level 0.05
The test claims the data are not correlated with each other with a p value of .490679. But the correlation is near 0, so there is almost no dependence on previous stock prices. Assuming \[Alpha] of .05, this means 95 % of the random samples that could be picked from this stock are certain to not be correlated with each other.
List Plot Analysis
The following are list plots.
Full Data Set
weekp1 = Table[logpbyweek1[[i]], {i, 1, 156}];
lpweek1 = ListPlot[prchbyweek1, Joined -> True,
AxesLabel -> {time, price in change}, BaseStyle -> {20}]
lpprweek1 =
ListPlot[weekp1, Joined -> True, AxesLabel -> {time, closing weekly price},
BaseStyle -> {20}]
Weekly Interval Changes
\!\(\*
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AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox["time", TraditionalForm],
FormBox[
RowBox[{"change", " ", "in", " ", "price"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
BaseStyle->{20},
PlotRange->Automatic,
PlotRangeClipping->True]\)
Closing Weekly Prices
\!\(\*
GraphicsBox[
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AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox["time", TraditionalForm],
FormBox[
RowBox[{"closing", " ", "price", " ", "weekly"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
BaseStyle->{20},
PlotRange->Automatic,
PlotRangeClipping->True]\)
The first list plot charts the incremental changes of log prices.The second list plot looks at closing log prices per week. It could be argued that this is a continuous process everywhere and that the increments behave independently from each other. There may be some heteroskedasticity at the end (variance is increasing over time), but that would fall in line with the idea that the market crisis upset the assumptions of Brownian motion. The second list plot is looking at closing log prices per week. This also could be continuous process following a Brownian motion path. Notice that there is a sharp drop off at the end, which represents the panic of the market with people selling in large amounts.
Short Data
weekpcut1 = Table[logpbyweek1[[i]], {i, 1, 119}];
lpweekcut1 =
ListPlot[prchbyweekcut1, Joined -> True, AxesLabel -> {time, price change},
BaseStyle -> {20}]
lpprweekcut1 =
ListPlot[weekpcut1, Joined -> True, AxesLabel -> {time, price change},
BaseStyle -> {20}]
Weekly Interval Changes
\!\(\*
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AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesLabel->{
FormBox["time", TraditionalForm],
FormBox[
RowBox[{"change", " ", "price"}], TraditionalForm]},
AxesOrigin->{0, Automatic},
BaseStyle->{20},
PlotRange->Automatic,
PlotRangeClipping->True]\)
Closing Weekly Prices
\!\(\*
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The excluding data set looks as it should, with no wild swings at the end of the timeline. So, it looks like it can be claimed that log stock price changes for Alcoa follow a continuous process and each interval Subscript[X, t-\[CapitalDelta]t] is N(\[Mu]\[CapitalDelta]t, \[Sigma]^2\[CapitalDelta]t) property satisfied.
Section 6 Data Analysis
The following table shows the results for all thirty stocks for each of the tests performed as above. Included are an average of the critical values and the critical values themselves to give the reader a better grasp of how an average stock would do under each test.
