Stock behaviour probability I found this question in a financial mathematics course exam, could anyone please help with a solution and some explanation? Thanks in advance :)


  A stock has beta of $2.0$ and stock specific daily volatility of $0.02$.
  Suppose that yesterday’s closing price was $100$ and today the market
  goes up by $1$%. What’s the probability of today’s closing price being
  at least $103$?
Expectation of the stock price is $100*(1+2*0.01)=102$

standard deviation is $2\%$, which is $2$

$103$ is $0.5$ standard deviation away,

$1-N(0.5)=31\%$
Assuming normality, vola being specified as standard deviation and assuming a risk free rate (r_risk_free) of zero the following reasoning could be applied:

1) From CAPM we see that the expected return of the stock (E(r)=r_risk_free+beta*(r_market-r_risk_free) here E(r)=0+2.0*.01=0.02

2) From a casual definition of beta we know that it relates stock specific vola to market vola (sd_stock_total=beta*sd_stock_specific) here sd_stock_total=2*0.02=.04

3) Since we have data on returns only, we transfer the value of interest (103) into returns space which is r_of_interest=(103-100)/100=0.03

4) Now we are looking for the probability that the return of interest occurs (P(x>=0.03)) given the probability distribution defined as N(0.02,0.04) or written differently P(x>=0.03)~N(0.02,0.04)

5) In R you could write 1-(pnorm(.03,mean=.02,sd=.04)) where 1-pnorm(...) is necessary because pnorm() returns P(X&lt;=x) of the cumulative distribution function (CDF) and you are interested in P(x>=0.03) - more pnorm() hints.