# Stock Probability

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?

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.
• 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 $\sigma$? Commented Apr 17, 2019 at 23:27