Chisquarevalues =
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18.7436, 13.7436, 33.359, 48.8718, 33.7436, 16.5641, 20.2821, 22.5897,
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30.7126, 45.4474, 57.1179, 61.8643, 35.2504, 38.2228, 30.2356, 38.6808,
32.0129, 56.6839, 16.7186, 39.1973, 24.8559, 25.884, 42.6413, 78.1899,
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2.45247, 2.10455, 3.30687, 2.38832, 2.48864, .83584, 3.44939, 2.21606,
5.19739, 1.53455, .321674, 1.11646, 1.13806, 1.52429, 1.78523, .554965,
1.17669, 1.56764, 1.08292, 1.53659, 1.8966, 1.52447, .885143, 1.80356,
1.20915, 1.59952, .834068, .454383, .74401,
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1.19781, .765402, .337372, .759857, 1.34818, .585836, 1.27156, .669893,
1.11646, 1.13806, 1.52429,
1.78253, .265918, .229714, .447055, .285095, .639768,
1.30645, .432368, .549639, .122869, .505043, .228387, .570759, .665769, \
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"AA", "AXP", "BA", "BAC", "C", "CAT", "CVX", "DD", "DIS", "GE", "GM",
"HD", "IBM", "INTC", "JNJ", "JPM", "KFT", "KO", "MCD", "MMM", "MRK",
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As you can see, the first column of values suggest that we can conclude strongly that stock prices do not follow GBM. This has implications for all types of investors, banks, and hedge funds. The second column contains critical Q values for Independence, and the full data sets clearly fail resoundingly. 6 stocks pass the Chi Square Distribution test, while 7 pass the independence test when using the full set of data. Disturbingly only 22 stocks pass the Distribution test and only 24 stocks pass the independence test when dealing with the shorter data set. This suggests that something is seriously off with the Black-Scholes assumptions. If you look at the critical values for the correlation tests, one sees lots of failure. One can only conclude we can not be certain that stock increments are independent from each other or that they are independent from closing prices. I constructed artificial category's of stocks which represents different sectors of the economy, testing to see if an explanation can be given for the irregularities. There was a strong connection between technology and financial stocks failing the tests more than any other sector. In aggregate, they accounted for 41% of all failed tests for the short data sets. If we use the full data set these two industries are 64% of all failed tests. For representing only 1/4 of all the stocks, they represent a large chunk of all the failures. For the correlation tests they were 33% of all failed tests for the full data, and 27% for the short data. It was random weather a stock would pass or fail the correlation tests for both sets of data. The Anova results were not expected. The test was performed to see if it was safe to assume that the mean was the same for each segment of time created by the Chi Square Tables. A total of 6 failures for 60 tests performed, all of which were from the full data set. So, the test did not leave one to conclude that the mean changed over time violating the Black-Scholes model.
Section 7 Conclusion
What does this all mean for the world of finance? A cheap, cursory look may suggest that the GBM is simple too non-predicative for major crises and thus loses its purpose in protecting gamblers from risk. The whole point of GBM is to find a safe way to bet on future values of underlying assets without taking on too much risk. With 24 failures out of 30 for the full data set that includes the market crash, stocks did not behave. Yet, we still receive 8 failures out of 30 for a more tranquil time. Independence held up better, but still 6 failures for the short data set. Possibly, the mean and variance changed for different segments of time and the model may only require an extra push to account for the wild data points. If this was so, we would expect the ANOVA test to say more. What we found is that all tests for the short data set have the same mean and 24 stocks for the full data set also passed the ANOVA quite strongly. Perhaps a few skewed data points for each stock is affecting the satisfaction of Brownian motion. As shown in the paper there were two significant observation more than 3 standard deviations away from the mean. This was generally true for most stocks. Some stocks had more than 2 observations that were significant. But again, the GBM model is supposed to account for wild fluctuations and still be functional. All this information leads one to conclude that some other model is needed for European options, because too much depends on a functioning financial system.
Bibliography
Roman, Steve. Introduction to the Mathematics of Finance. UnitedStates: Springer, 2004.
Hastings, Kevin. Probability and Statistics. Addison-Wesley: New York, 1997
Hastings, Kevin. Introduction to the Mathematics of Operations Research with Mathematica. Gales burg, Illinois: Chapman&Hall/CRC, 2006
Goodman, Victor., and Joseph Stampfli. The Mathematics of Financing: Modeling and Hedging. California: Brooks/Cole, 2001.
Devore, Jay., and Roxy Peck. Statistics The Exploration and Analysis of Data. United States: Thomson, 2005
Ang S., James., and David R. Peterson. "An Empirical Study of the Diffusion Process of Securities and Portfolios". The Journal of Financial Research VII.3 (1984): 219-229.
Black, Fischer., and Myron Scholes. "The Pricing of Options and Corporate Liabilities." The Journal of Political Economy 81.3 (1973): 637-654. 17 Apr. 2009 <http://www.jstor.org/stable/1831029>.
