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This appeared on a sample Quant Exam:

A stock has a beta of $2.0$ and a specific daily volatility of $2\%$. Yesterday's closing price was $100$ and today the market goes up by $1\%$. What is the probability of today's closing price being at least $110$ dollars?

In general, let $p_{T}$ be the closing price of today's stock, $p_{Y}$ be the closing price of yesterday's stock, $\beta$ be the beta of the stock, and $c$ be the percentage the stock market changed (went up or down), written as a decimal. From what understand about the question asked when the closing price was $103$ dollars, the expected price of today's closing price is the following: $$E(p_{T}) = p_{H} \cdot (1+\beta \cdot c )$$
This expected price then becomes the mean of a normal distribution, with standard deviation given by the daily volatility, $v_{d}$. In this case, we compute $$E(p_{T})=100\cdot(1+2\cdot 0.01)=100\cdot(1.02)=102$$ Since we are asked about the probability that $p_{T}$ is at least 110 dollars, we compute the $Z$ score for 100, which is $$Z=\frac{110-E(p_{T})}{v_{d}} = \frac{110-102}{2} = \frac{8}{2} = 4$$ Therefore, $P(p_{T} \geq 110) = P(Z \geq 4)$. Looking up $Z$ scores for a normal distribution with a table in the back of a stats textbook, I found that this probability is $0$ to the nearest hundredth. With a normal probability calculator at this site, I got the answer $0.000031671$ (to the nearest billionth). Somehow, this seems awfully low to me, but is this correct?

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Your calculation of the probability of a $4\sigma $ event in a normal distribution is correct. The problem is that the normal distribution falls to near zero so quickly that real world probabilities have larger tails. The $2\% \sigma$ may be a good fit for the variation that captures $68\%$ of the days, but large jumps will be more common than the normal distribution says. Maybe massive accounting fraud is discovered and the stock drops $50\%$ in a day. That is $25 \sigma$! Maybe the company was a dark horse for a large contract and managed to win it, making the stock shoot up. Maybe somebody launches a takeover bid. All of these make large jumps more common than the normal distribution would claim.

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  • $\begingroup$ Thank you for the insight, but in light of this, which distribution should I use for the problem? Is there a way I can take the normal distribution and other distributions into account? Or should I change the value of $\sigma$? $\endgroup$ Commented Apr 17, 2019 at 23:27
  • $\begingroup$ Lots of finance people have studied this. I don't know what they have come up with, I just know the tails of the normal distribution are too small. A great quote from Poincaré: "Everyone is sure of this [that errors are normally distributed], Mr. Lippman told me one day, since the experimentalists believe that it is a mathematical theorem, and the mathematicians that it is an experimentally determined fact." $\endgroup$ Commented Apr 17, 2019 at 23:37

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