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| I will make it simple. TARP was a bad bill. Not just a bad bill, but one that loaded 750 billion dollars of national debt on U.S. citizens. The only good news is that 350 billion of it hasn't been spent. But, that's not why economic disaster looms. The current stimulus bill is a good idea in spirit and perhaps theoretically. However, after Republicans dismantled many good parts of the bill, we are left with too many tax cuts, not enough spending on infrastructure, and cuts from education. Potentially, it will lose the green jobs parts, and state by state funding that should keep us floating. But that's not why economic disaster looms. We are losing so many jobs every month that I am not sure what the stimulus bill can do in its current form to save us from a deep and long depression. But, again, that is not why disaster looms.
I am hearing more and more prestigious right wing, super duper neoclassical economists saying we need to nationalize the banks, regulate the financial sector, and perhaps have a little protectionism. This is impossible. In fact it's such a contradiction, that I am left scratching my head, wondering what could possibly make them say such things. Ireland defaulted on national debt 10 times the size of its GDP. England may have to bailout the Bank of England, yeah the Bank of England. Spain saw its credit rating drop to AA, Japan claims manufacturing production levels will fall back to 1986, assuming something miraculous doesn't happen. China, who has had the best year out of everyone, is passing multiple trillion dollar stimulus packages to save itself from civil unrest. That is with 6.8% GDP last year. South America and Africa? I haven't heard any bad news down there, but who actually believes they walk away unscathed.
The good ole U.S.A. should be leading the pact making the big decisions that will save the world economy. Right?? What we have instead are Republicans claiming that we are the road to socialism (someone forgot to tell them it has arrived), that Democrats are burdening American with trillions in more debt (conveniently ignoring Iraq), and that this stimulus bill is worthless because it has money that goes into Pell Grants and flu shot research. I forgot to mention that they want tax cuts for big business and the top 1% income earners. Disgusting is not the word I am looking for, but malicious comes to mind. Malicious because they are betraying the people that elected them for a broken idea. The are informants for Reaganomics, you know that guy who didn't know anything about economics and believed tax cuts with increased government spending would balance the budget. What's worse about Republicans is that they know the 1980's model doesn't work, they know that doing nothing will bring economic ruin that none of us have ever seen.
I know they have convinced my boss, that if economic depression is necessary to teach regular, hard working people a lesson, and not the ones that got us into this catastrophe, than let us go all in. Let's vote against common sense and stability. My boss is a dimwit that does not understand what impact this will have on her life and her childrens' life. In fact., it has already affected her. She was concerned that banks would not loan out money to her son to finish his education. Now, imagine a year from now that the government has done nothing.
Economic disaster looms because too many people have bought into a broken ideology and hope that the stimulus bill fails. What does that say about our society and education system, when it is considered okay to hope millions of people suffer so a bad ideology remains untouched? It is a shame that first big thing Obama has to do, is fight a battle won years ago. If he does win, history will have a slightly different reflection of what Reagan accomplished, one in which he is seen as a villain who gave the keys to to Republicans to destroy common sense and decency.
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| Busy, busy, busy sums of my past few months with no end in sight. I could write about my thoughts on the Gaza conflict, but my opinion on the matter probably goes against everyone's on the planet. I could write about the financial crisis, but nothing new seems to be happening, but more layoffs. The worst thing to be write now is a recent graduate college or graduate school or an old person shy of retirement. To be blunt it looks like a slaughter house for those people when it comes to finding jobs. I could write about Obama and how he will save the galaxy, but I consider that self-evident.
How about I stick to the small things regular people are so obsessed about. I have talked to Brenda and all of my recommendations are sent in. I have mailed all my other materials. All I have left is my personal statement for ISU and I think it will be done shortly. That leaves me with the ever dreadful waiting part and crossing my fingers hoping they give me some money.
I really don't have time to worry because I am immersed with my studies, studying math. Doing such by yourself is hard. Latin is going well, especially after I got past the predicate nominative thing that I had so much trouble with last time we discussed grammar. The simplest of principles can allude me until that principle becomes self-evident. Econ will prove to be harder than expected and more time consuming. I have spent 6 hours this week reading. We know what our final papers will be and it will involve analyzing a phd level paper about Friedman's, The World is Flat. Basically this class is readings, writing, and oral presentations. And some of that group work I strive so hard to avoid.
I spoke to my mother and she told me that she will be laid off and Mark is turning 18. He is going to get a GED and do something after that. I tried some small talk stuff, but she was hell bent on talking about every big thing that came across her mind. She asked me how I kept myself from using drugs and alcohol. If you don't know, my mother has an exorbitantly high opinion of how I live my life and can't wait for me to be a success when I go out and work. Others of you think I need special attention for shopping or eating. It's wonderful how life works that way.
Speaking of small stuff, Sarah Patterson and I had exchanged sharp words on the merits of small talk and overall usefulness. I have found almost of my meaningful relationships come from people that do not engage in small talk and do their best to speak when important or interesting information is on their minds. I believe this policy with my friends holds true all the way back to kindergarten. I have been in atmospheres where small talk is the dominate form of communication. It doesn't take me long to conclude that the participants don't like each other much or that they have become haplessly indoctrinated by society to speak in a manner that leaves little room for intellectual growth between people. But I will not deny that small talk, as disgusting that it is, keeps the wheels on the car from falling off and thus, maintaining society. Colin, who is rather innocent in his views (like myself) of the world, has spent significant amounts of time with "small talkers" and it was quite clear that he hates it too and proceeds to spend not so much time with them. Anyway it doesn't matter. I am marrying a future rabbi who will deal with regular people on a daily basis. That means I have to be top notch when I communicate with them. I am spending my final 6 months here appreciating that I am not obligated to go out of my way for small talk. I already do so for work and some administrative stuff, but I prefer it stays out of my communications with friends.
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| Well, I have been spending a lot of time thinking about what my final classes should be. I need college writing and no matter what Brenda and Steve say, my writing is not good enough. But, I think I have found a way to make everyone happy. I secretly want to take Latin Composition because I want a mastery of the language and because it will greatly improve my technical understanding of the English language. I was looking at the course schedules and I discovered that College Writing is offered Spring term without causing a conflict. I will be taking Math 399 that term and I can guarantee you that Professor Hastings will be far more cruel about my writing than Professor Cohn (You don't want to see some of the comments he left on my Math 321 assignments). I think College Writing will do more for me Spring term than Winter, when I will be taking econ 399. Econ 399 can be passed by a rock, not matter how bad the writing is. I will take a writing course that helps with organization and structure alongside the most intense course I will take at Knox. That, along with Kat and Hastings bashing me over the head, will do a lot to help me become better writer. I greatly respect everyone who advises me (especially Kat), and I believe everyone is right for different reasons.
P.S. No semicolons, colons, question marks, or other peculiar means to express the spoken word is used in the above passage.
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| - I have a dentist appointment on Tuesday December 2nd at 10:20 A.M. I need to get x-rays sent to them to accelerate the process, so tomorrow I will pick up a copy from Carl Sandburg. Typical cost for a cavity is $150 dollars and there may be appointment fees. But it could be much more depending on how bad the cavities are.
- I have completed my resume, personal statement, revised writing sample, table of information, several grad school applications, and information on all my departments to hand in to my professors.
- I did well on my presentation in Latin, so much so, that Brenda felt the need to email me about it.
- Latin has taken over my life.
- My boss thinks I some super, crazy, smart person. She told me that she thinks I could hold my own at all those elite schools. Um...yeah right.
- I will take Latin Composition for two reasons: I asked Brenda what she thought grad schools would rather see on my transcript College writing or Latin Composition, to which she responded without hesitation: Latin Composition and the other reason is, I prefer for a more engaging class and nearly everyone I spoke to on the matter thinks it is a waste of time for me to take this course. Brenda told me that she thinks Latin has improved my grammatical understanding and the best way to keep that up is with more Latin. She was disappointed I was not signed up and she gave me this look of complete shock that it was possible for this to be so. That's the real reason I am in this class...
- Math makes a lot of sense right now, but the exam I completed may be bloody...stupid proofs.
- I need a break. This is the most stress I have ever had academically and I am becoming frayed. Thank HaShem I took the GRE already...unlike some people.
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